We have shown that a simple, modified version of the Magnetorotational Instability (MRI) can develop in the outer liquid core of the Earth, in the presence of a background shear (Petitdemange, Dormy, Balbus, GRL,35, 2008). It requires either a thermal wind, or a primary instability, such as convection, to drive a weak differential rotation within the core. The force balance in the Earth's core is unlike classical astrophysical applications of the MRI (such as gaseous disks around stars). Here, the weak differential rotation in the Earth core yields an instability by its constructive interaction with the planet's much larger rotation rate. The resulting destabilising mechanism is just strong enough to counteract stabilizing resistive effects, and produce growth on geophysically interesting timescales.
We refer to this instability as the magnetostrophic MRI (MS-MRI). We investigate linear and nonlinear developments of the MS-MRI. We present global numerical simulations both axisymmetric and weakly-three-dimensional (i.e. involving a limited number of azimuthal modes). We identify the 3D mechanism of this instability, and the role of a helical applied magnetic field. We discuss the possible signature of MS-MRI in recent geomagnetic simulations and address the connection between the MS-MRI and the planetary dynamos.