Planets - User contributions [en]

How to Cite

2025-12-03T11:52:11Z

SBhatnagar: /* Earth */

= Main reference to cite the Generic PCM =

* Forget et al., 2022-2023 (the documentation will be officially released at the time of the Forget et al. publication)

= How to cite? (process by process) =

== Slab ocean ==

Codron, Francis., Ekman heat transport for slab oceans, Climate Dynamics 38: 379-389 (2012)

Charnay, B.; Forget, F.; Wordsworth, R.; Leconte, J.; Millour, E.; Codron, F. and Spiga, A., Exploring the faint young Sun problem and the possible climates of the Archean Earth with a 3-D GCM, Journal of Geophysical Research (Atmospheres), vol.118, pp.10,414 (2013)

Bhatnagar, S.; Codron, F.; Millour, E.; Bolmont, E.; Brunetti, M.; Kasparian, J.; Turbet, M. and Chaverot, G., A Fast and Physically Grounded Ocean Model for GCMs: The Dynamical Slab Ocean Model of the Generic-PCM (rev. 3423), in review at Geoscientific Model Development, pp.1-41, https://doi.org/10.5194/egusphere-2025-3786 (2025)

== CO2 ice clouds ==

Forget, F.; Wordsworth, R.; Millour, E.; Madeleine, J. B.; Kerber, L.; Leconte, J.; Marcq, E. and Haberle, R. M., 3D modelling of the early martian climate under a denser CO2 atmosphere: Temperatures and CO2 ice clouds, Icarus, vol.222, pp.81 (2013)

== etc. ==

TBD

= List of all references (using the Generic PCM) =

=== terrestrial exoplanets ===

Wordsworth, R. D.; Forget, F.; Selsis, F.; Millour, E.; Charnay, B. and Madeleine, J.-B., Gliese 581d is the First Discovered Terrestrial-mass Exoplanet in the Habitable Zone, The Astrophysical Journal, vol.733, pp.L48 (2011)

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R.; Selsis, F.; Millour, E. and Spiga, A., 3D climate modeling of close-in land planets: Circulation patterns, climate moist bistability, and habitability, Astronomy and Astrophysics, vol.554, pp.A69 (2013)

Bolmont, E.; Libert, A. S.; Leconte, J.; & Selsis, F., Habitability of planets on eccentric orbits: Limits of the mean flux approximation, Astronomy & Astrophysics, vol.591, pp.A106 (2016)

Turbet, M.; Leconte, J.; Selsis, F.; Bolmont, E.; Forget, F.; Ribas, I.; Raymond, S. N. and Anglada-Escudé, G., The habitability of Proxima Centauri b. II. Possible climates and observability, Astronomy and Astrophysics, vol.596, pp.A112 (2016)

Turbet, M.; Forget, F.; Leconte, J.; Charnay, B. and Tobie, G., CO2 condensation is a serious limit to the deglaciation of Earth-like planets, Earth and Planetary Science Letters, vol.476, pp.11 (2017)

Turbet, M.; Bolmont, E.; Leconte, J.; Forget, F.; Selsis, F.; Tobie, G.; Caldas, A.; Naar, J. and Gillon, M., Modeling climate diversity, tidal dynamics and the fate of volatiles on TRAPPIST-1 planets, Astronomy and Astrophysics, vol.612, pp.A86 (2018)

Fauchez, T. J.; Turbet, M.; Villanueva, G. L.; Wolf, E. T.; Arney, G.; Kopparapu, R. K.; Lincowski, A.; Mandell, A.; de Wit, J.; Pidhorodetska, D.; Domagal-Goldman, S. D. and Stevenson, K. B., Impact of Clouds and Hazes on the Simulated JWST Transmission Spectra of Habitable Zone Planets in the TRAPPIST-1 System, The Astrophysical Journal, vol.887, pp.194 (2019)

=== super-Earths ===

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R.; Selsis, F.; Millour, E. and Spiga, A., 3D climate modeling of close-in land planets: Circulation patterns, climate moist bistability, and habitability, Astronomy and Astrophysics, vol.554, pp.A69 (2013)

=== mini-Neptunes ===

Benjamin's two papers on GJ1214b

Charnay, B.; Blain, D.; Bézard, B.; Leconte, J.; Turbet, M. and Falco, A., Formation and dynamics of water clouds on temperate sub-Neptunes: the example of K2-18b, Astronomy and Astrophysics, vol.646, pp.A171 (2021)

=== LES ===
Lefèvre, M.; Turbet, M. and Pierrehumbert, R., 3D Convection-resolving Model of Temperate, Tidally Locked Exoplanets, The Astrophysical Journal, vol.913, pp.101 (2021)

== paleoclimates ==

=== Earth ===

Charnay, B.; Forget, F.; Wordsworth, R.; Leconte, J.; Millour, E.; Codron, F. and Spiga, A., Exploring the faint young Sun problem and the possible climates of the Archean Earth with a 3-D GCM, Journal of Geophysical Research (Atmospheres), vol.118, pp.10,414 (2013)

Charnay, B.; Le Hir, G.; Fluteau, F.; Forget, F. and Catling, D. C., A warm or a cold early Earth? New insights from a 3-D climate-carbon model, Earth and Planetary Science Letters, vol.474, pp.97 (2017)

Turbet, M.; Bolmont, E.; Chaverot, G.; Ehrenreich, D.; Leconte, J. and Marcq, E., Day-night cloud asymmetry prevents early oceans on Venus but not on Earth, Nature, vol.598, pp.276 (2021)

Bhatnagar, S.; Codron, F.; Millour, E.; Bolmont, E.; Brunetti, M.; Kasparian, J.; Turbet, M. and Chaverot, G., A Fast and Physically Grounded Ocean Model for GCMs: The Dynamical Slab Ocean Model of the Generic-PCM (rev. 3423), in review at Geoscientific Model Development, pp.1-41, https://doi.org/10.5194/egusphere-2025-3786 (2025)

=== Mars ===

Forget, F.; Wordsworth, R.; Millour, E.; Madeleine, J. B.; Kerber, L.; Leconte, J.; Marcq, E. and Haberle, R. M., 3D modelling of the early martian climate under a denser CO2 atmosphere: Temperatures and CO2 ice clouds, Icarus, vol.222, pp.81 (2013)

Wordsworth, R.; Forget, F.; Millour, E.; Head, J. W.; Madeleine, J. B. and Charnay, B., Global modelling of the early martian climate under a denser CO2 atmosphere: Water cycle and ice evolution, Icarus, vol.222, pp.1 (2013)

Wordsworth, R. D.; Kerber, L.; Pierrehumbert, R. T.; Forget, F. and Head, J. W., Comparison of "warm and wet" and "cold and icy" scenarios for early Mars in a 3-D climate model, Journal of Geophysical Research (Planets), vol.120, pp.1201 (2015)

Kerber, L.; Forget, F. and Wordsworth, R., Sulfur in the early martian atmosphere revisited: Experiments with a 3-D Global Climate Model, Icarus, vol.261, pp.133 (2015)

Turbet, M.; Forget, F.; Head, J. W. and Wordsworth, R., 3D modelling of the climatic impact of outflow channel formation events on early Mars, Icarus, vol.288, pp.10 (2017)

Turbet, M. and Forget, F., The paradoxes of the Late Hesperian Mars ocean, Scientific Reports, vol.9, pp.5717 (2019)

Turbet, M.; Gillmann, C.; Forget, F.; Baudin, B.; Palumbo, A.; Head, J. and Karatekin, O., The environmental effects of very large bolide impacts on early Mars explored with a hierarchy of numerical models, Icarus, vol.335, pp.113419 (2020)

=== Venus ===

Turbet, M.; Bolmont, E.; Chaverot, G.; Ehrenreich, D.; Leconte, J. and Marcq, E., Day-night cloud asymmetry prevents early oceans on Venus but not on Earth, Nature, vol.598, pp.276 (2021)

=== Titan ===

Charnay, B.; Forget, F.; Tobie, G.; Sotin, C. and Wordsworth, R., Titan's past and future: 3D modeling of a pure nitrogen atmosphere and geological implications, Icarus, vol.241, pp.269 (2014)
DOI: 10.1016/j.icarus.2014.07.009

== future climates ==

=== Earth ===

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R. and Pottier, A., Increased insolation threshold for runaway greenhouse processes on Earth-like planets, Nature, vol.504, pp.268 (2013)

== Gas giants ==

=== Saturn ===

''' Saturn radiative transfer model '''

Guerlet, S.; Spiga, A.; Sylvestre, M.; Indurain, M.; Fouchet, T.; Leconte, J.; Millour, E.; Wordsworth, R.; Capderou, M.; Bézard, B. and Forget, F., Global climate modeling of Saturn’s atmosphere. Part I: Evaluation of the radiative transfer model, Icarus, vol.238, pp.110 (2014)
DOI: 10.1016/j.icarus.2014.05.010

''' Saturn's tropospheric dynamics with DYNAMICO Generic-PCM '''

Spiga, A.; Guerlet, S.; Millour, E.; Indurain, M.; Meurdesoif, Y.; Cabanes, S.; Dubos, T.; Leconte, J.; Boissinot, A.; Lebonnois, S.; Sylvestre, M. and Fouchet, T., Global climate modeling of Saturn's atmosphere. Part II: Multi-annual high-resolution dynamical simulations, Icarus, vol.335, pp.113377 (2020)
DOI: 10.1016/j.icarus.2019.07.011

''' Saturn's statistical analysis of the zonostrophic tropospheric dynamics with DYNAMICO Generic-PCM '''

Cabanes, S.; Spiga, A. and Young, R. M. B ., Global climate modeling of Saturn's atmosphere. Part III: Global statistical picture of zonostrophic turbulence in high-resolution 3D-turbulent simulations, Icarus, vol.345, pp. 113705 (2020)
DOI: 10.1016/j.icarus.2020.113705

''' Saturn's stratospheric dynamics with DYNAMICO Generic-PCM '''

Bardet, D.; Spiga, A.; Guerlet, S.; Cabanes, S.; Millour, E. and Boissinot, A., Global climate modeling of Saturn's atmosphere. Part IV: Stratospheric equatorial oscillation, Icarus, vol.354, pp. 114042 (2021)
DOI: 10.1016/j.icarus.2020.114042

Bardet, D.; Spiga, A. and Guerlet, S., Joint evolution of equatorial oscillation and inter-hemispheric circulation in Saturn’s stratosphere, Nature Astronomy (2022)
DOI: 10.1038/s41550-022-01670-7

''' Saturn's large-scale vortices analysis in its tropospheric dynamics with DYNAMICO Generic-PCM '''

Donnelly, P. T.; Spiga, A.; Guerlet, S.; James, M. K. and Bardet, D., Global climate modelling of Saturn's atmosphere, Part V: Large-scale vortices, Icarus, vol.425, pp.116302 (2025)
DOI: 10.1016/j.icarus.2024.116302

=== Jupiter ===
''' Jupiter radiative transfer model '''

Guerlet, S.; Spiga, A.; Delattre, H. and Fouchet, T., Radiative-equilibrium model of Jupiter's atmosphere and application to estimating stratospheric circulations, Icarus, vol.351, pp. 113935 (2020)
DOI: 10.1016/j.icarus.2020.113935

''' Jupiter's tropospheric circulation with DYNAMICO Generic-PCM '''

Boissinot, A.; Spiga, A.; Guerlet, S.; Cabanes, S. and Bardet, D., Global climate modeling of the Jupiter troposphere and effect of dry and moist convection on jets, Astronomy and Astrophysics, vol.687, pp.A274 (2024)
DOI: 10.1051/0004-6361/202245220

