The Irrelevance of Saturation: Why Carbon Dioxide Matters

© 2008 by Barton Paul Levenson

In 1896, the Swedish chemist Svante Arrhenius proposed that doubling the amount of carbon dioxide in Earth's atmosphere would raise the planet's surface temperature by five or six degrees Celsius.

In 1901, Swedish physicist Knut Ångström published a rebuttal based on lab work. He (and a lab assistant, J. Koch) measured the absorption of infrared light in a column of carbon dioxide equivalent to that in a column of real atmosphere. The amount absorbed changed only a little when the density of gas was reduced by one-third. Clearly, the carbon dioxide in the atmosphere was already absorbing all the infrared light it could, so adding to it couldn't possibly raise the absorption by a significant amount. The absorption lines of CO₂ in Earth's atmosphere were "saturated" -- no more absorption was possible.

As a result, for many years, physicists and students of climate in the early 20th century did not believe that rising carbon dioxide could warm the planet.

But high-altitude observations in the 1940s showed that the absorption properties of greenhouse gases changed significantly with pressure and temperature, and that plenty of absorption took place in the upper atmosphere where the absorption lines were narrower. Since the Canadian-American physicist Gilbert Plass nailed the problem for good in an article in 1956, no professional climate researcher has taken the saturation argument against carbon dioxide warming seriously.

One mechanical engineer has recently tried to revive it, and it may be instructive to see how and why he is mistaken. The concept that "absorption in the upper atmosphere is also important" is abstract and hard to grasp. An analysis of the arguments of American mechanical engineer Robert H. Essenhigh (2001) may lead to insight into how this concept actually applies.

Mr. Essenhigh produces a band scheme for water vapor, carbon dioxide, and methane, which is neatly tabulated in his article. He uses the data to determine the distance that infrared light could travel through each gas before being absorbed. The equation he applies is one of many possible variations of the Beer-Lambert-Bouguet law. The crucial section can be expressed this way:

This equation measures the transmissivity of a medium -- how much of a beam of light can get through that medium. T must logically fall between 0 and 1 in order for energy to be conserved, the complementary fraction going to absorption, scattering, or phase change. The other terms are:

Unit analysis shows that the exponentiated product is dimensionless, and thus so is the final result, T. Because of the negative-exponential form, T is always between 0 and 1.

It is simple to assume T = 0.01 (i.e., 99% absorption) and back-solve for the required column length. Essenhigh gives the following absorption coefficients for carbon dioxide:

Using the T = 0.01 criterion and a concentration of carbon dioxide of 0.0004 atmospheres, near-total saturation happens at 18, 82, 625, and 7800 meters in each respective band.

The troposphere is, of course, at a height ranging from 11 to 15 km depending on latitude (and daily and seasonal variations). Thus it would appear that even the most important carbon dioxide absorption band -- the 14.99 µ band (667 cm-1, to use the radiation physicist's preferred measure of wave number) saturates very close to the ground, and well before the total mass of CO₂ in the atmosphere must be taken into account.

Essenhigh's argument goes further than this. To demonstrate the relative unimportance of carbon dioxide, he uses band information for water vapor to show that water vapor absorbs most of the infrared radiation from the ground. He concludes that 90% of Earth's greenhouse effect is due to water vapor and 10%, or less, to carbon dioxide.

Climatologists, of course, find differently. According to Kiehl and Trenberth's 1997 energy budget for the Earth climate system, water vapor accounts for 60% of the clear-sky greenhouse effect and carbon dioxide for 26%. These are typical findings of such studies. See, for example, Ramanathan and Coakley (1978).

Why the discrepancy? Why are Kiehl and Trenberth, Ramanathan and Coakley right and Essenhigh wrong? Granted, they are climate scientists and Essenhigh is not, but that would be an argument from authority and wouldn't convince any red-blooded American who feels that one opinion on a scientific subject is as good as any other. Why does the absorption in upper layers of air matter when nearly all the infrared from the ground is absorbed by the lowest layer?

