PEEB6
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.
Radiative forcing and climate sensitivity
In PEEB5 we simulated the radiative forcing of CO2 and CH4 greenhouse gases
using MODTRAN. The figure below is a MODTRAN simulation of what would be
observed looking down from 70 km above Earth’s surface as an imaginary atmospheric
experiment is carried out. The blue emission curve is what would be seen when the
CO2 concentration is 280 ppm, i.e. preindustrial revolution. Then, without changing
anything else, including the atmospheric temperature profile, all other gas
concentrations, and the surface temperature, we imagine adding enough CO2 to double
its concentration to 560 ppm. The red emission curve is what would be observed from
this atmosphere with the doubled CO2 concentration.
At first glance, the overlaid red curve is seen to be a virtual duplicate of the blue
curve. Qualitatively, indeed there is little change in the shape of the CO2 absorption
band (centered at 667 cm-1); at such high CO2 concentrations, in this band the
atmosphere absorbs virtually all photons emitted from the surface. The band is said to
be saturated. Once band saturation is reached, the only photons detected from space
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originate from higher in the atmosphere where it is much colder, about 225K in this
simulation.
Although band-saturated in depth, the width of the CO2 band is however seen to
be slightly greater for the higher CO2 concentration. The band width originates primarily
from collisional broadening, involving energy transfer between vibrating molecules and
photons (PEEB5). The temperatures from which emission occurs in the collision
broadened wings are lower, hence higher in the atmosphere. As a result, there is
slightly less emission into space for the higher CO2 level. Less IR emission exacerbates
the energy imbalance due to more incoming than outgoing energy – hence the planet
warms further. CO2 concentration continues to be a positive radiative forcing on
temperature, even beyond band saturation.
For the particular atmospheric conditions chosen for this simulation, the doubled
CO2 concentration decreases the amount of IR emission energy by 2.86 W·m–2 (see
“Difference” in bottom of box in figure above). In other words, this imaginary
CO2 concentration doubling produces a radiative forcing of +2.86 W·m–2.
In PEEB5 we simulated the upward heat flux for a range of CO2 concentrations.
In the left-hand plot in the figure below, the difference in upward heat flux for 280 vs. 560
ppm corresponds to the radiative forcing of 2.86 W·m–2 just discussed. Band saturation
leads to a reduction in slope as the CO2 concentration increases, but the heat flux
continues to decrease due to collisional broadening. The trend fits a logarithmic
dependence. Although MODTRAN does not compute global warming, the model does
provide a way to simulate the warming effect. This is done by increasing the ground
temperature to restore the upward heat flux to its baseline (280 ppm) value. In other
words, as the greenhouse gas concentration increases, the upward heat flux escaping to
space decreases; in the absence of any change in Earth’s albedo, this can only be
countered by increasing the ground temperature. In the right-hand plot, the simulated
warming for a doubling of CO2 from 280 to 560 ppm is 0.9 K.
In PEEB5 we also simulated CH4, shown in the next figure. Here, CO2 is kept
fixed at its present 400 ppm level. Increasing CH4 from its current level of 1.8 ppm to
560 ppm results in a simulated warming of 4.4 K. For CH4, which at its present level is
not nearly band-saturated, we are currently on a much steeper part of the curve. As a
consequence, compared to CO2, CH4 is a much more potent greenhouse gas, i.e. for the
current atmospheric composition a small change in CH4 concentration will have a larger
effect than the same change in CO2 concentration.
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Ground Temp vs. CH4
Content
305
295
285
y = -2.437ln(x) + 279.34
275
265
255
0
500
1000
1500
CH4 concentra on (ppm)
2000
Ground temperature (K)
Upward heat flu (W/m2)
Heat Flux vs. CH4 Content
302
298
294
290
286
0
500
1000
1500
2000
CH4 concentra on (ppm)
Over the total range 0 – 2000 ppm, however, CO2 is seen to be a much stronger
greenhouse gas, i.e. the simulated reduction of upward heat flux is 34 W/m2 for CO2 vs.
20 W/m2 for CH4.
The consequence of a change in radiative forcing is illustrated in this figure, a
simple graphical representation of the forcing mechanism for planetary warming
associated with an increased concentration of atmospheric CO2.
