PEEB3
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.
Can Diet Alone Reduce Our CO2 Weight Gain?
In PEEB2 we calculated the equilibrium temperature of several planets based on a
Stefan-Boltzmann zero net power transfer: Power In (incoming photon flux from the Sun,
peaked in the visible) = Power Out (outgoing photon flux from the planet, peaked in the
infrared). Our results, in the table below, were compared to experimentally observed average
surface temperatures for several planets, derived from astronomical and planetary probe data
and direct thermometric data for Earth. As you found, the agreement between observation and
prediction is good for Mercury and Mars, not so good for Earth, and spectacularly bad for
Venus. Earth and Venus are both warmer than predicted by simple black body radiation
physics. (I have been unable to find an average measured surface temperature for the Moon - if
you find one, let me know.).
The rocky, inner planets of our solar system vary in sizes, atmospheres, and
temperatures. Mercury, the smallest and closest to the sun, has no atmosphere and extremes
of temperature that average close to that predicted by our simple black body model. Mars, the
next largest and farthest from the sun, has a tenuous atmosphere, less than 1% the density of
Earth’s, and an average surface temperature just a bit above that predicted by our simple black
body model. Venus is closest in size to Earth, but has an atmosphere nearly 100 times denser.
Venus is continuously shrouded in clouds that result in a very high albedo and make it
impossible to observe its surface at visible wavelengths from outside the atmosphere. The table
below provides a comparison of your predicted and the observed surface temperatures of the
planets, along with some essential attributes of their atmospheres.
The table provides evidence that an atmosphere has a pronounced effect on the
temperature at the planet surface, causing it to be warmer than predicted by the simple black
body model. Venus, with a thick atmosphere, has a surface temperature about 500 °K above
the prediction, despite the fact that 75% of the incoming photon flux is reflected back into space
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by that very same atmosphere. Earth, with a thinner atmosphere, has a mild 32 °K warming.
However, this warming of Earth above the freezing point of water (273 °K) has profound
consequences. Life, as we know it, would not be possible on a planet where the water is
permanently frozen - a snowball Earth instead of a “blue marble.” In short, planetary
atmospheric warming results from the one-way-street effect that certain gasses impose on the
photon traffic: incoming visible photons are transmitted while outgoing infrared photons are
absorbed.
It is important to note that the poor agreement between predicted and observed surface
temperatures is not the fault of our simple black body model. To be sure, there is nothing wrong
with our physics. Rather, it is our application of the model which is flawed. The model in fact
correctly predicts the planetary temperature as sensed from outer space, from which vantage
point only those photons which escape Earth can be sensed. For the case with an atmosphere,
the key point to realize is that some or most of the photons escaping into space originate not
from sea level or land, but rather from a gaseous “surface” higher in the atmosphere. On Earth,
this occurs at an altitude far above sea level, where the temperature is much colder. One need
only view a snow-capped mountain for evidence that temperature drops with increasing altitude.
It is the temperature at the top of the atmosphere, the temperature as sensed from outer space,
which is predicted (and correctly so) by our simple black body model.
Decreasing temperature with increasing altitude is a signature characteristic of the
greenhouse effect. As we will see later, the rate of change of temperature vs. altitude is a
constant, referred to as the lapse rate. With increasing greenhouse gas concentration, the
altitude from which photons can escape to space moves to higher altitude, which, given the
constant lapse rate, results in a higher surface temperature (but no change in the black body
temperature). This is the mechanism of the atmospheric greenhouse effect: for the same
planetary albedo and the same blackbody temperature at the point from which photons can
escape to space, the temperature at the bottom of the atmosphere is a sensitive function of the
greenhouse gas concentration.
This figure shows how the energy budget is balanced for Earth when the atmospheric
warming effect due to greenhouse gases is included.
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Figure from Trenberth, K. E., Fasullo, J. T., and Kiehl, J. T., “Earth's Global Energy
Budget,” Bull. Amer. Meteor. Soc., 2009, 90, 311-323. The data used in the analysis
include satellite measurements of incoming and outgoing radiation outside the
atmosphere, ground-based measurements of radiation reaching the surface, and models
of radiation interaction with atmospheric species. The article and figure are updates on
Kiehl, J. T. and Trenberth, K. E., “Earth's Annual Global Mean Energy Budget,” Bull.
Amer. Meteor. Soc., 1997, 78, 197-208.
The incoming solar energy, 341 Wm-2, is calculated based on the Sun’s black body
emission (close to our estimated value of 350 Wm-2). The reflected radiation, 102 Wm-2, shown
at the far left corresponds to the Earth’s albedo, 0.30. Of the remaining unreflected radiant
energy in the near ultraviolet, visible, and short wavelength infrared, about 30% (78 Wm-2 out of
239 Wm-2) is absorbed by gases in the atmosphere (mainly O3, O2, H2O, and CO2) and warms
the atmosphere. The remaining radiant energy, 161 Wm-2, reaches the surface and warms it.
In the middle of the diagram, “thermals” represent air warmed by contact with the warm surface.
