Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar

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 $〈\mathbf{v}(t,\mathbf{x})\mathbf{v}{(t}^{\prime },\mathbf{x}\left)〉-〈\mathbf{v}\right(t,\mathbf{x})\mathbf{v}{(t}^{\prime },{\mathbf{x}}^{\prime })〉\propto \delta \left(t-t^{\prime }\right)|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^{\varepsilon }.$ 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 certain essential or “dangerous” 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 ${\varepsilon }^2$ of the $\varepsilon$ expansion. Generalization of the results obtained to the case of a “slow” velocity field is also presented.