Passive advection of a vector field: effects of strong compressibility

The field theoretic renormalization group and the operator product expansion are applied to the stochastic model of a passively advected vector field. The advecting velocity field is generated by the stochastic Navier-Stokes equation with compressibility taken into account. The model is considered in the vicinity of space dimension $d=4$ and the perturbation theory is constructed within a double expansion scheme in $y$ and $\epsilon=4-d$, where $y$ describes scaling behaviour of the random force that enters a stochastic equation for the velocity field. We show that the correlation functions of the passive vector field in the inertial range exhibit anomalous scaling behaviour. The critical dimensions of tensor composite operators of passive vector field are calculated in the leading order of $y$,$\epsilon$ expansion.