In geophysical and astrophysical settings, rotating stratified flows often exhibit large-scale, nearly isolated vortices. Examples include the Mediterranean eddies in the Atlantic ocean, or the Great Red Spot (GRS) in Jupiter. These vortices have been widely studied using shallow-water or quasi-geostrophic models for decades. In particular, these models have successfully explained why these vortices maintain nearly lenticular shapes through time. However, prior reduced models have a blind spot when it comes to predicting the internal dynamics of such vortices, despite the fact that they are far from being motionless in their bulk (e.g. as observed for the GRS). Various instabilities may sustain small-scale turbulence and accelerate the decay of large-scale vortices on long time scales.

Here, I will present a reduced model accounting for the bulk dynamics of large-scale pancake-like vortices. This model, which is developed in the framework of an interdisciplinary collaboration between pure and applied mathematics, is largely inspired by some ideas and methods pioneered by astrophysicists (e.g. S. Chandrasekhar or N. Lebovitz). First, I will describe the properties of the normal modes, because wave motions are often key to understanding the transition to turbulence in geophysical flows. As in the rotating non-stratified case, it will be shown that the wave spectrum solely consists of eigenvalues, and that the eigenvectors are all smooth. Moreover, it will be explained why some low-frequency waves/modes, which are governed by a mixed hyperbolic-elliptic problem for the velocity, can exist below the usual cutoff frequency of inertia-gravity waves. Next, by combining local and global stability methods, I will discuss whether some bulk instabilities could sustain small-scale bulk turbulence in strongly deformed stratified vortices.