Anomalous scaling, nonlocality, and anisotropy in a model of the passively advected vector field

A model of the passive vector quantity advected by the Gaussian velocity field with the covariance proportional to delta (t-t')\x-x'\ (epsilon) is studied; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by an infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques. The exponents are organized in a hierarchical order according to their degree of anisotropy, with the spectrum unbounded from above and the leading (minimal) exponent coming from the isotropic sector. This picture extends to higher-order correlation functions. Like in the scalar model, the second-order structure function appears nonanomalous and is described by the simple dimensional exponent: S(2)proportional tor(2-epsilon). For the higher-order , structure functions, S(2n)proportional tor(n(2-epsilon)+Delta n), the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of "dangerous" composite operators, whose negative critical dimensions determine the anomalous exponents Delta (n)<0. A close formal resemblance of the model with the stirred Navier-Stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper. Using the renormalization group, the anomalous exponents are calculated in the O(<epsilon>) approximation, in large d dimensions, for the even structure functions up to the twelfth order.