Stability of a regular polygon of finite vortices

Abstract

It is well known that a system of N point vortices arranged in a circular row, so that the vortices are at the vertices of a regular polygon, is stable if N < 7, neutrally stable if N = 7 and unstable if N > 7 (Havelock 1931). The effect on this result of taking account of the finite size of the vortices is considered analytically. The vortices are considered to be uniform with small but finite core. Approximate equations for the shape and motion of a vortex subjected to an external velocity field are given and used to evaluate the shape and angular velocity of rotation of the system and to study its stability to plane infinitesimal disturbances. It is found that the system is stable if N < 7 and unstable if N ≥ 7. These asymptotic results for small core area are in general consistent with Dritschel (1985) where the motion and stability of up to N = 8 finite vortices is evaluated numerically; the steady configuration and the stability results for these values of N are in agreement except in a region of parameter space where a high degree of accuracy is required in the numerical calculation to resolve the growth rate of small disturbances. The case of a linear array of finite vortices is obtained as a special limiting case of the system. The growth rate of plane infinitesimal disturbances for this case is given.