Global Climate Modeling Project

For Summer 1997 Envision-It! Workshop


The goal of this model is to make some predictions about the global temperature of the earth and to see how this depends on a range of parameters. In particular, we will see how the "greenhouse effect" works and the impact this has on the global temperature. This is an example of an energy balance model which is very similar to the hot/cold can experiment discussed earlier.

We will first consider the following model for the average global temperature - we will treat the entire earth as a single point (as shown below).

Global Climate Model
Reprinted with permission from: A Climate Modeling Primer, A. Henderson-Sellers and K. McGuffie, Wiley, pg. 58, (1987)

Once we have familiarized ourselves with this model, we will work with a 1-Dimensional model of the earth which is broken up into 10o latitude regions. This will allow us to determine not just the global temperature, but the temperature in each of the 9 latitude regions in the Northern Hemisphere. For the independent phase of this project, you will refine the model to give more realistic estimates for the temperature distribution.

Much of the formalism (including a BASIC program for an Energy Balance Model) for this project is from the excellent book: A Climate Modeling Primer, 2nd Edition, K. McGuffie and A. Henderson-Sellers, Wiley, (1997). Other references include:
We will first consider the following model for the average global temperature - we will treat the entire earth as a single point.

In an energy balance model, the main goal is to account for all heat flows in (PGain) and out (PLoss) of the system. If these are balanced (PGain=PLoss), the system will be in a steady state and the system will be at a constant temperature. If the heat flows are not balanced, the temperature of the system will change.

The heat flow to the earth is from the sun. The Solar Constant (S) is the amount of energy arriving (during a 1 second period on a 1 square-meter area oriented perpendicular to the sun's rays) from the sun at the upper atmosphere. The annual average value of the solar constant is S=1370 W/m 2. This arrives primarily in the form of visible light, with some smaller amounts of infrared and ultraviolet radiation.

A fraction of the sun's radiation is immediately reflected back into space, either from the atmosphere, clouds or the earth's surface. The Albedo (α) of the earth is the fraction of the sun's radiation which is reflected back into space. Thus, the net amount of solar radiation arriving on a 1 m 2 area (perpendicular to sun) on the earth's surface is S(1- α).

From the point of view of the sun, the earth appears to be a disk with a radius R, so the total amount of power absorbed by the whole earth is the product of the arriving solar radiation times the area of a disk the size of the earth: PGain=πR2 S(1-α)

Any object at a temperature TK (in Kelvin) will emit thermal radiation at a rate given by: PLoss=ε σ TK4 times it surface area. The factor ε is the emissivity (approximately 1), σ is Stefan's constant, and the total surface area of the spherical earth (4πR2). Recall that a temperature in Kelvin is TK=T0+T where T is the temperature in centigrade and T0=273.15

In the steady state, the incoming radiation must balance the outgoing radiation. This leads to an energy balance equation for PGain=PLoss:

πR2 S(1-α) = (4πR2) σ (T+T0)4
where T is the average temperature of the earth in centigrade. Solving for T gives the following equation:

(Eqn. 1): T = [S(1-α)/4 / σ] 1/4-T0.

Where the symbols are defined as:
TThe Temperature of the Earth in Centigrade
SSolar Constant (1370 W/m2)
α Albedo - Fraction of incident solar radiation reflected (about 0.32)
σ Stefan's Constant (5.6696E-8 W/m2K4)
T0 Conversion from Kelvin to Centigrade (273.15)
To study this, we could develop a simple Matlab script to calculate T. To illustrate some of the GUI capabilities of Matlab, you may use the script ebmgui.m (along with ebmcalc.m) which allow you to easily change some of the parameters. To run this script in matlab, type ebmgui and enter an initial guess for the temperature in the upper left, and make sure that the "Blackbody" button in the lower left is on. Then click the "Calculate" button on the right.

EXERCISE 1: Using the ebmgui program, calculate T in Centigrade for the earth. Is this a reasonable value, or would we be mighty cold?
EXERCISE 2: Since the intensity of solar radiation is proportional to 1/r2, the solar radiation received at any other planet in the solar system can be calculated. Since Mars is about 1.5x further from the sun, it will receive roughly 1/(1.5)2=0.444 times the solar radiation on Earth. If it has an albedo of 0.16, what is the black-body temperature for Mars?

