Renormalization Group, Operator Product Expansion, and Anomalous Scaling in a Model of Advected Passive Scalar

Abstract

Field theoretical renormalization group methods are applied to the Obukhov--Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance $<{\bf v}(t,{\bf x}){\bf v}(t&#39;,{\bf x})> - < v(t,{\bf x}){\bf v}(t&#39;,x&#39;)> \proptoδ(t-t&#39;)| x-x&#39;|^{\eps}$. Inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of ``dangerous&#39;&#39; composite operators [powers of the local dissipation rate], whose negative critical dimensions determine anomalous exponents. The main technical result is the calculation of the anomalous exponents in the order $\eps^{2}$ of the $\eps$ expansion. Generalization of the results obtained to the case of a ``slow&#39;&#39; velocity field is also presented.