A new parameterization for momentum

It appears that the idea of Gent and McWilliams (1990) should apply to any quantity that is supposed to be conserved, so that every conservation equation in a non-eddy resolving model should include a new term to counteract the effect of eddies at the subgrid scale.

The idea should thus apply in particular to potential vorticity (PV). But, in general, PV is not a prognostic quantity –the model does not resolve the PV equation–, so that we would have to translare this into a new temr in the momentum equations. (Actually, Gent and McWilliams (1990) also translate the additional term that appears in the conservation equation of density into a term in the thickness equation. And it is the term in the latter that has the familiar form of Laplacian mixing. What would be the difference if we would have chosen the form (15) directly for the term in the density equation? After all, models do have the density equation as a prognostic equation.)

Another question: Do we actually want to be in the case where quantities are perfectly conserved (and eddies cancel exactly the mean-Eulerian flow), as Gent and McWilliams (1990) postulate? This is a fundamental assumption; it means that actual molecular dissipation does not matter. This is not true for instance for the meridional overturning circulation –we know that the mixing due to internal wave breaking, etc is supposed to maintain the stratification. Why would we do then such assumption for density, PV, etc?

Two questions from the literature:

• How the Gent and McWilliams’scheme has been shown to be a success compared to other schemes?
• Has the application to PV not already been done? Yes, see . I do not know if it would be worthy to read and revisit their study. In any case, their “analysis clearly implies that the horizontal viscosity coefficient used in the momentum equation should be set equal to kappa, which is the GM closure parameter.” Also, their parameterization is close to that of LANS-alpha. It would be thus interesting to see if the deep equatorial circulation appears better in models with these new parameterizations –although we do need PV not to be conserved to generate the zonal circulation.