Potential vorticity inversion and balanced equations of motion for rotating and stratified flows

Abstract

AbstractBalanced equations of motion based on potential vorticity evolution and inversion for the shallow water and stratified primitive equations are derived and, in some shallow‐water cases, numerically tested. The schemes are based on asymptotic expansions in Rossby or Froude number, or rational scaling‐based truncations of the equations of motion, assuming that the dynamics are determined by the advection of potential vorticity. Thus, regimes of validity are rapidly rotating and/or highly stratified flow. Both new and familiar results are straightforwardly obtained, in a unified framework in both height and isentropic coordinates. For both shallow‐water and stratified equations, Rossby number expansions 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 flow gives two‐dimensional flow, uncoupled in the vertical at lowest order. A single consistent inversion scheme can be derived that is valid at lowest order in Froude number for all Rossby numbers, for both shallow‐water and the stratified equations. It may be a particularly appropriate model for the atmospheric mesoscale and oceanic submesoscale, where rotation and stratification can both be important in defining balanced motion. A model is also proposed that is valid at both planetary and synoptic scales, combining the familiar planetary geostrophic and quasi‐geostrophic equations. Most of the models derived require the solution only of linear or near linear elliptic equations, possibly with varying coefficients.Numerical experiments indicate that a higher‐order inversion can be quantitatively better than quasigeostrophy, if Rossby number and divergence are sufficiently small. In some other cases, no noticeable improvement over quasi‐geostrophy is found, even when the Rossby number is quite small. However, the balanced model valid for both planetary and synoptic scales shows a significant qualitative and quantitative improvement over both planetary geostrophy and quasi‐geostrophy for large‐scale flows, and its evolution is in good agreement with a primitive equation model.