Planet Distance (AU) Albedo Ave. Temperature Atmosphere
Venus0.7230.76 425 oC 95 Atm, 96% CO2
Earth1.0000.32 15 oC 1 Atm, N2, O2, Trace H2O, CO2
Mars1.5240.16 -50 oC 0.02 Atm, 95% CO2
Europa (A Moon of Jupiter)5.2030.64 -145 oC Essentially None
For more info, see WWW pages at JPL


Obviously, the steady-state situation given by Eqn. 1 is not fully correct - it leads to a temperature of about -16 oC for the earth which is well below the mean global temperature of about 15oC. The major reason for this difference is the Greenhouse Effect in the atmosphere. The calculation for the temperature of Mars, with very little atmosphere, is in very good agreement with the observed temperature. Venus, on the other hand, with its very thick atmosphere of CO2 has a large greenhouse effect leading to a much higher temperature than predicted by Eqn. 1 On Europa gravitational tidal forces cause the interior to be warm enough that it appears that there may be liquid water beneath its ice crust.

The greenhouse effect is a due to absorption and re-emission of infrared radiation by the earth's atmosphere. As discussed earlier, the earth receives visible light from the sun and emits heat in the form of infrared radiation. The earth's atmosphere is nearly transparent in the visible region of the spectrum, but gases (such as water vapor, CO2 and others) in the atmosphere can absorb infrared radiation. When a gas molecule (such as CO2) absorbs some infrared radiation, it ends up in an excited state and eventually it decays to the ground state. In the process, it will re-emit energy in the infrared in a random direction (some of it back towards the earth). The net result is that some of the infrared radiation is "reflected" back towards the earth, thus reducing the loss of heat from the earth.

This is called the greenhouse effect because a similar mechanism occurs in a greenhouse (or a closed car in a parking lot on a sunny day). Visible light passes into the greenhouse and is absorbed by material inside. This material re-emits energy in the infrared, which is reflected back by the glass of the greenhouse (or car). Because of this trapping of heat, it is possible for a sealed greenhouse (or car) to reach a temperature far higher than the outside temperature.

The major contributor to the greenhouse effect in the atmosphere is water vapor (because it is the most abundant contributor). The next leading contributor is CO2, with others including ozone, N2O, methane and CFC's. Human activities are causing changes in the concentrations of these gases, which may lead to increased greenhouse effects.

Since all of our temperatures are within about +/- 20oC, it is possible to re-write the Stefan-Boltzmann equation using the binomial expansion: (1+x)n is approximately equal to (1+nx) if x is much less than 1. Therefore, we can write
TK4 = (T0+T)4 = T04(1 + T/T0)4 which is approximately T04(1 + 4T/T0).
This allows us to re-write the Stefan-Boltzmann equation as PLoss=(4πR2) (A + B*T) where the two constants are A= ε σ T04=315 W m-2 for a black body, and B=4ε σ T03=4.6 W m-2 oC-1 for a black body. The greenhouse effect can be included by modifying the values used for A and B.

In the steady state, the incoming radiation must balance the outgoing radiation. This leads to an energy balance equation for PGain=PLoss:

πR2 S(1-α) = (4πR2) (A + B*T)
where T is the average temperature of the earth in centigrade. Solving for T gives the following equation:

(Eqn. 2): T = [S(1-α)/4 - A]/B.

Where the symbols are defined as (see McGuffie and Henderson-Sellers, 1997 for details):
TThe Temperature of the Earth in Centigrade
SSolar Constant (1370 W/m2)
α Albedo - Fraction of incident solar radiation reflected (about 0.32)
A Constant Coefficient for Thermal Heat Flow (A=204 W m-2)
B Temperature Dependent Coefficient (B=2.17 W m-2 oC-1)

EXERCISE 3: Calculate T in Centigrade with the addition of the greenhouse effect. How much temperature change is due to the greenhouse effect? Other models use A=202 W m-2 and B=1.45 W m-2 oC-1 (Budyko:1969) or A=212 W m-2 and B=1.6 W m-2 oC-1 (Cess: 1976). How sensitive is the global temperature to these A & B parameters? Try changing the albedo to 0.31 or 0.33, or changing the multiplier for the solar constant to about 0.99 or 0.95.

The model described in Equation 2 is the steady state (equilibrium) temperature of the earth. Obviously, our experience indicates that the earth is never in equilibrium during the short term (the local weather changes on the scale of hours). However, if all of the factors in Equation 2 remained the same, the yearly average temperature would be the same.

In reality, due to various natural and man-made effects, the factors in Equation 2 can have long-term changes. A volcanic eruption or large fires can modify the the earth's albedo. Human activity has changed the amount of greenhouse gases in the atmosphere, thus changing the A & B factors. We could re-run the model to find the new equilibrium temperature after changing some or all of these factors.


