Alfv\'en Wave Conversion to Low Frequency Fast Magnetosonic Waves in Magnetar Magnetospheres

Rapid shear motion of magnetar crust can launch Alfv\'{e}n waves into the magnetosphere. The dissipation of the Alfv\'{e}n waves has been theorized to power the X-ray bursts characteristic of magnetars. However, the process by which Alfv\'{e}n waves convert their energy to X-rays is unclear. Recent work has suggested that energetic fast magnetosonic (fast) waves can be produced as a byproduct of Alfv\'{e}n waves propagating on curved magnetic field lines; their subsequent dissipation may power X-ray bursts. In this work, we investigate the production of fast waves by performing axisymmetric force-free simulations of Alfv\'{e}n waves propagating in a dipolar magnetosphere. For Alfv\'{e}n wave trains that do not completely fill the flux tube confining them, we find a fast wave dominated by a low frequency component with a wavelength defined by the bouncing time of the Alfv\'{e}n waves. In contrast, when the wave train is long enough to completely fill the flux tube, and the Alfv\'{e}n waves overlap significantly, the energy is quickly converted into a fast wave with a higher frequency that corresponds to twice the Alfv\'{e}n wave frequency. We investigate how the energy, duration, and wavelength of the initial Alfv\'{e}n wave train affect the conversion efficiency to fast waves. For modestly energetic star quakes, we see that the fast waves that are produced will become non-linear well within the magnetosphere, and we comment on the X-ray emission that one may expect from such events.