Torsional Oscillations of Relativistic Stars with Dipole Magnetic Fields II. Global Alfv\'en Modes

We investigate torsional Alfv\'{e}n modes of relativistic stars with a global dipole magnetic field. It has been noted recently (Glampedakis et al. 2006) that such oscillation modes could serve as as an alternative explanation (in contrast to torsional crustal modes) for the SGR phenomenon, if the magnetic field is not confined to the crust. We compute global Alfv\'{e}n modes for a representative sample of equations of state and magnetar masses, in the ideal MHD approximation and ignoring $\ell \pm 2$ terms in the eigenfunction. We find that the presence of a realistic crust has a negligible effect on Alfv\'{e}n modes for $B > 4\times 10^{15}$ G. Furthermore, we find strong avoided crossings between torsional Alfv\'{e}n modes and torsional crust modes. For magnetar-like magnetic field strengths, the spacing between consecutive Alfv\'{e}n modes is of the same order as the gap of avoided crossings. As a result, it is not possible to identify modes of predominantly crustal character and all oscillations are predominantly Alfv\'{e}n-like. Interestingly, we find excellent agreement between our computed frequencies and observed frequencies in two SGRs, for a maximum magnetic field strenght in the range of (0.8--1.2)$\times 10^{16}$ G.