By means of two dimensional , general-relativistic , magneto-hydrodynamical simulations we investigate the oscillations of magnetized neutron star models ( magnetars ) for one particular dipolar magnetic field configuration including the description of an extended solid crust .
The aim of this study is to understand the origin of the quasi-periodic oscillations ( QPOs ) observed in the giant flares of soft gamma-ray repeaters ( SGRs ) .
We confirm our previous findings which showed the existence of three different regimes in the evolution depending on the magnetic field strength : ( a ) a weak magnetic field regime B < 5 \times 10 ^ { 13 } G , where crustal shear modes dominate the evolution ; ( b ) a regime of intermediate magnetic fields 5 \times 10 ^ { 13 } G < B < 10 ^ { 15 } G , where Alfvén QPOs are mainly confined to the core of the neutron star and the crustal shear modes are damped very efficiently ; and ( c ) a strong field regime B > 10 ^ { 15 } G , where magneto-elastic oscillations reach the surface and approach the behavior of purely Alfvén QPOs .
When the Alfvén QPOs are confined to the core of the neutron star , we find qualitatively similar QPOs as in the absence of a crust .
The lower QPOs associated with the closed field lines of the magnetic field configuration are reproduced as in our previous simulations without crust , while the upper QPOs connected to the open field lines are displaced from the polar axis .
The position of these upper QPOs strongly depends on the magnetic field strength .
Additionally , we observe a family of edge QPOs and one new upper QPO , which was not previously found in the absence of a crust .
We extend our semi-analytic model to obtain estimates for the continuum of the Alfvén oscillations .
Our results do not leave much room for a crustal-mode interpretation of observed QPOs in SGR giant flares , but can accommodate an interpretation of these observations as originating from Alfvén-like , global , turning-point QPOs ( which can reach the surface of the star ) in models with mean surface magnetic field strengths in the narrow range of 3.8 \times 10 ^ { 15 } G \lesssim B \lesssim 1.1 \times { 10 ^ { 16 } } G ( for a sample of two stiff EoS and various masses ) .
This range is somewhat larger than estimates for magnetic field strengths in known magnetars .
The discrepancy may be resolved in models including a more complicated magnetic field structure or with models taking superfluidity of the neutrons and superconductivity of the protons in the core into account .