Several teams have reported peculiar frequency spectra for flows in a spherical shell. To address their origin, we perform numerical simulations of the spherical Couette flow in a dipolar magnetic field, in the configuration of the DT S experiment. The frequency spectra computed from time-series of the induced magnetic field display similar bumpy spectra, where each bump corresponds to a given azimuthal mode number m. The bumps show up at moderate Reynolds number (≃ 2 600) if the time-series are long enough (> 300 rotations of the inner sphere). We present a new method that permits to retrieve the dominant frequencies for individual mode numbers m, and to extract the modal structure of the full non-linear flow. The maps of the energy of the fluctuations and the spatio-temporal evolution of the velocity field suggest that fluctuations originate in the outer boundary layer. The threshold of instability if found at Re c = 1 860. The fluctuations result from two coupled instabilities: high latitude Bödewadt-type boundary layer instability, and secondary non-axisymmetric instability of a centripetal jet forming at the equator of the outer sphere. We explore the variation of the magnetic and kinetic energies with the input parameters, and show that a modified Elsasser number controls their evolution. We can thus compare with experimental determinations of these energies and find a good agreement. Because of the dipolar nature of the imposed magnetic field, the energy of magnetic fluctuations is much larger near the inner sphere, but their origin lies in velocity fluctuations that initiate in the outer boundary layer.
Steady dipolar vortices continuously driven by electromagnetic forcing in a shallow layer of an electrolytic fluid are studied experimentally and theoretically. The driving Lorentz force is generated by the interaction of a dc uniform electric current injected in the thin layer and the non-uniform magnetic field produced by a small dipolar permanent magnet (0.33 T). Laminar velocity profiles in the neighbourhood of the zone affected by the magnetic field were obtained with particle image velocimetry in planes parallel and normal to the bottom wall. Flow planes at different depths of the layer were explored for injected currents ranging from 10 to 100 mA. Measurements of the boundary layer attached to the bottom wall reveal that owing to the variation of the field in the normal direction, a slightly flattened developing profile with no shear stresses at the free surface is formed. A quasi-two-dimensional magnetohydrodynamic numerical model that introduces the non-uniformity of the magnetic field, particularly its decay in the normal direction, was developed. Vertical diffusion produced by the bottom friction was modelled through a linear friction term. The model reproduces the main characteristic behaviour of the electromagnetically forced flow.
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