The answer is that infrared doesn't only come from the ground. It comes from every layer of atmosphere as well. That warm lowest layer radiates both up and down, and if there is more CO₂ in the upper layers, more of the radiation from that lowest level will be absorbed and the world will be warmer. Every layer affects every other layer, and more absorption even in the highest level will wind up warming the ground.

How to illustrate this? I wouldn't be a climate science freak in good standing if I didn't try to create a model to show how this works. I'll use an extremely simple model which nonetheless captures most of the relevant physics. The model has four layers -- space, including the sun. Two layers of atmosphere. One layer of ground.

The layers of atmosphere are assumed to be completely transparent to sunlight. Sunlight zips right through them and warms the ground. The ground then radiates infrared light upward. The lower layer absorbs all the radiation from the ground. It radiates up and down. Radiation from the lower level heats the upper level, which in turn also radiates up and down. But the amount absorbed, and therefore radiated, by the upper layer is variable -- it only absorbs a fraction, a, of the radiation from the lower level. By Kirchhoff's Law, this means its own radiation must be multiplied by a; the layer does not radiate as much as an equivalent black body would.

Here's a diagram of the system:

All the layers radiate. Space, since it contains the sun, radiates downward, and we will call the amount of shortwave light it radiates into the system F (for absorbed Flux). The ground radiates an amount of longwave light, X, upward. Layer 1 (the lower layer) radiates longwave light Y both upward and downward. Layer 2 (the upper layer) radiates longwave light aZ both upward and downward. I am using the climate convention of calling the solar output of ultraviolet, visual and near-infrared light "shortwave" radiation and the terrestrial and atmospheric output of thermal infrared light "longwave" radiation.

The sources of each level's energy are then, for the ground, F and Y, for layer 1, X and aZ, for layer 2, aY, and for space, aZ and (1-a)Y (i.e., the output from the upper layer of air, and the output from the lower level that got past the upper level). The energy balance equations for each layer are then

where
a is the absorptivity of the upper level (which must fall between 0 and 1),
F is solar flux downward,
X is ground flux upward,
Y is Layer 1 flux in one direction, and
Z is what would be Layer 2 flux in one direction if a were equal to 1.

Straightforward algebra leads to a series of equations to solve for each variable, given a and F:

Now we can use the model for numerical examples. We will fix F at the actual level of flux absorbed by Earth's climate system:

where S is the solar constant and A is Earth's bolometric Bond albedo. According to the tabulations of Lean (2000), the mean solar constant from 1951 to 2000 was 1,366.1 watts per square meter. NASA's planetary fact sheets give the Earth's albedo as 0.306. Plugging these into equation (9), we find F = 237 W m^-2.

Let's assume layer 2's absorptivity is 0.5 -- it absorbs half the radiation from Layer 1; the rest goes through it and out to space. Knowing F and a, we then have X = 553 W m^-2, Y = 315 W m^-2, and Z = 158 W m^-2. Note that 237 W m^-2 comes from space and 237 W m^-2 ((1-a)Y + aZ) goes back out to space -- energy is conserved.

We can find the temperature of each layer (excluding space, which isn't really a body), by back-calculating according to the Stefan-Boltzmann law, as modified for a "gray" (equal emissivity at all wavelengths) radiator:

Here F is flux, ε is emissivity -- which must, by Kirchhoff's Law, equal absorptivity (a = ε) -- σ is the Stefan-Boltzmann constant, and T absolute temperature. The latest SI value for σ is 5.6704 x 10^-8 W m^-2 K^-4. The value of a for the ground and layer 1 is 1.0; they have been assumed all along to be perfect absorbers/ emitters. The value of a for layer 2 is 0.5. The corresponding temperatures are then 193 degrees Kelvin for layer 2, 273 K for layer 1, and 314 K for the ground.

Now the Earth, of course, has a mean global annual surface temperature of 287 or 288 K, depending on which study you go by. This is clearly not a very realistic model of Earth's climate system; it completely neglects band effects, convection, clouds, evaporation of seawater, etc. But we're just going for an illustrative effect here. The model is good enough for government work.

Let's put the results for a = 0.5 in a table:

Keep in mind that layer 1 is absorbing all the radiation from the ground. 100% of it.

