Radiative hydrodynamic simulations of acoustic waves in sunspots

We describe fully consistent NLTE radiation hydrodynamic simulations of the propagation of acoustic waves in sunspot umbrae. In contrast to the case of the quiet internetwork Sun, we find that the observed behavior of the resonance lines of singly ionized calcium (H and K lines) cannot be explained without adding non-acoustic heating. The best agreement with observations is obtained with an extra heating term as a function of height that is constant per volume.