Space-time correlations of passive scalar in Kraichnan model

Abstract

We consider the two-point, two-time (space-time) correlation of passive scalar R(r,τ) in the Kraichnan model under the assumption of homogeneity and isotropy. Using the fine-gird PDF method, we find that R(r,τ) satisfies a diffusion equation with constant diffusion coefficient determined by velocity variance and molecular diffusion. Its solution can be expressed in terms of the two-point, one time correlation of passive scalar, i.e., R(r,0). Moreover, the decorrelation of R^(k,τ), which is the Fourier transform of R(r,τ), is determined by R^(k,0) and a diffusion kernal.