Instability of two-dimensional vortices with a peripheral ring

We study the instability of piecewise-constant circular vortices in a two-dimensional incompressible fluid, focusing on configurations composed of a like-signed concentric core and annulus. First, the linear stability properties of these vortices are examined by identifying the possible resonances between Rossby waves supported on the three vorticity interfaces and classifying them according to the interface positions and vorticity jumps. The wavenumbers and growth rates of the most unstable normal modes are also determined as these parameters vary. Second, we simulate the finite-amplitude evolution of the linearly unstable vortices using a numerical solver for the nonlinear two-dimensional vorticity equation. Four main nonlinear regimes are identified: vortex core deformation, vortex core breaking, annulus breaking into a ring of vortices, and dipolar breaking of the entire structure. These regimes are mapped within parameter space, and the associated turbulent evolutions and underlying physical mechanisms are analysed. We further establish the conditions for the formation of vortex multipoles, characterise their steady-state configurations, and propose a simple stability criterion for stable multipoles These evolutions are contrasted with those previously reported for oceanic vortices composed of a core surrounded by an opposite-signed annulus. The results highlight the crucial role of like-signed vorticity regions, which often result in the spatial confinement of vorticity in the final states. Although highly idealised, our model results are compared with those of previous studies and with observations of Jupiter's polar vortices.