A New Equation for the Inversion of Ertel's Potential Vorticity to be Used for the Model Diagnosis of Group-Velocity-Based Energy Flux Associated with Waves at All Latitudes

By investigating the analytical solution of both equatorial waves and mid-latitude waves, we have derived an exact universal expression for the rotational flux which, after being added to pressure flux, is able to indicate the profile of the group velocity times wave energy for linear waves at all latitudes. This is what we call the level-0 expression of the energy flux. The level-0 energy flux is written using the solution of a new inversion equation for Ertel's potential vorticity (NIE-EPV), previously unmentioned in the literature. NIE-EPV is the cornerstone of the present study, because it suggests a possibility for the energy flux to be estimated (i) without using a Fourier analysis nor ray theory, and (ii) in the presence of coastal boundaries, which will allow for tropical-extratropical interactions in model output to be diagnosed in terms of an energy cycle in a future study. Presently, the level-0 energy flux is not practical for use as a model diagnostic, since the second-order time derivative term in NIE-EPV makes it difficult to solve for the streamfunction. Thus we hope that a future study is able to develop a numerical algorithm to solve NIE-EPV for the streamfunction. We also note the need to extend the theory to a continuously stratified ocean and also to test out the theory in the presence of a sheared mean flow, both of which topics await a future study.