Equilibrium singularity distributions in the plane

Abstract

We characterize all fixed equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary, or complex singularity strengths \Gamma . The dynamical system follows from the assumption that each of the N singularities moves according to the flowfield generated by all the others at that point. For strength vector {\Gamma} from R^N, the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from incompressible fluid flow. When {\Gamma} is purely imaginary, it corresponds to a system of sources and sinks, whereas when {\Gamma} from C^N the system consists of spiral sources and sinks discussed in Kochin et. al. (1964). We formulate the equilibrium problem as one in linear algebra, A\Gamma = 0, where A is a NxN complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel. {\Gamma} must then be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of {\Gamma} for which the system is a fixed equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We describe a method for classifying the equilibria in terms of the distribution of the non-zero eigenvalues (singular values) of A, or equivalently, the non-zero eigenvalues of the associated covariance matrix A'A, from which one can calculate the Shannon entropy of the configuration.