Cascade and dynamo action in a shell model of magnetohydrodynamic turbulence

A shell model of magnetohydrodynamic turbulence, which allows one to conserve all the integrals of motion in both two and three dimensions, is proposed and studied. We demonstrate that this model reproduces basic facts known in the small-scale turbulent dynamo theory. In particular, we consider a process of redistribution of magnetic helicity generated by the mean-field dynamo, described in the model as magnetic forcing, into a small-scale magnetic field. We argue that the resulting equilibrium magnetic field spectrum strongly depends on the level of magnetic helicity and cross helicity, introduced by the large scales. The spectra with spectral index "-5/3" dominate if the cross helicity vanishes. If the level of cross helicity is high (correlated velocity and magnetic field) the spectra depend on the magnetic helicity: the strong magnetic helicity suppresses any cascade providing steep spectra, while the vanishing helicity of turbulent magnetic fields results in the occurrence of Kraichnan-Iroshnikov spectral index "-3/2.".