7.6.1 General considerations

As shown in Section 6.1, without making the assumption that w(z=0)=0, the barotropic streamfunction provides an incomplete description of the external mode velocity since the vertically integrated velocity is no longer divergence-free. Instead, both a streamfunction and a velocity potential are necessary

$${\bf U} = \hat{z} \wedge \nabla_{h} \psi + \nabla_{h} \chi,$$ (7.64)

where $\chi$ is the velocity potential and ${\bf U} = (H+\eta) \, (\overline{u},\overline{v})$ is the vertically integrated horizontal velocity. Taking the horizontal divergence of both sides of this equation, and using the free surface height equation (7.18) renders

$$\nabla_{h} \cdot \nabla_{h} \chi -\eta_t + q_{w}.$$ = (7.65)

Only in the case of a steady state ocean with zero surface water fluxes can the velocity potential be ignored.