Point vortex equilibria on the sphere via Brownian ratchets

Abstract

We describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, A Γ =0, where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ ∈ N is the (unit) vector of vortex strengths that must lie in the null space of A . Existence of an equilibrium is expressed by the condition det( A T A )=0, while uniqueness follows if Rank( A )= N −1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A , and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N =4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy.