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Climate Science 101 appendix. Carbon Dioxide Spectral line Broadening, and the Continuum [1]

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Date: 2023-06-14  

Two weeks ago I wrote a diary Climate Science 101, which talked about how hot air rises. It wasn’t really about global warming, but It’s hard to ignore that subject. Robbruin commented that it was so incomplete that it did not capture the physics of the process. That seems a little harsh, but I did find some things about CO 2 ’s IR spectrum that I think is probably what he meant.

The physics in question relates to the “saturation” of the CO 2 ’s greenhouse effect. The argument for saturation is that if infrared radiation can’t even make it across a room in the absorption range of CO 2 , how is adding more of it to the atmosphere going to affect anything. Also, since the cause of radiative forcing is moving emission up to colder parts of the troposphere, there is a question how this can work, since the absorption band of CO 2 is so strong it must emit to space from in the stratosphere or above. Since the stratosphere warms as the altitude increases then thickening the atmosphere by adding CO 2 there should increase the outgoing radiation not reduce it.

Raymond T. Pierrehumbert tackled this problem in the linked article, which I recommend, Infrared Radiation and Planetary Temperature in Physics Today that explains why the CO 2 saturation idea was dismissed long ago. Since there is no way the magazine will be in creative commons, I can’t copy the figures, but Figure 2 is important.

In the title picture above, the power per unit area of long wave IR emitted radiation vs wavelength is plotted. The power per unit area of some black body radiators (a perfect emitter) at various temperatures are also plotted for comparison. The trench in emission at a wavelength of 15 microns is due to CO 2 . A plot of the absorption bands does not look like that “trench”. It contains a very strong peak rather than a mesa with a spike.

Figure 2 from Pierrehumburt, is a plot of carbon dioxides absorbance, which is how strongly the molecule absorbs energy vs the wavelength. Unfortunately that plot is in wavenumbers instead of wavelength, and the Y axis absorbance is a log scale. A linear version from nist shows a much thinner line.

But moving on. The absorbance of CO 2 is 10000M2/Kg at the 15 micron spike in the emission in the center of the CO 2 band. (Figure 2 from Pierrehumburt is plotted in wave-numbers, which is cm/ wavelength, so you take the reciprocal of the wavelength in cm.) The CO 2 emission band from the top of the atmosphere then features two wide areas with emission temperatures around 220K (-73C), which is the tropopause temperature, as cold as the lower atmosphere gets. I estimated from the title image the edges of the that flat bottomed trench as wavelengths of 16.66 microns and 13.33 microns (wave number 600cm-1 and 750cm-1). At those wavelengths, where the emission temperature is starting to increase, CO 2 absorbance is estimated 50M2/Kg so the absorption has already dropped by a factor of 200 from the center of the trench. If we look at the wavelengths were the CO 2 adsorption peak is just detectable in the title figure above, where the trench leaves the trend line the absorption there is 0.1M2/Kg, or a drop of a factor of 100,000 from the peak of the adsorption band (from Figure 2 Pierrehumburt again). So the CO 2 peak is wide and deep, and this has effects because the weak adsorption edges have a big effect on the climate.

Anyway, looking at the band shape, the spike at 15 microns is because the center of the band is saturated. CO 2 is so adsorbent at that point, that the last emission is in the stratosphere. The stratosphere is slightly warmer than the troposphere so the spike shows a slight increase in emission temperature. (The atmospheric temperature vs height is shown in Figure 3 of Pierrehumburt.) Adding more carbon dioxide will increase the temperature and intensity of that emission slightly, because the stratosphere slightly warms as it rises. The long flat areas with the lowest emission temperature emit at or near the tropopause, like in the basic theory of the climate. Because this region has a very low lapse rate increases in CO 2 will not affect it much. Changing the emission height won’t have a big effect either way. Where increases in CO 2 will mater is the sides of the trench where the adsorption is between 0.1 and 50. Increases in CO 2 will increase radiative forcing in those wavelengths. This is why the increase in CO 2 forcing with concentration is a, less than linear, logarithmic curve shown in the link from skeptical science. The effect is still there though. This Harvard article calculates the forcing from increasing CO 2 in Figure 5, but frankly the plots are confusing, and you have to read them carefully to understand what they did.

The reason that the CO 2 absorbance is so broad is due to vibrational rotational coupling. The molecule can absorb or emit both vibrational and rotational energy at the same time. This means that the number of allowed transitions is much larger than just the vibrational transition. The figures below shows some roto-vibrational spectra. The top figure shows only the allowed rotational bands, while the bottom one shows the central vibrational transition with other transitions where the rotational state also changes. The vibro-rotational bands are then further broadened by “collision”or pressure broadening, so that they overlap, making the adsorption continuous. In this case the energy of the transition is the sum of the changes of the vibrational, and rotational modes with the excess given to the kinetic energy of the air molecules in the collision. Once the lines overlap the spectral region is called the CO 2 continuum. There is another one for water vapor. These weak absorption bands in the continuum have a big effect on global warming, because even though they are very weak, they can still cause the atmosphere be opaque over the multiple kilometer distances in the atmosphere, even though they are difficult to detect in a laboratory. The bands that we are talking about are on the very small ones outside of these figures, where the rotational transitions are becoming very small.

The spectra from another CO 2 band. This one is the symmetric stretch, but the pure vibrational transition here is forbidden. Texts showing rotational spectra love this vibration, because the center line is missing, so you can only see the rotational structure. If you look for the IR spectra of CO 2 this one will often pop up.

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