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 Λ less than order unity. Attention is restricted to the mainstream outside the Ekman-Hartmann layers adjacent to the inner and outer boundaries. When Λ (Questioned equal to) 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(Questioned equal to) Λ (Questioned equal to) E^1/3, those quasi-geostrophic layers become thinner (E/Λ)^1/2 Hartmann layers (inside only when Λ > O(E^3/7)), across which the geostrophic shear is suppressed with increasing Λ; they blend with the E^1/3 Stewartson layer at Λ = O(E^1/3). When E^1/3 (Questioned equal to) Λ (Questioned equal to) 1, magnetogeostrophic adjustments are made in a thicker inviscid Λ-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/Λ³)^1/8) and width O((E³/Λ)^1/8). supporting strong zonal winds. With increasing Λ the main adjustment to the geostrophic flow occurs at Λ = O(E^1/2). When E^1/2 (Questioned equal to) A (Questioned equal to) 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/Λ)⁴ and (E^1/2/Λ)^4/7 respectively; the former is swallowed up by the Hartmann layer when Λ ≥ O(E^3/7).