29.6.4 Vertical velocities

Vertical velocities are diagnosed at baroclinic time steps using volume conservation within a grid cell. In particular, volume conservation over a surface cell indicates that the vertical velocity at the bottom face of this cell arises from the horizontal convergence of volume in this cell, time tendencies in the thickness of this cell, and volume passing across the top face from fresh water fluxes.

It is useful to see precisely how these velocities are determined. For this purpose, consider a vertical integral of the continuity equation $\nabla \cdot {\bf u} = 0$ over a rectangular top model grid cell

$$w_{k=1} = w(\eta) + \int^{\eta}_{z_{1}} \, dz \, \nabla_{h} \cdot {\bf u}_{h}.$$ (29.98)

Use of the surface kinematic boundary condition

$$(\partial_{t} + {\bf u}(\eta) \cdot \nabla_{h} ) \, \eta = w(\eta) + q_{w},$$ (29.99)

and Leibnitz's Rule with $\nabla_{h} \, z_{1} = 0$ leads to

$$\partial_{t} \eta - q_{w} + \nabla_{h} \cdot \int^{\eta}_{z_{1}} \, dz \, {\bf u}_{h}$$ w_k=1 =

$$-\nabla_{h} \cdot {\bf U} + \nabla_{h} \cdot \int^{\eta}_{z_{1}} \, dz \, {\bf u}_{h},$$ = (29.100)

where the last step used the vertically integrated continuity equation (7.18). This exact expression for w_k=1is approximated in the model with a discretized version of

$$w_{k=1} \approx -\nabla_{h} \cdot {\bf U} + \nabla_{h} \cdot (h \, {\bf u}_{h}),$$ (29.101)

where $h = \eta + \vert z_{1}\vert$ is the surface cell thickness, and the horizontal velocity ${\bf u}_{h}$ is that in the surface cell k=1. Notably, many implementations of the free surface linearize the surface kinematic boundary condition by dropping the surface height advection ${\bf u}(\eta) \cdot \nabla_{h} \eta$. No such approximation has been made here.

Given an expression for the convergence $-\nabla_{h} \cdot {\bf U}$, diagnosing w_k=1 in this manner allows for the remaining interior vertical velocities to be successively found through further integration of the continuity equation downward through a vertical column. On a B-grid, $-\nabla_{h} \cdot {\bf U}$ is centered on a tracer point. Hence, equation (29.102) yields the vertical velocity on the bottom face of the surface tracer cell. The vertical velocity on the surrounding velocity cells is constructed as a volume conserving average of the surrounding tracer cell vertical velocities. In MOM, the bottom of the ocean on tracer cells is a flat surface, representing the ``lopped off'' surfaces of topography^29.11. Hence, the vertical velocity must vanish at this location. A self-consistency check on how accurate the model's numerics conserve volume amounts to testing how well this property is satisfied when integrating downwards from the ocean surface, starting from the vertical velocity given by equation (29.102). Adding fresh water to the model provides a nontrivial test of these properties. The present scheme produces zero vertical velocities at the bottom of tracer cells, to within computer roundoff, regardless of the topography or surface forcing.