B. Finite difference equations of the model   a. Atmospheric model
 B.a.iii. Diagnostic equation of pressure function

Before making the finite difference equation, (A.8) are transformed as follows.

(definition of is shown in appendix B.a.i ). This equation is solved by using the dimension reduction method. The finite difference form of the pressure equation can be written in matrix form as follows.

 (B.19)

where are matrixes whose elements are finite difference form of following terms.

and are eigevalue and eigenvector of respectively. By using the eigenvalue matrix and the eigenvector matrix of , . Expanding , (B.19) can be rewritten as follows.

Therefore,

 (B.20)

The elements of matrix which is required to derive eigenvalue and eigenvector are evaluated by finite form of

The second and forth order centered schemes are used because the space differencing in the continuity equation is evaluated by the forth order centered scheme while that in the pressure gradient term is evaluated by the second order centered scheme. Therefore is quintdiagonal matrix. The element is represented as follows.

 (B.21) (B.22) (B.23) (B.24) (B.25)

The boundary conditions are at the lower and upper boundary.

The horizontal dependent terms are expanding by using some eigenfunction. In this model, we use Fourier series.

 (B.26) (B.27) (B.28)

 (B.29)

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