The Kraichnan-Kazantsev dynamo

The problem of the dynamo effect for a Kraichnan incompressible helicity-free velocity field is considered. Exploiting a quantum formalism first introduced by Kazantsev (A.P. Kazantsev, Sov. Phys. JETP 26, 1031-1034 (1968)), we show that a critical magnetic Reynolds number exists for the presence of dynamo. The value of the Prandtl number influences the spatial distribution of the magnetic field and its growth in time. The magnetic field correlation length is always the largest between the diffusive scale and the viscous scale of the flow. In the same way the field growth is characterized by a time scale that corresponds to the largest between the diffusive and the viscous characteristic time.