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Absorber Dispersion and the Absorption of Radiation

Gilbert N. Plass, one of the early writers on the role of carbon dioxide (CO2), in atmospheric warming, made the assertion that even if the absorption bands of CO2 and water vapor (H2O) completely overlapped CO2 could have a significant impact despite its near infinitesimal concentration (less than 0.04 of 1%) compared to that of water vapor (1 to 3%) because water vapor is more concentrated near the Earth's surface where CO2 is rather uniformly dispersed vertically in the atmosphere. There is a certain plausibility to this assertion but, as will be shown below, it runs counter to physical analysis.

The physical principle that applies concerning this question is the Beer-Lambert Law which says that if I(z) is the intensity of radiation at a distance z into an absorbing medium then

(1/I(z))dI/dz = −αρ(z)

where ρ(z) is the density of the absorbing medium at z and α is the radiative effectiveness of the absorbing medium. The value of α might depend upon the wavelength of the radiation but it does not depend upon the location in the medium. The solution to this familiar form differential equation can be written down simply from familiarity but let us go through its solution step by step.

The equation can be written as

d(ln(I(z))/dz = −αρ(z)

This equation may now be integrated from 0 to Z to obtain

ln(I(Z)) − ln(I(0)) = −α∫0Zρ(z)dz

which means that

I(Z) = I(0)e−D(Z)


D(Z) = α∫0Zρ(z)dz

The quantity D(Z) is known as the optical depth into the medium. The amount of radiation absorbed is

I(0) − I(Z) = I(0) − I(0)e−D(Z) = I(0)[1 − e−D(Z)]

The value of ρ(z) might, as is the case in the atmosphere, trail off to zero, rather than becoming zero at some finite z. That would not matter the the total optical thickness of the medium would still be a finite amount, D(∞). Its value depends only upon the total amount of the absorber in the medium ∫0ρ(z)dz and does not depend upon the dispersion of the absorber within that medium.

If there are two absorbing gases the optical depth is the sum of the optical depths due to each one independently; i.e.,

D = α10Zρ1(z)dz + α20Zρ2(z)dz

where αi and ρi are the radiative efficiency and linear density of the i-th gas, respectively. In other words,

D = D1 + D2

Therefore Gilbert Plass' assertion is wrong. If there were a complete overlap of the absorption spectra of CO2 and H2O then any difference in the vertical profiles of the two greenhouse gases is irrelevant. Furthermore it is worth noting that the radiative efficiency of H2O molecules appears to be significant larger than that of CO2 molecules. The values of the ratio that appear in the literature are 1.5 and 1.6. Suppose that 2% of the atmosphere is water vapor and 0.04 of 1% is CO2. This is a 50 to 1 ratio. However if H2O has a radiative efficiency 1.5 relative to CO2 the effective ratio in terms of the amount of radiation absorbed is 75 to 1. Thus a doubling of the amount of CO2 is only an increase in the share of greenhouse gases in the atmosphere from 2.04% to 2.08%, a relative increase of about 2%. However taking into account of the radiative efficiencies the increase is only 1.3%. So a 100% increase in CO2 in the atmosphere would produce only a 1.3% increase in the optical thickness of the atmosphere.

The principle involved above can be extended to spatial dispersion as well as vertical dispersion. A fixed number of H2O or CO2 molecules, say a trillion trillion, will absorb the same amount of radiant energy whether concentrated in small area or dispersed over a wide area. What matters is the total number of molecules in the atmosphere and the intensity of the radiation.

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