Balanced ellipsoidal vortex at finite Rossby number

Here we examine the motion of an isolated ellipsoidal vortex in a rotating stratified fluid. We derive an analytical solution to a set of balanced equations at the next order to quasi-geostrophic theory, providing insights into geophysical vortices at finite Rossby number epsilon. This is achieved through the solution of a set of complicated Poisson equations. Though complicated, the analytical solution give rise to a velocity field that depends linearly on the spatial coordinates inside the vortex, and, thus preserves the ellipsoidal form. From this general solution, we determine a number of equilibria where the vortex rotates steadily about the vertical axis and examine their stability. At the next order to QG, one finds asymmetry in the behaviour of cyclonic and anti-cyclonic vortices, with anti-cyclonic vortices rotating faster and generally more unstable than cyclonic vortices.