== Ice giants ==
Clément, N.; Leconte, J.; Spiga, A.; Guerlet, S.; Selsis, F.; Milcareck, G.; Teinturier, L.; Cavalié, T.; Moreno, R.; Lellouch, E. and Carrión-González, Ó., Storms and convection on Uranus and Neptune: Impact of methane abundance revealed by a 3D cloud-resolving model, Astronomy and Astrophysics, vol.690, pp.A227 (2024)
DOI: 10.1051/0004-6361/202348936

Milcareck, G.; Guerlet, S.; Montmessin, F.; Spiga, A.; Leconte, J.; Millour, E.; Clément, N.; Fletcher, L. N.; Roman, M. T.; Lellouch, E.; Moreno, R.; Cavalié, T. and Carrión-González, Ó., Radiative-convective models of the atmospheres of Uranus and Neptune: Heating sources and seasonal effects, Astronomy and Astrophysics, vol.686, pp.A303 (2024)
DOI: 10.1051/0004-6361/202348987

=== Uranus ===

=== Neptune ===

[[Category:Generic-Model]]

How to Cite

2025-12-03T11:50:17Z

SBhatnagar: /* Slab ocean */

= Main reference to cite the Generic PCM =

* Forget et al., 2022-2023 (the documentation will be officially released at the time of the Forget et al. publication)

= How to cite? (process by process) =

== Slab ocean ==

Codron, Francis., Ekman heat transport for slab oceans, Climate Dynamics 38: 379-389 (2012)

Charnay, B.; Forget, F.; Wordsworth, R.; Leconte, J.; Millour, E.; Codron, F. and Spiga, A., Exploring the faint young Sun problem and the possible climates of the Archean Earth with a 3-D GCM, Journal of Geophysical Research (Atmospheres), vol.118, pp.10,414 (2013)

Bhatnagar, S.; Codron, F.; Millour, E.; Bolmont, E.; Brunetti, M.; Kasparian, J.; Turbet, M. and Chaverot, G., A Fast and Physically Grounded Ocean Model for GCMs: The Dynamical Slab Ocean Model of the Generic-PCM (rev. 3423), in review at Geoscientific Model Development, pp.1-41, https://doi.org/10.5194/egusphere-2025-3786 (2025)

== CO2 ice clouds ==

Forget, F.; Wordsworth, R.; Millour, E.; Madeleine, J. B.; Kerber, L.; Leconte, J.; Marcq, E. and Haberle, R. M., 3D modelling of the early martian climate under a denser CO2 atmosphere: Temperatures and CO2 ice clouds, Icarus, vol.222, pp.81 (2013)

== etc. ==

TBD

= List of all references (using the Generic PCM) =

=== terrestrial exoplanets ===

Wordsworth, R. D.; Forget, F.; Selsis, F.; Millour, E.; Charnay, B. and Madeleine, J.-B., Gliese 581d is the First Discovered Terrestrial-mass Exoplanet in the Habitable Zone, The Astrophysical Journal, vol.733, pp.L48 (2011)

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R.; Selsis, F.; Millour, E. and Spiga, A., 3D climate modeling of close-in land planets: Circulation patterns, climate moist bistability, and habitability, Astronomy and Astrophysics, vol.554, pp.A69 (2013)

Bolmont, E.; Libert, A. S.; Leconte, J.; & Selsis, F., Habitability of planets on eccentric orbits: Limits of the mean flux approximation, Astronomy & Astrophysics, vol.591, pp.A106 (2016)

Turbet, M.; Leconte, J.; Selsis, F.; Bolmont, E.; Forget, F.; Ribas, I.; Raymond, S. N. and Anglada-Escudé, G., The habitability of Proxima Centauri b. II. Possible climates and observability, Astronomy and Astrophysics, vol.596, pp.A112 (2016)

Turbet, M.; Forget, F.; Leconte, J.; Charnay, B. and Tobie, G., CO2 condensation is a serious limit to the deglaciation of Earth-like planets, Earth and Planetary Science Letters, vol.476, pp.11 (2017)

Turbet, M.; Bolmont, E.; Leconte, J.; Forget, F.; Selsis, F.; Tobie, G.; Caldas, A.; Naar, J. and Gillon, M., Modeling climate diversity, tidal dynamics and the fate of volatiles on TRAPPIST-1 planets, Astronomy and Astrophysics, vol.612, pp.A86 (2018)

Fauchez, T. J.; Turbet, M.; Villanueva, G. L.; Wolf, E. T.; Arney, G.; Kopparapu, R. K.; Lincowski, A.; Mandell, A.; de Wit, J.; Pidhorodetska, D.; Domagal-Goldman, S. D. and Stevenson, K. B., Impact of Clouds and Hazes on the Simulated JWST Transmission Spectra of Habitable Zone Planets in the TRAPPIST-1 System, The Astrophysical Journal, vol.887, pp.194 (2019)

=== super-Earths ===

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R.; Selsis, F.; Millour, E. and Spiga, A., 3D climate modeling of close-in land planets: Circulation patterns, climate moist bistability, and habitability, Astronomy and Astrophysics, vol.554, pp.A69 (2013)

=== mini-Neptunes ===

Benjamin's two papers on GJ1214b

Charnay, B.; Blain, D.; Bézard, B.; Leconte, J.; Turbet, M. and Falco, A., Formation and dynamics of water clouds on temperate sub-Neptunes: the example of K2-18b, Astronomy and Astrophysics, vol.646, pp.A171 (2021)

=== LES ===
Lefèvre, M.; Turbet, M. and Pierrehumbert, R., 3D Convection-resolving Model of Temperate, Tidally Locked Exoplanets, The Astrophysical Journal, vol.913, pp.101 (2021)

== paleoclimates ==

=== Earth ===

Charnay, B.; Forget, F.; Wordsworth, R.; Leconte, J.; Millour, E.; Codron, F. and Spiga, A., Exploring the faint young Sun problem and the possible climates of the Archean Earth with a 3-D GCM, Journal of Geophysical Research (Atmospheres), vol.118, pp.10,414 (2013)

Charnay, B.; Le Hir, G.; Fluteau, F.; Forget, F. and Catling, D. C., A warm or a cold early Earth? New insights from a 3-D climate-carbon model, Earth and Planetary Science Letters, vol.474, pp.97 (2017)

Turbet, M.; Bolmont, E.; Chaverot, G.; Ehrenreich, D.; Leconte, J. and Marcq, E., Day-night cloud asymmetry prevents early oceans on Venus but not on Earth, Nature, vol.598, pp.276 (2021)

=== Mars ===

Forget, F.; Wordsworth, R.; Millour, E.; Madeleine, J. B.; Kerber, L.; Leconte, J.; Marcq, E. and Haberle, R. M., 3D modelling of the early martian climate under a denser CO2 atmosphere: Temperatures and CO2 ice clouds, Icarus, vol.222, pp.81 (2013)

Wordsworth, R.; Forget, F.; Millour, E.; Head, J. W.; Madeleine, J. B. and Charnay, B., Global modelling of the early martian climate under a denser CO2 atmosphere: Water cycle and ice evolution, Icarus, vol.222, pp.1 (2013)

Wordsworth, R. D.; Kerber, L.; Pierrehumbert, R. T.; Forget, F. and Head, J. W., Comparison of "warm and wet" and "cold and icy" scenarios for early Mars in a 3-D climate model, Journal of Geophysical Research (Planets), vol.120, pp.1201 (2015)

Kerber, L.; Forget, F. and Wordsworth, R., Sulfur in the early martian atmosphere revisited: Experiments with a 3-D Global Climate Model, Icarus, vol.261, pp.133 (2015)

Turbet, M.; Forget, F.; Head, J. W. and Wordsworth, R., 3D modelling of the climatic impact of outflow channel formation events on early Mars, Icarus, vol.288, pp.10 (2017)

Turbet, M. and Forget, F., The paradoxes of the Late Hesperian Mars ocean, Scientific Reports, vol.9, pp.5717 (2019)

Turbet, M.; Gillmann, C.; Forget, F.; Baudin, B.; Palumbo, A.; Head, J. and Karatekin, O., The environmental effects of very large bolide impacts on early Mars explored with a hierarchy of numerical models, Icarus, vol.335, pp.113419 (2020)

=== Venus ===

Turbet, M.; Bolmont, E.; Chaverot, G.; Ehrenreich, D.; Leconte, J. and Marcq, E., Day-night cloud asymmetry prevents early oceans on Venus but not on Earth, Nature, vol.598, pp.276 (2021)

=== Titan ===

Charnay, B.; Forget, F.; Tobie, G.; Sotin, C. and Wordsworth, R., Titan's past and future: 3D modeling of a pure nitrogen atmosphere and geological implications, Icarus, vol.241, pp.269 (2014)
DOI: 10.1016/j.icarus.2014.07.009

== future climates ==

=== Earth ===

Leconte, J.; Forget, F.; Charnay, B.; Wordsworth, R. and Pottier, A., Increased insolation threshold for runaway greenhouse processes on Earth-like planets, Nature, vol.504, pp.268 (2013)

== Gas giants ==

=== Saturn ===

''' Saturn radiative transfer model '''

Guerlet, S.; Spiga, A.; Sylvestre, M.; Indurain, M.; Fouchet, T.; Leconte, J.; Millour, E.; Wordsworth, R.; Capderou, M.; Bézard, B. and Forget, F., Global climate modeling of Saturn’s atmosphere. Part I: Evaluation of the radiative transfer model, Icarus, vol.238, pp.110 (2014)
DOI: 10.1016/j.icarus.2014.05.010

''' Saturn's tropospheric dynamics with DYNAMICO Generic-PCM '''

Spiga, A.; Guerlet, S.; Millour, E.; Indurain, M.; Meurdesoif, Y.; Cabanes, S.; Dubos, T.; Leconte, J.; Boissinot, A.; Lebonnois, S.; Sylvestre, M. and Fouchet, T., Global climate modeling of Saturn's atmosphere. Part II: Multi-annual high-resolution dynamical simulations, Icarus, vol.335, pp.113377 (2020)
DOI: 10.1016/j.icarus.2019.07.011

''' Saturn's statistical analysis of the zonostrophic tropospheric dynamics with DYNAMICO Generic-PCM '''

Cabanes, S.; Spiga, A. and Young, R. M. B ., Global climate modeling of Saturn's atmosphere. Part III: Global statistical picture of zonostrophic turbulence in high-resolution 3D-turbulent simulations, Icarus, vol.345, pp. 113705 (2020)
DOI: 10.1016/j.icarus.2020.113705

''' Saturn's stratospheric dynamics with DYNAMICO Generic-PCM '''

Bardet, D.; Spiga, A.; Guerlet, S.; Cabanes, S.; Millour, E. and Boissinot, A., Global climate modeling of Saturn's atmosphere. Part IV: Stratospheric equatorial oscillation, Icarus, vol.354, pp. 114042 (2021)
DOI: 10.1016/j.icarus.2020.114042

Bardet, D.; Spiga, A. and Guerlet, S., Joint evolution of equatorial oscillation and inter-hemispheric circulation in Saturn’s stratosphere, Nature Astronomy (2022)
DOI: 10.1038/s41550-022-01670-7

''' Saturn's large-scale vortices analysis in its tropospheric dynamics with DYNAMICO Generic-PCM '''

Donnelly, P. T.; Spiga, A.; Guerlet, S.; James, M. K. and Bardet, D., Global climate modelling of Saturn's atmosphere, Part V: Large-scale vortices, Icarus, vol.425, pp.116302 (2025)
DOI: 10.1016/j.icarus.2024.116302

=== Jupiter ===
''' Jupiter radiative transfer model '''

Guerlet, S.; Spiga, A.; Delattre, H. and Fouchet, T., Radiative-equilibrium model of Jupiter's atmosphere and application to estimating stratospheric circulations, Icarus, vol.351, pp. 113935 (2020)
DOI: 10.1016/j.icarus.2020.113935

''' Jupiter's tropospheric circulation with DYNAMICO Generic-PCM '''

Boissinot, A.; Spiga, A.; Guerlet, S.; Cabanes, S. and Bardet, D., Global climate modeling of the Jupiter troposphere and effect of dry and moist convection on jets, Astronomy and Astrophysics, vol.687, pp.A274 (2024)
DOI: 10.1051/0004-6361/202245220

== Ice giants ==
Clément, N.; Leconte, J.; Spiga, A.; Guerlet, S.; Selsis, F.; Milcareck, G.; Teinturier, L.; Cavalié, T.; Moreno, R.; Lellouch, E. and Carrión-González, Ó., Storms and convection on Uranus and Neptune: Impact of methane abundance revealed by a 3D cloud-resolving model, Astronomy and Astrophysics, vol.690, pp.A227 (2024)
DOI: 10.1051/0004-6361/202348936

Milcareck, G.; Guerlet, S.; Montmessin, F.; Spiga, A.; Leconte, J.; Millour, E.; Clément, N.; Fletcher, L. N.; Roman, M. T.; Lellouch, E.; Moreno, R.; Cavalié, T. and Carrión-González, Ó., Radiative-convective models of the atmospheres of Uranus and Neptune: Heating sources and seasonal effects, Astronomy and Astrophysics, vol.686, pp.A303 (2024)
DOI: 10.1051/0004-6361/202348987

=== Uranus ===

=== Neptune ===

[[Category:Generic-Model]]

Slab ocean model

2025-09-12T11:27:47Z

SBhatnagar: /* Notes for users */

== General description ==

''Note: This documentation reflects updates introduced in '''revision r3423''' of the Generic-PCM. Users who wish to include the dynamical slab ocean model in their simulations must use '''r3423 or later'''.''

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for modern Earth with the dynamical slab ocean takes around 50 years from a 290 K isothermal start.

-----

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-09-12T11:26:37Z

SBhatnagar: /* Notes for users */

== General description ==

''Note: This documentation reflects updates introduced in '''revision r3423''' of the Generic-PCM. Users who wish to include the dynamical slab ocean model in their simulations must use '''r3423 or later'''.''

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-09-12T11:25:58Z

SBhatnagar: /* General description */

== General description ==

''Note: This documentation reflects updates introduced in '''revision r3423''' of the Generic-PCM. Users who wish to include the dynamical slab ocean model in their simulations must use '''r3423 or later'''.''

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-09-12T11:25:11Z

SBhatnagar: /* General description */

== General description ==

''Note: This documentation reflects updates introduced in '''revision r3423''' of the ocean module. Users who wish to include the dynamical slab ocean model in their simulations must use '''r3423 or later'''.
''
The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-09-12T11:24:35Z

SBhatnagar: /* General description */

== General description ==

Note: This documentation reflects updates introduced in revision r3423 of the ocean module. Users who wish to include the dynamical slab ocean model in their simulations must use r3423 or later.

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-09-12T09:47:33Z

SBhatnagar: /* General description */

== General description ==

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in review). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-04-07T08:55:37Z

SBhatnagar: /* Notes for users */

== General description ==

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in prep). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

For historical reasons, we keep '''albedosnow''' (can be defined in callphys.def, default value=0.5), which is the maximum value that the albedo of sea ice can have.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-04-07T08:53:12Z

SBhatnagar: /* Notes for a user */

== General description ==

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in prep). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for users ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-19T18:13:40Z

SBhatnagar: /* General description */

== General description ==

The slab ocean model is based on the works of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)] and Bhatnagar et al. (in prep). The model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Physics of the Generic PCM

2025-03-19T18:12:33Z

SBhatnagar: /* Slab ocean */

This page describes the various physical parametrizations of the Generic PCM and the chronology of their call through the physical iteration. This chronology is important because some variables need to be updated by certain processes before others (examples).
During one physical iteration, the code passes through multiple sub-routines each encapsulating a parametrization. The sub-routines usually take as arguments:
*the dynamical values of the state variables
*the dynamical tendencies of the state variables
*any additional relevant variable
*any additional relevant tendency
and it returns in general tendencies (of the state variables as well as of any other relevant variable).

Work in progress. Need to add links.

=Initialization=
==First call==
Some initializations only need to be done at the very first iteration (e.g. examples).
==Each call==
Some other initializations need to be done at each iteration (e.g. examples).

=Radiative transfer=
==Correlated-k==
The main radiative transfer solver of the Generic PCM implements the correlated-k method, which provides a flexible and quick way to solve radiative transfer equations, particularly suited for GCMs. More informations here.
==Newtonian relaxation==
Newton was a very relaxed physicist, still inspiring us today.
==No atmosphere==
If you have no atmosphere, why do you need a GCM?

=Vertical diffusion=
Compute the vertical diffusion due to turbulence in the planetary boundary layer
==vdifc==
The "old" vertical diffusion routine.
==turbdiff==
The new (and improved!) vertical diffusion routine

=Convection=
Convective mixing in an atmosphere column involves non-resolved, sub-grid processes. These processes are parameterized using the following routines:

==Thermal plume==
This module implements a more accurate and sophisticated parametrization of convection. More info [[Thermal_plume_model_Generic_PCM|here]].

==Dry convection==
More info [[Convective_adjustment_scheme_in_the_generic_PCM|here]].
==Non-orographic gravity waves==
More info [[Non orographic gravity waves drag|here]].

=CO2 condensation=
Inheriting from the Mars PCM where it is the background species, CO2 condensation is treated as a dedicated step in the Generic PCM's physics. Does that only concern CO2 as a background gas? Or does it work also if CO2 is a non-background variable gas? In any case more info here.

=Tracers=
Many things can be advected by tracers in the Generic PCM, like chemical species or aerosols. Physical processes involving tracers are parameterized using the following routines:

==Volcano==
This routine parameterizes a source of tracers corresponding to volcanic eruption. More information here.

==Water/ice==
Water aerosols (liquid or solid) are created (resp. consumed) by condensation (resp. vaporization or sublimation), consuming (resp. releasing) latent heat in the atmosphere. In the Generic PCM, the (atmospheric part of the) water cycle is handled by various routines, as explained here.

==Photochemistry==
Chemistry can turn molecules into other molecules, by the action of temperature (thermochemistry) or UV light (photochemistry). This is handled by the photochemistry routine, described [[Photochemistry|here]].

==Generic condensation==
On Earth, only water condenses in the atmosphere, but on other planets (which the Generic PCM aims at simulating), many other chemicals can undergo state change. To take that into account, the Generic PCM has a flexible scheme to deal with any arbitrary species, as desbribed [[Radiative_Generic_Condensable_Specie|here]].

==Sedimentation==
What goes up must come down, as explained [[Sedimentation_of_tracers_in_the_generic_PCM|here]].

==Updates==
This section essentially takes care that the condensation of a major species (e.g. water vapor in steam-rich atmospheres) affects other species, as described here.

==Slab ocean==
This routine solves for big fish eating small fish, as described [[Slab_ocean_model|here]].

==Surface==
The surface part of the water cycle is handled here, as the hydrology page describes.

=Heat Conduction in the Subsurface=
The conduction of heat in the subsurface is solved as described [[Soil_Thermal_Conduction_in_the_Generic_PCM|here]].

=Diagnostics/write outputs=
Nothing very physical here, just writting the outputs! By the way, if you want to know how to output new variables in the [[diagfi.nc]] or [[XIOS]] file, check out [[Outputs|this page]].

Slab ocean model

2025-03-05T10:04:43Z

SBhatnagar: /* Notes for a user */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def so that the ocean is called every physics timestep.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:47:12Z

SBhatnagar: /* Notes for a user */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Diffusion / Advection (can only be true if slab_ekman = .true.):
slab_gm = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:41:48Z

SBhatnagar: /* Notes for a user */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:31:47Z

SBhatnagar: /* Notes for a user */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

If ok_slab_ocean = .false., the ocean can be modelled as a surface with a high thermal inertia.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:29:07Z

SBhatnagar: /* Variables of the slab ocean model which can be written as outputs in diagfi.nc: */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

Variables of the slab ocean model which can be written as outputs in diagfi.nc:

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:26:34Z

SBhatnagar: /* The 2024 version */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2025 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:26:15Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Notes for a user ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:25:57Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:14:20Z

SBhatnagar: /* Sea ice */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:14:07Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-03-05T09:12:02Z

SBhatnagar: /* General description */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by: $$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-02-13T16:30:14Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1 in callphys.def.

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2025-02-13T16:29:43Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day). It is recommended to force '''cpl_pas'''= 1

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-10-30T06:19:24Z

SBhatnagar: /* General description */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day)

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-10-30T06:07:18Z

SBhatnagar: /* General description */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

- 22:30 UT - The thorium calibration (COR_THA2) refused to start, with the UIF saying "la configuration n'a pas allume la lamp thorium!!!” I decided to restart the thorium calibration by adding a couple more THA2 entries. But when I ran this, it didn’t work either.
- 22:50 UT - Strangely, I saw that the THAR was displayed as ON on the Euler Status Panel. I put this OFF and then ON again on the Euler panel and repeated the above exercise, but this did not work either.
- 23:07 UT - Through the Utilities tab in the UIF, I clicked on Allume la lampe Thorium. Then repeated the same as above - did not work.
- 23:15 UT - Arrêt complet + Démarrage complet didn’t work
- 23:20 UT - T_reboot_plc_spectro + Arrêt complet + Démarrage complet didn’t work
- 23:40 UT - T_show_date showed that everything was online, except glsaux
- 23:41 UT - As a final resort, I rebooted glslogin1 with T_reboot_master.
- 00:30 UT - Since the monitor was still showing as “rebooting,” I manually switched OFF and ON the switch to the main screen. This brought back the computer to life.
- 00:35 UT - Switched on the EDP and associated menus again and attempted to relaunch the THA2 calibration. Same issue persisted as before.
- 00:40 UT - I checked the amps regulator machine in the pump room, and it is at a non-zero value (5 mA). So it seems like, in reality, the lamp is on.
- Assessment: The problem seems to be not with the lamp itself, but rather, the communication between the lamp and the software. The Euler Status Panel says that THAR is ON, the amps regulator machine is at 5 mA. But, somehow, the software/UIF/EDP is not recognising that the lamp is indeed ON.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day)

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-22T09:37:38Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day)

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.

-----

SB: Latest edits (r3397, 17 Jul 2024) need to account for tice in newstart and start2archive also.

