Circular Vortex Arrays in Generalised Euler’s and Quasi-geostrophic Dynamics

We investigate the stability of circular point vortex arrays and their evolution when their dynamics is governed by the generalised two-dimensional Euler's equations and the three-dimensional quasi-geostrophic equations. These sets of equations offer a family of dynamical models depending continuously on a single parameter $\beta$ which sets how fast the velocity induced by a vortex falls away from it. In this paper, we show that the differences between the stability properties of the \emph{classical} two-dimensional point vortex arrays and the \emph{standard} quasi-geostrophic vortex arrays can be understood as a bifurcation in the family of models. For a given $\beta$, the stability depends on the number $N$ of vortices along the circular array and on the possible addition of a vortex at the centre of the array. From a practical point of view, the most important vortex arrays are the stable ones, as they are robust and long-lived. Unstable vortex arrays can, however, lead to interesting and convoluted evolutions, exhibiting quasi-periodic and chaotic motion. We briefly illustrate the evolution of a small selection of representative unstable vortex arrays.