  The field theoretic renormalization group and the operator product expansion
are applied to two models of passive scalar quantities (the density and the
tracer fields) advected by a random turbulent velocity field. The latter is
governed by the Navier--Stokes equation for compressible fluid, subject to
external 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. The
original stochastic problems are reformulated as multiplicatively
renormalizable field theoretic models; the corresponding renormalization group
equations possess infrared attractive fixed points. It is shown that various
correlation functions of the scalar field, its powers and gradients,
demonstrate anomalous scaling behavior in the inertial-convective range already
for small values of $y$. The corresponding anomalous exponents, identified with
scaling (critical) dimensions of certain composite fields ("operators" in the
quantum-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 and
the perturbation theory presented here are manifestly Galilean covariant. The
validity of the one-loop approximation and comparison with Gaussian models are
briefly discussed.
