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The field theoretic renormalization group and the operator product expansionare applied to two models of passive scalar quantities (the density and thetracer fields) advected by a random turbulent velocity field. The latter isgoverned by the Navier--Stokes equation for compressible fluid, subject toexternal random force with the covariance $\propto \delta(t-t') k^{4-d-y}$,where $d$ is the dimension of space and $y$ is an arbitrary exponent. Theoriginal stochastic problems are reformulated as multiplicativelyrenormalizable field theoretic models; the corresponding renormalization groupequations possess infrared attractive fixed points. It is shown that variouscorrelation functions of the scalar field, its powers and gradients,demonstrate anomalous scaling behavior in the inertial-convective range alreadyfor small values of $y$. The corresponding anomalous exponents, identified withscaling (critical) dimensions of certain composite fields ("operators" in thequantum-field terminology), can be systematically calculated as series in $y$.The practical calculation is performed in the leading one-loop approximation,including exponents in anisotropic contributions. It should be emphasized that,in contrast to Gaussian ensembles with finite correlation time, the model andthe perturbation theory presented here are manifestly Galilean covariant. Thevalidity of the one-loop approximation and comparison with Gaussian models arebriefly discussed.

physical review. e, statistical, nonlinear and soft matter physics

Anomalous scaling of passive scalar fields advected by the Navier-Stokes  velocity ensemble: Effects of strong compressibility and large-scale  anisotropy