2. Numerical Model
 2.b. Simulation setups

Computational domain and spatial resolution of the model (Figure 1)
The computational domain of the model atmosphere extends 51.2 km horizontally and 20 km vertically. On the top of the model atmosphere, we have added a layer where only the temperature field is calculated to improve the accuracy of radiation flux. Both horizontal and vertical grid intervals are 100 m except in the lowermost 100 m height, where the vertical levels are located at z = 50, 25, 12.5, 6.25, and 3.125 m. Since the staggered grid is utilized, the lowest level at which horizontal wind is evaluated is located at about 1.5 m height. The value of grid interval, 100 m, is determined by some preliminary numerical experiments with varying the model resolution.

The vertical computational domain of the model ground surface layer extends to six times of diurnal skin depth δd. The value of δd is about 8 cm (appendix A.e ). The vertical grid interval increases with depth. The vertical grid points normalized by δd are located at -0.1, -0.2, -0.35, -0.53, -0.79. -1.2, -1.8, -2.7, -4.0, and -6.0.

Boundary conditions

The horizontal boundary condition of the model atmosphere is cyclic. The vertical wind velocity vanishes at the surface and upper boundary. Above 17 km height, the numerical diffusion is introduced to the horizontal and vertical momentum equations in order to attenuate gravity waves excited by the thermal convection. The value of numerical diffusion coefficient linearly increases from 0 to 1000 m2sec-1 between 17 and 19 km height.

The solar flux at the top of the model atmosphere changes diurnally under the condition of Ls = 100° at 20°N. The seasonal condition corresponds to the summer solstice of northern hemisphere (Ls = 90°). The latitudinal condition is close to that of Viking Lander 1 site (22.4°N).

Basic state and initial condition
The vertical temperature profile of the basic state of model atmosphere is given as that at local time (LT) 6:00 calculated by the 1D radiative convective model that has the same representations of radiative and ground surface processes as those described in Outline of the model. The profiles of pressure and density of the basic state are obtained from the temperature profile by the use of the hydrostatic equation and the equation of state for an ideal gas. Detailed mathematical expressions of the 1D model, calculation procedure of temperature profile, and the actual profile adapted for the basic state are shown in appendix C.

The initial condition for the dust-free case is a motionless atmosphere with horizontally uniform temperature. The vertical profile of the initial atmospheric temperature is the same as that of the basic state mentioned above. In order to facilitate the initial development of thermal convection, a random perturbation of potential temperature with the amplitude less than 3 K is imposed at the lowest level (z = 3.125 m). The vertical profile of the initial ground temperature is given as that calculated by the 1D model which is used for determining the temperature profile of the basic state. The initial condition for the dusty case is described in section 4.

Time step and computational resources
The time interval of integration is 0.5 or 1 second. Those values are determined by using CFL condition with the phase velocity of the fastest internal gravity wave in the model atmosphere, which is described as

where is buoyancy frequency and is the depth of the model atmosphere. The time interval for integrating the radiation process is 60 seconds, which is a duration short enough for radiation field to follow the temperature change associated with the thermal convection. This is because the temporal scale of temperature change associated with the thermal convection can be estimated as 100 to 1000 sec, provided that the magnitude of convective wind velocity is of the order of 10 msec-1 and the depth of convection layer ranges from 1 to 10 km.

Numerical integrations were performed by using the Fujitsu VPP 800 systems at Kyoto University Data Processing Center and Center for PLAnning and INformation Systems, Institute of Space and Astronautical Science. The size of necessary main memory was about 256M bytes. The CPU time for executing integration of 24 model hours with the time interval of 0.5 sec was about 8 hours.

A numerical simulation of thermal convection in the Martian lower atmosphere.
Odaka, Nakajima, Ishiwatari, Hayashi,   Nagare Multimedia 2001