In some cases, the system will not be in thermal equilibrium, namely the energy loss will not be equal to the energy gain. This will lead to a changing temperature in the system. We can also modify our model to allow energy to be stored or released, thus making a non-equilibrium model.

A large fraction of the thermal energy storage which effects the climate is due to the upper layer of mixed water in the ocean (about the upper 70 meters). We can write the effective heat capacity of a 1 square-meter area of the earth as: CE = f ρ ch, where:
fFraction of the Earth Covered by Water (0.7)
ρ Density of Sea Water (1023 kg/m3)
cSpecific Heat of Water (4186 J/kgoC)
h Depth of Mixed Layer of Sea Water (70 m)

Putting these together gives a heat capacity of CE= 2.08 × 108 J/m2 oC. In other words, to change the temperature of 1m2 of the earth by 1 oC will take on the average 2.08 × 108 J of energy. This will tend to moderate the effects of "rapid" changes in any of the parameters governing our global equilibrium temperature. Thus it may take many years for the temperature to converge to a new equilibrium.

As discussed in the previous model of the hot/cold cans, the temperature change which results from a difference in PGain and PLoss is given by:


(Eqn. 3): ΔT = (PGain-PLoss)Δt/ CE
In this case
PGain = S(1-α)/4
PLoss =A + BT

Where the constants are defined as:
TThe Temperature of the Earth in Centigrade
Δt Time interval (as measured in Seconds) for each iteration.
ΔT Temperature change during the time interval Δt.
SSolar Constant (1370 W/m2)
α Albedo - Fraction of incident solar radiation reflected (about 0.32)
A Constant Coefficient for Thermal Heat Flow (A=204 W m-2)
B Temperature Dependent Coefficient (B=2.17 W m-2 oC-1)
CE Global Heat Capacity of Earth (CE= 2.08 × 108 J/m2 oC)

You can develop a M-file to model this case (you may wish to use ploss.m as a starting point):

  1. Select a Time Interval Δt (such as Δt= (365*24*60*60)/20 to have a time interval of 1/20 of a year as measured in seconds) and the number of time intervals to use (such as 400 for a 20 year period at 20 intervals per year)
  2. Select a Starting Temperature TInit
  3. Include in your script the definitions of the other parameters CE, α, A and B (some or all of which may vary in time)
  4. Make a loop which varies over the number of time intervals (selected in Step 1.)
  5. Use the current value of all parameters and the temperature T from the previous iteration in Eqn. 3 to calculate a temperature change ΔT
  6. Add this temperature to the previous temperature to get a new value for the temperature.
  7. Store the current values of the time and temperature in an array (for plotting later)
  8. Repeat this loop starting from Step 4.
At the end of this process, you will have a list of the temperature as a function of time.

EXERCISE 4: Create a script file to calculate T in Centigrade as a function of time. What is the steady-state temperature. if there is a 1% or 5% decrease in the solar constant (caused by a volcanic eruption, for example), how much does the global temperature change from the steady-state of Exercise 3? Use an initial temperature from Exercise 3 and make a plot of the temperature as a function of time. How long does it take the earth to reach its new equilibrium?


There are many refinements we could make to this model. One of the more interesting is to put a temperature dependence on the albedo. As the temperature of the earth decreases, more of the water on the planet will freeze, which will increase its albedo (since the albedo of ice is about 0.6 compared to about 0.32 for land). We could account for this in the following way Where the constants are defined as
αIceAlbedo of Ice (0.6)
TIceGlobal Temperature at which entire Earth freezes (-10 C)
αLandAlbedo of Land (0.32)
TLandGlobal Temperature at which entire Earth melts to current state with small polar ice caps) (+10 C)
EXERCISE 5: Modify your script file to include a temperature dependent albedo as defined above. What is the final temperature if the initial temperature is 13 degrees? What about if the initial temperature is 8 degrees? If there is a 1% or 5% decrease in the solar constant (caused by a volcanic eruption, for example), how much does the global temperature change from the steady-state of Exercise 3? Use an initial temperature from Exercise 3 and make a plot of the temperature as a function of time. How long does it take the earth to reach its new equilibrium?
EXERCISE 6: Even more interesting, consider a 5% decrease in the solar constant during a 5 year period, after which it returns to its initial value - what happens to the temperature? How about if this lasts for 10 years? What if there is a 10% decrease for 5 years? What about a 20% decrease for 1 year?

Now we can proceed to develop a 1-Dimensional model of the earth which is broken up into 10o latitude regions.


Electronic Copy: http://physics.gac.edu/~huber/envision/instruct/ebm1doc.html
Revised: 14-JUL-97 by Tom Huber, Physics Department, Gustavus Adolphus College.