Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow

Field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the Gaussian non-solenoidal (``compressible'') velocity field with the covariance $\propto\delta(t-t')| x-x'|^{\epsilon}$. Convective range anomalous scaling for the structure functions and various pair correlators is established, and the corresponding anomalous exponents are calculated to the order $\epsilon^2$ of the $\epsilon$ expansion. These exponents are non-universal, as a result of the degeneracy of the RG fixed point. In contrast to the case of a purely solenoidal velocity field (Obukhov--Kraichnan model), the correlation functions in the case at hand exhibit nontrivial dependence on both the IR and UV characteristic scales, and the anomalous scaling appears already at the level of the pair correlator. The powers of the scalar field without derivatives, whose critical dimensions determine the anomalous exponents, exhibit multifractal behaviour. The exact solution for the pair correlator is obtained; it is in agreement with the result obtained within the $\epsilon$ expansion. The anomalous exponents for passively advected magnetic fields are also presented in the first order of the $\epsilon$ expansion.