7.2.3 Free surface height equation

Knowledge of the surface currents and fresh water flux q_w allow one to time step the free surface height through use of the surface kinematic boundary condition (7.16). However, because the motion of the free surface height is associated with fast barotropic motions, it is more useful algorithmically to determine $\eta$ within the barotropic system. Additionally, a direct discretization of the surface kinematic boundary condition (7.16) would require a discretization of the advective term, which is inconvenient at best.

Instead of directly discretizing the kinematic boundary condition, perform a vertical integral of the continuity equation over the full depth of the ocean to find

$$w(\eta) - w(-H) - \int^{\eta}_{-H} dz \; \nabla_{h} \cdot {\bf u}_{h}$$ =

$$- \nabla_{h} \cdot {\bf U} + {\bf u}(\eta) \cdot \nabla_{h} \eta + {\bf u}(-H) \cdot \nabla_{h} H.$$ = (7.17)

Use of the bottom and surface kinematic boundary conditions (7.15) and (7.16) yields

 

$$\eta_t = - \nabla_{h} \cdot {\bf U} + q_{w}.$$ (7.18)

This is a fundamental balance with the free surface. In words, the time tendency for the free surface height is determined by the convergence of the vertically integrated transport plus the fresh water flux through the sea surface. Note that no extra boundary conditions for $\eta$ are needed to solve this equation. Namely, the surface and bottom kinematic boundary conditions are implicitly fulfilled, and the lateral boundary conditions come from the boundary condition for the vertically integrated transport, which vanishes at the side walls. Note that in the rigid lid approximation, each of the three terms in this balance is individually set to zero.