MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field

Motion is generated in a rotating spherical shell, by a slight differential rotation of the inner core. We show how the numerical solution tends, with decreasing Ekman number, to the asymptotic limit of Proudman [J. Fluid Mech. 1 (1956) 505–516]. Starting from geophysically large values, we show that the main qualitative features of the asymptotic solution show up only when the Ekman number is decreased below 10−6. Then, we impose a dipolar and force-free magnetic field with internal sources. Both the inner core and the liquid shell are electrically conducting. The first effect of the Lorentz force is to smooth out the change in angular velocity at the tangent cylinder. As the Elsasser number is further increased, the Proudman–Taylor constraint is violated, Ekman layers are changed into Hartmann type layers, shear at the inner sphere boundary vanishes, and the flow tends to a bulk rotation together with the inner sphere. Unexpectedly, for increasing strength of the field, there is a super-rotation (the angular velocity does not reach a maximum at the inner core boundary but in the interior of the fluid) localized in an equatorial torus. At a given field strength, the amplitude of this phenomenon depends on the Ekman number and tends to vanish in the magnetostrophic limit.