Balance Equations Based on Momentum Equations with Global Invariants of Potential Enstrophy and Energy

Abstract

An approximate model for small Rossby number ? that is close to the balance equations (BE) but that is based on approximate momentum equations is formulated for a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, inviscid, adiabatic primitive equations with spatially variable Coriolis parameter. This model, referred to as BEM (balance equations based on momentum equations), conserves volume integrals of an appropriate energy density and also conserves potential vorticity on fluid particles and thus volume integrals of potential enstrophy density. The fact that, unlike the BE model which is derived from equations for the vertical component of vorticity and for the horizontal divergence, BEM is based on approximate momentum equations is important for two reasons. It allows the derivation of equations for the horizontal components of vorticity that are needed in the subsequent derivation of an equation for the potential vorticity and it allows the consistent formulation of boundary conditions at rigid surfaces. As is the case for BE, the BEM equations filter out high-frequency internal?gravity waves and remain valid for motion over O(1) variations in bottom topography and for flows with O(1) variations in the height of density surfaces. The governing equations for BEM may be conveniently expressed in a form similar to BE involving a vorticity and a divergence (balance) equation. In this formulation, the BE and BEM models involve identical equations for continuity, vorticity, and heat with differences represented only by the presence of additional higher order terms in the balance equation for BEM. Methods for the numerical solution of BEM and for the application of boundary conditions are presented.