Very Basic Climate Modeling
Spring 2012, Lecture 5
1
Radiation Balance
• The temperature of any body, including earth,
is determined by a balance between incoming
radiation and outgoing radiation
• This is similar to a bank account
• Fin = Fout
 Fin is the incoming energy flux
 Fout is the outgoing energy flux
2
Intensity of Incoming Energy
• Sunlight, at the earth’s distance from the sun,
has an intensity, I, given by:
Iin = 1350 w/m2
• Some of the incoming radiation is reflected
back into space – reflectivity is called the
albedo, denoted α (Greek alpha)
• For earth, average albedo is about 0.33
3
Corrected Iin
• Correcting for reflection, we get
Iin = 1350 w/m2 (1 – α) = 1000 w/m2
• We must correct this for the area of the earth
which receives solar illumination
• Only half of the earth is at any given time, and
sunlight near the poles is much weaker than near
the equator
4
Flux in Watts
• In order to measure the flux in watts, rather
than watts/m2, we need to multiply by the area
• The illuminated area is actually that of a circle
A[m2] = πr2earth
(Greek pi)
• So the incoming flux in watts becomes
 Fin = πr2earth (1-α) Iin
5
Computing Fout
• Next we need to compute the outbound energy
flux, Fout
• In order to do this, we use the StefanBoltzmann equation for a blackbody radiator
6
Stefan-Boltzmann Equation
• I = εσT4
 I is the Intensity of emitted radiation
 ε is the emissivity (Greek epsilon)
o ε is a number between 0 and 1
o If a blackbody is perfect, ε = 1
 σ is a fundamental constant of physics called the
Stefan-Boltzmann constant (Greek sigma)
 T is absolute temperature (K)
• Note that intensity varies as the fourth power of
the absolute temperature
7
Fout
• The earth radiates over the entire surface
• The area of the earth is given by
A = 4πr2earth
• Fout is computed by
Fout = AεσT4earth = 4πr2earth εσT4earth
8
Equating Incoming and Outgoing Flux
• Substituting in Fin = Fout gives
πr2earth (1-α) Iin = 4πr2earth εσT4earth
• Eliminating factors common to both sides gives:
(1-α) Iin = 4εσT4earth
• We know everything except T, so rearrange
T4earth = [(1-α) Iin]/[4εσ]
9
Tearth
• Solving for T:
Tearth = 4√[(1-α) Iin]/[4εσ]
10
Incoming vs. Outgoing Flux Diagram
11
Model Construction
• What we have done so far is to construct what
scientists call a “model”
• Models attempt to mimic the behavior of the
natural system we are studying
• Models can be constructed as computer
programs, which allow scientists to change
conditions, and to attempt to assess what
effects a given change, such as a doubling of
atmospheric carbon dioxide, will cause
12
Early Climate Models
• The earliest climate models were physical – a
dishpan full of water on a record turntable,
heated on the rim by a flame representing the
sun
• Such models were made to examine circulation
patterns in the atmosphere, and were called
Global Circulation Models
• Today this same name is used for far more
sophisticated models
13
Model Prediction
• If we calculate Tearth, we get a value of about
255 K, or about -18◦C
• We have seen before this is too cold
• Thus, we learn a very important lesson about
models – they are only as good as the
mathematical representation of the real world,
and the data that we put in
• Modelers have an acronym – GIGO – garbage
in, garbage out
14
Are Models Useful?
• Does this mean mathematical modeling is a waste
of time?
 No, because models do allow us to understand certain
phenomena, such as water vapor and sea ice
feedbacks fairly well
 They are relatively inexpensive, and often suggest
where new avenues of research are needed (clouds,
aerosol particles)
 However, they must be used with caution, and the
results understood in the context of what the model
was written to do
15
What Happens When a Model is Wrong?
• Observation: Our model result for worldwide
temperature is much too low
• Action
 Throw out the model?
 Improve the model?
o Add more features to it so it better mimics the real
world
o Attempt to collect better, or new, data so the model can
be improved
16
Improving Our Model
• Our model is known to predict a temperature
much lower than the observed global
temperature
• We also know that the earth has a greenhouse
effect, and this is not in our model
• We need to add atmospheric layers to allow for
the greenhouse effect
17
Adding a Layer
• Our new model uses a
hypothetical “pane of
glass” suspended above
the earth’s surface
• It is transparent to
visible light
• It acts like a blackbody
for infrared light,
absorbing all of it, and
reradiating it in all
directions
18
Model Changes
• Incoming radiation is unaffected
• Outgoing radiation must be changed
 Instead of Iout, we now have Iup, earth
 When infrared radiation strikes the glass pane, it
is absorbed and reradiated
o Half goes down, as Idown, atmosphere
o The other half goes up, as Iup, atmosphere
19
Atmospheric Energy Budget
• We can write an atmospheric energy budget
 Iup, earth = Iup, atmosphere + Idown, atmosphere
• Alternatively,
 2εσT4atmosphere = εσT4earth
20
Ground Energy Budget
• Iup, earth = Iin, solar + Idown, atmosphere
• εσT4earth = (1-α)/4*Isolar + εσT4atmosphere
21
Whole Earth Energy Budget
• The whole earth
budget can be
represented by
assuming that what
goes in must come out,
so that:
• Iin, solar = Iup, atmosphere
22
Solving the System
• εσT4atmosphere = (1-α)/4*Isolar
• We now have two unknowns, Tearth and Tatmosphere
• This equation has the same form as the one for the
bare earth
 Tearth = 4√[(1-α) Iin]/[4εσ]
• This is important! It means that the temperature is
controlled by the place at which earth radiates
into space
• We can call this the “skin” temperature, Tskin
23
Skin Temperature
• We now know that Tatmosphere = Tskin
• Plugging
 εσT4atmosphere = (1-α)/4*Isolar
• Into
 εσT4earth = (1-α)/4*Isolar + εσT4atmosphere
• Gives
 εσT4earth = 2εσT4atmosphere
24
Solving the Model
• Tground = 4√2 Tatmosphere
• The fourth root of 2 is 1.189
• Thus, the ground temperature of earth is about
19% warmer than it would be without the
greenhouse layer
• We got a value of 255 K from the bare earth
model – 1.189*255 = 303 K, which is a bit too
warm – actual global average temperature is
about 288, or 15ºC
25
Why is the Model Incorrect? - 1
• Like all models, this one does not fully
reproduce the natural system – in fact, it is
very simplified
• We have treated all wavelengths of radiation
the same, but we know that there are
“windows” through which some IR radiation is
barely absorbed, while other wavelengths are
fully absorbed
26
Why is the Model Incorrect? - 2
• There is nothing in the model that reflects to the
concentration of GHG
• It does not allow for clouds, or for different
degrees of surface reflectivity
• It does not allow the presence of aerosol particles
• Model assumes heat transfer from ground to
atmosphere is by radiation alone, ignoring
convection
• And many more….
27
Improving the Model
• We can further improve the
model by adding more
layers, such as the two layer
model seen here
• Layers might represent
troposphere and stratosphere,
for example
• Such models become more
complex, but are still soluble
28
Archer Sink
Model
29
Partially Plugged
Sink
The partial blockage of the sink
raises water levels, until the
increased pressure drains the
sink as fast as water flows in –
analogous to temperatures
increasing in the atmosphere
30