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Eulerian solutions of O(σNv) for the stochastic transport problem for conservative tracers coupled with O(σ4ƒ) solutions for the flow problem in an infinite domain

An O(σvN) Eulerian stochastic theory [Cushman and Hu, 1997] for conservative transport is coupled with an O(σf4) Eulerian solution to the flow problem. The theory provides a self‐consistent recursive solution to the closure problem associated with Eulerian methods. The stochastic concentration is given to arbitrary order in σv, the variance of fluctuating velocity. The balance law for mean concentration is not required in the analysis. Closed forms for mean concentration are given up to O(σv4). The mean concentration is transformed to wave‐vector and frequency domain via fast Fourier transform to calculate numerically the mean concentration as well as its spatial moments. The results show that the first‐order solution in σv2 is equivalent to the nonlocal theory ofDeng et al. [1993]. Second‐order corrections to flow and transport equations slightly decrease the second longitudinal moment but significantly increase the second transverse moment, which is consistent with Hsu et al.'s [1996] results. The influence of second‐order corrections to skewness is not clear. The second longitudinal moment obtained from the second‐order correction agrees with the Monte Carlo result, but second‐order results for the second transverse moment and skewness significantly differ from those given in Monte Carlo simulations. Coupling the transport correction models with the velocity covariance generated through Monte Carlo simulation gives second transverse spatial moments that are very close to Monte Carlo simulations, which suggests that the correction to flow is more important than the correction to transport. The results also bring into question the accuracy of Monte Carlo simulations for flow.