An exact steadily rotating surface quasi-geostrophic elliptical vortex

An elliptical region having a particular distribution of anomalous buoyancy or temperature at the surface of an otherwise unbounded rotating stratified fluid is shown to steadily rotate under the quasi-geostrophic approximation. The particular distribution is proportional to the vertical thickness of an ellipsoid, divided by its mean thickness, in the limit of vanishing thickness. The steady rotation of this structure or vortex is assured by the known steady rotation of any ellipsoid, and can be obtained by an appropriate limit. It is found by numerical experimentation that this vortex is stable if its minor to major aspect ratio lambda exceeds 0.427, approximately. Notably, a two-dimensional elliptical vortex (having uniform vorticity) is stable for lambda > 41/3. Instabilities of surface vortices are characterised by the ejection of a weak tongue of buoyancy, which subsequently rolls up into a street of weak vortices. The main vortex is thereby reduced in aspect ratio and remains robust for long times.