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, B 0 + b , is a combination of the strong, force-free background B 0 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 B 0 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 δ M 3 / 4 in the first case and ɛ M 3 / 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 B 0 that we chose the peak super-velocity scales like M 0.33 . This scaling is different from M 0.6 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].