B. Finite difference equations of the model
B. Finite difference equations of the model The outline of finite difference method adapted for our 2D model is as follows. Space differencing The finite difference form of governing equations of the model are considered on the Lorenz type staggered grid. Space differencing is evaluated by using the forth-order centered scheme for scalar advection terms (potential temperature, dust mixing ratio, turbulent kinetic energy) and the continuity equation. The second-order centered scheme is used to determine momentum advection, pressure gradient, turbulent diffusion, gravitational settling of dust. Numerical diffusion is introduced to the equation of motion, turbulent kinetic energy equation, and the advection diffusion equation of dust in order to suppress the 2-grid noise associated with central finite differencing. The numerical diffusion in equation of motion is proportional to the squared wind shear and that in turbulent kinetic energy equation and advection diffusion equation of dust is proportional to the third power of Laplace operator. Space differencing in the radiative transfer equation and the thermal conduction equation of ground temperature is also evaluated by the second-order centered scheme. The vertical integral, when calculating the infrared radiative flux of CO2, is evaluated by the trapezoidal rule. Time differencing Time integration is performed by the leap-frog scheme for advection and buoyancy terms, and the forward scheme is used for turbulent diffusion and forcing terms. For advection and the buoyancy terms, the forward scheme is also adopted once per 20 steps to obtain a stabilized numerical solution. The radiative flux associated with dust is given by an iteration method of the matrix equation, where the number of iteration is four. The time integration of 1D thermal conduction equation of ground surface is performed by the Crank-Nicolson scheme. In the following sections, the $i,j$ subscripts indicate horizontal and vertical grid point, and the $n,N$ superscripts indicate time step. $J$ is the number of vertical grid level. The scalar and basic state variables are evaluated on the grid point and the other variables are evaluated on the half grid point (see Figure 1). $\Delta x$ and $\Delta z_{j}$ are the horizontal and vertical grid intervals, and $\Delta t$ is the time interval.
A Numerical Simulation of Thermal Convection in the Martian Lower Atmosphere with a Two-Dimensional Anelastic Model