Posted: September 1st, 2023

SCI1300 – Climate Change: From Science to Society Assignment 2 – Energy Balance Model

SCI1300 – Assignment 2
Last Edit: September 2, 2019 Page 1 of 5
SCI1300 – Climate Change: From Science to Society
Assignment 2 – Energy Balance Model
Introduction to this Assignment
In this assignment, you will use an energy balance model to investigate some questions
related to the Earth’s energy budget. You will need to write down your answers to the
questions below and submit your solutions to your tutor in exactly two weeks after your Week
6 tutorial class (or by 5pm that same day). This assignment is worth 20% of the total
assessment for this course.
Radiation Budget and Energy Balance
In this assignment, you will use an Energy Balance Model from the book, A Climate
Modelling Primer (McGuffie and Henderson-Sellers,
2005) to investigate the energy balance of the planet
over different latitudes1. This is a one-dimensional
model that computes several variables in each latitude
band averaged over all longitudes (as opposed to a
three-dimensional model which computes the variables
in each latitude, longitude and height grid box). For example, we have one value for each of
the variables (e.g. surface temperature, incoming shortwave radiation etc.) for the latitude
band 0° to 10°N, and another set of values for 10°N and 20°N so on. This model treats both
hemispheres symmetrically, so there is no distinction between hemispheres (that is, the
variables for the 0° to 10° band represents both 0°N to 10°N and 0°S to 10°S).
The model allows you to adjust the albedo for each latitude band, the incoming
shortwave radiation, the outgoing longwave radiation, meridional (across-latitude) heat
transport, and takes into account whether the latitude is ice-covered. The equations used in
this model are summarised in the Appendix and further information can be found in McGuffie
and Henderson-Sellers (2005).
1 An executable program for the model can be found here: https://monkessays.com/write-my-essay/climatemodellingprimer.net/downloads.
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Group Work - Practice Problems
This is a practice question that will introduce you to the model and give you an idea of what
you are expected to do in this Assignment. Work in your group and answer the questions
below. At the end of your discussions, the tutors will take you through the answers.
(a) In the model, changing a factor that is multiplied to the incoming energy can change the
solar constant. The default setting is 1, representing the current solar constant. The global
mean temperature for this setting is 15.3°C. Lower the solar constant to 0.9 and press
‘Calculate’. What is the global mean temperature? What is the global mean temperature
when the solar constant is 1.1?
(b) The temperature at each latitude band is also given. Plot the temperature (y-axis) versus
latitude band (x-axis) for S = 0.9, 1.0 and 1.1 (one line each) in the same axes.
(c) The albedos for each latitude band are shown in the “ice-free albedos” column. Explain
why higher latitudes generally have higher albedos.
(d) The albedo will affect the temperature of the latitude band, but can it explain the variation
you see in (b)? To answer this question, reset S to 1.0 and replace all the albedos with 0.3
(planetary-average). You will need to select expert mode in ‘Mode > Expert’ to edit the
albedo values. Plot the temperature (y-axis) versus latitude band (x-axis) for the original
albedos and the new albedos (one line each) in the same axes. Does a constant albedo
remove the latitudinal variation in temperature? What other effect is at play here?
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Assignment Questions
1. Energy Balance Models – General [6 marks]
(a) Describe the basic idea behind any energy balance model. What is the variable calculated
in an energy balance model?
(b) What are the two main components of the energy balance at the top of Earth’s atmosphere
(that is, above planetary effects such as clouds and ozone)? Name the two main
characteristics of the Earth system that affect them?
2. Snowball Earth [13 marks]
(a) Lower the solar constant factor in steps of 0.02 and record the global mean temperature at
each value. Stop when the Earth is completely glaciated (hint: you can look at the albedo
of the latitude zones to see if the Earth is glaciated). At what value of the solar constant
factor does this occur?
(b) Now, we want to find the solar constant factor necessary for Earth to exit this glaciated
condition. To do so, set the calculation to begin from a glaciated initial condition. Select
‘Mode > Expert’, then under the ‘Initial conditions’ box, choose ‘Glaciated’.
Progressively increase the solar constant by 0.02, starting from the value in (a) to the
value when Earth is not fully glaciated anymore. Record the global mean temperature at
each step. At what solar constant factor does the Earth exit the full glaciation? Plot the two
temperatures from (a) and (b) on one graph (x-axis: solar constant; y-axis: temperature).
Label each curve clearly.
(c) Explain why the two curves are not equal. Why is there a jump when the Earth enters
glaciated or exits non-glaciated conditions? What is the physical effect responsible for
this?
(d) Currently, the critical temperature at which a latitude band becomes ice-covered is −10°C.
