Analytical solutions for rotating vortex arrays involving multiple vortex patches

Abstract

A continous two-parameter family of analytical solutions to the Euler equations are presented representing a class of steadily rotating vortex arrays involving $N\,{+}\,1$ interacting vortex patches where $N\,{\ge}\,3$ is an integer. The solutions consist of a central vortex patch surrounded by an $N$-fold symmetric alternating array of satellite point vortices and vortex patches. One of the parameters governs the size of the central patch, the other governs the size of the $N$ satellite patches. In the limit where the areas of the satellite vortex patches tend to zero, the solutions degenerate to the exact solutions of Crowdy (J. Fluid Mech. vol. 469, 2002, p. 209). Limiting states are found in which cusps form only on the central patch, only on the satellite patches, or simultaneously on both central and satellite patches. Contour dynamics simulations are used to check the mathematical solutions and test their robustness. The linear stability of a class of ‘point-vortex models’ (in which the patches are replaced by point vortices) are also studied in order to examine the stability of the distributed-vorticity configurations to pure-displacement modes. On the other hand, a desingularization of all point vortices to Rankine vortices leads to a class of ‘quasi-equilibria’ consisting purely of interacting vortex patches close to hydrodynamic equilibrium.