MEAN-FIELD THEORIES OF RANDOM ADVECTION

Two mean-field theories of random advection are formulated for the pur pose of predicting the probability density function (PDF) of a randoml y advected passive scalar, subject to an imposed mean scalar gradient. One theory is a generalization of the mean-field analysis used by Hol zer and Pumir [Phys. Rev. E 47, 202 (1993)] to derive the phenomenolog ical model of Pumir, Shraiman, and Siggia [Phys. Rev. Lett. 66, 2984 ( 1991)] governing PDF shape in the imposed-gradient configuration. The other theory involves a Langevin equation representing concentration t ime history within a fluid element. Predicted PDF shapes are compared to results of advection simulations by Holzer and Pumir. Both:theories reproduce gross trends, but the Langevin theory provides the better r epresentation of detailed features of the PDF's. An analogy is; noted between the two theories and two widely used engineering models of tur bulent mixing.