On the effect of mantle conductivity on the super-rotating jets near the liquid core surface

We consider hydromagnetic Couette flows in planar and spherical geometries with strong magnetic field (large Hartmann number, M≫1). The highly conducting bottom boundary is in steady motion that drives the flow. The top boundary is stationary and is either a highly conducting thin shell or a weakly conducting thick mantle. The magnetic field, B0+b, is a combination of the strong, force-free background B0 and a perturbation b induced by the flow. This perturbation generates strong streamwise electromagnetic stress inside the fluid which, in some regions, forms a jet moving faster than the driving boundary. The super-velocity, in the spherical geometry called super-rotation, is particularly prominent in the region where the 'grazing' line of B0 has a point of tangent contact with the top boundary and where the Hartmann layer is singular. This is a consequence of topological discontinuity across that special field line. We explain why the magnitude of super-rotation already present when the top wall is insulating [Dormy, E., Jault, D., Soward, A.M., 2002. A super-rotating shear layer in magnetohydrodynamic spherical Couette flow. J. Fluid Mech. 452, 263-291], considerably increases when that wall is even slightly conducting. The asymptotic theory is valid when either the thickness of the top wall is small, δ∼M−1 and its conductivity is high, ɛ∼1 or when δ∼1 and ɛ∼M−1. The theory predicts the super-velocity enhancement of the order of δM3/4 in the first case and ɛM3/4 in the second case. We also numerically solve the planar problem outside the asymptotic regime, for ɛ=1 and δ=1, and find that with the particular B0 that we chose the peak super-velocity scales like M0.33. This scaling is different from M0.6 found in spherical geometry [Hollerbach, R., Skinner, S., 2001. Instabilities of magnetically induced shear layers and jets. Proc. R. Soc. Lond. A 457, 785-802].