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Linear stability analysis of the Hall magnetorotational instability in a spherical domain

We investigate the stability of the Hall-MHD system and determine its importance for neutron stars at their birth, when they still consist of differentially rotating plasma permeated by extremely strong magnetic fields. We solve the linearised Hall-MHD equations in a spherical shell threaded by a homogeneous magnetic field. With the fluid/flow coupling and the Hall effect included, the magnetorotational instability and the Hall effect are both acting together. Results differ for magnetic fields aligned with the rotation axis and anti-parallel magnetic fields. For a positive alignment of the magnetic field the instability grows on a rotational time-scale for any sufficiently large magnetic Reynolds number. Even the magnetic fields which are stable against the MRI due to the magnetic diffusion are now susceptible to the shear-Hall instability. In contrast, the negative alignment places strong restrictions on the growth and the magnitude of the fields, hindering the effectiveness of the Hall-MRI. While non-axisymmetric modes of the MRI can be suppressed by strong enough rotation, there is no such restriction when the Hall effect is present. The implications for the magnitude and the topology of the magnetic field of a young neutron star may be significant.