Simple models of non-Gaussian statistics for a turbulently advected passive scalar

Abstract

We study the probability distribution of a passive scalar undergoing turbulent mixing in the presence of a mean scalar gradient. Kerstein’s model, which describes the turbulent mixing process as a collection of instantaneous local rearrangements of the scalar, is argued to be a plausible approximation when the size of the system is much larger than the velocity correlation length, and after the description has been coarse grained over a correlation volume. In the physical range of parameters, we find numerically that the fluctuations of the scalar θ are close to exponentially distributed when a linear mean scalar gradient is imposed. The phenomenological mean-field-like theory of Pumir, Shraiman, and Siggia [Phys. Rev. Lett. 66, 2984 (1991)] is derived heuristically beginning with the Kerstein model. This theory predicts strictly exponential tails for the probability distribution of the scalar fluctuations P(θ), i.e., P(θ)∼exp(‖θ‖). We also consider a simplified version of the Kerstein model which is analytically fully tractable and gives qualitatively similar results numerically. Under conditions of spatial homogeneity and imposed linear mean scalar gradient, this simplified model and the full Kerstein model are described by the same mean-field theory. However, for the simplified model, the large-‖θ‖ asymptotic behavior of P(θ) is Poisson-like, i.e., P(θ)∼exp[-‖θ‖ln‖θ/(const)‖].