Closures Used in Zonally Averaged Ocean Models

Abstract

There are at least three substantially different closures presently being used in two-dimensional ocean models. The main purpose of this paper is to clarify the assumptions that are implicit in these closures. Two of these formulations arise from zonally averaging the momentum equations: one has viscous damping represented by Fickian diffusion and the other by Rayleigh damping. Here a single equation is derived that includes both of these as special cases. The derivation shows that the Rayleigh damping term corresponds to horizontal diffusion of momentum into the western boundary and that this term is dominant for realistic parameter values. The vertical diffusion term can be neglected provided the Ekman transport is included in the surface layer, and the meridional diffusion term can be neglected if length scales less than 500 km are not resolved. If shorter length scales are considered, then the meridional diffusion term is required to avoid a numerical instability. The third zonally averaged model formulation is based on vorticity dynamics. This approach has the advantage that the large geostrophic terms are eliminated by cross-differentiation so that attention is focused on the important ageostrophic effects. Previous work has shown that this formulation results in an improved fit to general circulation model results, but this fit depends on the use of a boundary condition that is somewhat ad hoc. Here the authors present a derivation that suggests a consistent dynamical interpretation of this boundary condition.