Magnetic spherical Couette flow in linear combinations of axial and dipolar fields

We present axisymmetric numerical calculations of the fluid flow induced in a spherical shell with inner sphere rotating and outer sphere stationary. A magnetic field is also imposed, consisting of particular linear combinations of axial and dipolar fields, chosen to make $B_r=0$ at either the outer sphere, or the inner, or in between. This leads to the formation of Shercliff shear layers at these particular locations. We then consider the effect of increasingly large inertial effects, and show that an outer Shercliff layer is eventually de-stabilized, an inner Shercliff layer appears to remain stable, and an in-between Shercliff layer is almost completely disrupted even before the onset of time-dependence, which does eventually occur though.

💡 Research Summary

The paper investigates axisymmetric magnetohydrodynamic (MHD) flow in a spherical shell (the spherical Couette configuration) where the inner sphere rotates at a constant angular velocity and the outer sphere is stationary. In addition to the mechanical forcing, a magnetic field is imposed that is a linear combination of an axial (uniform) field and a dipolar field. By adjusting the relative amplitudes of the two components, the authors create three distinct magnetic configurations in which the radial component of the magnetic field, (B_r), vanishes either at the outer boundary, at the inner boundary, or at a prescribed intermediate radius. In each case a Shercliff shear layer—an MHD boundary layer that forms where magnetic tension suppresses motion across magnetic field lines—develops at the surface where (B_r=0).

The governing equations are the incompressible Navier‑Stokes equations coupled with the induction equation under the inductionless approximation (magnetic Reynolds number ≪ 1). The dimensionless parameters are the Reynolds number (\mathrm{Re}= \Omega_i r_i^2/\nu) (measuring inertial effects) and the Hartmann number (\mathrm{Ha}=B_0 r_i\sqrt{\sigma/(\rho\nu)}) (measuring magnetic damping). The magnetic field is written as
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