  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.
