A Comparison of Closures for Stochastic Advection-Diffusion Equations

Abstract

Perturbation-based moment equations for advection-diffusion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to the same equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches, using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, and prove that the resulting moment equations are in fact hyperbolic for the special case of pure advection b...