Eulerian solutions of O(sigma(N)(v)) for the stochastic transport problem for conservative tracers coupled with O(sigma(4)(f)) solutions for the flowproblem in an infinite domain

An O(sigma(nu)(N)) Eulerian stochastic theory [Cushman and Hu, 1997] for co nservative transport is coupled with an O(sigma(f)(4)) Eulerian solution to the flow problem. The theory provides a self-consistent recursive solution to the closure problem associated with Eulerian methods. The stochastic co ncentration is given to arbitrary order in sigma(nu), the variance of fluct uating velocity. The balance law for mean concentration is not required in the analysis. Closed forms for mean concentration are given up td O(sigma(n u)(4)). The mean concentration is transformed to wave-vector and frequency domain via fast Fourier transform to calculate numerically the mean concent ration as well as its spatial moments. The results show that the first-orde r solution in sigma(nu)(2) is equivalent to the nonlocal theory of Deng et al. [1993]. Second-order corrections to flow and transport equations slight ly decrease the second longitudinal moment but significantly increase the s econd transverse moment, which is consistent with Hsu et al.'s [1996] resul ts. The influence of second-order corrections to skewness is not clear. The second longitudinal moment obtained from the second-order correction agree s with the Monte Carlo result, but second-order results for the second tran sverse moment and skewness significantly differ from those given in Monte C arlo simulations. Coupling the transport correction models with the velocit y covariance generated through Monte Carlo simulation gives second transver se spatial moments that are very close to Monte Carlo simulations, which su ggests that the correction to flow is more important than the correction to transport. The results also bring into question the accuracy of Monte Carl o simulations for flow.