[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-16T11:53:26Z

SBhatnagar: /* Variables of the slab ocean model which can be written as outputs in diagfi.nc: */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# '''tslab1''', '''tslab2''': temperature of the surface and deep ocean layers respectively (K)
# '''pctsrf_sic''': grid fraction of sea ice
# '''sea_ice''': mass of sea ice (kg/m$$^2$$)
# '''tsea_ice''': the temperature of the surface in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# '''dt_hdiff1''', '''dt_hdiff2''': heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# '''dt_ekman1''', '''dt_ekman2''': heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# '''dt_gm1''', '''dt_gm2''': heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day)

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-16T11:48:42Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the ocean in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by '''cpl_pas''' (in physics timesteps; default value = once every Earth day)

'''rnat''' is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-16T11:48:04Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the ocean in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, '''tsea_ice''' was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it ('''tice''', which is a global variable within '''ocean_slab_mod'''), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow ('''tsea_ice''', in '''physiq_mod''', as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, '''tice''' and '''tsea_ice''' are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-16T11:44:27Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the ocean in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

In the old implementation, tsea_ice was always the temperature of the ice surface. It didn’t matter if snow was on top of it or not because previously, snow had zero heat capacity. But now, since snow has a finite heat capacity, there can be conductive fluxes between the top layer of the sea ice and snow on top of it (if the latter exists). This necessitates two different temperature variables: (1), for the sea ice temperature if snow exists on top of it (tice, which is a global variable within ocean_slab_mod), and is arguably not useful in physiq_mod and therefore not defined in it), and (2), the temperature of the surface in contact with the atmosphere, which can either be sea ice or snow (tsea_ice, in physiq_mod, as it is essentially a proxy for surface temperature, which is useful for the physics to compute interactions between the ocean and atmosphere). To summarise, tice and tsea_ice are only identical if snow does not exist on top of the ice.

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-07-16T11:41:03Z

SBhatnagar: /* Variables of the slab ocean model which can be written as outputs in diagfi.nc: */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the ocean in contact with the atmosphere (sea ice or snow on top of sea ice) (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Advanced Use of the GCM

2024-06-19T16:20:22Z

SBhatnagar: /* How to Change the Topography (or remove it) */

== Running in parallel ==

For large simulation (long run, high resolution etc...), the computational cost can be huge and hence the run time very long.
To overcome this issue, the model can be run in parallel. This however requires a few extra steps (compared to compiling and running the serial version of the code).
For all the details see [[Parallelism | the dedicated page]].

== Disambiguation between ifort, mpif90, etc. ==

For users not used to compilers and/or compiling and running codes in parallel, namely in MPI mode, there is often some confusion which hopefully the following paragraph might help clarify:
* the compiler (typically gfortran, ifort, pgfortran, etc.) is the required tool to compile the Fortran source code and generate an executable. It is strongly recommended that libraries used by a program are also compiled using the same compiler. Thus if you plan to use different compilers to compile the model, note that you should also have at hand versions of the libraries it uses also compiled with these compilers.
* the MPI (Message Passing Interface) library is a library used to solve problems using multiple processes by enabling message-passing between the otherwise independent processes. There are a number of available MPI libraries out there, e.g. OpenMPI, MPICH or IntelMPI to name a few (you can check out the [[Building an MPI library]] page for some information about installing an MPI library). The important point here is that on a given machine the MPI library is related to a given compiler and that it provides related wrappers to compile and run with. Typically (but not always) the compiler wrapper is '''mpif90''' and the execution wrapper is '''mpirun'''. If you want to know which compiler is wrapped in the '''mpif90''' compiler wrapper, check out the output of:
<pre>
mpif90 --version
</pre>
* In addition a second type of parallelism, shared memory parallelism known as OpenMP, is also implemented in the code. In contradistinction to MPI, OpenMP does not require an external library but is instead implemented as a compiler feature. At run time one must then specify some dedicated environment variables (such as OMP_NUM_THREADS and OMP_STACKSIZE) to specify the number of threads to use per process.
* In practice one should favor compiling and running with both MPI and OPenMP enabled.
* For much more detailed information about compiling and running in parallel, check out the [[Parallelism | the page dedicated to Parallelism]].

== A word about the IOIPSL and XIOS libraries ==
* The IOIPSL (Input Output IPSL) library is a library that has developed by the IPSL community to handle input and outputs of (mostly terrestrial) climate models. For the Generic PCM only a small part of this library is actually used, related to reading and processing the input [[The_run.def_Input_File | run.def]] file. For more details check out the [[The IOIPSL Library]] page.
* The [https://forge.ipsl.jussieu.fr/ioserver/wiki XIOS] (Xml I/O Server) library is based on client-server principles where the server manages the outputs asynchronously from the client (the climate model) so that the bottleneck of writing data in a parallel environment is alleviated. All aspects of the outputs (name, units, file, post-processing operations, etc.) are then controlled by dedicated XML files which are read at run-time. Using XIOS is currently optional (and requires compiling the GCM with the XIOS library). More about the XIOS library, how to install and use it, etc. [[The XIOS Library| here]].

== Playing with the output files ==

=== Changing the output temporal resolution and time duration ===

* To change the total time of a simulation, you need to open the 'For all the details see [[The_run.def_Input_File | run.def]]. file and change the variable 'nday':

<pre>
nday = 1000 # this means the simulation will run for 1000 days ; and that the associated output files will also be computed for a total duration of 1000 days
</pre>

Note: in the example, they are not necessarily 1000 Earth days, because it depends on the definition of the day duration that has been taken in the start files.

* To change the temporal resolution of the output files, you need to open the [[The_run.def_Input_File | run.def]] file and change the variable 'ecritphy':

<pre>
ecritphy = 200 # this means the simulation will write variables in the output files every 200 time steps of the simulation.
</pre>

Note: The output temporal resolution of the output files then depends also on the number of timestep per day ('day_step' variable in [[The_run.def_Input_File | run.def]] file). In this example:

<pre>
nday = 1000
daystep = 800
ecritphy = 200
</pre>

The output file will provide results every 0.25 days (800/200), and for a total duration of 1000 days (so 4000 time values in total).

=== Changing the output variable ===

To select the variable provided in the output file diagfi.nc, you simply need to add the list of variables needed in the [[The_diagfi.def_Input_File | diagfi.def]].

Note for experts: Some technical variables need to be de-commented in 'physiq_mod.F90' file to be written in the output files.

=== Spectral outputs ===

It is possible to provide spectral outputs such as the OLR (Outgoing Longwave Radiation, i.e. the thermal emission of the planet at the top of the atmosphere), the OSR (Outgoing Stellar Radiation, i.e. the light reflected by the planet at the top of the atmosphere), or the GSR (Ground Stellar Radiation, i.e. the light emitted by the star that reaches the surface of the planet).

For this, you need to activate the option 'specOLR' in the [[The_callphys.def_Input_File | callphys.def]] file, as follows:

<pre>
specOLR = .true.
</pre>

The simulations will then create diagspec_VI.nc and diagspec_IR.nc files (along with the standard diagfi.nc file), which contain the spectra of OLR, OSR, GSR, etc.

Note: The resolution of the spectra is defined by that of the correlated-k (opacity) files used for the simulation.

=== Statistical outputs ===

TBD (explain how to compute stats.nc files as well as what is inside)

== How to Change Vertical and Horizontal Resolutions ==

=== When you are using the regular longitude/latitude horizontal grid ===
To run at a different grid resolution than available initial conditions files, one needs to use the tools ''newstart.e'' and ''start2archive.e''

For example, to create initial states at grid resolution 32×24×25 from NetCDF files start and startfi at grid resolution 64×48×32 :

* Create file ''start_archive.nc'' with ''start2archive.e'' compiled at grid resolution 64×48×32 using old file ''z2sig.def'' used previously
* Create files ''restart.nc'' and ''restartfi.nc'' with ''newstart.e'' compiled at grid resolution 32×24×25, using a new file ''z2sig.def'' (more details below on the choice of the ''z2sig.def'').
* While executing ''newstart.e'', you need to choose the answer '0 - from a file start_archive' and then press enter to all other requests.

==== What you need to ''know'' about the ''z2sig.def'' file ====

For a model with Nlay layers, the [[The_z2sig.def_Input_File | z2sig.def]] file must contain at least Nlay+1 lines (the other not being read).

The first line is a scale height ($$H$$). The following lines are the target pseudo-altitudes for the model from the bottom up ($$z_i$$).
The units do not matter as long as you use the same ones for both.

The model will use these altitudes to compute a target pressure grid ($$p_i$$ ) as follows:
\begin{align}
\label{def:pseudoalt}
p_i &= p_s \exp(-z_i/H),
\end{align}
where $$p_s$$ is the surface pressure.

As you can see, the scale height and pseudo altitudes enter the equation only through their ratio. So they do not have to to be the real scale-height and altitudes of the atmosphere you are simulating.
So you can use the same [[The_z2sig.def_Input_File | z2sig.def]].def for different planets.

There is no hard rule to follow to determine the altitude/pressure levels you should use. As a rule of thumb, layers should be thiner near the surface to properly resolve the surface boundary layer. Then they should gradually increase in size over a couple scale heights and transition to constant thickness above that. Of course, some specific applications may require thinner layers in some specific parts of the atmospheres.

A little trick for those who prefer to think in terms of (log)pressure: if you use $$H= 1/\ln 10 \approx 0.43429448$$, then $$z_i=x$$ corresponds to a pressure difference with the surface of exactly x pressure decades (i.e. at $$z=1$$, $$p=0.1p_s$$). This is particularly useful for giant-planet applications.



'''WE SHOULD WRITE A PAGE (LINK HERE) ABOUT HYBRID VERTICAL COORDINATES'''

=== When you are using the DYNAMICO icosahedral horizontal grid ===

The horizontal resolution for the DYNAMICO dynamical core is managed from several setting files, online during the execution.
To this purpose, each part of the GCM managing the in/output fields ('''ICOSAGCM''', '''ICOSA_LMDZ''', '''XIOS''') requires to know the input and output grids:

'''1. ''context_lmdz_physics.xml'':'''

You can find several grid setup already defined:

<syntaxhighlight lang="xml" line>
<domain_definition>
<domain id="dom_96_95" ni_glo="96" nj_glo="95" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_144_142" ni_glo="144" nj_glo="142" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_512_360" ni_glo="512" nj_glo="360" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_720_360" ni_glo="720" nj_glo="360" type="rectilinear">
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_out" domain_ref="dom_720_360"/>
</domain_definition>
</syntaxhighlight>

In this example, the output grid for the physics fields is defined by

<syntaxhighlight lang="xml">
<domain id="dom_out" domain_ref="dom_720_360"/>
</syntaxhighlight>

which is an half-degree horizontal resolution. To change this resolution, you have to change name of the '''domain_ref''' grid, for instance:

<syntaxhighlight lang="xml">
<domain id="dom_out" domain_ref="dom_96_95"/>
</syntaxhighlight>

'''2. ''run_icosa.def'': setting file to execute a simulation'''

In this file, regarding of the horizontal resolution intended, you have to set the number of subdivision on the main triangle.
For reminder, each hexagonal mesh is divided in several main triangles and each main triangles are divided in suitable number of sub-triangles according the horizontal resolution

<syntaxhighlight lang="bash" line>
#nbp --> number of subdivision on a main triangle: integer (default=40)
# nbp = sqrt((nbr_lat x nbr_lon)/10)
# nbp: 20 40 80 160
# T-edge length (km): 500 250 120 60
# Example: nbp(128x96) = 35 -> 40
# nbp(256x192)= 70 -> 80
# nbp(360x720)= 160 -> 160
nbp = 160
</syntaxhighlight>

If you have chosen the 96_95 output grid in ''context_lmdz_physics.xml'', you have to calculate $$nbp = \sqrt(96x95) / 10 = 10$$ and in this case

<syntaxhighlight lang="bash">
nbp = 20
</syntaxhighlight>

After the number of subdivision of the main triangle, you have to define the number subdivision over each direction. At this stage you need to be careful as the number of subdivisions on each direction:
* needs to be set according to the number of subdivisions on the main triangle '''nbp'''
* will determine the number of processors on which the GCM will be most effective

<syntaxhighlight lang="bash" line>
## sub splitting of main rhombus : integer (default=1)
#nsplit_i=1
#nsplit_j=1
#omp_level_size=1
###############################################################
## There must be less MPIxOpenMP processes than the 10 x nsplit_i x nsplit_j tiles
## typically for pure MPI runs, let nproc = 10 x nsplit_i x nsplit_j
## it is better to have nbp/nsplit_i > 10 and nbp/nplit_j > 10
###############################################################
#### 40 noeuds de 24 processeurs = 960 procs
nsplit_i=12
nsplit_j=8

#### 50 noeuds de 24 processeurs = 1200 procs
#nsplit_i=10
#nsplit_j=12
</syntaxhighlight>

With the same example as above, the 96_95 output grid requires:
$$nsplit_i < 2$$ and $$nsplit_j < 2$$
We advise you to select:

<syntaxhighlight lang="bash">
## sub splitting of main rhombus : integer (default=1)
nsplit_i=1
nsplit_j=1
</syntaxhighlight>

and using 10 processors.

