We demonstrate with a nonlinear magnetohydrodynamic (MHD) code that angular momentum can be transported because of the magnetic instability of toroidal fields under the influence of differential rotation, and that the resulting effective viscosity may be high enough to explain the almost rigid-body rotation observed in radiative stellar cores. We only consider stationary, current-free fields, and only those combinations of rotation rates and magnetic field amplitudes which provide maximal numerical values of the viscosity. We find that the dimensionless ratio of the effective over molecular viscosity, ν T /ν, linearly grows with the Reynolds number of the rotating fluid multiplied by the square-root of the magnetic Prandtl number, which is approximately unity for the considered red subgiant star KIC 7341231. For the interval of magnetic Reynolds numbers considered -which is restricted by numerical constraints of the nonlinear MHD code -the magnetic Prandtl number has a remarkable influence on the relative importance of the contributions of the Reynolds stress and the Maxwell stress to the total viscosity, which is magnetically dominated only for Pm > ∼ 0.5. We also find that the magnetized plasma behaves as a non-Newtonian fluid, i.e., the resulting effective viscosity depends on the shear in the rotation law. The decay time of the differential rotation thus depends on its shear and becomes longer and longer during the spin-down of a stellar core.

The empirical finding with data from the Helioseismic and Magnetic Imager instrument onboard the Solar Dynamics Observatory of positive (negative) horizontal Reynolds stress at the northern (southern) hemisphere for solar giant cells is discussed for its consequences on the theory of solar/stellar differential rotation. By solving the nonlinear Reynolds equation for the angular velocity Ω when neglecting the meridional circulation, we show that the horizontal Reynolds stress of the northern hemisphere is always negative at the surface but positive in the bulk of the solar convection zone by the action of the Λ effect. The Λ effect, which describes the angular momentum transport of rigidly rotating anisotropic turbulence and which avoids a rigid-body rotation of the convection zones, is in the horizontal direction of cubic power in Ω and proves to be always positive (negative) in the northern (southern) hemisphere in both theory and simulations. In contrast, theories without the Λ effect, which may provide the observed solar rotation law only by the action of a meridional circulation, will lead to a horizontal Reynolds stress with signs that are not observed.