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

In this letter, we propose a phenomenological model to describe the dissipation of scalar fluctuations due to fluid turbulence, focusing on the time evolution of the probability distribution function of coarse-grained scalar concentration. This model is founded on a self-convolution process and is initially presented in the Batchelor regime, with empirical extensions to turbulent conditions. We examine its integration into broader transport equations, particularly in relation to two-dimensional turbulence and stratified flows. The turbulent transport of tracers like temperature or salinity holds significance for various applications; current models predominantly address mean quantities and their moments. Our approach, however, seeks to capture the full distribution evolution, acknowledging its relevance in scenarios such as reactive flows and the turbulent mixing of stably stratified fluids where sedimentation effects vary between fluid density types. We consider the impact of vorticity distributions in two-dimensional turbulence, where expected equilibrium states may be modified by turbulence-induced mixing effects. This study lays the groundwork for further investigations into the statistical mechanics of mixing in complex fluid environments.