Anomalous scaling, nonlocality, and anisotropy in a model of the passivelyadvected vector field - art. no. 046310

A model of the passive vector quantity advected by the Gaussian velocity fi eld with the covariance proportional to delta (t-t')\x-x'\ (epsilon) is stu died; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by a n infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional tec hniques. The exponents are organized in a hierarchical order according to t heir degree of anisotropy, with the spectrum unbounded from above and the l eading (minimal) exponent coming from the isotropic sector. This picture ex tends to higher-order correlation functions. Like in the scalar model, the second-order structure function appears nonanomalous and is described by th e simple dimensional exponent: S(2)proportional tor(2-epsilon). For the hig her-order , structure functions, S(2n)proportional tor(n(2-epsilon)+Delta n ), the anomalous scaling behavior is established as a consequence of the ex istence in the corresponding operator product expansions of "dangerous" com posite operators, whose negative critical dimensions determine the anomalou s exponents Delta (n)<0. A close formal resemblance of the model with the s tirred Navier-Stokes equation reveals itself in the mixing of relevant oper ators and is the main motivation of the paper. Using the renormalization gr oup, the anomalous exponents are calculated in the O(<epsilon>) approximati on, in large d dimensions, for the even structure functions up to the twelf th order.