Chapter 4 12/18/05 1
Chapter 4. Greenhouse Gases
Abstract
The layer model assumes that the atmosphere acts as a blackbody in the infrared,
absorbing and emitting all frequencies of IR light. In reality, gases absorb IR
light selectively, and most of the gas in the atmosphere doesn't interact with IR
light at all. The difference can be understood in terms of the effect of molecular
vibration on the electromagnetic field. Because gases absorb IR selectively, there
are some radiation bands that are completely absorbed (the gases are saturated),
and others such as the atmospheric window, where no gases absorb. This leads to
much higher greenhouse forcing per molecule from some trace gases, such as
freons, SF6, or to a lesser extent methane, than from more abundant gases such as
CO2. Some absorption bands fall in the middle of the IR emission spectrum of the
earth's surface, while other bands fall outside this spectrum and are therefore
irrelevant to the heat budget.
About Gases
The layer model is what we call an idealization of the real world. Now that we
understand the core mechanism of the greenhouse effect, by understanding the layer
model, we can add things one at a time from the real world, and see how they affect the
way that earth's temperature is controlled. The first modification we have to make to the
layer model is to think more about real gases in the atmosphere.
Let's begin by defining different ways of describing the amounts of gases in the
atmosphere. The word concentration means number of molecules within some volume.
The difficulty this raises for gases in the atmosphere is that there are fewer molecules per
volume overall as you go up in the atmosphere. The major gases in the atmosphere are
pretty well mixed, so that the concentrations of these gases go down proportionally with
altitude. This is why it's hard to breath oxygen quickly enough on Mount Everest. It is
often more convenient to talk about proportions of gases, like oxygen is about 20% of the
molecules of gas in the atmosphere, and nitrogen almost 80%. The proportion of CO2 is
currently 0.037%. We can express that in a more convenient way by saying 370 parts per
million or ppm. This number is called a mixing ratio. The mixing ratio of a gas is
numerically equal to the pressure exerted by the gas, denoted for CO2 as pCO2. In 2005
as I write this the pCO2 of the atmosphere is reaching 380 atm. It is rising by about 1.5
ppm per year, which is the same as saying 1.5 atm per year.
Gases, Vibrations, and Light
Most of the mass of an atom is in its nucleus, which resembles the massive sun at the
center of the solar system. Electrons float in ghostly quantum mechanical probability
clouds, called orbitals, around the nucleus. Two nuclei of two different atoms always
repel each other, because of their positive charges. The orbitals for the electrons fit
together better, however, with certain numbers of orbitals than with others. Electrons
Chapter 4 12/18/05 2
from two different atoms may be able to combine their orbitals in such a way that they
are lower energy, as if happier, when they share, a chemical bond. A chemical bond is
like a spring in that the two nuclei on either end of the bond have some freedom to move
closer or farther apart. There is an optimum distance for the nuclei to be from each other.
Closer, and the positive nuclei will start to repel each other. Farther, and you get less
energy gain from sharing the electrons. A bond vibrates when the distance between the
nuclei oscillates between the nuclei being too close together, then too far apart.
Gases are the simplest type of molecule, and they only vibrate in very particular
ways. Vibrations in a gas molecule are like vibrations of a piano string in that they are
fussy about frequency. This is because, like a piano string, a gas molecule will only
vibrate at its "ringing" frequency. The ringing frequency of an oscillator made of
weights and springs depends on two things: the amount of weight on the ends and the
strength of the spring holding them together. Heavy weights will have enough inertia to
keep a bond growing in the wrong direction for longer than will a pair of light weights, so
the frequency of the vibration will be slower. If the spring is very strong, it will reverse
the velocity of a vibration more quickly, and the frequency of the oscillation will be
higher. Vibrations in chemical bonds depend on the mass of the nuclei and on the energy
penalty for having the nuclei too close or too far apart: the springiness of the chemical
bond.
However, the vibrations of many gas molecules, such as the major gases in the
atmosphere oxygen and nitrogen, are invisible to the electromagnetic field. They don't
shine light or absorb infrared light; we say they are not infrared active. Oxygen and
nitrogen are not greenhouse gases, because they are transparent to infrared light. These
molecules are invisible because when you stretch one, it doesn't change the electric field.
These are symmetric molecules, made of two identical atoms whose electric fields just
cancel each other out. Neither atom can hold the electrons any more tightly than the
other. In general, symmetrical molecules with only two atoms are not greenhouse gases.
