Soil Thermal Conduction in the Generic PCM
Purpose
The goal of this page is to summarize key aspects, both theoretical and practical, of the computation of subsurface temperatures in the generic PCM.
Theoretical Considerations
The computation of the heat conduction in the soil was based on the 1D layer soil model originally described by Hourdin et al. (2013). Since that, E. Millour modified this scheme to include the possibility of varying the thermal properties with depth.
The PCM solves the time-dependent diffusion equation: \begin{equation} C \frac{\partial T}{\partial t} = - \overrightarrow{\nabla} \cdot \overrightarrow{F}_c \label{Eq:EqChaleur} \end{equation}
where $$C$$ is the volumetric specific heat (J m$$^{-3}$$ K$$^{-1}$$) such that $$C = \rho C_p$$ ($$\rho$$ is the material's density, in kg m$$^{-3}$$ and $$C_p$$ is its specific heat, in J kg$$^{-1}$$ K$$^{-1}$$), and $$F_c$$ is the conductive heat flux: $$\overrightarrow{F}_c = - \lambda \overrightarrow{\nabla}T$$ according to Fourrier's law ($$\lambda$$ is the solid’s heat conductivity, in J s$$^{-1}$$ m$$^{-1}$$ K$$^{-1}$$).
Thermal conduction is here considered as a one-dimensional (1D) process, lateral conduction is neglected (a reasonable assumption considering typical grid size in the Generic PCM). Temperature $$T$$ of the soil is thus a function of time $$t$$ and depth $$z$$, which must satisfy the following equation:
\begin{equation} C \frac{\partial T}{\partial t} = \frac{\partial}{\partial z}\left[ \lambda(z) \frac{\partial T}{\partial z} \right] \label{Eq:Chaleur1D} \end{equation}
The Mars PCM used to have the following boundary conditions:
- At the bottom (e.g.: $$z = H$$) of the layer of soil: No outgoing (or incoming) heat flux. This boundary condition is then simply: \begin{equation} \frac{\partial T}{\partial z} _{z= H} = 0 \end{equation}
- At the surface ($$z$$ = 0): Surface temperature $$T_S (t) = T (z = 0, t)$$ can be computed from the balance of heat fluxes through the surface and cooling thereof, which leads to:
\begin{equation} F_c + \sum F^\downarrow = \epsilon \sigma T_s^4 \end{equation} where $$F_c$$ is the flux of heat conduction given by Fourier's law, $$\sum F^\downarrow$$ is the sum of fluxes reaching the surface, $$\epsilon$$ is the emissivity of the ground and $$\sigma$$ the Stephan-Boltzman constant. In the case of condensation/sublimation, a latent heat term is added to this last equation.
Numerical approach
The spatial discretization of the unsteady-heat equation is based on a finite volume approach.
References
Hourdin, F., Van, P. L., Forget, F., and Talagrand, O. (1993). Meteorological variability and the annual surface pressure cycle on Mars. Journal of Atmospheric Sciences, 50(21):3625 – 3640.