The air expands and is buoyed up into the atmosphere where it delivers energy to the cooler
surroundings at higher altitudes. Likewise, energy absorbed by water as it evaporates or is
emitted as gas by plant transpiration is carried into the atmosphere as a gas, where it releases
energy to the surroundings when it condenses to form clouds.
The major players in the energy balance are shown at the right of the diagram where the
emission and absorption of energy as infrared radiation are represented. The surface radiation
emission, 396 Wm–2, is the energy flux calculated from the Stefan-Boltzmann equation for a
black body at a temperature of 288 °K, the observed average surface temperature of the Earth.
A small fraction of this energy is lost directly to space, but the great majority is absorbed by
gases and clouds in the atmosphere and re-emitted in all directions, including downward toward
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the surface, 333 Wm–2 warming the surface at the far right of the diagram. The result is a
greater warming of the surface than by the incoming solar radiation alone.
The rounded values at the top of the figure show the total incoming solar radiation flux,
341 Wm–2, equal to the total outgoing flux, the combination of the reflected short wavelength
radiation, 102 Wm–2, and the long wavelength radiation, 239 W·m–2:
(incoming) 341 W·m–2 = 102 Wm–2 + 239 Wm–2 (outgoing)
The outgoing infrared radiation flux, 239 Wm–2, from the atmosphere, clouds, and a small
amount from the surface, is essentially that calculated from the Stefan-Boltzmann equation for a
black body at a temperature of 255 °K, the predicted temperature of the Earth in the absence of
an atmospheric warming effect. This is the temperature that would be recorded by a probe
peering at Earth from outer space, i.e. the temperature at the top of the atmosphere.
Note, however, that the more precise values for the incoming and outgoing energy fluxes
do not exactly balance:
(incoming) 341.3 W·m–2 > 101.9 W·m–2 + 238.5 W·m–2 = 340.4 W·m–2(outgoing)
Earth in fact is not in equilibrium and is warming. The only way this can happen is for the
incoming energy to exceed the outgoing energy. This analysis indicates that the warming
planet is retaining the equivalent of 0.9 W·m–2 (very bottom of the figure).
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HyperGrade PEEB3
The physics and chemistry of Earth climate science is complex, primarily due to the
propensity of water vapor to change phase so readily over the temperature range of interest. In
PEEB3 we will consider a more tractable subset of the problem, the anthropogenic addition of
CO2 to the atmosphere. Over the last 250 years, human activity has shocked Earth’s
atmosphere with an injection of CO2 to a level approaching twice that experienced during
previous interglacial periods over the last several hundred thousand years. Between 1750 and
the present, the CO2 concentration in the atmosphere has increased from 280 to 400 ppm, at an
accelerating rate. Our purpose is to quantify this injection, in particular the extent to which it is
overwhelming nature’s capacity to reabsorb it.
Write a PEEB3 Java program to fill in the three tables below. Since carbon footprint is
most commonly expressed in the English unit of tons per person per year, use English units.
Top table: Given the atmospheric pressure and radius of Earth, calculate the weight of
the atmosphere. From the measured CO2 concentration in the atmosphere, calculate the weight
of CO2 in the atmosphere (be sure to convert from ppmv (parts per million by volume) to ppmw
(parts per million by weight)). Assume the atmosphere contains 78% N2, 21% O2, and 400 ppm
CO2 by volume, and that the atomic weights of atomic C, N, and O are 12, 14, and 16,
respectively. Compare this to the annual anthropogenic CO2 contribution calculated using the
current human population and the current per capita annual carbon footprint of four tons per
person. Express the anthropogenic fraction of the total as a percent.
Middle table: Calculate the CO2 weight gain between 1750 (280 ppmv) and 2015 (400
ppmv). Now put Earth on an aggressive diet: assuming zero CO2 emissions from all sources
going forward, calculate how many years it would take to for the atmosphere to lose its excess
CO2 weight. Assume oceans and forests will continue to reabsorb CO2 at an annual rate of 0.5
tons per person. Your answer represents the minimum recovery time.
Bottom table: Think of the anthropogenic percent calculated above as an annual interest
rate. Using the compound interest formula
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C2015 = C1900(1 + r)n,
where
C2015 = 400 ppmv
C1900 = 280 ppmv
n
= 115 years
final CO2 concentration (year 2015),
initial CO2 concentration (year 1900),
1900 to 2015,
calculate r, the average interest rate since 1900. How does this rate compare to the current
anthropogenic interest rate you calculated for the top table? Does it make sense?
Of course CO2 emissions will not be zero going forward, nor will population growth. At
least until China’s emissions are expected to peak, hopefully in the 2025 – 2030 time frame, the
interest rate will likely continue to increase from its already inflated value. Using the compound
interest formula, project the compounding factor assuming r = 0.82% for the next 20 years.
What would you project for the next 50 years?
Grading
This is an “extracurricular” activity which will be credited as a 15-point homework assignment: 5
points for HyperGrade and 10 points for your table. Hardcopy of your table is required. There
is a bonus opportunity for a critical analysis of your 50-year projection, i.e. your prediction of the
CO2 level and your thoughts on what impact your generation will have on the climate change
crisis.
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