7.3.2 The shallow water limit

The equations satisfied by the two fields ${\bf U}$ and ${\bf U}_0$are rewritten here for purposes of comparison

$$\partial_{t} \, {\bf U} + f\hat{z} \wedge {\bf U} - (H+\eta) \, \nabla_{h} (p_{a} + p_{s})/\rho_{o} + \eta_{t} \, {\bf u}_{h}(\eta) + {\bf X}$$ = (7.34)

$$\partial_{t} \, {\bf U}_{0} + f\hat{z} \wedge {\bf U}_{0} - H \, \nabla_{h} \, (p_{a} + p_{s})/\rho_{o} + {\bf X}_{0}.$$ = (7.35)

Notice that the equation satisfied by ${\bf U}_0$ contains nonlinearities only within the vertically integrated forcing term ${\bf X}_{0}$. Hence, the ``shallow water version'' of the ${\bf U}_0$ equation, defined by dropping ${\bf X}_{0}$, is linear. In contrast, the shallow water version of the ${\bf U}$ equation contains additional nonlinearities arising from the terms

$$\eta_{t} \, {\bf u}_{h}(\eta) - \eta \, \nabla_{h} \, p_{s}/\rho_{o} = \eta_{t} \, {\bf u}_{h}(\eta) - (g/2) \, \nabla_{h} \eta^{2}.$$ (7.36)

Each of these terms is very small in comparison to the other terms, except in those regions of the ocean where the depth H is on the order of the free surface height undulations $\eta$. It is in these regions that nonlinear wave steepening and breaking become important processes which the nonlinear shallow water equations admit.