Acoustic waves in a stratified atmosphere IV. Three-dimensional nonlinear hydrodynamics

Context. The quiet solar chromosphere in the interior of supergranulation cells is believed to be heated by the dissipation of acoustic waves that originate with a typical period of 3 min in the photosphere. Aims. We investigate how the horizontal expansion with height of acoustic waves traveling upward into an isothermal, gravitationally stratified atmosphere depends on the size of the source region. Methods. We have solved the three-dimensional, nonlinear, time-dependent hydrodynamic equations for impulsively-generated, upward-propagating acoustic waves, assuming cylindrical symmetry. Results. When the diameter of the source of acoustic waves is small, the pattern of the upward-propagating waves is that of a point source, for which the energy travels upward in a vertical cone, qualitatively matching the observed pattern of bright-point expansion with height. For the largest plausible size of a source region, i.e., with granular size of 1 Mm, wave propagation in the low chromosphere is approximately that of plane waves, but in the middle and upper chromosphere it is also that of a point source. The assumption of plane-wave propagation is not a good approximation in the solar chromosphere. The upward-directed energy flux is larger than that of the solar chromosphere, at least in the middle and upper chromosphere, and probably throughout. Conclusions. Simulations of impulsively generated acoustic waves emitted from source regions with diameters that are small compared to the pressure scale height of the atmosphere qualitatively reproduce the upward expansion observed in chromospheric bright points. The emission features in the cores of the H and K lines are predicted to be blueshifted for a pulse and redshifted for the waves in its wake. The contribution of internal gravity waves to the upward energy flux is small and decreases with increasing size of the source region.