1Mathematics Research Institute, School of Engineering, Computer Science & Mathematics, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, UK.
2MHD in Astro- & Geophysics (ENS/IPGP) & CNRS, LRA, Département de Physique, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France.
(Received 16 January 2009; revised 27 September 2009; accepted 30 September 2009; first published online 2 February 2010)
Abstract. We consider the steady axisymmetric motion of an electrically conducting fluid contained within a spherical shell and permeated by a centred axial dipole magnetic field, which is strong as measured by the Hartmann number M. Slow axisymmetric motion is driven by rotating the inner boundary relative to the stationary outer boundary. For M >> 1, viscous effects are only important in Hartmann boundary layers adjacent to the inner and outer boundaries and a free shear-layer on the magnetic field line that is tangent to the outer boundary on the equatorial plane of symmetry. We measure the ability to leak electric current into the solid boundaries by the size of their relative conductance ε. Since the Hartmann layers are sustained by the electric current flow along them, the current inflow from the fluid mainstream needed to feed them increases in concert with the relative conductance, because of the increasing fraction L of the current inflow leaked directly into the solids. Therefore the nature of the flow is sensitive to the relative sizes of ε-1 and M.
The current work extends an earlier study of the case of a conducting inner boundary and an insulating outer boundary with conductance εo = 0 (Dormy, Jault & Soward, J. Fluid Mech., vol. 452, 2002, pp. 263-291) to other values of the outer boundary conductance. Firstly, analytic results are presented for the case of perfectly conducting inner and outer boundaries, which predict super-rotation rates Ωmax of order M1/2 in the free shear-layer. Successful comparisons are made with numerical results for both perfectly and finitely conducting boundaries. Secondly, in the case of a finitely conducting outer boundary our analytic results show that Ωmax is O(M1/2) for εo-1<< 1 << M 3/4, O(εo2/3 M1/2) for 1 << εo-1 << M3/4 and O(1) for 1 << M3/4 << εo-1 On increasing εo-1 from zero, substantial electric current leakage into the outer boundary, Lo ~ 1, occurs for εo-1 << M3/4 with the shear-layer possessing the character appropriate to a perfectly conducting outer boundary. When εo-1 = O(M3/4) the current leakage is blocked near the equator, and the nature of the shear-layer changes. So, when M3/4 << εo-1, the shear-layer has the character appropriate to an insulating outer boundary. More precisely, over the range M3/4 << εo-1 << M the blockage spreads outwards, reaching the pole when εo-1 = O(M). For M << εo-1 current flow into the outer boundary is completely blocked, Lo << 1.