Using a 3D climate system model, I have studied the onset of moist greenhouse atmospheres for Earth as the Sun brightens over time.  (See figure at left)

For small (≤10%) increases in the solar constant (S), Earth's temperature warms nearly linearly with climate sensitivities of ~1 K/(Wm2) and global mean surface temperatures remaining below 310 K.  However, an abrupt shift in climate is found as the solar constant is increased to +12.5% S. Here climate sensitivity peaks at ~6.5 K/(Wm2), while global mean surface temperatures rise above 330 K.  This climatic transition is associated with a fundamental change to the radiative-convective state of the atmosphere. Hot, moist climates feature both strong solar absorption and inefficient radiative cooling in the low atmosphere, thus yielding net radiative heating of the near-surface layers. This heating forms an inversion that effectively shuts off convection in the boundary layer, which in turn reduces cloudiness.