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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 ψ₀(x) and distorting it locally in x space so that it exhibits the current scrP(pxi,{ξ}).
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Chen et al. (1989) studied this question.
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