PEEB5
Project Earth Energy Balance
Purpose: To develop a quantitative understanding of the temperature of the Earth, the
warming effect of the atmosphere, the anthropogenic impact on the atmosphere and
Earth-Sun energy balance, and how Earth’s temperature can be controlled.
A Multilayer Model of the Atmosphere
In PEEB4 we calculated the surface temperature of Earth as a function of atmospheric
emissivity (= absorptivity) for the single-layer atmosphere model:
Tp = {[2 (1 – α)Save]/ [σ(2 – ε)]}1/4, where α is albedo and ε is emissivity:
If the emissivity ε (and absorptivity) are zero, no radiation from the surface is absorbed.
This is identical to the energy balance for Earth acting as a black body in the absence of an
atmosphere, for which the planetary temperature is calculated to be 255°K (-18°C), the starting
point in the graph above. If the emissivity (and absorptivity) is unity, the atmosphere is a black
body and all radiation from the surface is absorbed. For this model, the graph shows that the
surface temperature is 303°K (30°C), while the temperature at the top of the atmospheric is
Ta = (1/2)1/4 Tp = 0.841 × (303°K) = 255°K,
exactly the same as the surface temperature in the absence of an atmosphere. These
conclusions should come as no surprise, since the limiting conditions are built into the model.
That is, the atmosphere emits outward into space as a black body at 255 K, just as it must to
balance the incoming solar energy absorbed by the planet.
The actual average temperature of the Earth’s surface is about 288°K. For the singlelayer atmosphere model, the graph shows that this temperature corresponds to an atmospheric
emissivity of about 0.77. Thus, the model succeeds in its main purpose, demonstrating how an
atmosphere that absorbs and re-emits some of the radiation from a planet’s surface results in a
surface that is warmer than if there were no atmosphere.
The single-layer atmosphere model captures the essence of the physical processes
governing atmospheric warming, but it is not adequate for use as a quantitative predictive tool.
Note that the energy emitted into space comes from two sources. If the emissivity is 0.8, part of
the energy, (1 – ε)σTp4, comes from the planetary surface at about 288°K; the rest of the
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emission, εσTa4, comes from the top of the atmosphere, which is at a temperature T a = 0.841 ×
(288°K) = 242°K. An observer (or satellite) outside the atmosphere will see an emission
spectrum that is the sum of the emissions from these two sources:
Assuming that emission from each source has the emission profile of a black body (true
for this model), the observed spectrum can be deconvoluted (broken into its components) to
give the temperatures and relative contributions of the two sources. The model demonstrates
that we can infer properties of both the surface and the atmosphere from observations of their
emissions from outside the atmosphere.
Building on this, we can start to account for the observations that the atmosphere is not
isothermal, and that the emission from Earth into space does not fit a simple grey body model,
shown by the experimental radiance data in the figure below. A multilayer atmosphere model
can qualitatively explain these observations.
A physical consequence of the warmed surface of Earth is warming of the atmosphere
near the surface by conduction. As it warms, the gas expands, becomes less dense, and is
buoyed up from the surface by convection. As the gas rises, it expands further due to
decreasing atmospheric pressure with altitude, cooling as it rises higher into the atmosphere.
The cooling expansion can cause water vapor to condense at higher altitudes, forming clouds of
tiny water droplets. Condensation releases the latent energy that was required to vaporize the
water at the surface and adds energy to the atmosphere. These conduction, convection,
evaporation, expansion and condensation processes produce movement of gases in the lowest
layer of the atmosphere that is responsible for winds and weather and give rise to its name,
troposphere, from the Greek tropos, mixing or turning.
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Theoretical (dashed) and experimentally measured (solid) radiance as a function of
wavenumber, showing absorption bands for greenhouse gasses H2O, CO2, O3, and CH4.
The Nimbus 4 satellite orbiting about 1000 km above the surface of the Earth obtained
this experimental data on the cloudless early afternoon of May 5, 1970, when the
satellite was over the Niger valley in northern Africa. Many other weather satellites have
obtained similar data over the past four decades. The dashed theoretical curves
represent the emissions that would be expected for black bodies at the temperatures
indicated.
