22.3.3.1 Summary of the continuum results

Before discussing the discrete results, it is useful to first recall the discussion of the vertical velocity within the context of the continuum equations as given in Chapter 7. The main result is that the vertical velocity at the ocean surface is given by (equation 7.39)

$$- \nabla_{h} \cdot {\bf U}_{0},$$ w(0) = (22.25)

which in words means that w(z=0) is due to the convergence of the vertically integrated velocity

$${\bf U}_{0} = \int^{0}_{-H} dz \, {\bf u}_{h}.$$ (22.26)

This result is general; i.e., it holds even when there is fresh water input to the free surface. For the rigid lid, there is identically zero convergence and so w(0) = 0. For the implicit free surface, only the case without fresh water is implemented in which $w(0) \approx \eta_{t}$ is assumed. For the explicit free surface, $w(0) = - \nabla_{h} \cdot {\bf U}_{0}$ is implemented. For the ocean bottom, recall the discussion from Section 4.3.1, in which it was shown that the vertical velocity at the ocean bottom is given through the kinematic boundary condition

 

$$- {\bf u}_{h} \cdot \nabla_{h} H \qquad z = -H(\lambda,\phi).$$ w = (22.27)

Each of these results will now be discussed within the context of the discrete ocean model in which nontrivial bottom topography is allowed.