We can break the symmetry, making a molecule of NO for example. This is a very
reactive molecule, an ingredient for producing urban smog, but that's another story. NO
has one atom of each element, and as a result has a slight imbalance in its distribution of
electrons. One side of the molecule will have a slight positive charge, and the other will
be slightly negative. We could oscillate the electric field simply by rotating an NO
molecule. Also, if we vibrate an NO molecule, the steepness of the transition from
slightly positive to slightly negative will oscillate with time. By these mechanisms, NO
could be a greenhouse gas, but it turns out not to be a very important one because there is
not very much of it.
Molecules with more than two atoms have more than one chemical bond. All of their
bonds ring together rather than each bond ringing with its own characteristic frequency.
Water, H2O, is a molecule that is bent in its lowest energy state (Figure 4-1). This is
because several of the electron orbitals stick off in the direction that appears in my
diagram to be empty space. Hydrogen atoms hold their electrons more loosely than
oxygen atoms, and so each hydrogen has a slightly positive charge (marked in Figure 4-1
Chapter 4 12/18/05 3
using the lowercase greek letter delta, as +). The oxygen end of the molecule has a slight
negative charge. Just as for the NO molecule, rotating an H2O molecule would oscillate
the electric field and generate light. Because the arrangement of the nuclei in H2O are
more complex than for NO, there are several modes of vibration of the water molecule,
including a symmetric stretch and a bend. These modes are also infrared active.
The CO2 molecule is shaped in a straight line with carbon in the middle (Figure 4-2).
It is a symmetric molecule; the oxygen atom on one end pulls the electrons just as tightly
as the other oxygen on the other end. Therefore rotating the molecule at rest has no effect
on the electric field. Nor does a symmetric stretch. However, there are two modes of
vibration which do generate an asymmetry in the electric field. One is an asymmetric
stretch, and the other is a bend. The bend is the most climatically important one, as we
shall see next.
How a Greenhouse Gas Interacts with Earth-Light
We have seen that gases are terrible blackbodies, because they are very choosy about
which frequencies they absorb and emit. What we will now see is that some frequency
bands are more important to the climate of the earth than others. There are two factors to
consider. One is the concentration of the gas, which we will discuss in a bit. The other is
the frequency of the absorption band relative to the blackbody spectrum for the earth.
Figure 4-3 shows blackbody spectra again for temperatures ranging from 300 K, a hot
summer day, down to 220 K, which is about the coldest it gets in the atmosphere, up near
the troposphere at about 10 km altitude. There is also a jagged-looking curve. This is the
intensity of light that an infrared spectrometer would see if it were in orbit over the earth,
looking down. Figure 4-3 is not data, but rather a model simulation from one of our online
models. You can point a web browser at
http://forecast.uchicago.edu/models/radiation.html to run this model yourself. We will do
so in the exercises.
The spectrum of the light leaving the earth going into space ranges between two
different blackbody spectra, a warmer one of about 270 K, and a colder one from about
220 K.
The parts of the spectra that seem to follow the colder blackbody curve come from
greenhouse gases in the upper atmosphere. They follow the colder blackbody curve
because it is cold in the upper atmosphere. The most pronounced of these absorption
bands, centered on a wave number of about 700 cycles/cm, comes from the bending
vibration of CO2. Light of this intensity that shines from the surface of the earth is
absorbed by the CO2 in the atmosphere (Figure 4-4). The CO2 in the atmosphere then
radiates its own light at this frequency. Remember from Chapter 1 that light emission
and absorption is a two-way street.
Other parts of the spectrum, most notably the broad smooth part around 1000
cycles/cm, follow a warmer blackbody spectrum. These come directly from the ground.
Chapter 4 12/18/05 4
The atmosphere is transparent to infrared light in these frequencies. This band is called
the atmospheric window.
The situation is analogous to standing on a pier and looking down into a pond of
water. If the water were very clear, you could see light coming from the bottom; you
would see rocks or old tires or whatever in the reflected light. If the water were murky,
the light you would see would be scattered light coming from perhaps just a few inches
down into the water. The old tires would be invisible, alas.
Remember we said that the total energy flux from one of these spectra can be
"eyeballed" as the total area under the curve. The areas of the pure blackbody curves are
going up proportionally to the temperature raised to the fourth power, because of the
Stefan-Boltzmann equation (our equation 2-1 in Chapter 2). The area trick works with
our new jagged spectrum as well. The effect of an atmospheric absorbtion band is to take
a bite out of the blackbody spectrum from the earth's surface, decreasing the area and
therefore decreasing the outgoing energy flux.
