Diffusion of passive scalar in a finite-scale random flow

We consider, a solvable model of the decay,of scalar variance in a single-scale random velocity field., We show. that if there is a separation between the flow scale k(flow)(-1) and the box size k(box)(-1), the decay rate lambda proportional to (k(box)/k(flow))(2) is determined by the turbulent diffusion of the box-scale mode. Exponential decay at the rate lambda is preceded by a. transient. powerlike decay (the total scalar variance similar tot(-5/2) if the Corrsin invariant is zero, t(-3/2) otherwise) that lasts a time tsimilar to1/lambda. Spectra are sharply peaked at k=k(box). The box-scale peak acts as a slowly decaying source to a secondary peak at the flow scale. The variance spectrum at scales intermediate between the two peaks (k(box)much less thankmuch less thank(flow)) is similar tok+ak(2)+... (a>0). The mixing of the flow-scale modes by the random flow produces, for the case of large Peclet number, a k(-1+delta) spectrum at kmuch greater thank(flow), where deltaproportional tolambda is a small correction. Our solution thus elucidates the spectral make up of thestrange mode,'' combining small-scale structure and a decay law set by the largest scales.