Two different rapid decorrelation in time limits for turbulent diffusion

Abstract

A turbulent diffusion model in which the velocity field is Gaussian and rapidly decorrelating in time (GRDT) has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations. The utility of the GRDT model is the fact that correlation functions of the passive scalar field solve closed partial differential equations; the usual moment closure obstacle is averted. We study here the sense in which the GRDT model describes turbulent diffusion by a general, non-Gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero. When the velocity field is rescaled in a particular manner in this rapid decorrelation limit, then a limit theorem of Khas'minskii indeed shows that the passive scalar statistics are described asymptotically by the GRDT Model for a broad class of velocity field models. We provide, however, an explicit example of a “Poisson blob model” velocity field which has two different well-defined rapid decorrelation in time limits. In one, the passive scalar correlation functions converge to those of the GRDT Model, and in the other, they converge to a distinct nontrivial limit in which the correlation functions do not solve closed PDE's. We provide both mathematical and heuristic explanations for the differences between these two limits. The conclusion is that the GRDT Model provides a universal description of the rapid decorrelation in time limit of general non-Gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process.