  A soft X-ray excess above the 2-10 keV power law extrapolation is generally
observed in AGN X-ray spectra. Presently there are two competitive models to
explain it: blurred ionized reflection and warm Comptonisation. In the latter
case, observations suggest a corona temperature $\sim$ 1 keV and a corona
optical depth $\sim$ 10. Moreover, radiative constraints from spectral fits
with Comptonisation models suggest that most of the accretion power should be
released in the warm corona. The disk below is basically non-dissipative,
radiating only the reprocessed emission from the corona. The true radiative
properties of such a warm and optically thick plasma are not well-known,
however. For instance, the importance of the Comptonisation process, the
potential presence of strong absorption/emission features or the spectral shape
of the output spectrum have been studied only very recently. We present in this
paper simulations of warm and optically thick coronae using the TITAN radiative
transfer code coupled with the NOAR Monte-Carlo code, the latter fully
accounting for Compton scattering of continuum and lines. Illumination from
above by a hard X-ray emission and from below by an optically thick accretion
disk is taken into account as well as (uniform) internal heating. Our
simulations show that for a large part of the parameter space, the warm corona
with sufficient internal mechanical heating is dominated by Compton cooling and
neither strong absorption nor emission lines are present in the outgoing
spectra. In a smaller part of the parameter space, the calculated emission
agrees with the spectral shape of the observed soft X-ray excess. Remarkably,
this also corresponds to the conditions of radiative equilibrium of an extended
warm corona covering almost entirely a non-dissipative accretion disk. These
results confirm the warm Comptonisation as a valuable model for the soft X-ray
excess.
