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.