Royal Society Publishing

Point vortex equilibria on the sphere via Brownian ratchets

Paul K Newton, Takashi Sakajo


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, AΓ=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 ΓEmbedded ImageN 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(ATA)=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.



  • An alternative way of conveying this same idea is contained in Shannon's original 1948 publication. He proves that any averaging operation on Embedded Image of the form Embedded Image, where Embedded Image, with all aij≥0, will increase S. Other properties of this logarithmic quantity are also discussed in Shannon's paper.

    • Received May 14, 2008.
    • Accepted September 23, 2008.
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