Perturbation theory for the delta-correlated model of passive scalar advection near the Batchelor limit

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 much less than 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(epsilon), implying an anomalous scaling of the third-order correlation function and the persistence of small scale anisotropy.