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The numerical scheme used

In the parameterization of vertical diffusion in a GCM equation 14 is discretized over the vertical and in time. In the following discussion we will assume that the calculation is performed over N levels. Variables are located at the full levels and fluxes are computed at intermediate levels, represented by dashed lines in Figure 2. Level 0 is the surface. The basic time step starts at time t when all variables are know and ends at time t+1 when all calculations are completed.

  figure453
Figure 2:   Levels used for the discretization of the equations.

When equation 14 is descretized in the vertical and an implicit time-stepping is used we obtain the following finite difference formula for level l:

  equation507

In order to solve this system of equations from the surface (l=0) to the top of the atmosphere or the planetary boundary (l=N) the method proposed by Richtmeyer and Morton (1967) is used. The aim is to reduce the system to a set of equation of the type:

  equation528

where the coefficients tex2html_wrap_inline1315 and tex2html_wrap_inline1317 can be computed in a descending order and then used in a back-substitution from bottom to top which yields the profile for X at time t+1. It is assumed here that the eddy-diffusivities tex2html_wrap_inline1323 are computed before using atmospheric conditions at time t.

To satisfy the zero flux condition at the top in equation 16 we have to set tex2html_wrap_inline1327 and tex2html_wrap_inline1329 . This allows to start an iteration from top to bottom which determines tex2html_wrap_inline1315 and tex2html_wrap_inline1317 over the entire column. The following iteration formulas are obtained :

    eqnarray551

In this set of equations only tex2html_wrap_inline1335 contains information from the surface. This implies that without any knowledge of the surface the downward iteration can only be performed up to l=2. The back substitution can not be performed independently from the land surface scheme for l=1, but once tex2html_wrap_inline1341 is known equation 16 can be solved for all tex2html_wrap_inline1343 .

In order to obtain a general interface between the land surface scheme and the vertical diffusion scheme of the GCM a formulation has to be derived for computing tex2html_wrap_inline1341 using only surface fluxes ( tex2html_wrap_inline1347 ).


next up previous
Next: The surface fluxes Up: The vertical diffusion and Previous: The vertical diffusion and

POLCHER Jan
Fri Mar 6 16:09:11 MET 1998