Influence of annular boundaries on Thomson's vortex polygon stability

The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle inside an annulus is presented. The problem is reduced to one of the equilibrium stability of the Hamiltonian system with a cyclic variable. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The stability of the stationary motion is interpreted as a Routh stability. The exponential instability is shown always to take place if N ≥ 7. For N = 2, 4, and 6, the parameter space is divided in two: a Routh stability domain in an exact nonlinear setting and an exponential instability domain. For N = 3 and 5, the parameter space consists of three domains. The stability of the third in an exact nonlinear setting is sandwiched between the Routh and exponential domains. Its analysis remains an open problem with its solution requiring nonlinear analysis.