Axisymmetric and non-axisymmetric magnetostrophic MRI modes

The magnetorotational-instability is central to the understanding of many astrophysical magnetohydrodynamic flows (in particular accretion discs). We have recently shown that a modified version of this instability, the magnetostrophic MRI (MS-MRI), is relevant to the dynamics of the Earth liquid core (Petitdemange et al., 2008). Our previous study used a purely axial imposed magnetic field and considered only axisymmetric instabilities. We investigate here the effects of a large scale toroidal magnetic field on the development and saturation of the MS-MRI both in an axisymmetric setup and in fully three-dimensional configurations. We use direct numerical modeling of the full MHD equations (both in the linear and non-linear regimes) in a spherical geometry. We interpret our results using WKB expansions and shearing coordinates. We find that three-dimensional MS-MRI modes exhibit a strong helical structure for parameters relevant to planetary interiors. Three-dimensional MS-MRI modes share some similarities with their axisymmetric counterparts. They are amplified on the same short timescale. When non-linear effects become significant, they act to decrease the shear. During saturation, the magnetic structure expands spatially, while drifting at a constant rate along the direction of the rotation axis. A striking result is that an m = 1 mode can dominate in the non-linear regime when a sufficiently large toroidal field is present. Three-dimensional MS-MRI modes could play an important role in planetary interior dynamics. They can cause rapid magnetic field variations and also act to limit the shear in the conducting liquid core.