Anomalous scaling for passively advected magnetic fields

The scaling behavior of the covariance of the magnetic field in the three-dimensional kinematic dynamo problem is considered. The velocity is Gaussian and δ-correlated in time; its structure function scales with a positive exponent ξ. No unbounded growth of the magnetic field, i.e., no dynamo effect, occurs for ξ<1. A statistically steady state is then attained in the presence of a homogeneous forcing concentrated at the scale L. In the limit of small molecular diffusivities and large L, the correlation function of the magnetic field has an inertial-range scaling exponent -[(3+ξ)2]+(3/2)√1-(1/3)ξ(ξ+2). This result is extended to the d-dimensional case. In all cases a crucial role is played by zero modes.