A Prognostic, One-Equation Model of Meso-Scale Eddy Momentum Fluxes

Abstract

We present a prognostic, one-equation model for eddy-mean flow interactions to parameterize the divergence of the Eliassen-Palm flux tensor (EPFT) that arises from thickness-weighted averaging (TWA) the hydrostatic Boussinesq equations. The TWA system of equations does not invoke approximations beyond those for which the hydrostatic Boussinesq equations are valid, constituting a mathematically consistent framework with clear physical interpretations. This model is intended for the adiabatic interior of zonally symmetric flows, in the absence of topographic features, where terms corresponding to eddy interfacial form drag in the EPFT dominate forces. We model eddy interfacial form drag terms for vertical flux of horizontal momentum using the gradient hypothesis, as the product of an eddy viscosity and the vertical gradient of horizontal momentum. We use mixing length theory to relate viscosity to an eddy length scale and an eddy velocity, which is proportional to the eddy energy in the TWA system. The eddy length scale is modeled as the first Rossby radius of deformation, which we calculate as a function of the mean flow. We use a prognostic equation for vertically integrated eddy energy at each horizontal location, which we derive from the TWA framework, and then simplify to the flows of interest by ignoring transport, redistribution and diabatic terms. The prognostic vertically integrated eddy energy is projected onto the water column using the eigenvalue of the first baroclinic mode to obtain the eddy energy at each vertical position. The eddy viscosity has horizontal as well as vertical structure. We diagnosed the model equations in an eddy resolving numerical simulation of a zonally re-entrant channel representative of the Southern Ocean. We have implemented the model parameterization in an ocean model and tested it to simulate a parameterized simulation of this flow.