  The near-surface layers of cool main-sequence stars are structured by
convective flows, which are overshooting into the atmosphere. The flows and the
associated spatio-temporal variations of density and temperature affect
spectral line profiles and thus have an impact on estimates of stellar
properties such as effective temperature, gravitational acceleration, and
abundances. We aim at identifying distinctive properties of the thermodynamic
structure of the atmospheres of different stars and understand their causes. We
ran comprehensive 3D radiation hydrodynamics simulations of the near-surface
layers of six simulated stars of spectral type F3V to M2V with the MURaM code.
We carry out a systematic parameter study of the mean stratifications, flow
structures, and the energy flux in these stars.\par Results: We find monotonic
trends along the lower main sequence in granule size, flow velocity, and
intensity contrast. The convection in the M-star models differs substantially
from that of the hotter stars, mainly owing to the more gradual transition from
convective to radiative energy transport. While the basic mechanisms driving
surface convection in cool stars are the same, the properties of the convection
vary along the main sequence. Apart from monotonic trends in rms velocity,
intensity contrast, granule size, etc., there is a transition between "naked"
and "hidden" granulation around spectral type K5V caused by the (highly
non-linear) temperature dependence of the opacity. These variations have to be
taken into account when stellar parameters are derived from spectra.
