Tangled magnetic field model of QPOs

The highly tangled magnetic field of a magnetar supports shear waves similar to Alfv\'en waves in an ordered magnetic field. Here we explore if torsional modes excited in the stellar interior and magnetosphere can explain the quasi-periodic oscillations (QPOs) observed in the tail of the giant flare of SGR 1900+14. We solve the initial value problem for a tangled magnetic field that couples interior shear waves to relativistic Alfv\'en shear waves in the magnetosphere. Assuming stellar oscillations arise from the sudden release of magnetic energy, we obtain constraints on the energetics and geometry of the process. If the flare energy is deposited initially inside the star, the wave energy propagates relatively slowly to the magnetosphere which is at odds with the observed rise time of the radiative event of $\lesssim 10$ ms. Nor can the flare energy be deposited entirely outside the star, as most of the energy reflects off the stellar surface, giving surface oscillations of insufficient magnitude to produce detectable modulations of magnetospheric currents. Energy deposition in a volume that straddles the stellar surface gives agreement with the observed rise time and excites a range of modes with substantial amplitude at observed QPO frequencies. In general, localized energy deposition excites a broad range of modes that encompasses the observed QPOs, though many more modes are excited than the number of observed QPOs. If the flare energy is deposited axisymmetrically, as is possible for a certain class of MHD instabilities, the number of modes that is excited is considerably reduced.