Double Albedo Model: Greenhouse Effect Reduced from 33 to 8 to 2 C

The albedo of a radiating body can be seen as a "transaction cost" as the fraction of incident radiation which is not absorbed and converted to heat for subsequent emission = absorption. The albedo of the Earth-atmosphere system subject to incident sun light mostly in the visible and ultraviolet frequency range, is about 0.3. The albedo of the Earth surface as emitter of infrared radiation is also about 0.3.

The "effective blackbody temperature" of the Earth+atmosphere system is 255 K, as the temperature of a blackbody emitting the 240 W/m2 absorbed by the Earth+atmosphere out of a total of 340 W/m2 incident sun light, thus with an albedo of 0.3.

The "effective emission altitude" is about 5 km as the altitude where the temperature is 255 K. The difference 33 K between the ground temperature of 288 K and 255 K is commonly termed the "total greenhouse effect" as the ground temperature difference of an Earth with and without an atmosphere.

But to compare Earths with and without atmosphere is not reasonable, unless you want to argue that with a "total greenhouse effect" as large as 33 C, a perturbation of 10% could represent an global warming of 3 C, which would seem very alarming.

It is more reasonable to compare Earths with different "atmospheric window", as the part of the total emission from the Earth-atmosphere system emitted directly from the ground surface. With a full window and a ground albedo of 0.3, the ground temperature emitting the 240 W/m2 would be about 280 K. The corresponding "total greenhouse effect" would thus be 8 K, to be compared with the standard estimate of 33 K. Closing the current window of 40 W/m2 could then increase the effect with 2 K.

With a "total greenhouse effect" of 8 K and "closing window effect" of 2 K,  instead of 33 K,  the effect of perturbations will be correspondingly smaller.  The 1 K upon doubling of CO2 in the standard perspective as no-feedback climate sensitivity, would be reduced by a factor 4 or 10 and thus not be measurable.

This analysis can be viewed to reflect a phenomenon of "double albedo", a first from absorption of sun light and a second from emission of infrared, as a double "transaction cost", in accordance with the analysis of blackbody radiation presented as Computational Blackbody Radiation.

We thus have the following emission scenarios with varying atmospheric windows and corresponding ground temperatures, all under absorption of  240 W/m2 by the Earth-atmosphere system:

1. Observed: Window 40 W/m2. Temperature 288 K. 
2. Estimated: Window 240 W/m2. Temperature 280 K.
3. Estimated: Window 0 W/m2. Temperature 290 K.

Doubled CO2 has been estimated to have a window effect of at most 4 W/m2 with a corresponding temperature effect of 10% of 2 K, thus not measurable.

The 2 K variation between 1. and 3. is consistent with the observed temperature variation since the end of the last ice age 10.000 years ago. Global cooling by more than 2 K would then seem to require increased system albedo, which could result from vast glaciation during an ice age, while a corresponding scenario for global warming larger than 2 K would seem to be difficult to dream up.

Listen to prof Richard Lindzen: The whole thing is pretty absurd.

PS1 More precisely, we are led to a 2-level system, where the Earth+atmosphere system absorbs 240 W/m2 by radiation from the Sun with an albedo of 0.3, which are then transferred thermodynamically to the Earth surface. With a fully open atmospheric window, these 240 W/m2 can then be emitted by radiation with an albedo of 0.3 at a temperature of 280 K.

PS2  Even more precisely, we are led to a 3-level system: radiative absorption - thermodynamic transfer - radiative emission, where both absorption and emission are subject to a "transaction cost" captured as albedo of about 0.3.  This gives a "greenhouse effect" of 8 K, to be compared with the standard 33 K resulting from a model with only absorption-emission. I will elaborate this idea in an upcoming post.