Generalized random forcing in random-walk turbulent dispersion models

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Abstract
The Kramers–Moyal expansion is used to derive an infinite hierarchy of Eulerian moment conservation equations from a random‐walk model with non‐Gaussian random forcing, thus generalizing the Langevin‐equation–Fokker–Planck analysis of van Dop et al. [H. van Dop, F. T. M. Nieuwstadt, and J. C. R. Hunt, Phys. Fluids 2 8, 1639 (1985)]. By imposing the condition that an initially well‐mixed state should remain so, equations for random forcing moments of arbitrary order are derived in terms of the Eulerian velocity moments of the turbulence. This procedure makes explicit the equivalence of the different procedures used by van Dop et al. and Thomson [D. J. Thomson, Q. J. R. Meteorol. Soc. 1 1 0, 1107 (1984)] to derive the first few forcing moments and extends their results. It is then shown that the random forcing approximation implies an infinite hierarchy of Eulerian closure assumptions, the first few of which were derived by van Dop et al. The analysis is extended to a class of rescaled random‐walk equations, and it is shown that the version developed by Wilson et al. [J. D. Wilson, B. J. Legg, and D. J. Thomson, Boundary‐Layer Meteorol. 2 7, 163 (1983)] is unique in that it alone is realizable for inhomogeneous Gaussian turbulence and then has Gaussian random forcing.