title: Diffusion of passive scalar in a finite-scale random flow.
abstract: 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) <<k<< k(flow) ) is approximately k+a k(2) +... (a>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.
authors.affiliation: DAMTP/CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. as629@damtp.cam.ac.uk
authors.forename: Alexander A
authors.initials: AA
authors.lastname: Schekochihin
authors.forename: Peter H
authors.initials: PH
authors.lastname: Haynes
authors.forename: Steven C
authors.initials: SC
authors.lastname: Cowley