Perturbation theory for the δ-correlated model of passive scalar advection near 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-ε, and in the presence of a mean gradient, is calculated perturbatively in ε≪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 O(ε), implying an anomalous scaling of the third-order correlation function and the persistence of small scale anisotropy.

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