Schematic showing atmospheric temperature decreasing with altitude at a
roughly linear rate, known as the lapse rate (sloping line, left). The
squiggly arrow represents the emission radiated to space and the altitude
from which it originates. If the CO2 concentration is suddenly increased,
an energy imbalance results due to less energy being radiated to space
(shorter squiggly arrow, middle). To restore balance, the lapse rate
remains constant but moves to a higher altitude , resulting in a warmer
atmosphere (longer squiggly arrow, right) and surface.
The decreasing concentration of CO2 with altitude is represented by the
decreasing intensity of color in the diagrams and the lines on the plots show the lapse
rate of atmospheric temperature. The initial (left) state represents the atmospheric
temperature and CO2 concentration profiles when the planet is in energy balance
between incoming solar radiation and outgoing thermal IR emission, represented by the
squiggly arrow.
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The middle diagram represents the situation after adding sufficient CO2 to double
its concentration throughout the atmosphere, but before the temperature or any other
variable has changed. That is, the lapse rate remains the same, but the top of the
CO2 column, from which emission to space occurs, is at a higher, colder altitude. Since
the emission occurs from colder layers, the amount of energy emitted is decreased and
the planet is no longer in energy balance. The amount of incoming solar radiation is
larger than the amount of outgoing thermal IR emission. This is a positive forcing and
the extra energy begins to warm the planet and its atmosphere.
A new energy balance is reached (right) when the warming has brought the
column of CO2 to a higher temperature, where its emission once again balances the
incoming solar radiation. The lapse rate at this new energy balance is the same as it
was initially, but moved to higher temperatures in each layer. The result of the
CO2 forcing at the top of the atmosphere is a warmer temperature at the surface. It is
the emission from the top of the atmosphere that controls the surface temperature.
When other variables are held constant, emission from the atmosphere is found
to be inversely proportional to the logarithm of the concentration of CO2, as we found
previously in our MODTRAN analysis. Our MODTRAN analysis, however, did not
include water vapor and clouds, and was constrained to mid latitudes. Using more
realistic atmospheric parameters, the following result is obtained:
Needs reference
How is slope = 5.35 calculated?
Why is slope different from MODTRAN slope?
Compare with MODTRAN for other atmospheres
Much of the contemporary interest in radiative forcing is concerned with the
effects of increasing atmospheric concentrations of greenhouse gases. The baselines
for comparison of the increases are usually taken as the concentrations in 1750, at the
beginning of the Industrial Revolution. For CO2, the baseline value, C0, is 278 ppm. The
slope of the line in the figure then gives the radiative forcing for C, another CO 2
concentration, as
radiative forcing for C ppm CO2, W·m–2 = (5.35 W·m–2) ln(C/C0).
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Using this relation, the radiative forcing for the 2016 CO2 concentration, 400 ppm, is 1.95
W·m–2. Similar analyses can be made for the radiative forcing by other greenhouse
gases and are included in the IPCC reports, summarized below. For the 2007 data
shown there, the CO2 forcing is about 1.7 W·m–2, slightly less than the 2016 value. In
2007, the total radiative forcing (CO2 + CH4 + N2O + O3 + stratospheric H2O) was about
3.1 W·m–2.
IPCC Fourth Assessment Report (2007), Chapter 2, Changes in Atmospheric
Constituents and in Radiative Forcing, Figure 2, FAQ 2.1
The complete story is, however, not just one of radiative forcing by greenhouse
gases. Other radiative forcings, albedo changes, and feedbacks, especially from
increasing water vapor, also occur. The effects of both positive and negative feedback
factors have to be accounted for in determining the climate sensitivity associated with an
increase in atmospheric CO2. Attaining the new energy balance involves some
processes that are relatively rapid, taking place over decades, and others that are
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slower, taking centuries, millennia, or longer to reach a balance. While the radiative
forcing induced temperature change due to greenhouse gasses is in itself significant,
physical and biological processes triggered by greenhouse gas warming have an even
stronger effect on temperature.
The concept of “climate sensitivity” is deceptively simple. How much would the
average surface temperature of the Earth increase (decrease) for a given positive
(negative) radiative forcing? The simplest approach to estimating climate sensitivity is to
combine the energy balance for the incoming and outgoing energies and a
simple atmospheric model to calculate how to counterbalance a given radiative forcing.
