Relative equilibria of vortices in two dimensions

Abstract

An old problem of the evolution of finitely many interacting point vortices in the plane is shown to be amenable to investigation by critical point theory in a way that is identical to the study of the planar n-body problem of celestial mechanics. For any choice of positive circulations of the vortices it is shown by critical point theory applied to Kirchhoff's function that there are many relative equilibria configurations. Each of these configurations gives rise to a stationary configuration of the vortices in a suitably chosen rotating coordinate system. A sharp lower bound on the number of stationary vortex configurations for the problem of point vortices interacting in the plane is given. The problem of point vortices in a circular disk is defined and it is shown that these estimates hold for stationary configurations of small size.An old problem of fluid dynamics that was introduced as a Hamiltonian system by Kirchhoff in 1876 is to determine the evolution of finitely many interacting point vortices in the plane (see ref. 1). A recent experiment on superfluid 4He has demonstrated the existence of stationary configurations of vortices in a rotating reference frame (see ref.2).Here a stationary vortex configuration is identified as a relative equilibrium configuration. A sharp lower bound is given on the numbers of relative' equilibria classes that exist for an arbitrary choice of positive circulations (K, Kn) E R + for which the Hamiltonian function is nondegenerate. These estimates hold in the plane as well as for the problem of the motion ofrectilinear vortices in a circular cylinder. These lower bounds are derived from the topology ofthe domain ofKirchhoff's function. Compare the statements in celestial mechanics (see refs. 3 and 4).The idea is to use topological methods to investigate the sta-*tionary vortex configurations in the same way that the relative equilibria ofpoint masses in the n-body problem is investigated in celestial mechanics. The reason that many ofthe results from the celestial mechanics setting are valid immediately in this fluid dynamics setting is that the topology of the domain of the respective functions is identical. In Section 1 we view the stationary vortex configurations as relative equilibria configurations and the way in which critical points of the Hamiltonian arise is shown. Main theorems are stated in Section 2. The problem of point vortices in a circular disk is defined and investigated in Section 3. Finally, the observations are summarized.

Integrable, Chaotic, and Turbulent Vortex Motion in Two-Dimensional Flows

“…This work was primarily motivated by experiments on vortices in superfluid He, performed by Yarmchuk et al (1979), in which uniformly rotating equilibria with N = 2, ... , 11 were visualized. Palmore (1982) considers the application of Morse theory to es timate the number of all equilibria as a function of N.…”

Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method

“…There has since then been extensive studies on relative equilibria in the planar N -vortex problem, see for example [30,31,34]. The study of relative equilibria is a subject in itself.…”

Relative equilibrium and collapse configurations of heterogeneous vortex triple rings

“…A lower bound on the number of relative equilibria can be established when all circulations are positive, by a simple extension of the topological arguments in [11]. Having circulations of differing sign opens up the possibility of vortex collapse [2], and indeed collapse configurations of two rings are known to exist [7].…”