Undulating coherent structures in two-dimensional turbulence: A quasi -equilibrium approach
The Great Dark Spot (GDS) on Neptune provides a particularly interesting example of a long-lived, large-scale coherent structure in geostrophic turbulence. This coherent structure exhibits periodic oscillations in the aspect ratio and the orientation with respect to the zonal direction of the GDS. Polvani et al.  proposed a simple deterministic model of the GDS as oval vortex patch in shear, having constant values of potential vorticity in the vortex and in the ambient zonal flow. Using an equivalent barotropic, quasi-geostrophic dynamics, they showed that the observed characteristics of GDS oscillation can be reproduced by their simple model. The primary goal of the present paper is to develop a statistical model of the GDS that is analogous to the deterministic model of Polvani et al. In order to take such an approach to model the GDS, which is a slowly-varying coherent structure, it is necessary to extend the statistical equilibrium theory based on exact invariants to a quasi-equilibrium theory, in which some further quasi-invariants constrain the most probable state. For an undulating oval vortex, these quasi-invariants are certain quadratic moments of the potential vorticity anomaly that effectively determine the aspect ratio and orientation of the vortex. We validate our quasi-equilibrium theory by reproducing the Kida ellipse, an undulating elliptical vortex patch that is an exact ideal fluid motion, through our quasi-equilibrium approach. In addition, we find new families of steady oval vortices in two-dimensional and quasi-geostrophic flow. The families of statistical equilibria extend the Moore-Saffman elliptical vortices, which are deterministic steady flows.