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(-1 )(flow ) and the box size k(-1 )(box ) , the decay rate lambda proportional, variant ( 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 approximately t(-5/2) if the Corrsin invariant is zero, t(-3/2) otherwise) that lasts a time t approximately 1/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) <0) . The mixing of the flow-scale modes by the random flow produces, for the case of large Péclet number, a k(-1+delta) spectrum at k>> k(flow) , where delta proportional lambda is a small correction. Our solution thus elucidates the spectral make up of the "strange mode," combining small-scale structure and a decay law set by the largest scales.