  Kraichnan's model of passive scalar advection is studied as a case model for
understanding the anomalous scaling in the anisotropic sectors. We show here
that the solutions of the Kraichnan equation for the $n$ order correlations
foliate into sectors that are classified by the irreducible representations of
the SO(d) group. We find a discrete spectrum of universal anomalous exponents
in every sector. Generically the correlations and structure functions appear as
sums over all the contributions, with non-universal amplitudes which 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 correlations of gradient fields. The
corresponding theory involves the control of logarithmic divergences which
translate into anomalous scaling with the ratio of the inner and the outer
scales appearing in the final result. In the second way we compute the
exponents from the zero modes of the Kraichnan equation for the correlations of
the scalar 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. We derive fusion rules which govern the small
scale asymptotics of the correlation functions in the sectors of the symmetry
group, in all dimensions.
