Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell

Constant-density electrically conducting fluid is confined to a rapidly rot ating spherical shell and is permeated by an axisymmetric magnetic field. S low steady nonaxisymmetric motion is driven by a prescribed non-axisymmetri c body force; both rigid and stress-free boundary conditions are considered . Linear solutions of the governing magnetohydrodynamic equations are deriv ed in the small Ekman number E limit analytically for values of the Elsasse r number Lambda less than order unity and they are compared with new numeri cal results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Tho ugh the ageostrophic layers correspond to those previously isolated by Klee orin et nl. (1997) for axisymmetric flows, the quasi-geostrophic layers hav e a new structure resulting from the asymmetry of the motion. In the absence of magnetic field, the inviscid limit exhibits a strong shea r singularity on the tangent cylinder only removeable by the addition of vi scous forces. With the inclusion of magnetic field, large viscous forces re main whose strength L was measured indirectly by Hollerbach (1994b). For ma gnetic fields with dipole parity, cf. Kleeorin ef al. (1997), L increases t hroughout the range Lambda much less than 1; whereas, for quadrupole parity , cf. Hollerbach (1994b), L only increases for Lambda much less than E-1/5. The essential difference between the dipole and quadrupole fields is the ma gnitude of their radial components in the neighbourhood of the equator of t he inner sphere. Its finite value for the quadrupole parity causes the inte rnal shear layer - the Hartmann-Stewartson layer stump - to collapse and me rge with the equatorial Ekman layer when Lambda = O(E-1/5). Subsequently th e layer becomes an equatorial Hartmann layer, which thins and spreads polew ards about the inner sphere surface as Lambda increases over the range E-1/ 5 much less than Lambda much less than 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined t hrough matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by L decreases with increasing Lambda. Sin ce L depends sensitively on the detailed boundary layer structure, it provi des a sharp diagnostic of new numerical results for Hollerbach's model; the realized L-values compare favourably with the asymptotic theory presented.