Dynamics of a passive tracer in a velocity field of four identical point vortices

Abstract

The dynamics of a passive tracer in the velocity field of four identical point vortices, moving under the influence of their self-induced advection, is investigated. Of interest is the change in mixing and transport properties of the tracer for the three different classes of vortex motion: periodic, quasi-periodic and chaotic. As a consequence of conservation laws, the vortex motion is confined to a finite region of phase space; therefore, the tracer phase space can be partitioned into an inner and an outer region. We find that in the case of quasi-periodic vortex motion the tracer phase space exhibits a well-defined barrier to transport between the central chaotic region and the outer region, where the trajectories are regular. In the case of chaotic vortex motion the barrier becomes permeable. The particle dynamics goes through an intermittent behaviour, where forays into the central region alternate with trapping in outer annular orbits. In the far field, an estimate of diffusion rates is made through a multipole expansion of the tracer velocity field. We make use of a specific stochastic model for the tracer velocity field which predicts no diffusion for the case of quasi-periodic vortex motion and, for the case of chaotic vortex motion, a diffusion rate that goes to zero at large distances from the vortex cluster.