Spectropolarimetric investigation of the propagation of magnetoacoustic waves and shock formation in sunspot atmospheres

Velocity oscillations in sunspot umbrae have been measured simultaneously in two spectral lines: the photospheric Si I lambda 10827 line and the chromospheric He I lambda 10830 multiplet. From the full Stokes inversion of temporal series of spectropolarimetric observations, we retrieved, among other parameters, the line-of-sight velocity temporal variations at photospheric and chromospheric heights. Chromospheric velocity oscillations show a 3 minute period with a clear sawtooth shape typical of propagating shock wave fronts. Photospheric velocity oscillations have basically a 5 minute period, although the power spectrum also shows a secondary peak in the 3 minute band that has been proven to be a predecessor for its chromospheric counterpart. The derived phase spectra yield a value of the atmospheric cutoff frequency around 4 mHz and give evidence for the upward propagation of higher frequency oscillation modes. The phase spectrum has been reproduced with a simple model of linear vertical propagation of slow magnetoacoustic waves in a stratified magnetized atmosphere that accounts for radiative losses through Newton's cooling law. The model explains the main features in the phase spectrum and allows us to compute the theoretical time delay between the photospheric and chromospheric signals, which happens to have a strong dependence on frequency. We find a very good agreement between this and the time delay obtained directly from the cross-correlation of photospheric and chromospheric velocity maps filtered around the 6 mHz band. This allows us to infer that the 3 minute power observed at chromospheric heights comes directly from the photosphere by means of linear wave propagation, rather than from nonlinear interaction of 5 minute (and/or higher frequency) modes.