If ΔF is the difference between incoming and outgoing energy flux (the equivalent
of radiative forcing), we have
ΔF = (1 – α)Save – εσTP4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
where α is the Earth’s albedo, Save is the average solar energy flux, 342 W·m–2, ε is the
effective emissivity of the planetary system, σ is the Stefan-Boltzmann constant, and
TP is the average planetary surface temperature. If ΔF is zero, the energies are
balanced:
(1 – α)Save = εσTP4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …. . . . . (2)
In the absence of any greenhouse effect, ε would be unity and TP would be 255 K.
Greenhouse gases in the atmosphere produce a lower effective emissivity that results in
an increase of TP to about 288 K to maintain energy balance (PEEB2).
For ΔF > 0, a positive radiative forcing, energy incoming energy is higher than
that outgoing. The surface temperature must increase by ΔT to counterbalance this
forcing. The required counterbalance, assuming no changes in other factors affecting
the climate, is represented by:
ΔF = εσ[TP + ΔT]4 – (1 – α)Save. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . (3)
An algebraic manipulation* (Algebra2/Precalc exercise) gives the relationship between
radiative forcing and the counterbalancing temperature change that would be required to
return the planet to energy balance:
ΔT ≈ Tp ΔF/[4(1 – α)Save]
≈ [0.3 K·(W·m–2)–1]ΔF
(for Tp ≈ 288 Κ). . . . (4)
To apply this approximation for climate sensitivity due to CO2 and CH4, we can examine
a case for which the change in concentration of greenhouse gases is reasonably well
known and whose temperature change from an initial constant temperature to a higher
constant temperature is also known. The figure below shows Antarctic ice core data that
spans the time from the end of the last glacial period to the beginning of the present
interglacial era. For our purposes, we need the initial and final concentrations of
CO2 and CH4, and the average global temperature change. For this test, we assume,
that radiative forcing by these gases is the only external forcing on the climate system.
(The detailed time course of the changes is interesting and can be correlated with
changes that are evident in other geological records from this time span, but are not
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relevant for our calculation.) Correlation does not prove causality – i.e. did greenhouse
gas increase cause temperature rise or did temperature rise cause greenhouse gas
increase?
The figure is based on a figure from the NOAA Paleoclimatology
Program website. The original reference is Eric Monnin, Andreas
Indermühle, André Dällenbach, Jacqueline Flückiger, Bernhard
Stauffer, Thomas F. Stocker, Dominique Raynaud, Jean-Marc
Barnola, “Atmospheric CO2 Concentrations Over the Last Glacial
Termination,” Science, 2001, 291, 112-114.
The increase in CO2 from about 185 to about 265 ppm gives a radiative forcing of
FCO2 = (5.35 W·m–2) ln(265/185) = 1.9 W·m–2
The radiative forcing for CH4 is determined in a way analogous to that for CO2. For the
increase of CH4 from about 375 to about 675 ppb, ΔFCH4 ≈ 0.3 W·m–2. Thus, the total
radiative forcing, ΔF, due to these two greenhouse gases is about 2.2 W·m–2. The
predicted change in the average planetary surface temperature is
ΔT ≈ [0.3 K·(W·m–2)–1] (2.2 W·m–2) ≈ 0.7 K
Analyses from multiple sites based on several different temperature proxies indicate that
Earth’s average surface temperature increased between 3 and 4 K during the change
from the last glacial period to the present era.
Our calculated temperature change, that includes only the radiative forcing from
increases in greenhouse gas concentrations, accounts for only 20 - 25% of this
observed temperature increase. Our analysis based only on greenhouse gas forcing
has not accounted for feedbacks in the planetary system triggered by increasing
temperature, including changes in the structure of the atmosphere. The implied climate
sensitivity factor, based on analysis of Antarctic ice core data, is perhaps four to five
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times greater, ∼1.3 K·(W·m–2)–1, than that obtained by simply balancing the radiative
forcing of the greenhouse gases.
For the first calculation of atmospheric warming by increased CO2, Arrhenius
(who in 1896 is acknowledged to be the first to ask, “Is the mean temperature of the
ground in any way influenced by the presence of heat- absorbing gases in the
atmosphere?”) chose to consider a doubling of its concentration and climate science has
stuck with this standard. Thus, most values for climate sensitivity are given today as the
temperature change predicted for doubling the CO2 concentration, ΔT2xCO2, or the
equivalent of its doubling, taking all the greenhouse gas radiative forcing into account.