Compare the CO2 absorption band at 700 cycles/cm with the absorption band of
methane at around 1300 cycles/cm. The CO2 band has a lot more room to change the
outgoing infrared energy flux than does the methane band, simply because the earth and
the atmosphere radiate a lot more energy near 700 cycle/cm than near 1300 cycles/cm.
Both blackbody spectra are pretty low intensity in the methane band.
Band Saturation
The core of the CO2 bend absorption band, between 600 and 800 cycles/cm, looks
smooth rather than jagged and it follows a blackbody spectrum from about 220 K. This
is about as cold as the atmosphere gets, and if we change the amount of CO2 in the
atmosphere, the intensity of light in this range does not get any lower (Figure 4-5). We
call this phenomenon band saturation. You can see it in a series of model runs in which
the CO2 concentration of the atmosphere goes up from zero to 1000 ppm. The current
concentration of CO2 in the atmosphere is about 370 ppm, as we will learn more in
Section II. If there were no CO2 in the atmosphere, the atmosphere would be transparent
to light of around 700 cycles/cm, as it is in the atmospheric window. Adding the first 10
ppm of CO2 has a fairly noticeable impact on the shape of the outgoing light spectrum,
but increasing CO2 from say 100 to 1000 has a somewhat subtler effect.
I have plotted the total energy intensity Iout in W/m2
as a function of the concentration
of CO2 in the atmosphere in Figure 4-6. Changes in CO2 concentration have the greatest
effect if we were starting out from no CO2 and adding just a bit. The first 10 ppm of
added CO2 changes Iout by as much as going from 10 to 100, or 100 to 1000 ppm. We can
understand why by analogy to our murky pond or by looking back at Figure 4-4. As we
increase the murkiness of the water, we decrease the distance that a photon of light can
travel before it is absorbed. It doesn't take much murk in the water to obscure the old tire
on the bottom, shifting the depth to which we can see from the bottom at say 3 meters to
maybe only one meter. If we make the pond a lot murkier we will only be able to see a
few cm down into the water. Making it murkier still will limit our view to only one cm.
Chapter 4 12/18/05 5
The change in depth is getting less sensitive to the murkiness of the pond. In the same
way, the changes in the temperature at which the atmosphere radiates to space get smaller
as the CO2 concentration of the air gets higher. You just see the coldest light that you can
get.
The band saturation for CO2 makes CO2 a less potent greenhouse gas than it would be
if we had no CO2 in the air to start with. Let's revisit our comparison of the CO2 and
methane as greenhouse gases. Methane had a disadvantage because its absorption band
sort of fell in the suburbs of the earth-light spectrum whereas CO2 fell right downtown.
Now we see the advantage shifting the other way. Methane has a much lower
concentration in the atmosphere. You can see from the jagged edges of the methane peak
in Figure 4-3 that the methane absorption band is not saturated. For this reason, in spite
of the suburban location of the methane band, a molecule of methane added to the
atmosphere is 20 times more powerful than is a molecule of CO2.
If the edges of the absorption bands were completely abrupt, as if CO2 absorbed 600
cycles/cm light completely and 599 cycles/cm light not at all, then once an absorption
band from a gas was saturated, that would be it. Further increases in the concentration of
the gas would have no impact on the radiation energy budget for the earth. CO2, the most
saturated of the greenhouse gases, would stop changing climate after it exceeded some
concentration. It turns out that this is not how it works. Even though the core of the CO2
band is saturated, the edges of the band are not saturated. When we increase the CO2
concentration, the bite that CO2 takes out of the spectrum doesn't get deeper, but it gets a
bit broader.
The bottom line is that the energy intensity Iout in units of W/m2
goes up
proportionally to the log of the CO2 concentration, rather than proportionally to the CO2
concentration itself (we would say linear in CO2 concentration). The logarithmic
dependence means that you get the same Iout change in W/m2
from any doubling of the
CO2 concentration. The radiative effect of going from 10 to 20 atm pCO2 is the same as
going from 100 to 200 atm, or 1000 to 2000 atm.
