99 07 03 8 v 1 2 9 Ju l 1 99 9 Topological fluid mechanics of point vortex motions

Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. The reduction of each system using its symmetries is described in detail. Restricting to three vortices with zero net circulation, each reduced system is described by a one degree of freedom Hamiltonian. The phase portrait of this reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston-Nielsen theory is then used to analyse these isotopy classes, and in certain cases strong conclusions about the dynamics of the advection can be made. §1 Introduction. The modelling of incompressible flow at high Reynolds number as a potential flow with embedded vortices has repeatedly proven useful both for analytical and numerical purposes. The subject has inspired numerous reviews, each stressing different aspects of the field. and Zabusky [Z] provide a representative sample. We are concerned here with the further simplification of modelling a two-dimensional flow by a finite collection of point vortices. While this system is admittedly highly idealised, it has found application and, to some extent, experimental verification starting with von Kármán's analysis in 1912 of the instability of the vortex street wake behind a cylinder and Onsager's 1949 explanation of the emergence of large coherent vortices in two-dimensional flow through an 'inverse cascade' of energy. (See the literature cited for further details.) In this paper we study the evolution of three kinds of point vortex systems, in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. An array of point vortices is useful in modelling flows in shear layers, wakes and jets. The motivation for studying vortex lattices comes from the problem of two-dimensional turbulence, a paradigm of atmospheric and oceanographic flows, and to a lesser extent from the study of vortex patterns formed in superfluid Helium. Kirchhoff recognised that the evolution of N point vortices could be formulated within the framework of classical mechanics as an N-degree of freedom Hamiltonian system. In many ways point vortices are the fluid mechanical analog of point masses evolving under the mutual interaction of Newtonian gravity. As with the N-body problem, point vortex systems have continuous symmetries that give rise to integrals of motion. In the …