Self-Consistent Mean Field Dynamo with Hall Effect

We investigate the influence of the Hall effect on the mean field dynamo action. As a starting point, we consider the theory developed by E. Blackman and G. Field, where the generation of a mean magnetic field is treated consistently by considering an evolution equation for the magnetic helicity and for the stochastic electromotive force, instead of prescribing its form as it is usually done. We found that the presence of the Hall effect, namely, the fact that the magnetic field is frozen in the electron flux and not in the bulk flux, introduces substantial changes in these evolution equations. We also present results of the numerical integration of the corresponding equations and compare them with previous results obtained without Hall effect. Our main result is that the dynamo effect also operates through an inverse cascade of magnetic helicity as in the standard dynamo theory, but now this inverse cascade is due to the electron fluid flux instead of the bulk fluid. We also find that the Hall effect can enhance or damp dynamo action, according to the sign of the terms in the evolution equation of the electromotive force, related to the cross-helicity