Approximating the equations governing rotating fluid motion: a case study based on a quasi-geostrophic model

Abstract

Nearly all theoretical work in geophysical fluid dynamics is based on approximate forms of the equations of motion, but the best ground-rules for deriving such approximate forms are not clear. Traditionally, scale analysis and global energy conservation have been the guiding principles. The existence of analogues of Lagrangian potential vorticity conservation has been seen as at least aesthetically desirable, but consequent improvements in practical accuracy have not often been demonstrated. A simple case study is here offered in order to illuminate these issues. The Type 1 quasi-geostrophic model (QG1) is adopted as a reference formulation and several approximations to it are examined. They are formally accurate to zeroth or first-order in a Burger number B, but may include some higher-order terms and may imply analogues of global energy conservation and Lagrangian potential vorticity conservation. The approximate forms are all characterized by exclusion of the external Rossby mode, and each is related to a certain geostrophic formulation which is familiar in dynamical oceanography. The various approximations are assessed by examining their behaviour in three test problems which may be treated analytically: finite-amplitude internal Rossby wave propagation, zonal flow stability criteria and linearized internal free waves on baroclinic zonal flows. Two of the problems yield support for the hypothesis that the practical accuracy of an approximation may be improved by including higher-order terms in such a way that a potential vorticity conservation analogue is implied. The validity of this hypothesis in the QG1 case could be further investigated by solving more complicated test problems. Its general applicability cannot of course be claimed on the basis of a single case study; but the results obtained here afford evidence in its favour.