A constraint on the magnitude and structure of eddy potential vorticity fluxes

Geostrophic eddy parameterization schemes based upon the down-gradient mixing of potential vorticity have previously been suggested. However, such schemes are susceptible to the introduction of physical inconsistencies. In particular, angular momentum and energy conservation impose strong constraints upon potential vorticity mixing, and hence impose constraints upon the eddy diffusivity.

A new framework for the parameterization of potential vorticity fluxes is described. This framework utilises a decomposition of the potential vorticity flux into the divergence of a stress tensor. We show that the components of this stress tensor can be re-expressed in terms of the eddy energy, eddy energy partitioning, eddy anisotropies (mixing efficiencies) and eddy mixing angles. The eddy anisotropies and angles are dimensionless and bounded. Angular momentum is conserved by construction, and energy conservation can be maintained. The popular Gent and McWilliams parameterization can be expressed in this formulation, by neglecting the momentum stress terms and assuming a down-gradient buoyancy closure. Expressing the Gent and McWilliams parameterisation in this manner yields a parameterised Eady growth rate that is consistent with the linear theory. A three layer quasi-geostrophic model is used to diagnose the eddy anisotropies and angles, and it is shown that the anisotropies are order ~0.1.