Dynamics of SQG Vortices

The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geophysical flows in which the dynamics are governed by potential temperature dynamics on the boundary. The model can be used to explore the transition from two-dimensional to three-dimensional mesoscale geophysical flows. We examine the dynamics of SQG vortices and the resulting flow in the entire fluid including at first order in Rossby number ($O(Ro)$). This requires solving an extension to the usual QG equation to compute the velocity corrections, and we demonstrate this mathematical procedure. As we show, it is simple to obtain the vertical velocity, but difficult to find the $O(Ro)$ horizontal corrections. We then consider the specific case of an exact SQG vortex solution developed by Dritschel (2011). We examine the interaction of two such vortices in both the infinite and doubly periodic domain.