The Dependence of Dynamo $α$-Effect on Reynolds Numbers, Magnetic Prandtl Number, and the Statistics of MHD Turbulence

Abstract

We generalize the derivation of dynamo coefficient $α$ of Field et al (1999) to include the following two aspects: first, the de-correlation times of velocity field and magnetic field are different; second, the magnetic Prandtl number can be arbitrary. We find that the contributions of velocity field and magnetic field to the $α$ effect are not equal, but affected by their different statistical properties. In the limit of large kinetic Reynolds number and large magnetic Reynolds number, $α$-coefficient may not be small if the de-correlation times of velocity field and magnetic field are shorter than the eddy turn-over time of the MHD turbulence. We also show that under certain circumstances, for example if the kinetic helicity and current helicity are comparable, $α$ depends insensitively on magnetic Prandtl number, while if either the kinetic helicity or the current helicity is dominated by the other one, a different magnetic Prandtl number will significantly change the dynamo $α$ effect.