Evolution of a scalar gradient's probability density function in a random flow

The evolution of the probability density function (PDF) of the passive scalar gradient is studied in the limit of large Peclet and Prandtl numbers in d dimensions. Without diffusion, the closed Fokker-Planck equation can be derived and solved analytically leading to a number of conclusions. In particular, it allows the description of the restoration of the rotational symmetry and enables one to distinguish different regimes of evolution and different intervals of PDF behavior.