# Effect of differential rotation on nonlinear mean electromotive force and stellar dynamos

7th September 2004

Authors: Igor Rogachevskii (Ben-Gurion University), Nathan Kleeorin (Ben-Gurion University)

### Abstract

An effect of the mean differential rotation on the nonlinear electromotive force 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 causes the "shear-current" effect. The ''shear-current" effect is associated with the $\bar{\bf W} {\bf \times} \bar{\bf J}$-term in the mean electromotive force and results in the generation of the mean magnetic field even in a nonhelical homogeneous turbulence (where $\bar{\bf W}$ is the mean vorticity caused by the differential rotation and $\bar{\bf J}$ is the mean electric current). The ''shear-current" effect changes its sign with the nonlinear growth of the mean magnetic field at some value $\bar{\bf B}_\ast$. The magnitude $\bar{\bf B}_\ast$ determines the level of the saturated mean magnetic field which is less than the equipartition field. However, there is no quenching of this effect. It is shown that the background magnetic fluctuations due to the small-scale dynamo enhance the "shear-current" effect, and reduce the magnitude $\bar{\bf B}_\ast$. When the level of the background magnetic fluctuations is larger than $1/3$ of the kinetic energy of the turbulence, the mean magnetic field can be generated due to the "shear-current" effect for an arbitrary exponent of the energy spectrum of the velocity fluctuations. These phenomena determine the nonlinear evolution of the stellar and solar large-scale magnetic fields. 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 $\alpha$ 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 $\tau$-approximation (the Orszag third-order closure procedure). An axisymmetric mean-field dynamo is considered. Applications of these effects to the stellar and solar large-scale magnetic fields are discussed.