Chaotic advection in the velocity field of leapfrogging vortex pairs

Abstract

The advection problem of passive tracer particles in the time-periodic velocity field of leapfrogging vortex pairs is investigated in the context of chaotic scattering. We numerically determine a few basic unstable periodic orbits of the tracer dynamics, and the non-attracting chaotic set responsible for the motion of particles injected in front of the vortex system. The latter consists of two parts: a hyperbolic component based on strongly unstable periodic orbits, and a non-hyperbolic component that is close to KAM surfaces, The invariant manifolds of the chaotic set are also plotted and their relevance for the particle dynamics is discussed. The tracer dynamics has one single dimensionless parameter: the energy of the vortex system. As a new phenomenon, we point out the existence of stable bounded trajectories between the vortex pairs at sufficiently large energies. A quantitative characterization of the tracer dynamics in terms of the so-called free energy function is given and the multifractal spectrum of Lyapunov exponents, the escape rate and other characteristics of the transient chaotic motion are determined.