The previous post discussed the fact that in a single-layer model a spectrum can saturate so that essentially 100% of the radiance available in a frequency range is absorbed. The single-layer model used here assumes that the sample is all at the same temperature and that the pressure is constant at one atmosphere.
Here I do a rough calculation of such an effect for the 15-micron band of carbon dioxide. The source spectra for this analysis come from the NIST web book, referenced below and explained more completely in a previous post. I downloaded the digitized data of the gas phase spectrum of carbon dioxide. There are several spectra included, the differences between the spectra are minute, and I decided to use the first spectrum (second column) for simplicity. The spectra were recorded at a concentration of 200 Torr diluted to 600 Torr with Nitrogen, i.e., a mixing ration of one third (~33% or 333,333 ppm). The cell length was 10 cm (0.1 m).
I converted the transmittance ( t ) data to absorbance (A) as follows:
A = -log ( t )
The spectrum is a transmittance spectrum; so I am free to use log-based 10 as long as I am consistent. I calculated the absorptivity as follows:
α = A/(0.1 m * 333,333 ppm)
So all units are in ppm-m. NB: to use these units in this manner, it is essential that I assumed that the layer is at constant pressure.
The current atmospheric mixing ratio of carbon dioxide is about 380 ppm. I can now calculate and plot the transmittance spectra of carbon dioxide for various pathlengths. I chose pathlengths of 10 m, 100 m, 1 km, 10 km, and 100 km and focused attention on the 15-micron band.
As the pathlength increases, note that the spectrum flattens out at essentially zero transmittance. Somewhere between 10 km and 100 km the spectrum can be considered saturated (note that because of the logarithmic form of transmittance that the precise point of saturation depends on an arbitrary definition of saturation, i.e., the spectra approach 100% absorbance asymptotically). If the atmosphere were really a single layer, increasing the concentration would not increase the absorption of IR radiation in the atmosphere.
At a pathlength of 100 km, there is not effectively any difference between 380 ppm and 560 ppm in terms of transmittance for this single-layer model.
Considerations along these lines have sometimes been used erroneously to imply that carbon dioxide increase is not a concern for global warming. The sophisticated reader should immediately see the problem with such an analysis. The atmosphere is not a single-layer system. The temperature and pressure of the atmosphere change significantly with altitude. One might naively think that simply taking the average pressure and temperature would address this issue, but that is not the case. Unfortunately, it is necessary to understand the atmosphere and atmospheric processes at a more detailed level.
Before we congratulate ourselves too much in this realization, it is worth noting that no less of a scientist than Knut Ångström made this error when considering the pioneering work of Svante Arrhenius.
The most important consideration here is that layers of the atmosphere do not just absorb radiation; they also emit radiation. The emission of radiation from a layer in the atmosphere depends on its temperature. When I introduced Beer's Law, I treated the transmittance as a simple ratio between the spectral radiance of the sample and the spectral radiance of a source or radiation (Is/I0). This assumption was based upon the source of radiation being much hotter than the sample (as would be expected under laboratory conditions.). Once this assumption no longer holds, the transmittance is no longer equal to the indicated ratio. We have to take into account each layer in the model and its thermal radiance.
The next post in this series will examine a two-layer model of infrared radiation and start to introduce the concept of a radiative-transfer model.
- NIST Webbook
- HITRAN data for Carbon Dioxide
- Knut Ångström
- Svante Arrhenius
- Real Climate: A Saturated Gassy Argument