The sensitivities of climate models are often compared as the average equilibrium
temperature change from doubling CO2, a diagnostic number that is called T2x. Most
models have T2x between 2 and 5 K, which is the same as 2 to 5°C. You can use T2x to
estimate a temperature change resulting from some change in CO2. Note that this is the
ultimate temperature change, after hundreds or even thousands of years have passed (see
Chapters 7 and 12). The equation is

T = T2x ×
ln
new pCO2
orig.pCO2






ln 2( )
(4.1)
where ln is the natural log, the reverse operation of the exponential function ex
, The
symbol e denotes a number which has no name other than simply e. We will meet e
again in Chapter 5. The exponential function is to raise e to a power of x. If
Chapter 4 12/18/05 6
ex
= y
then
y = ln(x)
Equilibrium temperature changes from changes in CO2, assuming various T2x values,
are shown in Figure 4-7.
What happens to the energy balance of the earth if we add a greenhouse gas to its
atmosphere? If the energy budget was in equilibrium before, it isn't any more, because
the greenhouse gas has decreased the amount of energy leaving the earth to space. We
can see this visually as the big bite out of the spectrum going from the top to the middle
diagram in Figure 4-8. The decrease in energy flux is proportional to the area of that bite,
the difference between the top and middle figures. Remember back to chapter 2, the
premise of the layer model is that the energy coming into and going out of the planet
must balance, and the planet accomplishes this feat by adjusting its temperature. If we
want to re-balance the energy flux after kicking it by adding CO2, we do that by
increasing the temperature of the ground. Using the on-line model, we find that a
temperature change of 8.5 K brings us back to the same energy output Iout as we had
before. Looking at the bottom result in Figure 3-8, we see that the new, warmer output
spectrum has risen everywhere compared to the middle figure. Visually, we have cut
some area out of the CO2 absorption band, and added it in the atmospheric window and
other parts of the spectrum, until the overall area under the curve is the same as it was
initially. Adding the CO2 caused the planet to warm.
Take-Home Points
Gases absorb / emit infrared light if they vibrate at the frequency of the light, and if
its vibration has a dipole moment that affects the electric field. O2 and N2 are not
greenhouse gases. All molecules of three or more atoms are infrared active.
A greenhouse gas has a stronger impact on the radiative balance of the earth if it
interacts with light in the middle of the earth-light spectrum.
Band saturation: A greenhouse gas at relatively high concentration like CO2 will be
less effective, molecule per molecule, than a dilute gas like methane.
Further Reading
The Discovery of Global Warming (2003) by Spencer Weart. This is a historical
account of the science and the scientists who discovered global warming including my
favorite, Svante Arrehnius, who used the infrared spectrum of moonlight, in 1896, to
predict that doubling CO2 would raise global temperature by 3-6° C (whereas the modern
prediction is 2-5°C). There is a good discussion of piecing together the band saturation
effect in this book.
Chapter 4 12/18/05 7
IPCC Scientific Assessment 2001, from Cambridge University Press or downloadable
from http://www.grida.no/climate/ipcc_tar/. Chapter 6 Radiative Forcing of Climate
Change.
Figure Captions
1. Vibrational modes of a water molecule that interact with infrared light in the
atmosphere.
2. Vibrational modes of a CO2 molecule that interact with infrared light in the
atmosphere.
3. The solid line is a model-generated spectrum of the infrared light escaping to space at
the top of the atmosphere. For comparison, the broken lines are blackbody spectra at
different temperatures. If the earth had no atmosphere, the outgoing spectrum would
look like a blackbody spectrum for 270 K, between the 260 K and 280 K spectra shown.
The atmospheric window is between about 900 - 1000 cm-1
, where no gases absorb or
emit infrared light. CO2, water vapor, ozone, and methane absorb infrared light emitted
from the ground, and emit lower-intensity infrared from high altitudes where the air is
colder than at the surface.
4. A comparison of the fate of infrared light in the optically thick CO2 bend frequency
(left) versus the optically thin atmospheric window (right).
5. A demonstration of band saturation by CO2. The addition of 10 ppm CO2 (upper
right) makes a huge difference to the outgoing infrared light spectrum relative to an
atmosphere that has no CO2 (upper left). Increasing CO2 to 100 and 1000 ppm (lower
panels) continues to affect the spectrum, but you get less bang for your CO2 buck as CO2
concentration gets higher.
6. Band saturation viewed in a different way from Figure 4-5. This is a plot of the total
energy flux carried by all infrared light, which is proportional to the area under the
spectrum curves in Figure 4-5. The outgoing energy flux is less sensitive to CO2 when
CO2 concentration is high.
7. The average temperature of the earth as a function of atmospheric CO2 concentration
and the climate sensitivity parameter, T2x.
