Chaotic Mixing in a Diffeomorphism of the Torus

The advection and diffusion of a passive scalar is investigated for a diffeomorphism of the 2-torus. The map is chaotic and the limit of weak nonuniformity of stretching is considered. This allows an analytic understanding of the transition between the superexponential and exponential phases of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the variance is concentrated at small scales, even though the large scale sets the decay rate. The duration of the superexponential phase is proportional the the logarithm of the exponential decay rate, which means that the the decay must be very fast for the superexponential phase to be observable.