Mixing in frozen and time-periodic two-dimensional vortical flows

In the first part of this paper, we investigate passive scalar or tracer advection-diffusion in frozen, two-dimensional, non-circular symmetric vortices. We develop an asymptotic description of the scalar field in a time range 1 much less than t/T much less than Pe(1/3), where T is the formation time of the spiral in the vortex and Pe is a Peclet number, assumed much larger than 1. We derive the leading-order decay of the scalar variance E(t) for a singular non-circular streamline geometry, E(0)-E(t) proportional to (t(3)/T(3)Pe)(2+mu /2 beta /1+beta) The variance decay is solely determined by a geometrical parameter U and the exponent beta describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snapshots of the advected scalar field by reconstructing the streamlines and their period from just two snapshots of the advected scalar field. In the second part of the paper, we investigate variance decay in a periodically moving singular vortex. We identify three different regions (core, chaotic and KAM-tori). We find fast mixing in the chaotic region and investigate a conjecture about mixing in the KAM-tori region. The conjecture enables us to use the results from the first section and relates the Kolmogorov capacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numerical simulation of advection-diffusion of scalar in such a flow.