Quasigeostrophic, ellipsoidal vortices in a stratified fluid

As a very simple model of some of the lens-like vortices of Mediterranean origin that drift in the eastern North Atlantic, we consider an ellipsoidal volume of uniform potential vorticity embedded in an otherwise quiescent, uniformly stratified fluid of infinite extent. We show that, under the quasigeostrophic approximation, such a vortex is a steadily rotating equilibrium solution and obtains its rotation rate as a function of the lengths of the principal axes. We also determine the dispersion relations for normal modes propagating on the ellipsoid. We find that unstable eigenmodes exist over a considerable range of the geometrical parameters that characterise the ellipsoid.