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The numerical $\sigma $-coordinate model

The governing equations form a set of simultaneous partial differential equations which cannot be solved using known analytic methods. Therefore the equations have been discretized using finite difference methods. The routines called to approximate the solution of the governing equations in each time step of the model are described in this section. The horizontal finite difference scheme is staggered, and the Arakawa C-grid [34] has been used, see figures 1, 2 and 3.

\begin{picture}(130,77)
% The coordinate axis
\put(20,10){\vector(1,0){90}}
\put...
...he central variables
\put(58,40){W(I,J,K)}
\put(58.5,45){S(I,J,K)}
\end{picture}

Figure 1. Horizontal view of the location of 3-D variables in the staggered grid. T, RHO and other scalar fields are defined in S-points. KM, KH, Q2 and Q2L are defined in W-points.


\begin{picture}(130,80)
% The coordinate axis
\put(20,10){\vector(1,0){90}}
\put...
...values
\put(110,19){Z(K+1)}
\put(110,43){ZZ(K)}
\put(110,69){Z(K)}
\end{picture}


Figure 2. Vertical view of the location of 3-D variables in the staggered grid. T, RHO and other scalar fields are defined in S-points. KM, KH, Q2 and Q2L are defined in W-points.


\begin{picture}(130,80)
% The coordinate axis
\put(20,10){\vector(1,0){90}}
\put...
...,50){\line(0,1){20}}
% the central variables
\put(58,43){ETA(I,J)}
\end{picture}


Figure 3. Horizontal view of the location of 2-D variables in the staggered grid.

The model is written in FORTRAN 90 and the discrete versions of the state variables and parameters are gathered in a module, STATE, that may be addressed by all subroutines. Equations (36) - (41) are stepped forward in time using the same time step for all equations. The method of fractional steps is applied. That is a sequence of subroutines is called to perform specific subtasks and update the corresponding variables in MODULE STATE in each time step. After all subroutines are called the effects of all terms in the governing equations are included.

A description of the variables in MODULE STATE is given in the appendix.


Subsections
next up previous contents
Next: The time stepping algorithm Up: report Previous: Mode splitting   Contents
Helge Avlesen 2004-05-10