The Conservation of Pseudoenergy in Lagrangian Time-Mean Flow

Abstract

Abstract When the time-averaging operator is applied to the Generalized Lagrangian Mean equations of motion there results a conservation law involving a total static energy invariant which contains the so-called “pseudoenergy”. This invariant is analogous to the Kelvin or Bjerknes circulation which is conserved as an invariant in the zonal averaging case. An approximate pseudoenergy is also derived which is applicable in cases where quadratic “available” potential energy is of interest. In the small-amplitude limit, the pseudoenergy may be evaluated as an Eulerian diagnostic in terms of the perturbation potential vorticity and entropy fields. As in the zonal averaging case, the Lagrangian time mean leads to conservation laws not containing any kind of artificial conversion of energy which appears in the conventional Eulerian mean formulation. Hence the Lagrangian mean provides a static energy invariant analogous to the Kelvin or Bjerknes circulation which may be of use in the study of nonlinear waves on t...