36.2.3 xlandmix_eta

Consider a model in which the Mediterranean Sea is artificially land-locked due to the use of coarse resolution. When using MOM's explicit free surface with explicit fresh water fluxes, the net evaporation over the Mediterranean Sea will cause the free surface height to decrease without bound. In a model resolving the Straits of Gilbraltor, there will be a transfer of volume across the Strait from the Atlantic. This volume transfer will create a change in the height of the free surface, and the transfer time will be determined by the speed of external gravity waves. The purpose of this section is to describe a means of parameterizing this effect in MOM when employing the explicit free surface and explicit fresh water fluxes.

As discussed in Section 7.3.3, MOM's free surface tendency is given by

$$\eta^{0}_{t}$$ = w(z=0) + q_w, (36.7)

where $\eta^{0}$ is the linearized free surface height, $w(z=0) = -\nabla_{h} \cdot {\bf U}_{0}$ is the vertical velocity at the ocean surface, as determined by the convergence of the vertically integrated flow, and q_w is the fresh water input through evaporation, precipitation, and river runoff. This equation results from neglecting the advection of free surface height by the surface currents. In the model, if $\eta$ lives at a grid point identified as one of a cross-land mixing pair, and option xlandmix_eta is enabled, the time tendency for these points takes the modified form

$$\eta^{here}_{t} w(0) + q_{w} + {\cal S}^{here}/A^{here},$$ = (36.8)

where ${\cal S}^{here}$ is the volume mixing term

$${\cal S}^{here} = \left(\frac{\eta^{there}-\eta^{here}}{\tau}\right) \left(\frac{A^{here}+A^{there}}{2}\right).$$ (36.9)

The mixing term ${\cal S}^{here}$ nudges the free surface height $\eta^{here}$ towards $\eta^{there}$. Note that the straightforward volume difference form ${\cal S}^{here} = (A^{there} \, \eta^{there}- A^{here} \, \eta^{here})/\tau$ will lead to a nonzero source when $\eta^{there} = \eta^{here}$ but if $A^{there} \ne A^{here}$. Such mixing is not desirable, hence motivating the chosen form. Originally, the time scale $\tau$ for the mixing was determined by an estimate of the time it takes an external gravity wave to cross between the two points:

$$\tau = L \, (g \, H)^{-1/2},$$ (36.10)

where H is set to the averaged depth of the two cross-land points, and L is the horizontal distance between them. However, this time scale gave very noisy results. Consequently, the model currently has the value $\tau=3$days hardwired. Changes to this time scale may be necessary depending on grid resolution. It is useful to discuss what cross-land $\eta$ mixing implies about tracers. For this purpose, consider the case in which one starts with two basins connected by a strait, where the basins have differing surface heights, yet let the temperature and salinity be the same uniform values. Also, remove all surface heat and water fluxes. In a model with strait resolved by at least one velocity point (which, for a B-grid, means at least two tracer points), there will be an adjustment process in which the free surface height equilibrates to the same value across the two basins. In contrast, the tracer concentrations, as there are no surface fluxes, remain unchanged throughout the adjustment. In a model with an unresolved strait, one should achieve the same equilibrated solution. Cross-land mixing of $\eta$ clearly yields a uniform free surface height at equilibrium. In order to preclude affecting the tracer concentrations through the $\eta$ mixing, it is necessary to not include the cross-land mixing term ${\cal S}^{here}$ in the tracer budget. Similar considerations apply to the baroclinic velocity tendency. Note that this result does not depend on having made the linearized free surface approximation. Rather, it is simply a result of tracer conservation. The result is convenient, since it means that no model changes are required for the tracer or baroclinic equations when adding cross-land $\eta$ mixing.