Space-time correlations of passive scalar in Kraichnan model

We consider the two-point, two-time (space-time) correlation of passive scalar $R(r,\tau )$ in the Kraichnan model under the assumption of homogeneity and isotropy. Using the fine-gird PDF method, we find that $R(r,\tau )$ 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 $\hat{R}(k,\tau )$, which is the Fourier transform of $R(r,\tau )$, is determined by $\hat{R}(k,0)$ and a diffusion kernal.