Jean Reinaud

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.

We determine and characterise relative equilibria for arrays of point vortices in a three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal rings whose centre lies on the same vertical axis. An additional vortex may be placed along this vertical axis. Depending on the parameters defining the array, the vortices on the two rings are of equal or opposite sign. We address the linear stability of the point vortex arrays. We find both stable equilibria and unstable equilibria, depending on the geometry of the array. For unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the point vortices. The linear stability of the vortex arrays depends on the number of vortices in the array, on the radius ratio between the two rings, on the vertical offset between the rings and on the vertical offset between the rings and the central vortex, when the latter is present. In this case the linear stability also depends on the strength of the central vortex. The nonlinear evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular motion and of chaotic motion.