Anisotropic non-perturbative zero modes for passively advected magnetic fields

A first analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the $d$-dimensional kinematic magneto-hydrodynamics problem in the presence of a mean magnetic field. The velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent $\xi$. Inertial-range anisotropic contributions to the scaling exponents, $\zeta_j$, of second-order magnetic correlations are associated to zero modes and have been calculated non-perturbatively. For $d=3$, the limit $\xi\mapsto 0$ yields $\protect{\zeta_j=j-2+ \xi (2j^3 +j^2 -5 j - 4)/[2(4 j^2 - 1)]}$ where $j$ ($j\geq 2$) is the order in the Legendre polynomial decomposition applied to correlation functions. Conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made.