  Three-dimensional numerical simulations with CO5BOLD, a new radiation
hydrodynamics code, result in a dynamic, thermally bifurcated model of the
non-magnetic chromosphere of the quiet Sun. The 3-D model includes the middle
and low chromosphere, the photosphere, and the top of the convection zone,
where acoustic waves are excited by convective motions. While the waves
propagate upwards, they steepen into shocks, dissipate, and deposit their
mechanical energy as heat in the chromosphere. Our numerical simulations show
for the first time a complex 3-D structure of the chromospheric layers, formed
by the interaction of shock waves. Horizontal temperature cross-sections of the
model chromosphere exhibit a network of hot filaments and enclosed cool
regions. The horizontal pattern evolves on short time-scales of the order of
typically 20 - 25 seconds, and has spatial scales comparable to those of the
underlying granulation. The resulting thermal bifurcation, i.e., the
co-existence of cold and hot regions, provides temperatures high enough to
produce the observed chromospheric UV emission and -- at the same time --
temperatures cold enough to allow the formation of molecules (e.g., carbon
monoxide). Our 3-D model corroborates the finding by Carlsson & Stein (1994)
that the chromospheric temperature rise of semi-empirical models does not
necessarily imply an increase in the average gas temperature but can be
explained by the presence of substantial spatial and temporal temperature
inhomogeneities.
