Complete Eulerian-mean tracer equation for coarse resolution OGCMs

Abstract

McDougall and McIntosh showed that the adiabatic mesoscale mixing is represented incompletely in the tracer Eulerian-averaged equation (EAE) of coarse resolution OGCMs. We show that completing EAE requires an adequate decomposition of the mesoscale tracer flux which is achieved by means of transforming mesoscale fields to isopycnal coordinates (IC) where mesoscale dynamics has the simplest form. The transformation results in splitting F τ into two components and : the former is determined by buoyancy mesoscale dynamics only and has a trivial kinematic dependence on the mean tracer field, the latter is determined by mesoscale tracer dynamics. Thus, the problem of modelling (parameterizing) F τ in ZC is divided in two stages which can be termed kinematic and dynamic. The kinematic stage consists in adequate decomposing F τ, and the result is expressed in terms of mesoscale fields. The dynamic stage consists in applying a specific dynamic mesoscale model to parameterize the components of F τ. In this article, we show that some components of F τ are missing in ZC-OGCMs tracer equation and that their contribution is of the same order of magnitude as the mesoscale contribution itself. We also show that F τ has components across mean isopycnals and that their existence is consistent with the adiabatic approximation which requires vanishing all fluxes across isopycnal surfaces. As for practical results, we derive the complete equation for the large scale tracer in ZC-OGCMs and present the parameterization of the terms which have been missing thus far.