Probability distribution of a stochastically advected scalar field

A systematic method is outlined for constructing workable approximations to the joint probability distribution scrP(ψ,ξ) of amplitude ψ and spatial gradient ξ of an active or passive scalar field that is advected by a prescribed isotropically distributed stochastic velocity field and subjected to molecular diffusion. scrP(ψ,ξ) is sampled along fluid-element paths, and closure is obtained by taking a multivariate-Gaussian reference field ${\mathrm{\psi }}_0$(x) and distorting it locally in x space so that it exhibits the current scrP(pxi,ξ).