The IPCC’s analysis gives a very likely (> 90% probability) value of 3 K with a likely (>
66% probability) range from 2 to 4.5 K. Our radiative forcing for doubling CO2 from 280
to 560 ppm is 3.53 W·m–2 [= (5.35 W·m–2) ln(2)], which gives ΔT2xCO2 = 4.6 K (= [1.3
K·(W·m–2)–1][ 3.53 W·m–2]). Although on the high side, this first level approximation is
not wildly amiss and provides some insight into the factors that affect climate sensitivity.
Water Vapor and Clouds
One of the most important sources of feedback in the planetary system, shown
graphically below, is the increase in the vapor pressure of water as the ocean’s
temperature increases. The vapor pressure increases by about 7% per degree
Kelvin. Warming oceans evaporate more water and a warmer atmosphere can
accommodate more water vapor, the most important greenhouse gas. This feedback
amplifies the warming effect of the non-condensable greenhouse gases and is
responsible for a good part of the multiplier effect on climate sensitivity just noted.
An increase in atmospheric water vapor also affects cloud formation. The effect
of clouds on the energy balance between incoming solar radiation and outgoing thermal
IR radiation depends on the kinds of clouds and can result in either positive or negative
feedback for planetary warming. Clouds are composed of tiny water droplets or ice
crystals, which makes them very good black bodies for absorption and re-emission of
thermal IR radiation. Unlike greenhouse gases that absorb and emit only at discrete
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wavelengths, clouds absorb and emit like black bodies throughout the thermal IR. The
higher the top of the cloud, the lower the temperature from which emission takes place
and the lower the energy emitted. Thus, the higher the cloud, the greater its positive
feedback effect on planetary warming. Thin, wispy cirrus clouds very high in the
troposphere near the stratosphere have the strongest warming effect while low-lying
layers of stratus clouds have a weaker warming effect.
Cumulous and stratus clouds in the lower troposphere are opaque – we can’t see
through them. The tiny water droplets or ice crystals in these clouds scatter visible light
in all directions, including back into space, so they reduce the amount of solar energy
that reaches the surface. That is, they increase the Earth’s albedo and therefore have a
negative feedback effect on planetary warming. The very high cirrus clouds contain very
little water (as ice) and are not opaque – we can see the sky through them. They do not
scatter very much solar radiation and have only a weak negative feedback effect.
Since clouds have both positive and negative feedback effects, which
predominates and how will changing global temperature affect this balance? These are
very uncertain aspects of climate science. Increasing sea surface temperature
intensifies the evaporation-condensation-precipitation water cycle, but the factors that
control where clouds form, what kinds are formed, and how increased temperature and
atmospheric water vapor affect their formation are complex. Computer modeling of the
turbulence, condensation, and growth of water droplets in cloud formation requires large
amounts of computer time and capacity. At present, even the fastest computers, running
general circulation models (GCM) of the climate, require so much computer capacity that
the models cannot incorporate the further complexity of cloud formation. Thus, the
GCMs incorporate algorithms that relate cloud formation to other parameters, such as
relative humidity, to estimate their formation and effects. These lead to great variation in
the model predictions, depending on the algorithms and parameters used, but generally
suggest that, as the planet warms, clouds will be a positive feedback, although perhaps
relatively weak.
Aerosol Radiative Forcing
Aerosol particulate matter, tiny particles or liquid droplets suspended in the
atmosphere, generally scatter and absorb incoming solar radiation, thus contributing to
the Earth’s albedo. Naturally occurring aerosol particles are mainly picked up by the
wind as dust and water spray or produced by occasional volcanic eruptions. Poor landuse practices by humans can make dust storms worse and intensify the natural effects.
Human activities do, however, add significantly to aerosol sulfate particles as well as
producing black carbon (soot) particles. Fossil fuels often contain sulfur which is
oxidized to SO2 in the atmosphere, ultimately forming hygroscopic sulfuric acid
molecules and salts that act as nucleation sites for tiny water droplets. These aerosol
sulfate particles tend to be quite small, so, for a given amount of emission, the number of
particles is large and scatters a good deal of solar radiation. This scattering increases
the albedo and produces negative radiative forcing by reducing the amount of solar
radiation reaching the surface.