== How to Change the Topography (or remove it) ==

The generic model can use in principle any type of surface topography, provided that the topographic data file is available in the right format, and put in the right place. The information content on the surface topography is contained in the ''startfi.nc'', and we do have developed tools (see below) to modify the ''startfi.nc'' to account for a new surface topography.

To change the surface topography of a simulation, we recommend to follow the procedure detailed below:

* Create file ''start_archive.nc'' with ''start2archive.e'' compiled at the same (horizontal and vertical) resolution than the ''start.nc'' and ''startfi.nc'' files.
* Create files ''restart.nc'' and ''restartfi.nc'' with ''newstart.e'' compiled again at the same (horizontal and vertical) resolution.
* While executing ''newstart.e'', you need to choose the answer '0 - from a file start_archive' and then press enter to all other requests.
* At some point, the script ''newstart.e'' asks you to chose the surface topography you want from the list of files available in your 'datagcm/surface_data/' directory.

We do have a repository of for Venus, Earth and Mars through time available at https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/surface_data/. You can download the surface topography files and place them in your 'datagcm/surface_data/' directory.

We also offer a tutorial to design new topography maps here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Surface_Topography_Files

----

Special note: To remove the topography, you can simply add the following tag in callphys.def (but currently, this only works if ''callsoil=.false.''):

<pre>
nosurf = .true.
</pre>

== How to Change the Stellar Spectrum ==

To simulate the effect of the star's radiation on a given planetary atmosphere, it is necessary to accurately represent the stellar spectrum (spectral shape and total bolometric flux) at the top of this atmosphere. In the model, we have set up two different options to model the stellar spectra of any star.

=== Black Body Stellar Spectra ===

First, it is possible to simply use a black body. In this case, the stellar spectrum depends only on the effective temperature of the star which is provided to the model.

For this, you need to activate the option 'stelbbody' in the [[The_callphys.def_Input_File | callphys.def]] file, as follows:

<pre>
stelbbody = .true.
</pre>

and then add, also in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
stelTbb = 3500.0
</pre>

to specify the effective temperature of the host star. (in this example, we have chosen a M-star with an effective temperature of 3500K)

=== Pre-Tabulated spectra ===

Second, the model can read a file containing any pre-computed stellar spectrum. Traditionally, we have used synthetic spectra from the PHOENIX database, that we adapt to the Generic PCM by decreasing the spectral resolution (use 10000 points with a fixed spectral resolution of 0.001 micron) and by adapting the units (in W/m2/micron). This is the option that is generally preferred to better represent the effect of the star (whose real spectrum can strongly deviate from the black body approximation).

For this, you need to make sure the option 'stelbbody' in the [[The_callphys.def_Input_File | callphys.def]] file is equal to .false. (it not specified, by default stelbbody is assumed to be .false.).

Then you need to add in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
startype = 1
</pre>

and change the value of startype depending on the star you want to model. (here 1 means we use the solar spectrum)

To know which stellar spectra are available, you need to open the file LMDZ.GENERIC/libf/phystd/ave_stelspec.F90 and adapt the value of startype accordingly. You also need to make sure the spectra are available in your /datadir/stellar_spectra (or /datagcm/stellar_spectra) directory.

To calculate the true stellar spectrum at the top of the atmosphere, the Generic PCM renormalizes the stellar spectrum by the bolometric flux at 1 Astronomical Unit (AU) provided by the user, which it then converts into the true stellar spectrum by using the star-planet distance.

To specify the flux at 1 AU, you need to add in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
Fat1AU = 1366.0
</pre>

(here 1366W/m2 corresponds to the flux at 1AU for the Sun)

1st NOTE: We will improve this second part (Martin and Mathilde) by the end of 2022.

2nd NOTE: The Generic PCM eventually has the capability to run without any stellar flux. To do that, you can simply put Fat1AU = 0. (@LUCAS_TEINTURIER, could you check that?)

== How to Change the Opacity Tables ==

'''SECTION ET PAGES EN COURS D'ÉCRITURE'''

The model uses opacity tables to compute heating rates throughout the atmosphere. These opacity tables are generated "offline" for a given set of pressures, temperatures, a given composition and a specific spectral decomposition.

=== Getting opacity tables for your desired atmospheric composition ===

* You should first check our common repository here (https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/corrk_data/) to check whether your desired opacity table is not already included.
We have a dedicated page on how to build new opacity tables here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Opacity_Tables ; N

=== Implementing your opacity tables in the Generic PCM ===

, follow these steps:

* copy your directory containing your opacity tables in .... It has to contain... [[Building_Opacity_Tables | you need to build opacity tables]]
* change corrkdir = ... in [[The_callphys.def_Input_File | callphys.def]] with the name of that directory.
* change gases.def: it has to be consistent with Q.dat. A special case concerns...
* change -b option when compiling the model with makelmdz_fcm: it has to correspond to the number of bands (in the IR x in the visible) of the new opacity tables. For instance, compile with -b 38x26 if you used 38 bands in the infrared and 26 in the visible to generate the opacity tables.

== How to Change the Aerosols Optical Properties ==

Aerosol optical properties are represented using three distinct properties: the extinction coefficient (Q_ext), the single scattering albedo (omega) and the asymmetry factor (g).

The Generic PCM can compute the radiative effects of any aerosol, provided that they optical properties (Q_ext, omega, g) are tabulated and provided in the right format.

=== Getting optical properties for your aerosols ===

* You should first check our common repository here (https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/aerosol_properties/) to check whether your favorite aerosol is not already included. The optical properties of each aerosol is built using two distinct files: one in the 'visible' (which is used in the visible part of the radiative transfer, to compute the fate of stellar radiation) and one in the 'infrared' (which is used in the thermal infrared part of the radiative transfer, to compute the fate of thermal emission by the surface and atmosphere).

For instance, if you want to include the radiative effect of CO2 ice clouds, then you just need the files:

- optprop_co2ice_vis_n50.dat

- optprop_co2ice_ir_n50.dat

* Otherwise, you can create your own tables of optical properties, using existing databases. Check this page to learn how to do this: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Tables_Of_Aerosol_Optical_Properties

=== Implementing your aerosols in the Generic PCM ===

Before including the aerosol scheme you want to use, you need to indicate the number of aerosol layers in callphys.def with the option '''naerkind=#number_of_aerosol_layers'''.

There are two available aerosol schemes you can use to add a new aerosol in the Generic PCM:

====The n-layer aerosol scheme====

You can use the n-layer scheme (implemented by Jan Vatant d'Ollone) to easily prescribe an aerosol vertical distribution. The scheme is activated by adding ''aeronlay=.true''. in callphys.def.

In this scheme, each layer can have different optical properties, particle sizes, etc. You can use different options (e.g. to fix the aerosol distribution between two atmospheric pressures) of the scheme with ''aeronlay_choice = 1'', 2, etc.

Firstly, you have to precise the number of aerosols of this scheme. For instance,:
<pre>
nlayaero = 3
</pre>

Then, you can indicate the properties of your aerosol layer(s) one after the other on the same line. For 3 aerosol layers, we have:

<pre>
aeronlay_tauref = 1.0 0.05 0.03
aeronlay_lamref = 0.8e-6 0.8e-6 0.8e-6
aeronlay_choice = 2 2 2
aeronlay_pbot = 2.0e5 1.6e5 0.2e5
aeronlay_ptop = 0.10e5 2.0e5 1.
aeronlay_sclhght = 0.1 2.0 0.1
aeronlay_size = 0.8e-6 0.05e-6 2.5e-6
aeronlay_nueff = 0.3 0.3 0.3
</pre>

''aeronlay_tauref'' is the optical depth at the reference wavelength ''aeronlay_lamref'' (in metres). The ''aeronlay_choice'' allows you to choose the aerosol distribution between the bottom pressure (''aeronlay_pbot'') and the top pressure (''aeronlay_ptop'') of the aerosol layer (''aeronlay_choice=1'') or from a bottom pressure (''aeronlay_pbot'') with a fractional scale height (''aeronlay_sclhght'') with the ''aeronlay_choice=2''. For the corresponding layer, if you use the choice=2, the ''aeronlay_ptop'' is deactivated. In the same way, for the choice=1, the ''aeronlay_sclhght'' is deactivated for the corresponding layer. You can choose the mean radius of the particles with the ''aeronlay_size'' (in metres) and the corresponding effective standard deviation with ''aeronlay_nueff''.

And finally, you will need to provide the name of your aerosols optical properties tables in callphys.def. For instance:

<pre>
optprop_aeronlay_vis = optprop_neptune_n2_vis_n30.dat optprop_neptune_n3_vis_n30.dat optprop_ch4_vis.dat
optprop_aeronlay_ir = optprop_neptune_n2_ir_n30.dat optprop_neptune_n3_ir_n30.dat optprop_ch4_ir.dat
</pre>
(here, to use the optical properties of aerosols for Neptune)

We encourage you to search for the keyword "aeronlay" in the source code to use more specific options of the scheme.

====The generic condensable scheme====

You can use the generic condensable scheme (implemented by Lucas Teinturier) to easily compute the radiative effect of cloud particles formed by condensation.
The scheme has to be used conjointly to the generic condensation scheme.
To activate it, one needs to add the '''aerogeneric=n'''( with n>0 the number of condensable species handled in the scheme) in callphys.def. On top of that, one needs to add in traceur.def, on the solid/ice traceur the option '''is_rgcs = 1'''.

In the model itself, one needs then to manually add your aerosol by modifying the suaer_corrk.F90 routine (if your favorite aerosol is not there yet already) to specify the correct names for your condensing species. A more dynamical/flexible approach will be added at some point, so one won't need to directly modify the code.

More information on how to use the scheme is provided here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Radiative_Generic_Condensable_Specie

Don’t forget to add your new aerosol species in traceur.def and adapt the -t option (with the correct number of radiatively active aerosols) at the compilation stage.

== How to Manage Tracers ==

Tracers are managed thanks to the [https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/The_traceur.def_Input_File ''traceur.def''] file.

Specific treatment of some tracers (e.g., water vapor cycle) can be added directly in the model and an option added in [https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/The_callphys.def_Input_File ''callphys.def''] file.

== Use the Z of LMDZ : Zoomed version ==

Do we need this? Has anyone already made use of the zoom module?

