[Vision2020] Undermining Science: Examining Saturation, Absorption of CO2 in Earth's Atmosphere

[Ted Moffett]

I was researching the climate science question of saturation,
regarding the absorption of CO2 in Earth's atmosphere, when I found a
draft I wrote to the following post from April 2010, that raised this
issue, that I never sent.

I know millions are hanging on every word, still waiting... Ha... So
I'll post part of my response to the saturation issue and
Beer-Lambert's law, regarding the absorption and of CO2 in Earth's
atmosphere.

The objection raised below to the magnitude of the climate forcing
from increasing CO2 levels is a worn out skeptics talking point, that
a cursory high school level research effort would reveal has been
examined by climate scientists and addressed in detail, and found to
be faulty.

Lower down I offer a source that explains why the following post is
scientifically misleading, if not outright false ("Undermining
Science," as the subject heading indicates), as I offered in my never
sent draft from April 2010.

The science involved in this issue is very difficult, and I don't
claim to understand all complexities invovled.  But given the extent
to which the issue has been examined by competent scientists, I am
confident that the following post offers an appallingly incomplete
examination of this issue, that could easlily hoodwink the gullible
into thinking anthropogenic global warming is a highly questionable
scientific theory, when in fact it is not.

On 4/5/10, Paul Rumelhart <godshatter at yahoo.com> wrote:

http://mailman.fsr.com/pipermail/vision2020/2010-April/069593.html

> Another point that I don't see stressed by AGW proponents is that CO2
> temperature increases are a diminishing return, it's logarithmic.
> There's an upper limit to what CO2 can do temperature-wise, at least if
> you only look at the radiative component.  It's similar to covering a
> window with shades.  The first one stops a lot of light, the next one a
> bit more.  Eventually, it won't matter how many shades you tack up over
> the window, it won't get appreciably darker in the room.  CO2 covers a
> certain range of the infrared spectrum, and won't stop the other parts
> no matter how much you put in the air.  Look into the Beer-Lambert law
> to see why it's logarithmic.
>

The Irrelevance of Saturation: Why Carbon Dioxide Matters
© 2008 by Barton Paul Levenson

http://bartonpaullevenson.com/Saturation.html

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:

    T = exp-k p L (1)

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:

k the absorption coefficient, which Essenhigh measures in reciprocal
meter-atmospheres (m-1 atm-1).

p the "concentration" of the gas (actually, for the units to make
sense, the partial pressure of the gas in atmospheres).

L the distance traveled (in meters).

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:

--------------------------------------------------------------------------------
Table 1. Infrared Absorption Coefficients of Carbon Dioxide (after
Essenhigh 2001)

--------------------------------------------------------------------------------

Band (μ) k (m-1 atm-1)
 1.9 -  2.1 656
 2.6 -  2.9 139.4
 4.1 -  4.5  18.37
13   - 17   1.48

--------------------------------------------------------------------------------

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:

--------------------------------------------------------------------------------
Table 2. Layers of the Simple Model

--------------------------------------------------------------------------------

Layer Energy input Energy output
Space aZ, (1-a)Y F
Air Layer 2 aY 2aZ
Air Layer 1 X, aZ 2Y
Ground F, Y X

--------------------------------------------------------------------------------
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

Space:     a Z + (1 - a) Y = F (2)
Layer 2:     a Y = 2 a Z (3)
Layer 1:     X + a Z = 2 Y (4)
Ground:     F + Y = X (5)

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:

    Z = F / (2 - a) (6)
    Y = 2 Z (7)
    X = F + Y (8)

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:

    F = (S / 4) (1 - A) (9)

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:

    F = ε σ T4 (10)

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:
--------------------------------------------------------------------------------
Table 3. Model Fluxes and Temperatures at a = 0.5.

--------------------------------------------------------------------------------

Layer Flux (W m-2) Temperature (°K.)
Layer 2 158 193
Layer 1 316 273
Ground 553 314

--------------------------------------------------------------------------------

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

Now, let's try a = 0.6:

--------------------------------------------------------------------------------
Table 4. Model Fluxes and Temperatures at a = 0.6.

--------------------------------------------------------------------------------

Layer Flux (W m-2) Temperature (°K.)
Layer 2 169 206
Layer 1 339 278
Ground 576 317

--------------------------------------------------------------------------------
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:

--------------------------------------------------------------------------------
Table 5. Model Fluxes and Temperatures at a = 1.0.

--------------------------------------------------------------------------------

Layer Flux (W m-2) Temperature (°K.)
Layer 2 237 254
Layer 1 474 302
Ground 711 335

--------------------------------------------------------------------------------

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.

Page created: 12/29/2009
Last modified:   02/13/2011
Author: BPL
------------------------------------------
Vision2020 Post: Ted Moffett