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

Field theoretical renormalization group (RG) methods are applied to a simpl e model of a passive scalar quantity advected by the Gaussian nonsolenoidal ("compressible") velocity field with the covariance proportional to delta( t - t')\x - x'\(epsilon). Convective-range anomalous scaling for the struct ure functions and various pair correlators is established, and the correspo nding anomalous exponents are calculated to the order epsilon(2) Of the eps ilon expansion. These exponents are nonuniversal, as a result of the degene racy 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 a nontrivial dependence on both the IR and UV characte ristic scales, and the anomalous scaling appears already at the level of th e pair correlator. The powers of the scalar field without derivatives, whos e critical dimensions determine the anomalous exponents, exhibit multifract al behavior. The exact solution for the pair correlator is obtained; it is in agreement with the result obtained within the epsilon expansion. The ano malous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion. [S1063-651X(98)06412-5].