Torsional oscillations of a magnetar with a tangled magnetic field
We propose a scenario for the quasi-periodic oscillations observed in magnetar flares wherein a tangled component of the stellar magnetic field introduces nearly isotropic stress that gives the fluid core of the star an effective shear modulus. In a simple, illustrative model of constant density, the tangled field eliminates the problematic Alfv\'en continuum that would exist in the stellar core for an organized field. For a tangled field energy density comparable to that inferred from the measured dipole fields of $\sim 10^{15}$ G in SGRs 1806-20 and 1900+14, torsional modes exist with fundamental frequencies of about 20 Hz, and mode spacings of $\sim 10$ Hz. For fixed stellar mass and radius, the model has only one free parameter, and can account for {\em every} observed QPO under 160 Hz to within 3 Hz for both SGRs 1806-20 and 1900+14. The combined effects of stratification and crust stresses generally decrease the frequencies of torsional oscillations by $<10$% for overtones and increase the lowest-frequency fundamentals by up to 50%, and so the star can be treated as having constant density to a generally good first approximation. We address the issue of mode excitation by sudden readjustment of the stellar magnetosphere. While the total energy in excited modes is well within the energy budget of giant flares, the surface amplitude is $< 10^{-3}$ of the stellar radius for global oscillations, and decreases strongly with mode frequency. The 626 Hz QPO reported for SGR 1806-20 is particularly problematic to excite beyond a surface amplitude of $10^{-6}$ of the stellar radius.