Motion of three point vortices in a periodic parallelogram

Abstract

The motion of three interacting point vortices with zero net circulation in a periodic parallelogram defines an integrable dynamical system. A method for solving this system is presented. The relative motion of two of the vortices can be ‘mapped’ onto a problem of advection of a passive particle in ‘phase space’ by a certain set of stationary point vortices, which also has zero net circulation. The advection problem in phase space can be written in Hamiltonian form, and particle trajectories are given by level curves of the Hamiltonian. The motion of individual vortices in the original three-vortex problem then requires one additional quadrature. A complicated structure of the solution space emerges with a large number of qualitatively different regimes of motion. Bifurcations of the streamline pattern in phase space, which occur as the impulse of the original vortex system is changed, are traced. Representative cases are analysed in detail, and a general procedure is indicated for all cases. Although the problem is integrable, the trajectories of the vortices can be surprisingly complicated. The results are compared qualitatively to vortex paths found in large-scale numerical simulations of two-dimensional turbulence.