Greenhouse effect in semi-transparent planetary atmospheres

In this work the theoretical relationship between the clear-sky outgoing infrared radiation and the surface upward radiative flux is explored by using a realistic finite semi-transparent atmospheric model. We show that the fundamental relationship between the optical depth and source function contains real boundary condition parameters. We also show that the radiative equilibrium is controlled by a special atmospheric transfer function and requires the continuity of the temperature at the ground surface. The long standing misinterpretation of the classic semi-infinite Eddington solution has been resolved. Compared to the semi-infinite model, the finite semi-transparent model predicts much smaller ground surface temperature and a larger surface air temperature. The new equation proves that the classic solution significantly overestimates the sensitivity of greenhouse forcing to optical depth perturbations. In Earth-type atmospheres sustained planetary greenhouse effect with a stable ground surface temperature can only exist at a particular planetary average flux optical depth of 1.841. Simulation results show that the Earth maintains a controlled greenhouse effect with a global average optical depth kept close to this critical value. The broadband radiative transfer in the clear Martian atmosphere follows different principle resulting in different analytical relationships among the fluxes. Applying the virial theorem to the radiative balance equation, we present a coherent picture of the planetary greenhouse effect.