A framework for understanding wave-vortical mode interactions in rotating, stratified flows

We derive a hierarchy of new PDE models intermediate between quasi-geostrophic (QG) models and the full dynamical equations for Rotating Shallow Water (RSW) and Rotating Boussinesq (RB) flows. The new models are constructed by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that 2DQG and 3DQG result from interactions between the vortical eigenmodes only. Continuing beyond QG in a non-perturbative manner, the simplest new model results by adding all interactions between two vortical modes and one inertial-gravity wave.
We illustrate how the new PDEs can be used to identify the nonlinear interactions primarily responsible for observed phenomena not captured by QG dynamics, such as anticyclone dominance in RSW decay, and the dramatic change in behavior of RB with respect to Burger number unity. Our aim is a complete understanding of wave-vortical interactions in rotating stratified flows.