Anomalous scaling of a passive scalar in the presence of strong anisotropy

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity θ(t,x), advected by the Gaussian strongly anisotropic velocity field with the covariance ~δ(t-t')\|x- x'\|^ɛ. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions S_n(r)≡<[θ(t,x+r)- θ(t,x)]ⁿ> are obtained. They are represented by superpositions of power laws; the corresponding anomalous exponents, which depend explicitly on the anisotropy parameters, are calculated to the first order in ɛ in any space dimension d. In the limit of vanishing anisotropy, the exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. For the finite anisotropy, the exponents cannot be associated with individual operators (which are essentially ``mixed'' in renormalization), but the aforementioned hierarchy survives for all the cases studied. The second-order structure function S₂ is studied in more detail using the renormalization group and zero-mode techniques; the corresponding exponents and amplitudes are calculated within the perturbation theories in ɛ, 1/d, and in the anisotropy parameters. If the anisotropy of the velocity is strong enough, the skewness factor S₃/S^3/2₂ increases going down towards the depth of the inertial range; the higher-order odd ratios increase even if the anisotropy is weak.