Note: The relation between wavelength λ and frequency ν for electromagnetic
radiation is λν = c, where c is the speed of light. Frequency ν = c/λ, the number of cycles
per unit time, is measured in Hz with units of sec-1. Often in the physical sciences, one
uses the wavenumber ũ = 1/λ = ν/c, the number of cycles per unit distance in units of
cm-1. For example, the CO2 absorption band at 15 μm corresponds to a frequency ν = 2
x 1013 Hz (20 THz) and a wavenumber û = 667 cm-1. In this scheme, wavelength
increases to the left, unlike our original (PEEB2) description of Planck’s Law where
wavelength increases to the right. This can be confusing at first, but both descriptions
represent the same relation - Planck’s Law.
These processes also produce the temperature profile for the troposphere - temperature
decreasing with altitude - illustrated in the next figure. The observed change of temperature with
altitude is called the lapse rate. For example, the top of the troposphere is at an altitude of
about 15 km in the figure and the atmospheric temperature at that altitude is colder than at the
surface by about 115 K (or °C), so the lapse rate is (115 K)/(14 km) ≈ 8.2 K·km–1. Based on this
lapse rate, as a modern jetliner ascends to a cruising altitude of 30,000 feet (9.1 km), a
temperature drop of 75C (135F) is predicted.
At some altitude, the convection processes driven largely by energy transported from the
surface finally reach a limit and the atmospheric temperature stops dropping. This limit, the top
of the troposphere, is higher in the warm-surface tropics, up to about 20 km, than over coldsurface polar regions, about 7 km. If there were no sources of energy above the limiting
altitude, the atmospheric temperature would remain roughly constant or drop slowly with
increasing altitude.
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However, in Earth’s oxygen-containing atmosphere, ultraviolet (UV) radiation from the
sun drives photochemical reactions whose net result is to add energy to the atmosphere above
the troposphere. At these high altitudes, ozone, O3, is formed and decomposed by high and
medium energy UV radiation, respectively. These are the processes that protect life on the
planet by absorbing damaging UV wavelengths and at the same time warming the surrounding
atmosphere, so that temperature increases with altitude, as shown in the figure. Because
warmer, less dense gas is layered over cooler, denser gas, there is little vertical mixing and this
part of the atmosphere is stratified - hence the name stratosphere.
An improved model treats the troposphere as a stack of layers of decreasing
temperature and pressure. Within each layer the temperature and pressure are constant and
the gases are assumed to be in thermal equilibrium and in equilibrium with respect to absorption
and emission of thermal infrared radiation, just as in the single-layer model. Furthermore, the
presence of trace amounts of IR-absorbing greenhouse gases introduces wavelength regions
with higher atmospheric emissivities and absorptivities, so the grey-body assumption of
constant emissivity is not valid. Instead, thermal IR emissivity is not treated as constant but is
governed by the wavelength and temperature dependent Planck black-body function for each
layer.
In the three-layer atmosphere model illustrated here, squiggly arrows represent
radiation emitted by the body from which the arrow originates. Straight-line arrows
represent radiation transmitted without absorption through a layer. The temperaturedependent Planck black-body function for this wavelength is represented by B with a
subscript denoting the temperature of the layer to which it applies. The emissivity of
each layer is similarly labeled. The temperatures are TP at the surface and T1 > T2 >
T3 for the temperatures of the layers. Decreasing pressure is represented by the
decreasing depth of color of the layers.
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To interpret this model, begin with the radiation from the Earth’s surface at the bottom
left of the diagram. Since the surface emits like a black body, the energy emitted at the
wavelength in question is the emission for this wavelength from a black body at the temperature
of the surface, BP. When this radiation interacts with IR-absorbing molecules in layer 1, some of
it is absorbed, ε1BP, and the remainder, (1 – ε1)BP, passes through, as shown by the straight
arrow emerging from the layer. The absorbed energy is not labeled in the diagram, because the
focus of the model is on the radiation that has to leave the atmosphere in order to maintain (or
try to maintain) the energy balance of the planet.
Radiation from the surface and each layer is, of course, emitted in all directions; in this
one-dimensional model, emission is shown only up and down relative to the Earth’s surface.
