Equilibration of baroclinic eddies in a primitive equation model

Very roughly speaking, theories for the equilibration of baroclinic eddies fall in or between two classes: so-called adjustment theories and diffusive theories. In the former, the mean flow adjusts to some preferred equilibrium that is typically only slightly supercritical (in some appropriate sense) and the eddy fluxes then adopt whatever value is required to maintain this. In diffusive theories, the fluxes are a smooth function of the mean state, which implies that the mean state must vary continuously with the forcing. We explore these issues using primitive equation numerical models and scaling theories based on quasi-geostrophy.

In the simplest case - quasi-geostrophic turbulence in two layers - theory suggests that a diffusive theory may approximately hold, with a diffusivity that increases rapidly with supercriticality so resembling in some ways an adjustment. However, in the earth's atmosphere the eddy scale is barely larger than the deformation scale and an inverse cascade, if it exists at all, is evidently of limited extent. It is not clear whether this is a general result, or is particular to a certain set of parameters, including those of the Earth's atmosphere. We have investigated this problem in perhaps the simplest framework that allows changes in stratification, namely a two-level primitive-equation model on the beta plane. Our results suggest that quasigeostrophic theory works reasonably well, at least in some parameter regimes, provided that the stratification is diagnosed from the primitive-equation model. In particular, for some parameter regimes the mean flow is found to be supercritical and eddies are created that are considerably larger than the deformation radius, suggesting a vigorous inverse cascade. To go beyond quasi-geostrophic theory one also needs a theory for the stratification and we propose a closure for the eddy vertical heat flux assuming that the eddy mixing slope scales with the isentropic slope. This captures reasonably well the empirical dependency of the flux, and provides full closure to the non-geostrophic problem.