[[Category:Generic-Model]]

Advanced Use of the GCM

2024-06-19T16:20:03Z

SBhatnagar: /* How to Change the Topography (or remove it) */

== Running in parallel ==

For large simulation (long run, high resolution etc...), the computational cost can be huge and hence the run time very long.
To overcome this issue, the model can be run in parallel. This however requires a few extra steps (compared to compiling and running the serial version of the code).
For all the details see [[Parallelism | the dedicated page]].

== Disambiguation between ifort, mpif90, etc. ==

For users not used to compilers and/or compiling and running codes in parallel, namely in MPI mode, there is often some confusion which hopefully the following paragraph might help clarify:
* the compiler (typically gfortran, ifort, pgfortran, etc.) is the required tool to compile the Fortran source code and generate an executable. It is strongly recommended that libraries used by a program are also compiled using the same compiler. Thus if you plan to use different compilers to compile the model, note that you should also have at hand versions of the libraries it uses also compiled with these compilers.
* the MPI (Message Passing Interface) library is a library used to solve problems using multiple processes by enabling message-passing between the otherwise independent processes. There are a number of available MPI libraries out there, e.g. OpenMPI, MPICH or IntelMPI to name a few (you can check out the [[Building an MPI library]] page for some information about installing an MPI library). The important point here is that on a given machine the MPI library is related to a given compiler and that it provides related wrappers to compile and run with. Typically (but not always) the compiler wrapper is '''mpif90''' and the execution wrapper is '''mpirun'''. If you want to know which compiler is wrapped in the '''mpif90''' compiler wrapper, check out the output of:
<pre>
mpif90 --version
</pre>
* In addition a second type of parallelism, shared memory parallelism known as OpenMP, is also implemented in the code. In contradistinction to MPI, OpenMP does not require an external library but is instead implemented as a compiler feature. At run time one must then specify some dedicated environment variables (such as OMP_NUM_THREADS and OMP_STACKSIZE) to specify the number of threads to use per process.
* In practice one should favor compiling and running with both MPI and OPenMP enabled.
* For much more detailed information about compiling and running in parallel, check out the [[Parallelism | the page dedicated to Parallelism]].

== A word about the IOIPSL and XIOS libraries ==
* The IOIPSL (Input Output IPSL) library is a library that has developed by the IPSL community to handle input and outputs of (mostly terrestrial) climate models. For the Generic PCM only a small part of this library is actually used, related to reading and processing the input [[The_run.def_Input_File | run.def]] file. For more details check out the [[The IOIPSL Library]] page.
* The [https://forge.ipsl.jussieu.fr/ioserver/wiki XIOS] (Xml I/O Server) library is based on client-server principles where the server manages the outputs asynchronously from the client (the climate model) so that the bottleneck of writing data in a parallel environment is alleviated. All aspects of the outputs (name, units, file, post-processing operations, etc.) are then controlled by dedicated XML files which are read at run-time. Using XIOS is currently optional (and requires compiling the GCM with the XIOS library). More about the XIOS library, how to install and use it, etc. [[The XIOS Library| here]].

== Playing with the output files ==

=== Changing the output temporal resolution and time duration ===

* To change the total time of a simulation, you need to open the 'For all the details see [[The_run.def_Input_File | run.def]]. file and change the variable 'nday':

<pre>
nday = 1000 # this means the simulation will run for 1000 days ; and that the associated output files will also be computed for a total duration of 1000 days
</pre>

Note: in the example, they are not necessarily 1000 Earth days, because it depends on the definition of the day duration that has been taken in the start files.

* To change the temporal resolution of the output files, you need to open the [[The_run.def_Input_File | run.def]] file and change the variable 'ecritphy':

<pre>
ecritphy = 200 # this means the simulation will write variables in the output files every 200 time steps of the simulation.
</pre>

Note: The output temporal resolution of the output files then depends also on the number of timestep per day ('day_step' variable in [[The_run.def_Input_File | run.def]] file). In this example:

<pre>
nday = 1000
daystep = 800
ecritphy = 200
</pre>

The output file will provide results every 0.25 days (800/200), and for a total duration of 1000 days (so 4000 time values in total).

=== Changing the output variable ===

To select the variable provided in the output file diagfi.nc, you simply need to add the list of variables needed in the [[The_diagfi.def_Input_File | diagfi.def]].

Note for experts: Some technical variables need to be de-commented in 'physiq_mod.F90' file to be written in the output files.

=== Spectral outputs ===

It is possible to provide spectral outputs such as the OLR (Outgoing Longwave Radiation, i.e. the thermal emission of the planet at the top of the atmosphere), the OSR (Outgoing Stellar Radiation, i.e. the light reflected by the planet at the top of the atmosphere), or the GSR (Ground Stellar Radiation, i.e. the light emitted by the star that reaches the surface of the planet).

For this, you need to activate the option 'specOLR' in the [[The_callphys.def_Input_File | callphys.def]] file, as follows:

<pre>
specOLR = .true.
</pre>

The simulations will then create diagspec_VI.nc and diagspec_IR.nc files (along with the standard diagfi.nc file), which contain the spectra of OLR, OSR, GSR, etc.

Note: The resolution of the spectra is defined by that of the correlated-k (opacity) files used for the simulation.

=== Statistical outputs ===

TBD (explain how to compute stats.nc files as well as what is inside)

== How to Change Vertical and Horizontal Resolutions ==

=== When you are using the regular longitude/latitude horizontal grid ===
To run at a different grid resolution than available initial conditions files, one needs to use the tools ''newstart.e'' and ''start2archive.e''

For example, to create initial states at grid resolution 32×24×25 from NetCDF files start and startfi at grid resolution 64×48×32 :

* Create file ''start_archive.nc'' with ''start2archive.e'' compiled at grid resolution 64×48×32 using old file ''z2sig.def'' used previously
* Create files ''restart.nc'' and ''restartfi.nc'' with ''newstart.e'' compiled at grid resolution 32×24×25, using a new file ''z2sig.def'' (more details below on the choice of the ''z2sig.def'').
* While executing ''newstart.e'', you need to choose the answer '0 - from a file start_archive' and then press enter to all other requests.

==== What you need to ''know'' about the ''z2sig.def'' file ====

For a model with Nlay layers, the [[The_z2sig.def_Input_File | z2sig.def]] file must contain at least Nlay+1 lines (the other not being read).

The first line is a scale height ($$H$$). The following lines are the target pseudo-altitudes for the model from the bottom up ($$z_i$$).
The units do not matter as long as you use the same ones for both.

The model will use these altitudes to compute a target pressure grid ($$p_i$$ ) as follows:
\begin{align}
\label{def:pseudoalt}
p_i &= p_s \exp(-z_i/H),
\end{align}
where $$p_s$$ is the surface pressure.

As you can see, the scale height and pseudo altitudes enter the equation only through their ratio. So they do not have to to be the real scale-height and altitudes of the atmosphere you are simulating.
So you can use the same [[The_z2sig.def_Input_File | z2sig.def]].def for different planets.

There is no hard rule to follow to determine the altitude/pressure levels you should use. As a rule of thumb, layers should be thiner near the surface to properly resolve the surface boundary layer. Then they should gradually increase in size over a couple scale heights and transition to constant thickness above that. Of course, some specific applications may require thinner layers in some specific parts of the atmospheres.

A little trick for those who prefer to think in terms of (log)pressure: if you use $$H= 1/\ln 10 \approx 0.43429448$$, then $$z_i=x$$ corresponds to a pressure difference with the surface of exactly x pressure decades (i.e. at $$z=1$$, $$p=0.1p_s$$). This is particularly useful for giant-planet applications.



'''WE SHOULD WRITE A PAGE (LINK HERE) ABOUT HYBRID VERTICAL COORDINATES'''

=== When you are using the DYNAMICO icosahedral horizontal grid ===

The horizontal resolution for the DYNAMICO dynamical core is managed from several setting files, online during the execution.
To this purpose, each part of the GCM managing the in/output fields ('''ICOSAGCM''', '''ICOSA_LMDZ''', '''XIOS''') requires to know the input and output grids:

'''1. ''context_lmdz_physics.xml'':'''

You can find several grid setup already defined:

<syntaxhighlight lang="xml" line>
<domain_definition>
<domain id="dom_96_95" ni_glo="96" nj_glo="95" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_144_142" ni_glo="144" nj_glo="142" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_512_360" ni_glo="512" nj_glo="360" type="rectilinear" >
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_720_360" ni_glo="720" nj_glo="360" type="rectilinear">
<generate_rectilinear_domain/>
<interpolate_domain order="1"/>
</domain>

<domain id="dom_out" domain_ref="dom_720_360"/>
</domain_definition>
</syntaxhighlight>

In this example, the output grid for the physics fields is defined by

<syntaxhighlight lang="xml">
<domain id="dom_out" domain_ref="dom_720_360"/>
</syntaxhighlight>

which is an half-degree horizontal resolution. To change this resolution, you have to change name of the '''domain_ref''' grid, for instance:

<syntaxhighlight lang="xml">
<domain id="dom_out" domain_ref="dom_96_95"/>
</syntaxhighlight>

'''2. ''run_icosa.def'': setting file to execute a simulation'''

In this file, regarding of the horizontal resolution intended, you have to set the number of subdivision on the main triangle.
For reminder, each hexagonal mesh is divided in several main triangles and each main triangles are divided in suitable number of sub-triangles according the horizontal resolution

<syntaxhighlight lang="bash" line>
#nbp --> number of subdivision on a main triangle: integer (default=40)
# nbp = sqrt((nbr_lat x nbr_lon)/10)
# nbp: 20 40 80 160
# T-edge length (km): 500 250 120 60
# Example: nbp(128x96) = 35 -> 40
# nbp(256x192)= 70 -> 80
# nbp(360x720)= 160 -> 160
nbp = 160
</syntaxhighlight>

If you have chosen the 96_95 output grid in ''context_lmdz_physics.xml'', you have to calculate $$nbp = \sqrt(96x95) / 10 = 10$$ and in this case

<syntaxhighlight lang="bash">
nbp = 20
</syntaxhighlight>

After the number of subdivision of the main triangle, you have to define the number subdivision over each direction. At this stage you need to be careful as the number of subdivisions on each direction:
* needs to be set according to the number of subdivisions on the main triangle '''nbp'''
* will determine the number of processors on which the GCM will be most effective

<syntaxhighlight lang="bash" line>
## sub splitting of main rhombus : integer (default=1)
#nsplit_i=1
#nsplit_j=1
#omp_level_size=1
###############################################################
## There must be less MPIxOpenMP processes than the 10 x nsplit_i x nsplit_j tiles
## typically for pure MPI runs, let nproc = 10 x nsplit_i x nsplit_j
## it is better to have nbp/nsplit_i > 10 and nbp/nplit_j > 10
###############################################################
#### 40 noeuds de 24 processeurs = 960 procs
nsplit_i=12
nsplit_j=8

#### 50 noeuds de 24 processeurs = 1200 procs
#nsplit_i=10
#nsplit_j=12
</syntaxhighlight>

With the same example as above, the 96_95 output grid requires:
$$nsplit_i < 2$$ and $$nsplit_j < 2$$
We advise you to select:

<syntaxhighlight lang="bash">
## sub splitting of main rhombus : integer (default=1)
nsplit_i=1
nsplit_j=1
</syntaxhighlight>

and using 10 processors.

== How to Change the Topography (or remove it) ==

The generic model can use in principle any type of surface topography, provided that the topographic data file is available in the right format, and put in the right place. The information content on the surface topography is contained in the ''startfi.nc'', and we do have developed tools (see below) to modify the ''startfi.nc'' to account for a new surface topography.

To change the surface topography of a simulation, we recommend to follow the procedure detailed below:

* Create file ''start_archive.nc'' with ''start2archive.e'' compiled at the same (horizontal and vertical) resolution than the ''start.nc'' and ''startfi.nc'' files.
* Create files ''restart.nc'' and ''restartfi.nc'' with ''newstart.e'' compiled again at the same (horizontal and vertical) resolution.
* While executing ''newstart.e'', you need to choose the answer '0 - from a file start_archive' and then press enter to all other requests.
* At some point, the script ''newstart.e'' asks you to chose the surface topography you want from the list of files available in your 'datagcm/surface_data/' directory.

We do have a repository of for Venus, Earth and Mars through time available at https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/surface_data/. You can download the surface topography files and place them in your 'datagcm/surface_data/' directory.

We also offer a tutorial to design new topography maps here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Surface_Topography_Files

----

Special note: To remove the topography, you can simply add the following tag in callphys.def (but currently, this only works if callsoil=.false.):

<pre>
nosurf = .true.
</pre>

== How to Change the Stellar Spectrum ==

To simulate the effect of the star's radiation on a given planetary atmosphere, it is necessary to accurately represent the stellar spectrum (spectral shape and total bolometric flux) at the top of this atmosphere. In the model, we have set up two different options to model the stellar spectra of any star.

=== Black Body Stellar Spectra ===

First, it is possible to simply use a black body. In this case, the stellar spectrum depends only on the effective temperature of the star which is provided to the model.

For this, you need to activate the option 'stelbbody' in the [[The_callphys.def_Input_File | callphys.def]] file, as follows:

<pre>
stelbbody = .true.
</pre>

and then add, also in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
stelTbb = 3500.0
</pre>

to specify the effective temperature of the host star. (in this example, we have chosen a M-star with an effective temperature of 3500K)

=== Pre-Tabulated spectra ===

Second, the model can read a file containing any pre-computed stellar spectrum. Traditionally, we have used synthetic spectra from the PHOENIX database, that we adapt to the Generic PCM by decreasing the spectral resolution (use 10000 points with a fixed spectral resolution of 0.001 micron) and by adapting the units (in W/m2/micron). This is the option that is generally preferred to better represent the effect of the star (whose real spectrum can strongly deviate from the black body approximation).

For this, you need to make sure the option 'stelbbody' in the [[The_callphys.def_Input_File | callphys.def]] file is equal to .false. (it not specified, by default stelbbody is assumed to be .false.).

Then you need to add in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
startype = 1
</pre>

and change the value of startype depending on the star you want to model. (here 1 means we use the solar spectrum)

To know which stellar spectra are available, you need to open the file LMDZ.GENERIC/libf/phystd/ave_stelspec.F90 and adapt the value of startype accordingly. You also need to make sure the spectra are available in your /datadir/stellar_spectra (or /datagcm/stellar_spectra) directory.

To calculate the true stellar spectrum at the top of the atmosphere, the Generic PCM renormalizes the stellar spectrum by the bolometric flux at 1 Astronomical Unit (AU) provided by the user, which it then converts into the true stellar spectrum by using the star-planet distance.

To specify the flux at 1 AU, you need to add in the [[The_callphys.def_Input_File | callphys.def]] file, the following line:

<pre>
Fat1AU = 1366.0
</pre>

(here 1366W/m2 corresponds to the flux at 1AU for the Sun)

1st NOTE: We will improve this second part (Martin and Mathilde) by the end of 2022.

2nd NOTE: The Generic PCM eventually has the capability to run without any stellar flux. To do that, you can simply put Fat1AU = 0. (@LUCAS_TEINTURIER, could you check that?)

== How to Change the Opacity Tables ==

'''SECTION ET PAGES EN COURS D'ÉCRITURE'''

The model uses opacity tables to compute heating rates throughout the atmosphere. These opacity tables are generated "offline" for a given set of pressures, temperatures, a given composition and a specific spectral decomposition.

=== Getting opacity tables for your desired atmospheric composition ===

* You should first check our common repository here (https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/corrk_data/) to check whether your desired opacity table is not already included.
We have a dedicated page on how to build new opacity tables here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Opacity_Tables ; N

=== Implementing your opacity tables in the Generic PCM ===

, follow these steps:

* copy your directory containing your opacity tables in .... It has to contain... [[Building_Opacity_Tables | you need to build opacity tables]]
* change corrkdir = ... in [[The_callphys.def_Input_File | callphys.def]] with the name of that directory.
* change gases.def: it has to be consistent with Q.dat. A special case concerns...
* change -b option when compiling the model with makelmdz_fcm: it has to correspond to the number of bands (in the IR x in the visible) of the new opacity tables. For instance, compile with -b 38x26 if you used 38 bands in the infrared and 26 in the visible to generate the opacity tables.

== How to Change the Aerosols Optical Properties ==

Aerosol optical properties are represented using three distinct properties: the extinction coefficient (Q_ext), the single scattering albedo (omega) and the asymmetry factor (g).

The Generic PCM can compute the radiative effects of any aerosol, provided that they optical properties (Q_ext, omega, g) are tabulated and provided in the right format.

=== Getting optical properties for your aerosols ===

* You should first check our common repository here (https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/datagcm/aerosol_properties/) to check whether your favorite aerosol is not already included. The optical properties of each aerosol is built using two distinct files: one in the 'visible' (which is used in the visible part of the radiative transfer, to compute the fate of stellar radiation) and one in the 'infrared' (which is used in the thermal infrared part of the radiative transfer, to compute the fate of thermal emission by the surface and atmosphere).

For instance, if you want to include the radiative effect of CO2 ice clouds, then you just need the files:

- optprop_co2ice_vis_n50.dat

- optprop_co2ice_ir_n50.dat

* Otherwise, you can create your own tables of optical properties, using existing databases. Check this page to learn how to do this: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Building_Tables_Of_Aerosol_Optical_Properties

=== Implementing your aerosols in the Generic PCM ===

Before including the aerosol scheme you want to use, you need to indicate the number of aerosol layers in callphys.def with the option '''naerkind=#number_of_aerosol_layers'''.

There are two available aerosol schemes you can use to add a new aerosol in the Generic PCM:

====The n-layer aerosol scheme====

You can use the n-layer scheme (implemented by Jan Vatant d'Ollone) to easily prescribe an aerosol vertical distribution. The scheme is activated by adding ''aeronlay=.true''. in callphys.def.

In this scheme, each layer can have different optical properties, particle sizes, etc. You can use different options (e.g. to fix the aerosol distribution between two atmospheric pressures) of the scheme with ''aeronlay_choice = 1'', 2, etc.

Firstly, you have to precise the number of aerosols of this scheme. For instance,:
<pre>
nlayaero = 3
</pre>

Then, you can indicate the properties of your aerosol layer(s) one after the other on the same line. For 3 aerosol layers, we have:

<pre>
aeronlay_tauref = 1.0 0.05 0.03
aeronlay_lamref = 0.8e-6 0.8e-6 0.8e-6
aeronlay_choice = 2 2 2
aeronlay_pbot = 2.0e5 1.6e5 0.2e5
aeronlay_ptop = 0.10e5 2.0e5 1.
aeronlay_sclhght = 0.1 2.0 0.1
aeronlay_size = 0.8e-6 0.05e-6 2.5e-6
aeronlay_nueff = 0.3 0.3 0.3
</pre>

''aeronlay_tauref'' is the optical depth at the reference wavelength ''aeronlay_lamref'' (in metres). The ''aeronlay_choice'' allows you to choose the aerosol distribution between the bottom pressure (''aeronlay_pbot'') and the top pressure (''aeronlay_ptop'') of the aerosol layer (''aeronlay_choice=1'') or from a bottom pressure (''aeronlay_pbot'') with a fractional scale height (''aeronlay_sclhght'') with the ''aeronlay_choice=2''. For the corresponding layer, if you use the choice=2, the ''aeronlay_ptop'' is deactivated. In the same way, for the choice=1, the ''aeronlay_sclhght'' is deactivated for the corresponding layer. You can choose the mean radius of the particles with the ''aeronlay_size'' (in metres) and the corresponding effective standard deviation with ''aeronlay_nueff''.

And finally, you will need to provide the name of your aerosols optical properties tables in callphys.def. For instance:

<pre>
optprop_aeronlay_vis = optprop_neptune_n2_vis_n30.dat optprop_neptune_n3_vis_n30.dat optprop_ch4_vis.dat
optprop_aeronlay_ir = optprop_neptune_n2_ir_n30.dat optprop_neptune_n3_ir_n30.dat optprop_ch4_ir.dat
</pre>
(here, to use the optical properties of aerosols for Neptune)

We encourage you to search for the keyword "aeronlay" in the source code to use more specific options of the scheme.

====The generic condensable scheme====

You can use the generic condensable scheme (implemented by Lucas Teinturier) to easily compute the radiative effect of cloud particles formed by condensation.
The scheme has to be used conjointly to the generic condensation scheme.
To activate it, one needs to add the '''aerogeneric=n'''( with n>0 the number of condensable species handled in the scheme) in callphys.def. On top of that, one needs to add in traceur.def, on the solid/ice traceur the option '''is_rgcs = 1'''.

In the model itself, one needs then to manually add your aerosol by modifying the suaer_corrk.F90 routine (if your favorite aerosol is not there yet already) to specify the correct names for your condensing species. A more dynamical/flexible approach will be added at some point, so one won't need to directly modify the code.

More information on how to use the scheme is provided here: https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/Radiative_Generic_Condensable_Specie

Don’t forget to add your new aerosol species in traceur.def and adapt the -t option (with the correct number of radiatively active aerosols) at the compilation stage.

== How to Manage Tracers ==

Tracers are managed thanks to the [https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/The_traceur.def_Input_File ''traceur.def''] file.

Specific treatment of some tracers (e.g., water vapor cycle) can be added directly in the model and an option added in [https://lmdz-forge.lmd.jussieu.fr/mediawiki/Planets/index.php/The_callphys.def_Input_File ''callphys.def''] file.

== Use the Z of LMDZ : Zoomed version ==

Do we need this? Has anyone already made use of the zoom module?

[[Category:Generic-Model]]

Slab ocean model

2024-06-04T06:14:14Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-06-04T06:13:55Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

#Ekman transport: (default=.false.)
slab_ekman = .true.
#Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-06-04T06:13:44Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

#Ekman transport: (default=.false.)
slab_ekman = .true.

#Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-06-04T06:13:32Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

#Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-06-04T06:13:22Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-06-04T06:13:05Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

#Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-05-22T09:44:57Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-05-22T09:44:43Z

SBhatnagar: /* Technical aspects */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

== References ==

Slab ocean model

2024-04-03T12:12:32Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is also seen in other exoplanet-based GCMs like ROCKE-3D and ExoCAM. It is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-03T12:10:17Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice documented by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, $$A^\rm{max}_\rm{ice}$$ is found to be 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands, respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances and is typical of other exoplanet-based GCMs like ROCKE-3D and ExoCAM. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-02T15:57:18Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, we found $$A^\rm{max}_\rm{ice}$$ values of 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances and is typical of other exoplanet-based GCMs like ROCKE-3D and ExoCAM. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-02T15:56:07Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [https://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, we found $$A^\rm{max}_\rm{ice}$$ values of 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-02T15:55:36Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by [hhttps://ui.adsabs.harvard.edu/abs/2009JGRD..114.8101P/abstract Pedersen et al. (2009)]. With this, we found $$A^\rm{max}_\rm{ice}$$ values of 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-02T15:53:53Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005)]]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by Pedersen et al. (2009). With this, we found $$A^\rm{max}_\rm{ice}$$ values of 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]

Slab ocean model

2024-04-02T15:51:48Z

SBhatnagar: /* Sea ice evolution */

== General description ==

The slab ocean model is primarily based on the work of [https://ui.adsabs.harvard.edu/abs/2012ClDy...38..379C/abstract Codron (2012)]. This model uses the same horizontal grid as the GCM and is composed of two layers. The first layer (50 m depth) represents the surface mixed layer, where the exchanges with the atmosphere take place. The second layer (150 m depth) represents the deep ocean. Each oceanic grid point of the model is defined by the sea ice fraction & thickness and snow thickness.

