Nonperturbative spectrum of anomalous scaling exponents in the anisotropicsectors of passively advected magnetic fields

We address the scaling behavior of the covariance of the magnetic field in the three-dimensional kinematic dynamo problem when the boundary conditions and/or the external forcing are not isotropic. The velocity field is Gauss ian, space homogeneous, and delta correlated in time, and its structure fun ction scales with a positive exponent xi. The covariance of the magnetic fi eld is naturally computed as a sum of contributions proportional to the irr educible representations of the SO(3) symmetry group. The amplitudes are no nuniversal, determined by boundary conditions. The scaling exponents are un iversal, forming a discrete, strictly increasing, spectrum indexed by the s ectors of the symmetry group. When the initial mean magnetic field is zero, no dynamo effect is found, irrespective of the anisotropy of the forcing. The rate of isotropization with decreasing scales is fully understood from these results.