Compressible advection of a passive scalar: Two-loop scaling regimes
  The influence of compressibility on the stability of the scaling regimes of the passive scalar advected by a Gaussian velocity field with finite correlation time is investigated by the field theoretic renormalization group within two-loop approximation. The influence of compressibility on the scaling regimes is discussed as a function of the exponents $\epsilon$ and $\eta$, where $\epsilon$ characterizes the energy spectrum of the velocity field in the inertial range $E\propto k^{1-2\epsilon}$, and $\eta$ is related to the correlation time at the wave number $k$ which is scaled as $k^{-2+\eta}$. The restrictions given by nonzero compressibility on the regions with stable infrared fixed points which correspond to the stable infrared scaling regimes are discussed in detail. A special attention is paid to the case of so-called frozen velocity field, when the velocity correlator is time independent. In this case, explicit inequalities which must be fulfilled in the plane $\epsilon-\eta$ are determined within two-loop approximation. The existence of a "critical" value $\alpha_c$ of the parameter of compressibility $\alpha$ at which one of the two-loop conditions is canceled as a result of the competition between compressible and incompressible terms is discussed. Brief general analysis of the stability of the scaling regime of the model with finite correlations in time of the velocity field within two-loop approximation is also given. 
M. Hnatich, E. Jurcisinova, M. Jurcisin and M. Repasan