"Our results suggest that currently favored greenhouse [gas] solutions could be in conflict with constraints emerging for the middle and late Archean [young Earth]."However, if one instead simply assumes CO2 concentrations had no significant influence on Earth temperature, the faint young Sun problem can be resolved with the basic thermodynamics of the lapse rate, as shown in this post today by Dr. Claes Johnson, Professor of Applied Mathematics, KTH:
The analysis of the lapse rate in earlier posts suggests the following resolution of the paradox:
A reduction of 30% of the insolation could mean a reduction from 180 to 125 W/m2 absorbed by the Earth surface, and a reduction of 140 to 100 W/m2 to be radiated from the tropopause, assuming 40 W/m2 directly radiated through the atmospheric window in both cases.
This would require a drop in the temperature of the tropopause from - 50 C to - 68 C (from 223 K to 205 K by Stefan-Boltzmann with 223 =(140/5.66)^0.25 x 100). The Earth surface temperature could then remain at + 15 C if the lapse rate increased from the present 6.5 C/km to 8.3 C/km.
The maximal lapse rate is 10 C/km and could be attained in an atmosphere without thermodynamics, in an atmosphere at rest without motion of the air, with heat transfer by conduction and radiation but no thermodynamics of convection and evaporation/condensation.
An effect of thermodynamics in the present atmosphere can thus be viewed as a reduction of the lapse rate from 10 to 6.5 C/km with the difference increasing with the vigor of the thermodynamics. With a less vigorous atmosphere the lapse rate could thus increase from 6.5 to 8.3 C/km and thus sustain the same surface temperature with only 70% of the input from the Sun of today.
In the extreme case of an atmosphere without motion with a lapse rate of 10 C/km, a 50% Sun would thus be enough to sustain comfortable organic life at + 15 C, thus very early in the history of the solar system. Organic life is supposed to be 4 billion years old, apparently ignited by a young 50% Sun.
The argument supports the idea of the thermodynamics of the atmosphere as an air conditioner acting to reduce the lapse rate and thus cool the Earth surface as the inside, with the the tropopause as the outside with a temperature set by the input via Stefan-Boltzmann.
Abstract from Geophysical Research Letters:
Faint young Sun problem more severe due to ice-albedo feedback and higher rotation rate of the early Earth
During the Archean (3.8–2.5 billion years ago), the Sun was up to 25% less luminous than today, yet there is strong evidence that the Earth's ocean surface was not completely frozen. The most obvious solutions to this ‘faint young Sun problem’ demand high concentrations of greenhouse gases such as carbon dioxide. Here we present the first comprehensive 3-dimensional simulations of the Archean climate that include processes as the sea-ice albedo feedback and the higher rotation rate of the Earth. These effects lead to CO2 partial pressures required to prevent the Earth from freezing that are significantly higher than previously thought. For the early Archean, we find a critical CO2 partial pressure of 0.4 bar in contrast to 0.06 bar estimated in previous studies with 1-dimensional radiative-convective models. Our results suggest that currently favored greenhouse solutions could be in conflict with constraints emerging for the middle and late Archean.