POTENTIAL VORTICITY INVERSION AND BALANCED EQUATIONS OF MOTION FOR ROTATING AND STRATIFIED FLOWS

Balanced equations of motion based on potential vorticity evolution an d inversion for the shallow water and stratified primitive equations a re derived and, in some shallow-water cases, numerically tested. The s chemes are based on asymptotic expansions in Rossby or Froude number, or rational scaling-based truncations of the equations of motion, assu ming that the dynamics are determined by the advection of potential vo rticity. Thus, regimes of validity are rapidly rotating and/or highly stratified flow. Both new and familiar results are straightforwardly o btained, in a unified framework in both height and isentropic coordina tes. For both shallow-water and stratified equations, Rossby number ex pansions schemes give quasi-geostrophy at lowest order. Both gradient- wind balance and the nonlinear terms in the potential vorticity enter at next order. A low Froude number expansion for non-rotating now give s two-dimensional now, uncoupled in the vertical at lowest order. A si ngle consistent inversion scheme can be derived that is valid at lowes t order in Froude number for all Rossby numbers, for both shallow-wate r and the stratified equations. It may be a particularly appropriate m odel for the atmospheric mesoscale and oceanic submesoscale, where rot ation and stratification can both be important in defining balanced mo tion. A model is also proposed that is valid at both planetary and syn optic scales, combining the familiar planetary geostrophic and quasi-g eostrophic equations. Most of the models derived require the solution only of linear or near linear elliptic equations, possibly with varyin g coefficients. Numerical experiments indicate that a higher-order inv ersion can be quantitatively better than quasi-geostrophy, if Rossby n umber and divergence are sufficiently small. In some other cases, no n oticeable improvement over quasi-geostrophy is found, even when the Ro ssby number is quite small. However, the balanced model valid for both planetary and synoptic scales shows a significant qualitative and qua ntitative improvement over both planetary geostrophy and quasi-geostro phy for large-scale flows, and its evolution is in good agreement with a primitive equation model.