- Published 2003 in Chaos

The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.

@article{Thiffeault2003ChaoticMI,
title={Chaotic mixing in a torus map.},
author={Jean-Luc Thiffeault and Stephen Childress},
journal={Chaos},
year={2003},
volume={13 2},
pages={502-7}
}