The Mean Electromotive Force for Mhd Turbulence: the Case of a Weak Mean Magnetic Field and Slow Rotation

The mean electromotive force that occurs in the framework of mean–field magnetohydrodynamics is studied for cases in which magnetic field fluctuations are not only due to the action of velocity fluctuations on the mean magnetic field. The possibility of magnetic field fluctuations independent of a mean magnetic field, as they may occur as a consequence of a small–scale dynamo, is taken into account. Particular attention is payed to the effect of a mean rotation of the fluid on the mean electromotive force, although only small rotation rates are considered. Anisotropies of the turbulence due to gradients of its intensity or its helicity are admitted. The mean magnetic field is considered to be weak enough to exclude quenching effects. A τ -approximation is used in the equation describing the deviation of the cross–helicity tensor from that for zero mean magnetic field, which applies in the limit of large hydrodynamic Reynolds numbers. For the effects described by the mean electromotive force like α–effect, turbulent diffusion of magnetic fields etc. in addition to the contributions determined by the velocity fluctuations also those determined by the magnetic field fluctuations independent of the mean magnetic field are derived. Several old results are confirmed, partially under more general assumptions, and quite a few new ones are given. Provided the kinematic helicity and the current helicity of the fluctuations have the same signs the α–effect is always diminished by the magnetic fluctuations. In the absence of rotation these have, however, no influence on the turbulent diffusion. Besides the diamagnetic effect due to a gradient of the intensity of the velocity fluctuations there is a paramagnetic effect due to a gradient of the intensity of the magnetic fluctuations. In the absence of rotation these two effects compensate each other in the case of equipartition of the kinetic and magnetic energies of the fluctuations of the original turbulence, i.e. that with zero mean magnetic field, but the rotation makes the situation more complex. The Ω× J–effect works in the same way with velocity fluctuations and magnetic field fluctuations. A contribution to the electromotive force connected with the symmetric parts of the gradient tensor of the mean magnetic field, which does not occur in the absence of rotation, was found in the case of rotation, resulting from velocity or magnetic fluctuations. The implications of the results for the mean electromotive force for mean–field dynamo models are discussed with special emphasis to dynamos working without α-effect. The results for the coefficients defining the mean electromotive force which are determined by the velocity fluctuations in the case of vanishing mean motion agree formally with the results obtained in the kinematic approach, specified by second–order approximation and high–conductivity limit. However, their range of validity is clearly larger.