Generalized dispersion relation for a conservative, finite-amplite Rossby wave in slowly varying barotropic shear flow

We propose a form of dispersion relation for a conservative, finite-amplitude, near-plane barotropic Rossby wave in slowly varying parallel shear flow. The relation is expressed in terms of pseudomomentum and pseudoenergy densities of the wave whose exact conservation laws are known. The zonal phase speed is given by the functional derivative of pseudoenergy density with respect to pseudomomentum density, wherein the effects of wave-mean flow interaction and the amplitude dependence of the phase speed are implicit. This theoretical prediction is evaluated against the observed phase speed in an idealized numerical simulation of nonlinear barotropic decay on a sphere. The theory agrees well with the observed phase speed of the wave except in regions where the meridional eddy momentum flux vanishes and/or where the phase speed and phase tilts change abruptly. In the generalized dispersion relation critical lines (at which the zonal phase speed equals the zonal mean zonal wind speed) are no longer singular, which suggests that a finite-amplitude Rossby wave can enter a region where the phase speed is EASTWARD with respect to the mean flow. Multiple critical lines identified on each flank of the jet in the barotropic decay simulation corroborates this prediction.