When the radiation transmitted through layer 1 interacts with the IR-absorbing molecules in
layer 2, some of it is absorbed, ε2(1 – ε1)BP (not labeled), and the remainder, (1 – ε2)(1 – ε1)BP,
passes through. This process is repeated in layer 3, so the atmospheric layers have attenuated
energy that started from the surface and only (1 – ε3)(1 – ε2)(1 – ε1)BP leaves the top layer.
Similarly, radiation emitted upwards from any lower layer leaves the atmosphere from the top
layer after attenuation by the intervening layers. From the point of view of an observer outside
the atmosphere, all the radiation at this wavelength is seen as coming from the topmost, coldest
atmospheric layer. The absorptivity (and therefore the emissivity) of the layers decreases with
altitude for several reasons, the most obvious being the decrease in numbers of IR-absorbing
molecules as the density of the atmosphere decreases. Emissivity is also temperature
dependent, decreasing with decreasing temperature.
We can apply this (still rudimentary) three-layer model to observations of the Earth’s
actual atmosphere. For IR-absorbing gases in the Earth’s atmosphere, we have to account for
absorptions of different strengths and for condensable and non-condensable gases. For
absorption strengths, there are, at one extreme, wavelengths where none of the atmospheric
gases absorb IR radiation, that is, where all the emissivities in the diagram above are zero. In
this case, the radiation leaving the atmosphere all comes from the surface and is characterized
by its temperature, TP. For wavelengths where atmospheric gases absorb, but relatively
weakly, emissivities in the higher, colder layers are zero and the radiation leaving the
atmosphere comes from a layer of intermediate temperature. In the diagram, this might
correspond to a wavelength for which ε3 is zero, so the radiation observed outside the
atmosphere would be coming from the layer with the temperature T2.
For a condensable gas, most notably water vapor, there are very few molecules in the
higher, colder layers of the atmosphere. Thus, emissivities at the wavelengths where water
vapor absorbs are zero in the higher atmospheric layers. Radiation from water vapor comes
from lower, warmer layers where the partial pressures and emissivities of water vapor are high
enough to contribute.
Now we can use concepts derived from the multilayer model to interpret the emission
spectrum of the Earth observed from space (previous graph of radiance vs. wavenumber). The
dashed-line, black body emission curves are a temperature scale included to show the intensity
of the emission that would be produced by black bodies of the designated temperatures. These
temperatures, from top to bottom, correspond to the temperatures of the surface and
atmospheric layers of increasing altitude and, hence, decreasing temperature.
The CO2 molecule has four main groups of absorption features in the thermal IR, of
which the most important for this discussion is the one with wavenumber 667 cm-1 (wavelength
15μm). This feature arises from the vibrational bending modes of the linear triatomic molecule,
which are given a fine structure by mixing with rotational transitions. Water vapor is a polar
molecule and its richer set of vibrational and rotational modes allows it to absorb effectively over
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a much broader range of frequencies than CO2. All of these spectral features acquire a finite
width by virtue of a number of processes that allow a molecule to absorb a photon even if the
energy is slightly detuned from that of an exact transition. For reasonably dense atmospheres,
the most important of these processes is collisional broadening, which borrows kinetic energy
from collisions to make up the difference between the absorbed photon’s energy and a
transition. Any collision can cause this perturbation, so the amount of the line broadening is a
function of the total pressure or density of the gas, including both IR-absorbing and nonabsorbing molecules. Thus, line broadening for IR-absorbing gases is larger at lower altitudes
and decreases with altitude. A wavelength that is absorbed relatively strongly in the broadened
wavelength range at lower altitudes will absorb more weakly at higher altitudes where the
broadening is less. Thus the absorptivity and emissivity at this wavelength will decrease with
altitude.
From the graph of radiance vs. wavenumber, we observe that the CO2 emission band
around 15 μm (see scale on top edge of graph) comes from a layer at about 220K, near the top
of the troposphere. The emissivities (and absorptivities) corresponding to these wavelengths
are so large that this radiation escapes into space only from extremely high altitude, where the
CO2 partial pressure has dropped low enough to no longer absorb these wavelengths.
