A cylindrical model for rotational MHD instabilities in aluminum reduction cells

Large-scale horizontal vortices associated with deformations of the aluminum-electrolyte interface have been observed in operating aluminum reduction cells as well as in physical and numerical models. To expose their importance, we analyze a particular class of magnetohydrodynamic (MHD) interfacial instabilities which are induced by rotation. As we focus on a single vortex, a cylindrical geometry is preferred. Two analytical models are proposed. In a first model based on the MHD shallow-water approximation, we consider a vortex that has a solid rotation profile to obtain a wave equation and a dispersion relation. A more realistic second model includes a viscous rotation profile and the treatment of the base-state interface deformation. Energetics of the flow gives further insight on how an initial perturbation evolves as an oscillatory or a non-oscillatory instability, depending on the direction of rotation. We find that the mechanism at the very origin of these instabilities is neither due to a shear between the two layers—and are therefore not Kelvin–Helmholtz instabilities—nor simply due to magnetic force alone, but rather to the indirect action of the centripetal pressure due to the rotation induced by magnetic force.