Statistical properties of point vortex equilibria on the sphere
Preprint Series # 877 NEWTON, Paul and SAKAJO, Takashi Statistical properties of point vortex equilibria on the sphere. (2007); AbstractWe describe a Brownian ratchet scheme which 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 linear algebra problem $A\Gamma = 0$ where $A$ is a $N \times N(N − 1)/2$
nonnormal configuration matrix obtained by requiring that all intervortical distances
on the sphere remain constant, and $\Gamma \in R^N$ is the (unit) vector of vortex strengths
which must lie in the nullspace of $A$. Existence of an equilibrium is expressed by the
condition $det($A^TA) = 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 nullspace, yielding all
values of the vortex strengths for which the configuration is an equilibrium. To home in
on an equilibrium, we allow the point vortices to undergo a random walk on the sphere
and after each random step we compute the smallest singular value of the configuration
matrix, keeping the new arrangement only if it decreases. When the singular value drops
below a predetermined convergence threshold, an equilibrium configuration is achieved
and we find a basis set for the nullspace of A by calculating the right singular vectors
corresponding to the singular values that are zero. For each $N = 4 \rightarrow 10$, we generate an
ensemble of 1000 equilibrium configurations which we then use to calculate statistically
averaged singular value distributions in order to obtain the averaged Shannon entropy
and Frobenius norm of the collection. We show that the statistically averaged singular
values produce an average Shannon entropy that closely follows a powerlaw scaling of
the form $< S > \sim N^\beta$, where $\beta \sim 2/3$. We also show that the length of the conserved
centerofvorticity vector clusters at a value of one and the total vortex strength of the
configurations cluster at the two extreme values ±1, indicating that the ensemble average
produces a single vortex of unit strength which necessarily sits at the tip of the centerofvorticity
vector. The Hamiltonian energy averages to zero reflecting a relatively uniform
distribution of points around the sphere, with vortex strengths of mixed sign.
