Subgrid Slopes in the Mars PCM

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Purpose

In order to study surface processes in the Mars Planetary Climate Model (e.g., gully formation, glacier evolution, ...), a representation of the sub-grid slopes has been introduced in the Mars PCM. This parametrization is detailed in Lange et al., 2023, "Modelling Slope Microclimates in the Mars Planetary Climate Model" https://arxiv.org/pdf/2306.12449.pdf. Contact: lucas.lange@lmd.ipsl.fr

For each GCM mesh, we decompose the cell as a distribution of sloped terrains (defined by characteristic slopes) and a flat terrain. On each sub-grid terrain, we let the microclimates evolve so that the slope-specific features (e.g., condensation of volatiles, formation of glaciers, migration of subsurface ice, etc.) can be simulated. The portion of the atmosphere above the ground within the cell sees an average of these surface microclimates, and all sub-grid terrains see the same ”shared atmosphere”.

Practical Implementation

The practical implementation is detailed in section 3. of Lange et al., 2023. In short:

  • The mesh can be divided in 1, 5, or 7 sub-grid slopes that are North - South facing. Then, using MOLA data (resolution: 64 pixel per degree), the distribution of sub-slope within the mesh is computed.
  • All the surface quantities have now a dimension nslope: qsurf, watercap, tsurf, tsoil, albedo, emis, inertiesoil
  • The radiative transfer is computed with grid-box albedo, emissivity, temperature. Then, the values are adapted for-each sub-grid slopes following Spiga & Forget 2008 (section 2. of physiq_mod, param_slope).
  • The turbulent exchanges (vdifc_mod) are solved with the predominant sub-grid surface (i.e., the sub-grid surface which has the largest cover within the grid box).
  • Water and CO2 ice condensation are computed for each sub-grid slopes (vdifc_mod, co2_condens)
  • Grid box averaged values are then computed using the specific computation detailed in the dedicated section.


How to run with sub-grid slopes ?

By default, the model runs with only 1 sub-slope (i.e., a flat surface). If one wants to use this sub-grid parameterization, then one must use newstart and choose the option to add sub-slopes. FYI: The CPU time is increased by ~20% with 7 sub-slopes (64x48x26 grid).

Post Processing

In order to go from the sub-grid surface quantity to the grid-box value, the following computations are done:

  • For albedo and emissivity:

\begin{align} X = \sum_{i=1}^7 X_i \delta_i \label{eq:opticalproperties} \end{align} where $$X$$ refers to the grid parameter, $$X_i$$ the sub-grid parameter of slope $$i$$, $$\delta_i$$ the cover fraction (subslope_dist in the code).

  • For surface tracers, to ensure mass conservation:

\begin{align} X = \sum_{i=1}^7 X_i \frac{\delta_i}{\cos(\mu_i)} \label{eq:traceurs} \end{align} where $$\mu_i$$ is the slope angle of the sub-grid slope (def_slope_mean in the code)

  • For surface temperature, we average the surface flux:

\begin{align} \epsilon T_{\rm{surf,grid}} = \sum_{i=1}^7 \epsilon_i T_{\rm{surf,i}}^4 \delta_i \label{eq:Tsurf} \end{align} where $$\epsilon$$ (1) is the grid emissivity, $$T_{\rm{surf,grid}}$$ (K) the grid surface temperature


  • This new parameterization enable, for each grid point, to determine the surface fields for the seven types of slopes modeled here. The question is now: for any point on the Martian surface, knowing the characteristics of the terrain (slope and azimuth), can we determine the values of the surface quantities using the GCM outputs that are discretized in space and slopes (i.e., build high resolution map of surface quantities)?

I (LL) propose to use the following method, inspired from the MCD. Let us consider a point (noted $M$) of the surface of Mars, placed on a surface of slope angle $$\theta$$ and azimuth $$\psi$$. We compute $\mu$ and find the index $$i,i+1$$ of the two sub-grid slopes such that: \begin{align} \mu_i \leq \mu < \mu_{i+1} \end{align}

The point $$M$$ is surrounded by four GCM meshes. We can interpolate the values obtained at the GCM points for the sub-grid slope $$\mu_i$$ and $$\mu_{i+1}$$ at point $$M$$. This is performed by using a bilinear interpolation in space. After that, the value for slope $$\mu$$ at $$M$$ is obtained with linear interpolation between the fields obtained after the bilinear interpolation for $$\mu_i$$ and $$\mu_i+1$$. If $$\mu < \mu_0$$ or $$\mu > \mu_7$$, an extrapolation is computed. Thus, we have virtually access to any surface for any kind of slopped surface on Mars using this complete climate model. WARNING: 1) we assume linear interpolation on processes that are not necessarily linear (although we interpolate neighbouring points); 2) we neglect the time-shift (generally lower than 1-2~hours) for slopes oriented East-West that are not well represented by this parameterization.

I (LL) have a python algorithm to do that, feel free to ask me!

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