Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell

Abstract

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.