  Although observational data unequivocally point out to the presence of
chromospheres in red giant stars, no attempts have been made so far to model
them using 3D hydrodynamical model atmospheres. We therefore compute an
exploratory 3D hydrodynamical model atmosphere for a cool red giant in order to
study the dynamical and thermodynamic properties of its chromosphere, as well
as the influence of the chromosphere on its observable properties. 3D radiation
hydrodynamics simulations are carried out with the CO5BOLD model atmosphere
code for a star with the atmospheric parameters (Teff=4010 K, log g=1.5,
[M/H]=0.0), which are similar to those of the K-type giant star Aldebaran
(alpha Tau). ... we compute the emergent continuum intensity maps at different
wavelengths, spectral line profiles of Ca II K, the Ca II infrared triplet line
at 854.2nm, and H alpha, as well as the spectral energy distribution (SED) of
the emergent radiative flux. The initial model quickly develops a dynamical
chromosphere characterised by propagating and interacting shock waves. The peak
temperatures in the chromospheric shock fronts reach values on the order of up
to 5000 K although the shock fronts remain quite narrow. Like for the Sun, the
gas temperature distribution in the upper layers is composed of a cool
component due to adiabatic cooling in the expanding post-shock regions and a
hot component due to shock waves. For this red giant model, the hot component
is a rather flat high-temperature tail, which nevertheless affects the
resulting average temperatures significantly. The simulations show that the
atmospheres of red giant stars are dynamic and intermittent. Consequently, many
observable properties cannot be reproduced with one-dimensional static models
but demand for advanced 3D HD modelling. Furthermore, including a chromosphere
in the models might produce significant contributions to the emergent UV flux.
