Transport in the cat's eye flow on intermediate time scales

TL;DR

This paper investigates how tracers move in a flow with cat's eye patterns, revealing complex transport behaviors like subdiffusion and super-ballistic motion at intermediate times, using simulations and Lévy walk theory.

Contribution

It introduces a comprehensive analysis of tracer transport in cat's eye flows, connecting flow parameters with diverse transport regimes through numerical simulations and Lévy walk modeling.

Findings

Transport varies from subdiffusive to super-ballistic depending on conditions.

Intermediate time scales exhibit non-monotonic aging effects.

Lévy walk theory effectively describes the ballistic regime.

Abstract

We consider the advection-diffusion transport of tracers in a one-parameter family of plane periodic flows where the patterns of streamlines feature regions of confined circulation in the shape of "cat's eyes", separated by meandering jets with ballistic motion inside them. By varying the parameter, we proceed from the regular two-dimensional lattice of eddies without jets to the sinusoidally modulated shear flow without eddies. When a weak thermal noise is added, i.e. at large P\'eclet numbers, several intermediate time scales arise, with qualitatively and quantitatively different transport properties: depending on the parameter of the flow, the initial position of a tracer, and the aging time, motion of the tracers ranges from subdiffusive to super-ballistic. Extensive numerical simulations of the aged mean squared displacement for different initial conditions are compared to a theory based on a L\'evy walk that describes the intermediate-time ballistic regime. Interplay of the walk process with internal circulation dynamics in the trapped state results at intermediate time scale in non-monotonic characteristics of aging.