Magneto-acoustic wave energy from numerical simulations of an observed sunspot umbra

We aim at reproducing the height dependence of sunspot wave signatures obtained from spectropolarimetric observations through 3D MHD numerical simulations. A magneto-static sunspot model based on the properties of the observed sunspot is constructed and perturbed at the photosphere introducing the fluctuations measured with the \SiI\ $\lambda$ 10827 \AA\ line. The results of the simulations are compared with the oscillations observed simultaneously at different heights from the \HeI\ $\lambda$ 10830 \AA\ line, the \CaIIH\ core and the \FeI\ blends in the wings of the \CaIIH\ line. The simulations show a remarkable agreement with the observations. They reproduce the velocity maps and power spectra at the formation heights of the observed lines, as well as the phase and amplification spectra between several pair of lines. We find that the stronger shocks at the chromosphere are accompanied with a delay between the observed signal and the simulated one at the corresponding height, indicating that shocks shift the formation height of the chromospheric lines to higher layers. Since the simulated wave propagation matches very well the properties of the observed one, we are able to use the numerical calculations to quantify the energy contribution of the magneto-acoustic waves to the chromospheric heating in sunspots. Our findings indicate that the energy supplied by these waves is too low to balance the chromospheric radiative losses. The energy contained at the formation height of the lowermost \SiI
$\lambda$ 10827 \AA\ line in the form of slow magneto-acoustic waves is already insufficient to heat the higher layers, and the acoustic energy which reaches the chromosphere is around 3-9 times lower than the required amount of energy. The contribution of the magnetic energy is even lower.