Stability analysis of a class of two-dimensional multipolar vortex equilibria

Abstract

The stability properties of a class of explicit multipolar vortex solutions of the two-dimensional Euler equations found in Crowdy [Phys. Fluids. 11, 2556 (1999)] are studied. While the tripole solutions are linearly unstable in all configurations, it is found that the exact quadrupolar vortices have distinguished linear stability properties revealing them to be neutrally stable in all configurations. This result is consistent with observations by previous investigators on the general robustness of quadrupolar vortex structures. Higher-order multipolar structures are linearly unstable when the satellites are too close together, but become neutrally stable when the satellites are far enough apart and the ambient vortex patch sufficiently distorted. The nonlinear evolution of perturbed solutions is investigated numerically using contour dynamics methods. Some new results concerning limiting states involving cusp singularities in the vortex patch boundaries are also presented.