A super-rotating shear layer in magnetohydrodynamic spherical Couette flow

Abstract

We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M^−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M^−2/3 ([Lt ]M^−1/2) and latitudinal extent M^−1/3. We construct its analytic solution and so determine an inward displacement width O(M^−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 10⁴. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.