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

Abstract

A model of the passive vector quantity advected by the Gaussian velocity field with the covariance approximately 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: S2 approximately r(2-epsilon). For the higher-order structure functions, S(2n)approximately r(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.