The atmospheric models described here have been one-dimensional. They have
focused on the properties of an atmospheric column only as a function of altitude. Two-
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dimensional models can aggregate the properties of one-dimensional models over many
different locations to get a more realistic average over the planet. However, a twodimensional model still lacks an essential characteristic of the climate system –
continuous exchange of matter and energy among the one-dimensional atmospheric
cells. Atmospheric circulation, the winds, must be accounted for to make any attempt to
capture the observable characteristics of climate changes – past, present, or future.
Similarly, the vast ocean currents, albeit much slower than atmospheric circulation, must
be accounted for over long modeling periods. Developing, testing, refining, and
comparing general circulation models of the climate are vitally important to further our
understanding of the climate and make predictions of its future more reliable
Secondarily, the high concentration of these tiny aerosol sulfate particles leads to
the formation of clouds with high concentrations of tiny water droplets that scatter more
solar radiation than a lower concentration of larger droplets. This change in the
composition of clouds also increases the albedo and produces further negative radiative
forcing, sometimes called the indirect aerosol effect. Also, these clouds are more stable
against formation of precipitation, so can have a longer lifetime to reflect sunlight. The
uncertainties surrounding the modeling of both the direct and indirect effects of aerosol
particulate matter, especially those involving cloud formation, are large and add further
to the uncertainty in predicting the climate sensitivity resulting from human activities.
This uncertainty is captured in the error bars associated with aerosols in the IPCC
graphic above, and is also the impetus for increased research to better understand
aerosols and clouds.
* Solve for ΔT (see p6 above):
ΔF = εσ[TP + ΔT]4 – (1 – α)Save. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . (3)
ΔF = εσ[TP (1 + ΔT/ TP)]4 – (1 – α)Save
ΔF = εσTP4(1 + ΔT/ TP)4 – (1 – α)Save
Substituting εσTP4 = (1 – α)Save, gives
ΔF = [(1 – α)Save ](1 + ΔT/ TP)4 – (1 – α)Save
The factor (1 + ΔT/ TP)4 can be expanded and approximated:
(1 + ΔT/ TP)4 = 1 + 4(ΔT/ TP) + 6(ΔT/ TP)2+ 4(ΔT/ TP)3 + (ΔT/ TP)4
(1 + ΔT/ TP)4 ≈ 1 + 4(ΔT/ TP)
Because ΔT is small, ΔT/ TP << 1, and the higher order terms in the expansion
are negligible. Substituting in the expression for ΔF, gives
ΔF ≈ [(1 – α)Save ] [1 + 4(ΔT/ TP)] – (1 – α)Save
ΔF ≈ (1 – α)Save + 4(1 – α)Save (ΔT/ TP) – (1 – α)Save
ΔF ≈ 4(1 – α)Save (ΔT/ TP)]
Solving for ΔT, gives the climate sensitivity based on this simple approach.
ΔT ≈ Tp ΔF/[4(1 – α)Save]
ΔT ≈ Tp ΔF/[4(1 – α)Save] ≈ [0.3 K·(W·m–2)–1] ΔF (for Tp ≈ 288 Κ). . . . (4)
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PEEB6 – a few sentences per answer
1. Explain qualitatively the effect of clouds on surface temperature during the day and
during the night.
2. Following 9-11, the nation’s major airports were temporarily closed. Airports
continued to monitor the local weather, however, and it was observed that the diurnal
temperature range, (daily high) – (daily low), increased. Why? Hint: consider contrails.
3. What are some differences between the troposphere and the stratosphere? Why is it
that the temperature decreases (increases) with altitude in the troposphere
(stratosphere)? Why does this lead to a natural stratification of the atmosphere (hence
the name, stratosphere)?
4. What is the name for the boundary between the troposphere and the stratosphere?
What is the approximate altitude of this boundary at the equator? At the poles?
Qualitatively rank the atmospheric residence time for an aerosol particle (e.g., sulfate,
carbon soot, aluminum oxide) injected from (i) a smokestack, (ii) a jetliner, or (iii) a
volcano capable of injecting into the stratosphere.
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