B. Finite difference equations of the model
B.e. Ground and surface layer Time integration of the 1D thermal conduction equation (A.55) in Appendix A.e is performed by the Crank-Nicolson scheme. Space differencing for Equation (A.55) is evaluated by the second-order centered scheme. Ground temperature and the vertical grid interval are evaluated on the grid point and the heat flux is evaluated on the half grid point. The number of vertical grid points is and the suffix varies, , from the lowest grid point. The temperature at highest grid point, is assumed to be surface temperature, . Where . If the terms at are moved to the left-hand side and the terms at are moved to the right-hand side, then Where $\overline{\Delta z}_{j+\frac{1}{2}}=(\Delta z_{j+1}+\Delta z_{j})/2$. When , this equation can be represented in matrix form. The matrix is a J'-th order square matrix and its elements are represented as follows. Considering the upper boundary condition (A.56) and the insulation lower boundary, (B.61) can be modified as follows. Therefore, where the first and th diagonal element of are represented as follows. is a column vector whose dimension is and its elements are represented as follows.
A Numerical Simulation of Thermal Convection in the Martian Lower Atmosphere with a Two-Dimensional Anelastic Model