Linear magnetoconvection in a rotating spherical shell, incorporating a finitely conducting inner core

Abstract

Abstract The problem of the onset of convection in a rotating spherical shell with an imposed magnetic field is studied. This problem is relevant to understanding the dynamics of the Earth's outer core. The finite conductivity of the inner core is taken into account and no-slip boundary conditions are assumed at the inner-core and core-mantle boundaries. The problem is investigated numerically, using values of the Ekman number down to 10−6. Models in which the toroidal magnetic field vanishes near the core-mantle boundary, as expected in the Earth, are considered. We find the preferred non-axisymmetric wavenumber, m, of modes proportional to exp (imφ) as a function of Elsasser number A. We also find that toroidal fields with A Λ ≥ 10 are unstable due to magnetic instability even when there is no thermal driving, i.e. at zero Rayleigh number. In the range of Elsasser number appropriate to the geodynamo, convective motions in the interior of the outer core in our model have azimuthal velocities which are only weakly dependent on the coordinate parallel to the rotation axis. We have also compared the fields and fluid velocities arising from our model with those deduced from geomagnetic data, to the extent possible in our very simplified models. We find that solutions with the m = 2 mode best resemble published maps of the geomagnetic field at the core surface. Our calculations generally support the hypothesis that large scale convection is occurring in the Earth's outer core.