We consider the motion of a spherical annulus of fluid driven by the slightly differential rotation of the confining spherical shells. The novelty of our analysis is the imposition of a strong axisymmetric potential magnetic field on a conducting annulus bounded by insulators. The strength of the field is measured by its ability to dominate the dynamics of the fluid shell, being much larger than the viscous and Coriolis forces. The fluid flow is structured such as to minimize the effect of the strong field and the associated induction currents. The thrust of the analysis is the identification of regions of the fluid shell in which the azimuthal flow and generated field are relatively uniform. These regions are bounded by shear layers in which the azimuthal flow and field vary rapidly. Thickness of the shears is about a square root of a Hartmann layer thickness. These Hartmann boundary layers control main streams in the magnetohydrodynamic flow. An imposed axial magnetic field produces two rigid-body rotating regions that have different rotation rates outside and inside the axial cylinder tangent to the inner sphere. The azimuthal magnetic field is proportional to the cylindrical radius and is expelled from the outside of the tangent cylinder. For an imposed dipole magnet, similar regions with rigid-body rotation are divided by a shear that has a shape of a lobe touching the outer equator. Very weak meridional fluid flux is inversely proportional to a cube of the imposed magnetic field for these degenerated cases. In a quadrupole magnetic field, two lobe-like regions are symmetric with respect to the shell’s equator, touch the outer sphere, and rotate with the inner sphere. Outside these regions, differential rotation is valuable and a weak meridional fluid flux is inversely proportional to a square of the imposed magnetic field as in the general case. ©1998 American Institute of Physics.