Anomalous exponents in the rapid-change model of the passive scalar advection in the order $ε^{3}$

Abstract

Field theoretic renormalization group is applied to the Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance $<{\bf v}(t,{\bf x}){\bf v}(t',{\bf x})> - <{\bf v}(t,{\bf x}){\bf v}(t',{\bf x'})> \proptoδ(t-t')|{\bf x}-{\bf x'} |^ε$. Inertial-range anomalous exponents, related to the scaling dimensions of tensor composite operators built of the scalar gradients, are calculated to the order $ε^{3}$ of the $ε$ expansion. The nature and the convergence of the $ε$ expansion in the models of turbulence is are briefly discussed.