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 $〈\mathbf{v}(t,\mathbf{x})\mathbf{v}{(t}^{\ensuremath{'}},\mathbf{x})〉\ensuremath{-}〈\mathbf{v}(t,\mathbf{x})\mathbf{v}{(t}^{\ensuremath{'}},{\mathbf{x}}^{\ensuremath{'}})〉\ensuremath{\propto}\ensuremath{\delta}(t\ensuremath{-}{t}^{\ensuremath{'}})|\mathbf{x}\ensuremath{-}{\mathbf{x}}^{\ensuremath{'}}{|}^{\ensuremath{\varepsilon}}.$ 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 ${\ensuremath{\varepsilon}}^{2}$ of the $\ensuremath{\varepsilon}$ expansion. Generalization of the results obtained to the case of a ``slow'' velocity field is also presented.