The Stability of Vortex Arrays on Double Concentric Circles

Abstract

The stability of double circular arrays of vortices is investigated where each array consists of N point vortices of equal strength. The permanent configurations of the arrays are found to need the vortices of different strength on a different array depending on the distance between the arrays. The linear stability of the configurations is investigated for several values of N, and it is found that the configuration is stable for N=2 and 3 and there exist almost stable arrangements for N=4, 5 and 6. The nonlinear stability of the vortex configurations is numerically studied, and it is found that linearly stable configurations are stable against finite amplitude disturbances and the almost linearly stable configurations are also stable for small amplitude disturbances.