A dynamical equation for the distribution of a scalar advected by turbulence

A phenomenological model for the dissipation of scalar fluctuations due to the straining by the fluid motion is proposed in this letter. An explicit equation is obtained for the time evolution of the probability distribution function of a coarse-grained scalar concentration. The model relies on a self-convolution process. We first present this model in the Batchelor regime and then extend empirically our result to the turbulent case. The inclusion of this model in more general transport equations, including spatial gradients, is discussed in relation with 2D turbulence and stratified flows.