Having scrolled through G&Y's "Atmospheric Radiation" some more times, I thought it might be time to peel of another layer of the theory, which is somewhat related to my most recent posts. The key chapters I think are Chapter 2: "Theory of radiative transfer" and Chapter 9: "Atmospheres in radiative equilibrium".
Chapter 2 contains a long and complicated discussion about radiative transfer, that, after the dust is cleared (ie Taylor expansion to first order), I believe boils down to the following relation (using a slightly different notation):
B(a(z)) = sigma*T^4/pi (1)
The right hand side is simply S-B's law divided by pi, and the left hand side is a source function, derived from a mish-mash of other considerations, whose main property I guess is that it increses with the optical parameter a(z). To a first approximation:
B(a(z)) = F(1 + 3a(z)/2)/(2*pi) (2)
I guess the line of reasoning is something like this: The radiation must be a function of the heat source F and some optical absorption/emission coefficient a(z), but since the radiation must also equal the radiation given by S-B's law we now have a relationship between the parameters F and a(z) on the one hand and the temperature T on the other.
Assuming also that a(z) is altitude dependent: a(z) = a*exp(-z/H), probably reflecting the fact that the density decreses roughly like that, we now have a relation between the temperature T, the altitude z and a. Differentiating both sides of equation (1) yields after some manipulation:
-dT/dz = (T/H)a(z)/(1 + 3a(z)/2) (3)
Hence, we arrive at a lapse rate that depends explicitly neither on the heat source F nor the atmospheric mass, but instead solely on the optical parameter a(z). The heat source F is instead reintroduced in a rather ad-hoc fashion when fixing the ground temperature. This is peculiar. Furthermore, there is something odd about the assumption on the S-B law. Wouldn't it be more reasonable to assume that
Total radiation = F(a(z), T) ,
Where F is a function dependent on both the temperature and a(z).
A model should not be dismissed just because it is simplistic. However, when studying these equations and comparing them with the equally simplistic formulas that can be derived from conventional thermodynamics there is reason to raise your eye-brows.
G&Y themselves write the following about radiative equilibrium considerations:
"In addition to their value in examining general principles, there is a recurrent, although disputed theme that radiative equilibrium has direct relevance to the observed atmospheric structure"
This was in 1989. Twenty years later, "although disputed" has suddenly turned into "widely acknowledged", at least according to Saint Lindzen. So what happened in the mean time? Well, probably not any major educational effort on the part of established climatology, since very few people seem to know anything in particular about these matters.
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