The essential step in this derivation is to assume no motion in the deep layers, so the unknown pressure Pa, which represents the contribution from the atmospheric pressure and the free surface elevation, can be eliminated. Without assuming a stagnant deep layer, the rigid-lid pressure Pa will remain as part of the pressure expression. In such cases one may have to use other approaches to eliminate Pa. For example, in numerical models based on the rigid-lid approximation, Pa can be eliminated by cross-differentiating the horizontal momentum equations, and the problem is reduced to solving an elliptic equation for the barotropic streamfunction.
The same expressions can be derived for a model including the free surface explicitly (Fig. 4.2b). We start from the sea surface z = Z and integrate the hydrostatic relation downward
where Pa,0 is the sea-level atmospheric pressure. Assuming Pa,0 = const, the pressure gradient in each layer can be written as
VhPi = PigVhZ
Dividing by pi and using the reduced-gravity notation, we obtain 1
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