The Fourth-Order Correlation Function of a Randomly Advected Passive Scalar

Advection of a passive scalar $\theta$ in $d=2$ by a large-scale velocity field rapidly changing in time is considered. The Gaussian feature of the passive scalar statistics in the convective interval was discovered in \cite{95CFKLa}. Here we examine deviations from the Gaussianity: we obtain analytically the simultaneous fourth-order correlation function of $\theta$. Explicit expressions for fourth-order objects, like $⟨({\theta }_1-{\theta }_2{)}^4⟩$ are derived.