A super-rotating shear layer
in magnetohydrodynamic
spherical Couette flow
Emmanuel Dormy, Dominique Jault, Andrew Soward
1Institut de Physique du Globe de Paris / C.N.R.S.,
4, place Jussieu, 75252 Paris cedex 05, France.
2L.G.I.T./C.N.R.S., Université Joseph Fourier BP53,
38041 Grenoble Cedex 9, France.
3 School of Mathematical Sciences, University of Exeter, Exeter,
EX4 4QE, UK.
J. Fluid Mech., vol. 452,
pp. 263-291, 2002.
Abstract. We consider axisymmetric magnetohydrodynamic motion in a spherical shell
driven by rotating the inner boundary relative to the stationary
outer boundary - spherical Couette flow. The inner solid sphere
is rigid with the same electrical conductivity as the surrounding
fluid; the outer rigid boundary is an insulator. A force free dipole
magnetic field is maintained by a dipole source at the centre.
For strong imposed fields (as measured by the Hartmann number M),
the numerical simulations of Dormy et al. (1998) showed
that a super-rotating shear layer (with angular velocity about
50% above the angular velocity of the inner core)
is attached to the magnetic magnetic field line C tangent
to the outer boundary at the equatorial plane of symmetry.
At large M, we obtain analytically the mainstream solution
valid outside all boundary layers by application of Hartmann jump
conditions across the inner and outer sphere boundary layers.
We formulate the large M boundary layer problem for the free
shear layer of width M-1/2 containing C and
solve it numerically. The super-rotation can be understood
in terms of the nature of the meridional electric current flow
in the shear layer, which is fed by the outer sphere
Hartmann layer. Importantly, a large fraction of the current entering
the shear layer is tightly focused and effectively released from a
point source at the equator triggered by the tangency of the
C-line. The current injected by the source follows the C-line closely
but spreads laterally due to diffusion. In consequence, a strong
azimuthal Lorentz force is produced, which takes opposite signs either
side of the C-line; order unity super-rotation results on the
equatorial side. In actuality, the point source is the small
Equatorial Hartmann layer of radial width M-2/3 (<< M-1/2
)
and latitudinal extent M-1/3. We construct its analytic solution
and so determine an inward displacement width
O(M-2/3) of the free shear layer. We compare our numerical
solution of the free shear layer problem with our numerical
solution of the full governing equations for M in excess of 104.
We obtain excellent agreement. Some of our more testing
comparisons are significantly improved by incorporating
the shear layer displacement caused by the Equatorial Hartmann layer.