Anomalous scaling in statistical models of passively advected vector fields

We use the methods of the renormalization group and the operator product expansion to consider the problem of the stochastic advection of a passive vector field with the most general form of the nonlinear term allowed by the Galilean symmetry. The external velocity field satisfies the Navier–Stokes equation. We show that the correlation functions have anomalous scaling in the inertial range. The corresponding anomalous exponents are determined by the critical dimensions of tensor composite fields (operators) built from only the fields themselves. We calculate the anomalous dimensions in the leading order of the expansion in the exponent in the correlator of the external force in the Navier–Stokes equation (the one-loop approximation of the renormalization group). The anomalous exponents exhibit a hierarchy related to the anisotropy degree: the lower the rank of the tensor operator is, the lower its dimension. The leading asymptotic terms are determined by the scalar operators in both the isotropic and the anisotropic cases, which completely agrees with Kolmogorov's hypothesis of local isotropy restoration.