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

In the first part of this paper, we investigate passive scalar or tracer ad vection-diffusion in frozen, two-dimensional, non-circular symmetric vortic es. We develop an asymptotic description of the scalar field in a time rang e 1 much less than t/T much less than Pe(1/3), where T is the formation tim e of the spiral in the vortex and Pe is a Peclet number, assumed much large r 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 th e exponent beta describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snaps hots of the advected scalar field by reconstructing the streamlines and the ir period from just two snapshots of the advected scalar field. In the second part of the paper, we investigate variance decay in a periodi cally moving singular vortex. We identify three different regions (core, ch aotic and KAM-tori). We find fast mixing in the chaotic region and investig ate a conjecture about mixing in the KAM-tori region. The conjecture enable s us to use the results from the first section and relates the Kolmogorov c apacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numeric al simulation of advection-diffusion of scalar in such a flow.