Onset of oscillatory magnetoconvection under rapid rotation and spatially varying magnetic field

The onset of periodic time-varying magnetoconvection in a regime relevant to the Earth's outer core is investigated in this study. A rapidly rotating plane fluid layer subject to an axially varying horizontal magnetic field is considered under the dynamical regimes of stronger magnetic diffusion compared to thermal and viscous diffusion rates. Dynamically specific convective instabilities, both inside and outside the tangent cylinder regions of the Earth's core, have been investigated by using appropriate patterns of the imposed mean magnetic field. The hallmark of convection onset, with such axially varying mean field, is a viscous oscillatory mode weakly modified by the magnetic field. This modified viscous oscillatory (mVO) mode is observed to exist over a wide range of the strength of the imposed field, making it a dynamically appropriate flow structure with characteristics of outer core convection. An optimal Prandtl number $P r *$ is found through numerical experiments where the mVO mode can be the most unstable to infinitesimal perturbations. It is further shown that this optimal state admits oscillations for earthlike regimes where thermal diffusion is much less than magnetic diffusion. Also, the formation of columnar convection rolls from isolated vortices is demonstrated as a result of combinations of the classical viscous oscillatory and mVO modes in the rapidly rotating limit (Ekman number $E → 0$⁠). Overall, the qualitative characteristics of magnetoconvection modes for the various imposed patterns are found to be similar despite representing distinct regions in the Earth's outer core.