Finite-core quasi-geostrophic circular vortex arrays with a central vortex

We numerically determine equilibrium states for three-dimensional quasi-geostrophic vortex arrays. The vortex arrays consist of five or eight equal volume and equal uniform potential vorticity peripheral eddies whose center lies equally spaced along a circle and of a similar vortex at the center of the array. We are specifically interested in the influence of the height-to-width aspect ratio of the vortices on the arrays’ properties. The linear stability of the vortex arrays is addressed and the vortex arrays are shown to be sensitive to instabilities when the vortices are close enough together. The onset of instability corresponds to a threshold for the distance between the peripheral vortices and the center of the array. Measured as a fraction of the mean vortex horizontal radius, the stability threshold increases as the height-to-width aspect ratio of the vortices increases. For a separation larger than the stability threshold between the vortices, the arrays are linearly stable, hence robust and long-lived in the nonlinear regime. We also show that prolate peripheral vortices exhibit a concave outer side when they are close to the center of the array, while it is convex otherwise. The nonlinear evolution of a selection of unstable vortex arrays is examined. In such cases, the vortices deform and some vortices merge, breaking the symmetry of the vortex array. The later evolution of the unstable vortex arrays can be convoluted.