Dissipation statistics of a passive scalar in a multidimensional smooth flow

Abstract

We compute analytically the probability distribution function ${\cal P}(ε)$ of the dissipation field $ε=(\nabla θ)^{2}$ of a passive scalar $θ$ advected by a $d$-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime). The tail of the distribution is a stretched exponential: for $ε\to \infty$, $\ln {\cal P}(ε)\sim -(d^2ε)^{1/3}$.