In Liou's book we are provided the following formula for the calculation of a steady state "radiative equilibrium":
In Goody and Yung they present the following formula:
The latter I guess corresponds to choosing the kernel function:
However, Liou is not very explicit on how to construct this function and GY do not motivate their choise either. So where do we go from here? Maybe some expert on radiative transfer could help us here.
One of the most important observations that I would like to point out is that, if using the GY formula the lapse rate becomes independent of the heat source Fs, in contrast to what would be the default assumption in conventional thermodynamics:
Where k is the conductivity. Regarding the "convective adjustment" Liou writes:
"For applications to one-dimensional climate models, the critical lapse rate,.., is usually assumed to be 6.5 K /km for the globally averaged condition. This number is based on the fact that the climatological atmospheric temperature profile in the troposphere has a lapse rate close to this value."
If this is the level of sophistication we are working at here, I wonder why they don't just let the computer adjust the lapse rate to the one they want from the very beginning.