Passive scalar advected by a rapidly changing random velocity field: Probability density of scalar differences

The problem of a passive scalar advected by a correlated-in-time Gaussian random velocity field is considered. The equation for the probability density P(ΔT,r) of the scalar differences is derived. This equation resembles the nonlinear Boltzmann equation reflecting the fact that the scalar fluctuations carried by the velocity field along adjacent trajectories can interact due to the action of finite diffusivity. It is shown that the probability density of the scalar differences has algebraic tails in the scale-invariant range and that the moments ${\mathrm{S}}_{\mathrm{n}}$ =〈(ΔT${)}^{\mathrm{n}}$ 〉 with n