Anomalous scaling in the anisotropic sectors of the Kraichnan model of passive scaler advection

Kraichnan's model of passive scalar advection in which the driving (Gaussia n) velocity field has fast temporal decorrelation is studied as a case mode l for understanding the anomalous scaling behavior in the anisotropic secto rs of turbulent fields. We show here that the solutions of the Kraichnan eq uation for the n-order correlation functions foliate into sectors that are classified by the irreducible representations of the SO(d) symmetry group. We find a discrete spectrum of universal anomalous exponents, with a differ ent exponent characterizing the scaling behavior in every sector. Generical ly the correlation functions and structure functions appear as sums over al l these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always charact erized by the smallest exponent, and therefore for sufficiently small scale s local isotropy is always restored. The calculation of the anomalous expon ents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory o f these functions involves the control of logarithmic divergences that tran slate into anomalous scaling with the ratio of the inner and the outer scal es appearing in the Anal result. In the second method we compute the expone nts from the zero modes of the Kraichnan equation for the correlation funct ions of the scaler field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer a nd inner scales as renormalization scales. In addition we derive exact fusi on rules, which govern the small scale asymptotics of the correlation funct ions in all the sectors of the symmetry group and in all dimensions.