Climate Modelling Practical
A Zonal Energy Balance Model
Model Description
The zonal energy balance model is based on those developed by Sellers (1969) and Budyko (1969).
The world is broken up into bands spanning 10o latitude. The energy balance of each band is used
to infer its temperature:
RSN = RLN+H
(1)
where RSN is the absorbed solar radiation, RLN is the net outgoing longwave radiation, and H is the
heat transported out of the layer. The absorbed solar is dependent on the incident solar radiation,
RS and the albedo of the latutudinal band,  :
RSN = RS {1 - }
(2)
The incident solar radiation depends on the solar constant and the latitude of the band, :
RS = 1370 fS / 4 . w()
(3)
where fS is the solar constant as a fraction of its present day value of 1370 W m-2 and w() is the
weighting factor which accounts for the variation of solar radiation with latitude. The albedo of
each band is a function of temperature which is designed to capture the growth of bright ice-sheets
at low temperatures:
 = L + (I - L)/(1 + exp{T-Tc})
(4)
where L is the albedo of the ice-free planet and I is the albedo of the ice-covered planet. This
function approximates a step function at the critical temperature for ice formation, Tc, such that  ~
L for T > Tc, and  ~ I for T < Tc. The net outgoing longwave depends on surface temperature
and the concentrations of greenhouse gases. We also follow Budyko (1969) in making a linear
assumption:
RLN = A + BT
where A and B are constants and T is the temperature in oC. However, we slightly modify this form
to make the dependence on CO2 explicit:
RLN = A0 – B T2x log{FCO2}/log{2} + BT
(5)
where T2x is a model parameter representing the climate sensitivity to doubling CO2 in the
absence of albedo feedbacks, and FCO2 is the CO2 concentration relative to its current value. We
follow Budyko (1969) in assuming A0=204 Wm-2 and B=2.17 W m-2 oC-1. The heat transport is
parameterised relative to the global mean temperature, <T>
H = K (T - <T>)
Equation 1 to 6 define the zonal energy balance model.
(6)
The Excel Spreadsheet
The Excel spreadsheet version of this model (“ZEBM.xls”) is based on one that is available
on the CDROM which comes with the book “A Climate Modelling Primer” by McGuffie and
Henderson-Sellers. Modifications to this include
 correction of a bug
 an improved interface including some graphics
 inclusion of an explicit dependence on CO2 (see equation 5)
 a revised continuous albedo formulation to improve the iteration of the model to
equilibrium (see equation 4).
The model consists of 3 sheets:
1. Graphs : contains tables for input of parameters for 2 runs; outputs from these runs
in the form of global mean albedo and temperature, and some graphics; a work area
for plotting a phase space diagram.
2. EBM : is the spreadsheet for run 1.
3. EBM2 : is the spreadsheet for run 2.
Under most circumstances it should only be necessary to modify the “Graphs'” sheet.
Through this the user can change the parameters shown in table 1.
Parameter
Climate Sensitivity to CO2 (in the absence of albedo feedback), T2x
Fractional Increase in CO2, FCO2
Heat Diffusion Coefficient, K
Ice Albedo, I
Land Albedo, L
Temperature for Ice Formation, Tc
Fraction of the Solar Constant, fS
Initial Ice-Edge
Table 1: Model Parameters and their default values.
Default Value
2.0 oC
1.0
3.57 Wm-2 oC-1
0.6
0.3
-10 oC
1.0
60o latitude
The “Initial Ice-Edge” determines the initial albedo field. This is especially important since the
model permits multiple equilibrium solutions and this initial state controls which of these is
produced (see the “Exercises”).
Exercises
1. Set the model parameters for Runs 1 and 2 to the default parameters given in table 1.
a) where is the final ice-edge?
b) what are the mean temperature and mean albedo?
c) what are the approximate temperatures at the pole and the equator?
d) does this look like the real Earth?
2. How do you expect the ice-edge, mean albedo and mean temperature to vary with the parameter
K? Run the model with different values of K to check your intuition.
a) what is the mean temperature and albedo in the absence of heat transport?
b) at what K value does the model become completely ice-free?
c) what are the implications of this sensitivity to K with regard to variations in poleward heat
transports (e.g. variations in the Gulf stream)? Is this realistic?
3. Return parameters to their default values in table 1.
a) By reducing the “Fraction of Solar Constant” in Run 1 determine the value at which the
Earth becomes completed ice-covered.
b) Set the initial ice-edge to 0o (i.e. “Snowball Earth'”) and increase the solar constant from the
value found in (a). At which value does the ice-sheet completely melt?
c) Set the initial ice-edge to 90o (i.e. ice-free Earth) and reduce the solar constant from the
value found in (b). At which value does the ice begin to reappear?
d) Sketch phase diagrams of mean albedo versus fractional solar constant, and mean
temperature versus fractional solar constant. Initialise Run 1 and Run 2 with ice-edges at 0o
and 90o, and increase the solar constant values from 0.8 to 1.2 in steps of 0.05. Fill in the
mean values of temperature and albedo in the workspace table at the bottom of the
“Graphs” sheet. This will produce phase space diagrams below the table.
e) Given that the Sun was fainter in the past, what do these diagrams “predict” the current
Earth state should be? What missing processes might resolve this “faint young Sun
paradox”?
4. Return parameters to their default values in table 1.
a) What is the impact of doubling CO2 on global temperature and albedo?
b) Why does this differ from the T2x input parameter?
c) The global mean temperature is given by :
<T> = [<RS> {1 - <>} – A] / B
where <RS> =1370 fS / 4 is the mean incident solar radiation, <> is the mean albedo, and
B=2.17 W m-2 oC-1. Derive a relation between the increment to the mean temperature due to
albedo changes, Tsnow , and the albedo change itself, . Does this fit with the answers
given in (a) and (b) above?
d) Now try 4xCO2 in Run 1. How large are the mean temperature and mean albedo changes in
going from 2xCO2 to 4xCO2? Why are they lower than in (a)?