Passive Advection of a Vector Field by Compressible Turbulent Flow: Renormalizations Group Analysis near d = 4 ^†

The renormalization group approach and the operator product expansion technique are applied to the model of a passively advected vector field by a turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. 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 the Navier-Stokes equation. The properties of the correlation functions are investigated, and anomalous scaling and multifractal behaviour are established. All calculations are performed in the leading order of y, $\epsilon$ expansion (one-loop approximation).