Space-time correlations of passive scalar in Kraichnan model

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

Automated summary

This study investigates the two-point, two-time correlation of passive scalar in the Kraichnan model. By employing the fine-grid PDF method, it is found that the passive scalar satisfies a diffusion equation, and its solution can be expressed in terms of the two-point, one-time correlation of passive scalar. Furthermore, the decorrelation of the passive scalar is determined by a diffusion kernel and the one-point, one-time correlation.