Convection enhanced di usion for random ows

We analyze the e ective di usivity of a passive scalar in a two dimensional, steady, incompressible random ow that has mean zero and a stationary stream function. We show that in the limit of small di usivity or large Peclet number, with convection dominating, there is substantial enhancement of the e ective di usivity. Our analysis is based on some new variational principles for convection di usion problems [5,9] 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 in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the e ective di usivity and substantiate the result of Isichenko and Kalda [15] that the e ective di usivity behaves like 3=13 when the molecular di usivity is small, assuming some percolation theoretic facts. We also analyze the e ective di usivity for a special class of convective ows, random cellular ows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random ows. Department of Mathematics, University of California at Davis, Davis, CA 95616. Internet: [email protected]. This work was mostly done when A. Fannjiang was visiting Department of Mathematics, UCLA. and was supported by grants URI #N00014092-J-1890 from DARPA and NSF #DMS-9306720 from the National Science Foundation Department of Mathematics, Stanford University, Stanford, CA 94305. Internet: [email protected]. The work of G. Papanicolaou was supported by NSF grant DMS 9308471 and by AFOSR grant F49620-94-1-0436