Axisymmetric flow between differentially rotating spheres in a dipole magnetic field

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric potential magnetic field with dipole parity; the regions outside the shell are rigid insulators. Slow steady axisymmetric motion is driven by rotating the inner sphere at a slightly slower rate. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E-limit for values of the Elsasser number Lambda less than order unity. Attention is restricted to the mainstream outside the Ekman-Hartmann layers adjacent to the inner and outer boundaries. When Lambda much less than E-1/2, MHD effects only lead to small perturbations of the well-known Proudman-Stewartson solution. Motion is geostrophic everywhere except in the E-1/3 shear layer containing the tangent cylinder to the inner sphere; that is embedded in thicker E-2/7 (interior), E-1/4 (exterior) viscous layers in which quasi-geostrophic adjustments are made, When E-1/2 much less than Lambda much less than E-1/3, those quasi-geostrophic layers become thinner (E/Lambda)(1/2) Hartmann layers (inside only when Lambda > O(E-3/7)), across which the geostrophic shear is suppressed with increasing ii; they blend with the E-1/3 Stewartson layer at Lambda = O(E-1/3). When E-1/3 much less than Lambda much less than 1, magnetogeostrophic adjustments are made in a thicker inviscid Lambda-layer. Viscous effects are confined to the shrinking (blended) Hartmann-Stewartson layer; it consists of a column (stump) aligned to the tangent cylinder, attached to the equator, height O((E/Lambda(3))(1/8)) and width O((E-3/Lambda)(1/8)), supporting strong zonal winds. With increasing Lambda the main adjustment to the geostrophic flow occurs at Lambda = O(E-1/2). When E-1/2 much less than Lambda much less than 1, the mainstream analogue to the non-magnetic Proudman solution is a state of rigid rotation, except for large quasi-geostrophic shears in (magnetic-Proudman) layers adjacent to but inside both the tangent cylinder and the equatorial ring of the outer sphere of widths (E-1/2/Lambda)(4) and (E-1/2/Lambda)(4/7) respectively;the former is swallowed up by the Hartmann layer when Lambda greater than or equal to O(E-3/7).