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Field theoretical renormalization group methods are applied to a simple modelof 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 structurefunctions and various pair correlators is established, and the correspondinganomalous exponents are calculated to the order $\epsilon^2$ of the $\epsilon$expansion. These exponents are non-universal, as a result of the degeneracy ofthe RG fixed point. In contrast to the case of a purely solenoidal velocityfield (Obukhov--Kraichnan model), the correlation functions in the case at handexhibit nontrivial dependence on both the IR and UV characteristic scales, andthe anomalous scaling appears already at the level of the pair correlator. Thepowers of the scalar field without derivatives, whose critical dimensionsdetermine the anomalous exponents, exhibit multifractal behaviour. The exactsolution for the pair correlator is obtained; it is in agreement with theresult obtained within the $\epsilon$ expansion. The anomalous exponents forpassively advected magnetic fields are also presented in the first order of the$\epsilon$ expansion.Comment: 31 pages, REVTEX file. More detailed discussion of the  one-dimensional case and comparison to the previous paper [20] are given;  references updated. Results and formulas unchange

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