In general for a slab ocean, the temperature at the ocean surface evolves as per the following equation:
 
$$\rho c_p H \frac{\partial T}{\partial t} = F_{surf} + Q_{flux}$$

The evolution of temperature is driven by surface fluxes ($$F_{surf} = F_{rad} + F_{sens} + F_{lat}$$) arising from the atmosphere model and a prescribed / parameterised flux ($$Q_{flux}$$) that can represent the effect of geothermal heating / ocean circulation (depends on the kind of planet that is being simulated). Both, the surface fluxes and the conductive fluxes coming from within interact with snow or ice that can be present at the ocean surface. Over each grid point, sea ice is represented by a uniform layer of depth $$H$$ and fractional area $$f$$. It may be covered by a layer of snow, which has its own heat capacity. The temperature at the base of the ice layer is always made to be equal to the freezing temperature of sea water $$T_o$$, for reasons shown below.

In its current form in the Generic-PCM, the slab ocean model handles two key processes, (a) '''Sea ice evolution''', and (b) '''Heat transport by ocean circulation''':

==== Sea ice evolution ====

Sea ice develops when the ocean temperature falls below the freezing point of water ($$T_0=–1.8°C$$, the freezing point of Earth's sea water; function of salinity) and melts when its temperature rises above freezing. The changes in ice extent and thickness are driven by energy conservation, such that the ocean temperature is kept at $$T_0$$ as long as ice is present. Specifically, if the temperature of the surface layer instantaneously falls below its freezing point, its temperature is set back to $$T_0$$, and the resulting energy difference is used to create ice. A framework of rules determines how these energy disparities can make ice thickness and coverage evolve. Conversely, if the ocean temperature becomes positive when ice is already present, its temperature is set back to $$T_0$$, and the resulting energy difference is used to melt ice. The same arguments also apply to the melting of snow if it has precipitated onto the ice.

[[File:Albedo vs ice thickness spectral.png|thumb|Variation of bare sea ice albedo as a function of ice thickness for the VIS (250-690 nm; dark green line) and NIR (690-4000 nm; red line) spectral bands. The curves have been fit to long-term Antarctic sea ice observations made by Brandt et al. (2005).]]

Further, a planet's simulated climate (and consequently, its observability) is highly sensitive to the albedo parameterisations of sea ice and snow of the GCM. [https://ui.adsabs.harvard.edu/abs/2013JGRD..11810414C/abstract Charnay et al. (2013)] addressed this by treating the albedo of bare sea ice as a function of ice thickness. The oceanic surface albedo ($$A$$) could be that of snow (if any) or bare sea ice. In the model, the bare sea ice albedo increases between the minimum and maximum albedo values as a function of ice thickness (see the light green curve in the figure on the right) and is given by:

$$A = A^\rm{max}_\rm{ice} – (A^\rm{max}_\rm{ice} – A^\rm{min}_\rm{ice} ) e^{\left ( \frac{–h_\rm{ice}}{h^0_\rm{ice}} \right )}$$

Here, we improve upon the work of Charnay et al. (2013) by instituting a spectral dependence of albedo, complementary to the existing ice thickness dependence. In the above equation, $$A^\rm{min}_\rm{ice}$$ is the minimal albedo (that of liquid water), equal to 0.06 irrespective of the incoming spectra. This is because the albedo of liquid water is almost constant between the Visible (VIS) and Near-Infrared (NIR) bands. $$A^\rm{max}_\rm{ice}$$ is the maximal albedo, which we computed by fitting the above equation to long-term albedo observations of Antarctic sea ice made by [https://ui.adsabs.harvard.edu/abs/2005JCli...18.3606B/abstract Brandt et al. (2005]; see the figure on the right) and summarised by Pedersen et al. (2009). With this, we found $$A^\rm{max}_\rm{ice}$$ values of 0.69 and 0.31 in the VIS (250-690 nm; dark green curve) and NIR (690-4000 nm; red curve) bands respectively. Further, $$h_\rm{ice}$$ is the ice thickness (in m) and $$h^0_\rm{ice}$$ is the minimum threshold of ice, taken to be 0.3 m.

While the bare sea ice albedo is resolved up to 4000 nm, the new model only distinguishes between the VIS and NIR bands concerning radiation extinction with depth. This extinction assumes solar-type surface irradiance fractions in these bands, implying that radiation beyond a certain wavelength [(> 1200 nm ?!)] is not transmitted. This solar assumption is expected to be revised later to incorporate sensitivity to different stellar spectral irradiances. Nevertheless, the new spectral dependency of albedo provides a reasonable initial approximation.

It should be noted that changes in the sea-ice model, or in parameters such as the snow albedo, can have considerably large impacts on the simulated climate. This happens through changes in the ice extent and the global temperature through the ice-albedo feedback. This effect is, however, largely independent of the heat transport schemes presented in this page.

==== Heat transport by ocean circulation ====

The transport of heat by the ocean circulation is given by four components:
# The impact of subgrid-scale eddies is represented by horizontal diffusion, with a uniform diffusivity in both layers.
# [SB WILL ADD FIGURE] The mean wind-driven circulation is computed by calculating the Ekman mass fluxes in the surface layer from the wind stress, and taking an opposite return flow at depth. We then use this mass flux to advect the slab temperature horizontally. If we have a divergence in the horizontal component of the transport, upwelling is induced (and therefore, temperature advection) between the two layers. Conversely, convergence implies downwelling. The integrated oceanic horizontal transport (mass flux) due to wind stress in the Ekman layer is given by $$ \mathbf{V_E} = - \frac{1}{f} \mathbf{k} \times \mathbf{\tau} $$, where $$\mathbf{\tau}$$ is the net surface wind stress and $$f$$ is the Coriolis parameter (orthogonal to the wind stress). This simplified model reproduces the global meridional oceanic heat transport quite closely compared to an Oceanic-GCM.
# The two-layer nature of the ocean model also facilitates a convective adjustment. For instance, if the surface layer is colder than the deep layer (for e.g., in winter at mid-high latitudes), the vertical convective adjustment ensures that the heat stored at depth is then restored to the surface. In effect, this simulates denser colder water at the surface descending to the depths.
# [SB WILL ADD FIGURE] We parameterise ocean eddies with the Gent-McWilliams scheme, which is based on oceanic isotherms. These act to mix temperatures horizontally. Figure XX represents a simplified cross-sectional view of the ocean horizontally (across latitude) and vertically (across depth). The top and bottom halves represent the mixed and deep layers of the ocean respectively. One example of an isotherm has been overplotted. The figure illustrates how the ocean surface temperature changes from warmer to cooler waters from the equator to the pole. Eddies act to reduce the slope of the isotherms, which results in high-latitude surface heating and low-latitude depth cooling. This in turn re-stratifies the vertical ocean layers as shown in the figure, which increases the temperature difference between the surface and deep ocean, making Ekman transport larger. The horizontal advective velocity is directly proportional to the slope of the isotherms.

Other ocean circulation features like density-driven / thermohaline circulation and horizontal gyres are not present in the model. While these features play an important role in shaping regional climates on modern Earth, their effect is weaker on the global average; gyres would be absent in the case of a global ocean.

== The 2024 version ==

Compared to the old model, the new model has the following changes (non-exhaustive):
# It can be used in parallel mode.
# It has a more realistic description of sea ice creation and evolution - simultaneous surface, side and bottom melting / freezing depending on fluxes.
# There is a spectral dependence of bare sea ice albedo as a function of ice thickness.
# Snow has an effective heat capacity.
# Snow has "weight"; it can sink an ice block if there is too much of it.
# Snow can be blown off by wind.
# The two-layer ocean facilitates convective adjustment.
# Diffusion can follow the Gent-McWilliams scheme + Eddy diffusivity.
# Since Ekman Transport doesn't work at the equator (Coriolis term = 0), we prescribe an alternative "Sverdrup scheme" for regions close to the equator which relies on direct friction-driven wind transport.

==== Variables of the slab ocean model which can be written as outputs in diagfi.nc: ====

# tslab1, tslab2: temperature of the surface and deep ocean layers respectively (K)
# pctsrf_sic: grid fraction of sea ice
# sea_ice: mass of sea ice (kg/m$$^2$$)
# tsea_ice: temperature of the sea ice surface (K)
# dt_hdiff1, dt_hdiff2: heating of the surface and deep ocean layers by horizontal diffusion (W/m$$^2$$)
# dt_ekman1, dt_ekman2: heating of the surface and deep ocean layers by Ekman transport (W/m$$^2$$)
# dt_gm1, dt_gm2: heating of the surface and deep ocean layers by Gent-McWilliams eddy transport (W/m$$^2$$)

== Technical aspects ==
The slab ocean model is activated in callphys.def with:
ok_slab_ocean = .true.

There are a host of transport schemes that one can switch on/off in regard to oceanic heat transport. In general, if one wants to use all schemes of ocean heat transport, they can keep the settings detailed below. '''Note that the following can only be set to true if ok_slab_ocean = .true.'''

Ekman transport: (default=.false.)
slab_ekman = .true.

Ekman zonal advection: (default=.false. ; can only be true if slab_ekman = .true.)
slab_ekman_zonadv = .true.

Gent-McWilliams Scheme (can only be true if slab_ekman = .true.):
slab_gm = .true.

Horizontal diffusion (default=.false.):
slab_hdiff = .true.

Convective adjustment (0 - no, 1 - yes):
slab_cadj = 1

The timestep for updating the slab temperature and ice fraction is given by cpl_pas (in physics timesteps; default value = once every Earth day)

rnat is a standard diagfi output, which either represents 0 for oceanic grid points or 1 for continent grid points.

The model doesn't permit surface fluxes to enter the deep ocean. The latter is present in the model for exchanges and transport to take place.

The model also does not compute dynamical sea ice drift.

A word of caution: Ekman transport doesn’t work at the equator (Coriolis parameter = 0), we institute an alternative "Sverdrup scheme”, which relies on pressure-driven transport specifically at the equator. This scheme is also important for the case of slow-rotating planets when the Coriolis parameter is also low (?!). 

QUESTION FOR FRANCIS: It seems that the Sverdrup scheme might only be beneficial for the special case of the Earth? Need to confirm.

The model follows energy conservation. However, if an upper limit on sea ice thickness is prescribed (for e.g., 10 m, for faster convergence), energy conservation can be violated. If one wants to purely conserve energy (at the cost of computation time), no limit on sea ice thickness can be prescribed. It is OK to prescribe an upper limit for the case of modern Earth (however, if your prescription is too small, it can lead to anomalous behaviour like stronger seasonal cycles). However, it might not be accurate to do this for more "exotic" cases like Snowball Earth and tidally locked ocean worlds.

Convergence - As far as the ocean is concerned, convergence timescales are governed by the thermal inertia and depth of the mixed layer. As an order of magnitude, convergence for Modern Earth with the dynamical slab ocean takes around 50 years.
[[Category:Generic-Model]]