This means that, when the temperature of the latitude band drops to −10°C or lower, it
will be considered ice-covered and its albedo is that of ice (default = 0.62). Now, reset all
parameters to default values and then change the critical temperature from −10°C to 0°C.
Find the solar constant factors to fully glaciate Earth starting from present day conditions,
and to thaw a glaciated Earth. Explain why these values are different from the ones found
in (a) and (b).
3. Changing Earth’s Energy Loss [9 marks]
The energy loss of each latitude zone by radiation in the model is approximated by the
formula (see Appendix for details):
(a) Reset all the values to default and switch back to ‘Present day’ under ‘Initial conditions’.
Now, keeping the solar constant factor at one, lower A by steps of 1 from 204 to 196. For
each step, record the temperatures of four latitude zones: 0° – 10°, 10° – 20°, 70° – 80°,
80° – 90°, as well as the global mean temperature. Plot T0–10, T10–20, and Tmean as a
Ri = A+ BTi
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function of the value of A in the one diagram, and T70–80 and T80–90 in another. Put A on the
x-axis and the temperatures on the y-axis.
(b) Explain the change in global mean temperature as A decreases. What physical mechanism
does lowering A represent in the real world?
(c) Compare how the temperature changes with A for the tropical latitudes and the polar
latitudes. Explain the difference in behaviour at low values of A.
4. Heat Transport [8 marks]
The simple model used here allows for heat transfer between the latitude zones, described by
the simple equation (see Appendix for more detail):
Reset all values to the defaults. In a single graph plot the temperature in each zone versus
latitude for C = 3.3, C = 3.8 (default), C = 4.3 and C = 4.8 (x-axis: latitude; y-axis:
temperature; you will have one line for each C). Also, for each value of C, write the global
mean temperature. Describe the change in global mean temperature and the temperature
distribution with latitude with changing C and use your results and your physical
understanding of the parameter C to explain the temperature behaviour.
5. Modelling Climate Change [5 marks]
The Energy Balance Model provides many parameters that can be changed to affect the
overall climate of the model Earth. Below we list some possible influences on global climate.
Using the knowledge gained in the exercise above, write down the parameter that you should
modify to simulate the climate change in the list and identify the sign of the parameter change
(increase or decrease). For example, to simulate an increasing solar input into the climate
system, we would increase the solar constant S. Choose only one parameter for each change.
(a) Increased greenhouse effect.
(b) Melting sea ice.
(c) Weakening ocean circulation.
(d) Injecting aerosols into the stratosphere to reduce shortwave radiation (geoengineering).
(e) Placing mirrors on major deserts (geoengineering).
References
McGuffie, K., and A. Henderson-Sellers (2005), A Climate Modelling Primer, John Wiley & Sons, Ltd.
Fi = C(Ti −T)
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Appendix
The Energy Balance Model is a one-dimensional model divided into nine latitude zones
between 0° and 90°. For all latitude zones at equilibrium, the incoming shortwave radiation is
balanced with the outgoing longwave radiation and the loss by heat transport. This is
expressed as
,
where is the mean annual shortwave radiation, is the albedo, is the longwave
radiation and is the loss by heat transport. The subscript i denotes the latitude zone.
is a function of latitude, but a simple cosine function cannot be applied because of
the axial tilt of the planet. As such, the model manually assigns a ‘weight’ for each zone (from
pole to equator): 0.5, 0.531, 0.624, 0.77, 0.892, 1.021, 1.12, 1.189, 1.219. This weight is then
multiplied to 1370 / 4 W m-2 to obtain .
and are given by the formulae,
,
,
where A, B and C are positive-valued parameters that can be changed, and and are the
zonal and global mean temperatures respectively. It is important to remember that the
averaging needs to be weighted by the area of the latitude zone (effectively the cosine of the
midpoint latitude of the zone). Note that if the zone is colder than the global mean ( ),
the heat loss is negative, implying that heat is transported to that zone.
The albedo can take two values. If is less than the critical temperature (default is
−10°C), the zone is considered ice-covered and takes the ice albedo (default is 0.62). If
is higher than the critical temperature, the zone is considered ice-free and takes a predefined
value which can be changed.
The energy balance equation can be solved for ,
.
This equation, together with the calculation of and the albedos of each zone, can be
computed iteratively until the solutions converge.
Si (1− α
i ) = Ri + Fi
Si α
i Ri
Fi
Si
Si
Ri Fi
Ri = A+ BTi
Fi = C(Ti −T)
Ti T
Ti < T
Fi
αi
Ti
αi
Ti
αi
Ti
Ti =
Si (1− α
i )+CT − A
B+C
T