Nonlinear turbulent magnetic diffusion and mean-field dynamo

The nonlinear coefficients defining the mean electromotive force (i.e., the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, the nonlinear $\kappa$ tensor, etc.) are calculated for an anisotropic turbulence. A particular case of an anisotropic background turbulence (i.e., the turbulence with zero-mean magnetic field) with one preferential direction is considered. It is shown that the toroidal and poloidal magnetic fields have different nonlinear turbulent magnetic diffusion coefficients. It is demonstrated that even for a homogeneous turbulence there is a nonlinear effective velocity that exhibits diamagnetic or paramagnetic properties depending on the anisotropy of turbulence and the level of magnetic fluctuations in the background turbulence. The diamagnetic velocity results in the field being pushed out from the regions with stronger mean magnetic field, while the paramagnetic velocity causes the magnetic field to be concentrated in the regions with stronger field. Analysis shows that an anisotropy of turbulence strongly affects the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, and the nonlinear $\alpha$ effect. Two types of nonlinearities (algebraic and dynamic) are also discussed. The algebraic nonlinearity implies a nonlinear dependence of the mean electromotive force on the mean magnetic field. The dynamic nonlinearity is determined by a differential equation for the magnetic part of the $\alpha$ effect. It is shown that for the $\alpha \Omega$ axisymmetric dynamo the algebraic nonlinearity alone (which includes the nonlinear $\alpha$ effect, the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, etc.) cannot saturate the dynamo generated mean magnetic field while the combined effect of the algebraic and dynamic nonlinearities limits the mean magnetic field growth.