Hamiltonian fractals and chaotic scattering of passive particles by a topographical vortex and an alternating current

Abstract

We investigate the dynamics of passive particles in a two-dimensional incompressible open flow composed of a fixed topographical point vortex and a background current with a periodic component, the model inspired by the phenomenon of topographic vortices over mountains in the ocean and atmosphere. The tracer dynamics is found to be typically chaotic in a mixing region and regular in far upstream and downstream regions of the flow. Chaotic advection of tracers is proven to be of a homoclinic nature with transversal intersections of stable and unstable manifolds of the saddle point. In spite of simplicity of the flow, chaotic trajectories are very complicated alternatively sticking nearby boundaries of the vortex core and islands of regular motion and wandering in the mixing region. The boundaries act as dynamical traps for advected particles with a broad distribution of trapping times. This implies the appearance of fractal-like scattering function: dependence of the trapping time on initial positions of the tracers. It is confirmed numerically by computing a trapping map and trapping time distribution which is found to be initially Poissonian with a crossover to a power-law at the PDF tail. The mechanism of generating the fractal is shown to resemble that of the Cantor set with the Hausdorff fractal dimension of the scattering function to be equal to d≃1.84.