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 (Gaussian) velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation 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 different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the Anal result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions 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 and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions.