  The field-theoretic renormalization group (RG) and the operator product
expansion are applied to the model of a divergence-free vector quantity,
passively advected by the ``synthetic'' turbulent flow with a finite
correlation time. The vector field is described by the stochastic
advection-diffusion equation with the most general form of the inertial
nonlinearity. The statistics of the advecting velocity field is Gaussian with
powerlike energy spectrum and dispersion law. The inertial-range behavior of
the model is described by seven regimes that correspond to nontrivial fixed
points of the RG equations and exhibit anomalous scaling. The corresponding
anomalous exponents are associated with the critical dimensions of tensor
composite operators built solely of the passive vector field, which allows to
construct a regular perturbation expansion. The actual calculation is performed
in one-loop approximation, including the anisotropic sectors. Universality of
the exponents, their (in)dependence on the forcing, effects of the large-scale
anisotropy, compressibility and pressure are discussed. In particular, for all
the scaling regimes the exponents obey a hierarchy related to the degree of
anisotropy: the more anisotropic is the contribution of a composite operator to
a correlation function, the faster it decays in the inertial-range. Relevance
of the results for the real developed turbulence described by the stochastic
Navier-Stokes equation is discussed.
