On the Kraichnan Model of Passive Scalar Advection Near the Batchelor limit

TL;DR

This paper calculates the third order correlation function of a passive scalar in a Gaussian velocity field near the Batchelor limit, revealing anomalous scaling and persistent small-scale anisotropy.

Contribution

It provides a perturbative calculation of the third order correlation function close to the Batchelor limit, highlighting anomalous scaling behavior.

Findings

Scaling exponent is approximately 1 in 2D and 3D.

Third order correlation function exhibits anomalous scaling.

Small scale anisotropy persists at the Batchelor limit.

Abstract

The third order correlation function of the scalar field advected by a Gaussian random velocity, with a spatial scaling exponent $2 - \epsilon$, and in the presence of a mean gradient, is calculated perturbatively in $\epsilon << 1$. This expansion corresponds to the regime close to Batchelor's advection by linear diffeomorphisms. The scaling exponent is found to be equal to 1 in dimensions 2 and 3, up to corrections smaller than $ { \cal O } ( \epsilon ) $, implying an anomalous scaling of the third order correlation function and the persistence of small scale anisotropy.