Now, let's try a = 0.6:

Look at that. By increasing the absorption in the upper layer, layer 2, the ground became warmer by 3 K -- even though all its thermal radiation output was absorbed in layer 1. How does that happen?

You can think of it in stages. Layer 2 absorbs more infrared light. It heats up. It radiates more. Some of the radiation goes down and heats layer 1. It absorbs more infrared light (100% of what it's getting from layer 2). It heats up. It radiates more. And some of the radiation goes down and heats the ground.

So increasing radiation even at the highest level of the atmosphere can warm the ground, and that is just what adding additional CO₂ to the atmosphere does.

In our model, there's an upper limit to how far this process can go -- a = 1.0 for layer 2. When we do that, we get the results in Table 5:

A pretty hot Earth, since it has no convection to lower the surface temperature. But an upper limit. In a case where every layer of atmosphere is a black body, there seems to be a limit to how high greenhouse warming can go.

But this is true only when every level is a blackbody radiator, and that can't happen. Absorption lines narrow with decreasing pressure, and in any real terrestrial-planet atmosphere, the upper layers will always be under less pressure than the lower layers. Gravity and all that. Air has mass. Thus you will always be able to raise the temperature of the system by adding more carbon dioxide.

Eventually things would get so hot that the thermal radiation being put out would shift its maximum to shorter wavelengths, where there are gaps in the absorption bands, and that would be a true maximum temperature for a planet. But Earth is nowhere near that limit. Even Venus is nowhere near that limit. It has 4.67 x 10²⁰ kilograms of CO₂ in its atmosphere compared to only 3.04 x 10¹⁵ kg for Earth -- more than 10,000 times more carbon dioxide. Its surface temperature is an oven-like 735 K. (Make that one of those self-cleaning ovens that can get up that high.) And it still isn't saturated with respected to carbon dioxide; modeling by Bullock and Grinspoon (2001) shows that surface temperatures on Venus may have hit 900 K at times of high volcanism. Saturation of carbon dioxide lines just does not prevent global warming on the terrestrial planets we know of.

What of Essenhigh's assertion that water vapor, and not carbon dioxide, accounts for almost all the absorption in Earth's atmosphere? Again, he has mistaken what happens in the lowest layer of air for what happens in the whole atmosphere. Carbon dioxide is a well-mixed gas, roughly the same mass fraction or volume fraction at each level of the troposphere. But water vapor is not well-mixed. It has a very shallow "scale height" of about 1.8 kilometers -- the local mass of water vapor declines by a factor of e (2.7182818...) for every 1.8 km higher you go. But the scale-height for the well-mixed gases averages 7 km. Water vapor peters out quickly with height; carbon dioxide does not. So carbon dioxide is more important at higher levels, and it is more important in Earth's greenhouse effect than Essenhigh gives it credit for. Kiehl and Trenberth's 26% really does make more sense than Essenhigh's 10% or less.

The Earth has enjoyed a fairly stable climate for something like 10,000 years. That stability is rapidly being eroded as we pump more and more carbon dioxide into the atmosphere. Our agriculture and our economy depend on operating in a certain climate, but the side-effects of our technology are changing that climate. The consequences may be very bad.

References

Ångström, Knut 1901. "The Dependence of the Absorption of Gases, Particularly of Carbonic Acid, on the Density." Annalen der Physik 6, 163-173.

Arrhenius, Svante 1896. "On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground." Philosophical Magazine and Journal of Science, 5th Series. 41, 237-275.

Bullock, Mark A. and Grinspoon, David H. 2001. "The Recent Evolution of Climate on Venus." Icarus 150, 19-37.

Essenhigh, Robert H. 2001. "In Box: Robert Essenhigh Replies." Chemical Innovation, 31, 62-64.

Lean, Judith 2000. "Evolution of the Sun's Spectral Irradiance Since the Maunder Minimum." Geophysical Research Letters, 27, 2425-2428.

Plass, Gilbert Norman 1956. "Effect of Carbon Dioxide Variations on Climate." American Journal of Physics 24, 376-387.

Ramanathan, V. and Coakley, J. A. 1978. "Climate Modeling through Radiative-Convective Models." Review of Geophysics and Space Physics, 16, 465-490.