A. Governing equations of the model   d. Radiation
A.d.iv. Dust opacity The monochromatic optical depth for wave number is represented by using the extinction coefficient per unit volume as follows. where is the altitude at the top of the atmosphere. is given as follows. where is the extinction cross section and is the size distribution of a scattering particle (cf. Liou, 1980; Shibata, 1999). By using the extinction coefficient per unit mass , Equation (A.40) is written as follows. where is atmospheric dencity, and is the mass mixing ratio of a scattering particle. Similarly, the scattering and absorption coefficient per unit volume are represented by using the scattering cross section and the absorption cross section as follows. and the single scattering albedo is given as follows. The extinction efficiency is defined as the ratio of the extinction cross section to the geometric cross section. Similarly, the scattering efficiency and absorption efficiency are defined as follows. In the present study, dust opacity is derived from the mass mixing ratio of atmospheric dust . The provided parameters include the cross section weighted mean extinction efficiency , the single scattering albedo , the size distribution function of dust , the effective (i.e., cross section weighted mean) radius , and the density of a dust particle . , are defined as follows. Assuming the shape of a scattering particle is spherical, the extinction coefficient per unit mass is given as follows. where is atmospheric density. Therefore, optical depth can be represented as follows.
A Numerical Simulation of Thermal Convection in the Martian Lower Atmosphere with a Two-Dimensional Anelastic Model