Correlation functions of a passive scalar as a measure of the statistics of the velocity gradient

The mixing of a passive impurity in a random flow in the limit of weak molecular diffusion, when there is a range of scales between a small diffusion scale and a relatively large correlation length of the gradient of the flow velocity field, is considered. In this range of scales, to describe the evolution of the initial distribution of the impurity in space near a certain Lagrangian trajectory, it is sufficient to approximate the velocity field by a linear profile. Second- and fourth-order correlation functions of the impurity concentration are related to the statistics of the affine deformation of a small volume element of the liquid. Meanwhile, the pair correlation function reflects the extension statistics only in the principal direction, and the fourth-order correlation function reflects in its angular singularities the complete extension statistics in a three-dimensional flow. In a two-dimensional flow, the behavior of the fourth order correlation function in angular singularities has different properties and significantly depends on the diffusion coefficient. The conclusions are valid for any time-uniform gradient of the velocity field and are practically significant for the measurement of this statistics.