Renormalization group, operator expansion, and anomalous scaling in a simple model of turbulent diffusion

Using the renormalization group method and the operator expansion in the Obukhov–Kraichnan model that describes the intermixing of a passive scalar admixture by a random Gaussian field of velocities with the correlator $\langle\mathbf v(t,\mathbf x)\mathbf v(t',\mathbf x)\rangle- \langle\mathbf v(t,\mathbf x)\mathbf v(t',\mathbf x')\rangle\propto\delta(t-t')|\mathbf x-\mathbf x'|^{\varepsilon}$, we prove that the anomalous scaling in the inertial interval is caused by the presence of “dangerous” composite operators (powers of the local dissipation rate) whose negative critical dimensions determine the anomalous exponents. These exponents are calculated up to the second order of the $\varepsilon$ expansion.