8. A demonstration of the greenhouse effect of CO2. In the top panel, we begin with no
CO2. Let's assume that the energy budget of the earth was in balance at a ground
temperature of 270 K. In the middle panel, we add 1000 ppm CO2, decreasing the
outgoing energy flux. The ground and the atmosphere above it respond by warming up
8.5 K. The total outgoing energy flux is restored to its initial value. The total energy flux
is proportional to the area under the curves. CO2 takes a bite out of the top curve to
generate the middle curve, but then the bottom curve bulks up everywhere to compensate.
Chapter 4 12/18/05 8
Projects
Answer these questions using the on-line model at
http://forecast.uchicago.edu/Projects/radiation.html . The model takes CO2 concentration
and other environmental variables as input, and calculates the outgoing IR light spectrum
to space, similarly to Figures 3-3, 3-5, and 3-7. The total energy flux from all IR light is
listed as part of the model output, and was used to construct Figure 3-6.
1. Methane. Methane has a current concentration of 1.7 ppm in the atmosphere, and it's
doubling at a faster rate than is CO2.
a) Is ten additional ppm of methane in the atmosphere more or less important than ten
additional ppm of CO2 in the atmosphere at current concentrations?
b) Where in the spectrum does methane absorb? What concentration would it take to
begin to saturate the absorption in this band? (How do you identify saturation of a band,
on a spectrum plot?)
c) Would a doubling of methane have as great an impact on the heat balance as a
doubling of CO2?
d) What is the "equivalent CO2" of doubling atmospheric methane? That is to say, how
many ppm of CO2 would lead to the same change in outgoing IR radiation energy flux as
doubling methane? What is the ratio of ppm CO2 change to ppm methane change?
2. CO2.
a) Is the direct effect of increasing CO2 on the energy output at the top of the atmosphere
larger in high latitudes or in the tropics?
b) Set pCO2 to an absurdly high value of 10,000 ppm. You will see a spike in the CO2
absorption band. What temperature is this light coming from? Where in the atmosphere
do you think this comes from?
3. Earth Temperature. Our theory of climate presumes that an increase in the
temperature at ground level will lead to an increase in the outgoing IR energy flux at the
top of the atmosphere.
a) How much extra outgoing IR would you get by raising the temperature of the ground
by one degree? What effect does the ground temperature have on the shape of the
outgoing IR spectrum and why?
b) More water can evaporate into warm air than cool air. By setting the model to hold the
water vapor at constant relative humidity rather than constant vapor pressure (the default)
calculate again the change in outgoing IR energy flux that accompanies a 1 degree
temperature increase. Is it higher or lower? Does this make the earth more sensitive to
CO2 increases or less sensitive?
Chapter 4 12/18/05 9
c) Now see this effect in another way. Starting from a base case, record the total outgoing
IR flux. Now increase pCO2 by some significant amount, say 30 ppm. The IR flux goes
down. Now, using the constant vapor pressure of water option, increase the Temperature
Offset until you get the original IR flux back again. What is the change in T required?
Now repeat the calculation but at constant relative humidity. Does the increase in CO2
drive a bigger or smaller temperature change? This is the water vapor feedback.
O
H H
++
2Figure
4-1
Resting State
O
H H
O
H H
Symmetric Stretch Bend
3657 cm-1 1594 cm-1
CO O
Symmetric Stretch
Asymmetric Stretch Bend
Figure 4-2
CO O
C
O OCO O
Resting State
No Resting Dipole IR Inactive
3760 cm-1 1595 cm-1
Wavenumber cycles / cm
CO2
Ozone
H2O
Methane
H2O
Figure 4-3
CO2
CO2
CO2
CO2
900 cycles
cm
700 cycles
cm
Atmospheric WindowCO2 Bend
Temperature 270 K
Temperature 220 K
Figure 4-4
Wavenumber cycles / cm
Figure 4-5
0 ppm CO2 10 ppm CO2
100 ppm CO2 1000 ppm CO2
220
225
230
235
240
245
250
255
0 200 400 600 800 1000
Atmospheric CO2 Concentration, ppm
Figure 4-6
0
1
2
3
4
5
6
7
250 350 450 550 650 750
T2x = 4°C
3°C
2°C
pCO2
Figure 4-7
No CO2
Iout = 249 W/m2
1000 ppm CO2
Iout = 223 W/m2
1000 ppm CO2
8.5 K warmer
Iout = 249 W/m2
Figure 4-8
Wavenumber cycles / cm