Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity

Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical hows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasi-geostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi-geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.