CONVECTION-ENHANCED DIFFUSION FOR RANDOM FLOWS

We analyze the effective diffusivity of a passive scalar in a two-dime nsional, steady, incompressible random flow that has mean zero and a s tationary stream function. We show that in the limit of small diffusiv ity or large Peclet number, with convection dominating, there is subst antial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show i n detail how the variational principles convert information about the geometry of the level lines of the random stream Function into propert ies of the effective diffusivity and substantiate the result of Isiche nko and Kalda that the effective diffusivity behaves like epsilon(3/13 ) when the molecular diffusivity epsilon is small, assuming some perco lation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in th e variational principles is more direct than for general random flows.