A Framework for Mesoscale Eddy Parameterization Based on Density-Weighted Averaging at Fixed Height

Abstract

A framework for mesoscale eddy parameterization based on density-weighted averaging at fixed height is developed. The method uses the fully non-Boussinesq equations of motion and is connected to the equations carried by Boussinesq ocean models only after the averaged equations have been developed. The framework applies to the continuity, tracer, and momentum equations within a single formalism. Two methods for applying parameterizations in ocean models are obtained. The first, based on the tracer equation, corresponds to the approach commonly taken when including eddy effects in ocean models. The second puts the forcing for the eddy-induced transport into the averaged momentum equation where it appears as the divergence of a generalized Eliassen?Palm flux. It is then shown how to solve for the tracer transport velocity. The solutions form a family closely related to the temporal residual mean (TRM) velocity of McDougall and McIntosh, valid to O(α3), where α is perturbation amplitude. The analysis is extended to obtain a family of exact solutions for the eddy-induced mass transport, valid at any order in perturbation amplitude. It is also shown how to obtain a generalization of the TRM to take account of diffusion and time dependence in the instantaneous equations. The solution suggests that the tracer transport velocity could be different for different tracers, depending primarily on the structure of the mean field. This conclusion also applies in the case of isopycnal averaging; it is not a result that is peculiar to averaging at fixed height. Finally, it is shown how the non-Boussinesq analysis presented in the paper can be modified to analyze output from eddy-resolving, Boussinesq ocean models.