MHD flow in a slightly differentially rotating
spherical shell, with conducting inner core, in a dipolar magnetic field
Emmanuel Dormya, Philippe Cardinb,c
and Dominique Jaulta,c
a IPGP, 4 place Jussieu, F-75252, Paris, France
b ENS, 24 rue Lhomond, F-75231, Paris, France
c CNRS, Paris, France
Earth And Planetary Science Letters Vol. 160 (1-2) pp. 15-30
Abstract. Motion is generated in a rotating spherical shell, by
a slight differential rotation of the inner core. We show how the
numerical solution tends, with decreasing Ekman number, to the
asymptotic limit of Proudman [J. Fluid Mech. 1 (1956)
505-516]. Starting from geophysically large values, we show that the
main qualitative features of the asymptotic solution show up only when
the Ekman number is decreased below 10-6. Then, we impose a
dipolar and force-free magnetic field with internal sources. Both the
inner core and the liquid shell are electrically conducting. The first
effect of the Lorentz force is to smooth out the change in angular
velocity at the tangent cylinder. As the Elsasser number is further
increased, the Proudman-Taylor constraint is violated, Ekman layers
are changed into Hartmann type layers, shear at the inner sphere
boundary vanishes, and the flow tends to a bulk rotation together with
the inner sphere. Unexpectedly, for increasing strength of the field,
there is a super-rotation (the angular velocity does not reach a
maximum at the inner core boundary but in the interior of the fluid)
localized in an equatorial torus. At a given field strength, the
amplitude of this phenomenon depends on the Ekman number and tends to
vanish in the magnetostrophic limit.
Keyword(s): Ekman layer; Stewartson layer; magnetic
induction.