Combined effects of compressibility and helicity on the scaling regimes of a passive scalar advected by turbulent velocity field with finite correlation time

The influence of compressibility and helicity on the stability of the scaling regimes of a 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 helicity and 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 and helicity on the regions with stable infrared fixed points which correspond to the stable infrared scaling regimes are discussed. 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.