Oscillations and instabilities in neutron stars with poloidal magnetic fields

We study the time evolution of non-axisymmetric linear perturbations of a rotating magnetized neutron star, whose magnetic field is purely poloidal. The background stellar configurations are generated self-consistently, with multipolar field configurations and allowing for distortions to the density distribution from rotational and magnetic forces. The perturbations split into two symmetry classes, with perturbations in one class being dominated by an instability generic to poloidal fields, which is localized around the 'neutral line' where the background field vanishes. Rotation acts to reduce the effect of this instability. Perturbations in the other symmetry class do not suffer this instability and in this case we are able to resolve Alfvén oscillations, whose restoring force is the magnetic field; this is the first study of non-axisymmetric Alfvén modes of a star with a poloidal field. We find no evidence that these modes form a continuum. In a rotating magnetized star, we find that there are no pure Alfvén modes or pure inertial modes, but hybrids of these. We discuss the nature of magnetic instabilities and oscillations in magnetars and pulsars, finding the dominant Alfvén mode from our simulations has a frequency comparable with observed magnetar quasi-periodic oscillations (QPOs).