FRACTAL TRACER PATTERNS IN OPEN HYDRODYNAMICAL FLOWS: THE CASE OF LEAPFROGGING VORTEX PAIRS

Abstract

Patterns obtained by using tracer particles to visualize open hydrodynamical flows are investigated. As an example, the advection problem of passive tracers in the time-periodic velocity field of leapfrogging vortex pairs is considered. We show that the patterns are fractal if the tracer dynamics is chaotic, i.e., there exists a chaotic saddle, and if the initial conditions overlap with the stable manifold of this chaotic set (that is itself a fractal). Dye particles not advected away very quickly then accumulate on the unstable manifold of the chaotic saddle and the fractal dimension of such patterns is thus independent of the initial conditions. The time evolution of the fractal patterns appearing on snaphsots is followed within one period of the velocity field. We investigate which features of the chaotic dynamics determine the fractal properties. They turn out to be independent of when the snapshots are taken. The entropy function of the local Lyapunov exponents is computed based on the scattering motion of the tracer particles, along with the f(α) spectrum of the chaotic saddle.