Stability of 3D Gaussian vortices in an unbounded, rotating, vertically-stratified, Boussinesq flow: Linear analysis

The linear stability of three-dimensional (3D) vortices in rotating, stratified flows has been studied by analyzing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely-used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby number (\(-0.5 < Ro < 0.5\)) and Burger number (\(0.02 < Bu < 2.3\)) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of \(Ro-Bu\). We have found neutrally-stable vortices only over a small region of the \(Ro-Bu\) parameter space: cyclones with \(Ro \sim 0.02-0.05\) and \(Bu \sim 0.85-0.95\). However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g., \(Ro