### abstract

- An effect of the differential rotation on the nonlinear electromotive force in MHD turbulence is found. It includes a nonhelical $\alpha$ effect which is caused by a differential rotation, and it is independent of a hydrodynamic helicity. There is no quenching of this effect contrary to the quenching of the usual $\alpha$ effect caused by a hydrodynamic helicity. The nonhelical $\alpha$ effect vanishes when the rotation is constant on the cylinders which are parallel to the rotation axis. The mean differential rotation creates also the shear-current effect which changes its sign with the nonlinear growth of the mean magnetic field. However, there is no quenching of this effect. These phenomena determine the nonlinear evolution of the mean magnetic field. An effect of a uniform rotation on the nonlinear electromotive force is also studied. A nonlinear theory of the ${\bf \Omega} {\bf \times} \bar{\bf J}$ effect is developed, and the quenching of the hydrodynamic part of the usual $\alpha$ effect which is caused by a uniform rotation and inhomogeneity of turbulence, is found. Other contributions of a uniform rotation to the nonlinear electromotive force are also determined. All these effects are studied using the spectral $\tau$ approximation (the third-order closure procedure). An axisymmetric mean-field dynamo in the spherical and cylindrical geometries is considered. The nonlinear saturation mechanism based on the magnetic helicity evolution is discussed. It is shown that this universal mechanism is nearly independent of the form of the flux of magnetic helicity, and it requires only a nonzero flux of magnetic helicity. Astrophysical applications of these effects are discussed.