Dynamical aspects of a restricted three-vortex problem

Abstract

Point vortex systems that include vortices with constant coordinate functions are largely unexplored, even though they have reasonable physical interpretations in the geophysical context. Here, we investigate the dynamical aspects of the restricted three-vortex problem when one of the point vortices is assumed to be fixed at a location in the plane. The motion of the passive tracer is explored from a rotating frame of reference within which the free vortex with non-zero circulation remains stationary. By using basic dynamical system theory, it is shown that the vortex motion is always bounded, and any configuration of the three vortices must go through at least one collinear state. The present analysis reveals that any non-relative equilibrium solution of the vortex system either has periodic inter-vortex distances or it will asymptotically converge to a relative equilibrium configuration. The initial conditions required for different types of motion are explained in detail by exploiting the Hamiltonian structure of the problem. The underlying effects of a fixed vortex on the motion of vortices are also explored.