Water vapor has strong absorptivities and emissivities for wavelengths at both ends of
the thermal IR, but its emission comes only from the relatively warmer atmospheric layers
nearer the surface. Higher layers are too cold to sustain adequate partial pressure of water
vapor to absorb the emissions from the lower, warmer layers. Thus, the emissions come from
layers not too much below the freezing point (273K) of water.
The graphic below is a visual reminder of the different distributions of CO2 and H2O(g) in
the troposphere. The squiggly arrows show the emission observed from each gas and the
altitude from which it originates. In the atmosphere, these gases are mixed of course, but, for
clarity, are shown separately here. In order for Earth’s temperature to remain constant, the
outgoing energy radiated by the planet and its atmosphere must equal the incoming energy
from the sun that is absorbed by the planet. To balance the present incoming energy, the planet
must radiate to space an amount of energy equivalent to the emission from a black body at 255
K. Since the surface of the planet is, on the average, at about 288 K, it emits too much energy
to maintain the energy balance. The observed emission spectrum shows that the IR-absorbing
gases in the atmosphere reduce the temperatures at which emissions to space occur. This
lowers the effective emission temperature of the planet and maintains its energy balance, so
long as the atmospheric composition, surface features, and energy from the sun do not change.
With the atmospheric composition now changing so rapidly, it behooves us to attempt to
quantify the resultant radiative forcing, feedback, and climate sensitivity consequences.
Radiative forcing by a climate variable is a change in Earth’s energy balance between
incoming solar radiation energy and outgoing thermal IR emission energy when the variable is
changed while all other factors are held constant. The best understood radiative forcings are
due to variations in solar energy input and changes in greenhouse gas atmospheric
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concentrations, because their radiative forcings can be computed using experimentally verified
concepts.
MODTRAN (MODerate resolution atmospheric TRANsmission) is a computer program
designed to model atmospheric propagation of electromagnetic radiation in the 100-50,000 cm-1
(0.2 to 100 um) spectral range. This covers the spectrum from middle ultraviolet to visible light
to far infrared. Developed and patented by Spectral Sciences Inc. under the auspices of the US
Air Force since 1987, MODTRAN is written entirely in FORTRAN. One result obtained using
MODRAN is shown here:
Comparison of experimental (satellite, 1970, jagged black) and theoretical (MODTRAN,
jagged red) spectra of infrared light detected at the top of the atmosphere looking down.
The smooth curves are theoretical emission spectra of blackbodies at different
temperatures. The model demonstrates the effect of wavelength-selective greenhouse
gases on Earth's outgoing IR energy flux.
MODTRAN has been continuously refined and currently is administered by the
University of Chicago as part of its course on climate science. In the public domain, a webbased version can be found at :
http://climatemodels.uchicago.edu/modtran/
Once in MODTRAN, enter the default values shown in the control panel below, for which the
simulated Upward IR Heat Flux, Iout, should be 278.769 W/m2, and the Ground Temperature
should be 294.2 K:
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MODTRAN Control Panel
Note that MODTRAN does not compute global warming, that is, it doesn't change
Earth's temperature in response to changes in the atmosphere. However, the model does
provide a way to simulate the ground temperature. This is done using the Ground Temperature
offset, which by trial and error can be adjusted to restore the upward heat flux to its baseline
value. In other words, as the greenhouse gas concentration increases, the upward heat flux
escaping to space decreases, but energy balance can be restored by increasing the ground
temperature.
For PEEB5, use MODTRAN to explore these questions:
1. Demonstrate the band saturation effect for CO2 and CH4 by making plots of Iout as a function
of greenhouse gas concentration, say from 0 to 2000 ppm, with more points at low
concentrations where Iout is varying most rapidly. For the CO2 plot, fix the CH4 concentration at
1.8 ppm. For the CH4 plot, fix the CO2 concentration at 400 ppm.
2. Compare CO2 vs. CH4 as greenhouse gases. For the same concentration, which is
stronger? Which is stronger given their current concentrations?
3. Simulate the temperature response to CO2 and CH4 greenhouse gas IR forcing by adjusting
the ground temperature to maintain a constant Iout as the gas concentration is increased. Using
the pre-industrial CO2 level of 280 ppm as a baseline, what is the simulated temperature rise for
the current CO2 level of 400 ppm? 560 ppm (twice the pre-industrial level)?
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