## Abstract

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/Λ^{3})^{1/8}) and width O((E^{3}/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}).

Original language | English |
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Pages (from-to) | 213-244 |

Number of pages | 32 |

Journal | Journal of Fluid Mechanics |

Volume | 344 |

DOIs | |

State | Published - 10 Aug 1997 |

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering