After the his famous paper in 1896, where Arrhenius did the first calculations of the CO2 greenhouse effect, his theory was dismissed by Angstrom with a simple experiment. He let an infrared beam pass through a tube filled with CO2 and measured the emerging light intensity. Upon reducing CO2 concentration in the tube, only a tiny difference could be found and he concluded that even a small number of CO2 molecules is sufficient to completely absorb the IR beam. The conclusion was that a CO2 increase could not matter. This was the birth of the first skeptic of what was then called "CO2 theory" and of the more recent skeptic argument "the CO2 effect is saturated".

Thirty years later, E. O. Hulburt (Phys. Rev. 38, 1876–1890 (1931)) considered convection in addition to the purely radiative equilibrium assumed by Arrhenius. He found that convective equilibrium holds in the lower part of the troposphere up to about 10 Km, while radiative equilibrium holds above. The important consequence is that the details of the absorption in the lower troposphere do not matter since heat "is spread around and transferred upward by convection". In other words, what governs the energy balance of the earth is the radiative balance in the upper troposphere and CO2 concentration there does matter.

Hulburt was very prudent in his conclusions:

"The agreement is no doubt better than is warranted by the accuracy of the data on which the calculations are based. Apparently the uncertainties and omissions have conspired to counteract each other to some extent."

Nevertheless, his work is definitely a milestone in the understanding of our atmosphere.

Hulburt's work should have put the controversy on the CO2 theory to an end, since "objections which have been raised against it by some physicists are not valid". Unfortunately, this paper passed almost unnoticed, I guess because meteorologists and geologists do not read Physical Review so often.

At the time of Hulburt the CO2 absorption coefficient was not known very accurately and even less its line shape, forcing Hulburt to use a "box-like" shape. We may now build a simple model with a more realistic line shape and show that we get an increased absorption with increasing CO2 concentration in any case.

Consider the CO2 absorption band around 15 ?m (about 650 cm-1), it is strong enough to not let any light go through after a few tens of meters at surface temperature and pressure. Did this energy disappear forever? Surely not, radiatively or convectively this energy "is spread around and transferred upward". But on the way up this light will encounter fewer and fewer CO2 molecules. There will be a point where the light can escape to the outer space. The intensity of the emerging light will be appropriate to the temperature of this "last" layer.

We can crudely model this behavior using the Planck law and a gaussian-shaped absorption coefficient. We consider just two layers, the surface and the "last" layer, and the emissivity of this outer layer is modulated between 0 and 1 according to the absorption coefficient ?. The result is shown in the figure below.

In the calculations, I focused on absorption near the wavenumber of 650 cm-1 and tuned the optical depth to reach saturation. The two dashed lines correspond to the Planck law for T=300 K and T=220 K. The red curve is the calculated emission; it follows the 300 K curve but deviates from it near the absorption band. This bite taken out of the 300 K curve represents the energy prevented fron reaching outer space, i.e. the greenhouse effect.

This graph can be *qualitatively* compared with real measurements (from G.W. Petty 2006) to be sure we're not too far off.

We can now look at what happens when we increase ?. Following Angstrom (and many others in his times) the energy absorbed should not change. On the contrary, if we recall that the absorption coefficient is gaussian we would expect an increase in the energy retained by our layer along the wings. The effect is shown in the figure below.

We can see that although the absorption dip cannot fall below the 220 K curve, it becomes wider and the absorbed energy increases accordingly. This is as far as we can get with this simple model. Needless to say that there's much more than what can be done with the very crude model presented here. We know, for example, that the line shape of the absorption coefficient changes with both pressure and temperature due to what are called pressure and Doppler broadening. In the upper layers of the atmosphere the band initially gets narrower and then splits into several narrow bands (the roto-vibrational spectrum) leaving more room for the increase in CO2 concentration being more effective. We also know that there are weaker absorption peaks other than the stronger one quoted above which are not saturated.

Gilbert Plass in 1956 used these words:

One further objection has been raised to the carbon dioxide theory: the atmosphere is completely opaque at the center of the carbon dioxide band and therefore there is no change in the absorption as the carbon dioxide amount varies. This is entirely true for a spectral interval about one micron wide on either side of the center of the carbon dioxide band. However, the argument neglects the hundreds of spectral lines from carbon dioxide that are outside this interval of complete absorption. The change in absorption for a given variation in carbon dioxide amount is greatest for a spectral interval that is only partially opaque; the temperature variation at the surface of the Earth is determined by the change in absorption of such intervals.

There's one more subtle effect related to increased absorption. Upon increasing CO2 concentration, the layer at which the absorption coefficient at each wavelength is low enough to let the IR light escape will be found higher in the atmosphere. The emitting layer will then have a lower temperature, at least until the tropopause is reached, and hence a lower emitting power.

Clearly there's a world behind the absorption of IR light by CO2 in the atmosphere which I omitted. The physics behind it is now solid thanks to the decades of work of many different scientists, and despite the first highly respected skeptic ever who put the CO2 theory on hold for half a century. But you know, this is how science works.

*Note: I cannot conclude without acknowledging the fundamental role of Spencer Weart "The Discovery of Global Warming" from which I borrowed (and learned) a lot. His book and the supporting website are a treasure cove for anyone interested in how our current knowledge has been built step by step over time.*

This post is the *Advanced* version (written by Riccardo) of the skeptic argument "CO_{2} effect is saturated".

Posted by Riccardo on Saturday, 9 October, 2010

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