** **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: http://www.climatemodellingprimer.net/downloads.

## 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.

- In the model, changing a factor that is multiplied to the incoming energy can change the solar 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?
- The temperature at each latitude band is also 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. - The albedos for each latitude band are shown in the “ice-free albedos” Explain why higher latitudes generally have higher albedos.
- 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 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?

**Assignment Questions**

## 1. Energy Balance Models – General [6 marks]

** **

- Describe the basic idea behind any energy balance What is the variable calculated in an energy balance model?
- 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]

** **

- Lower the solar constant factor in steps of 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?
- 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 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. - Explain why the two curves are not Why is there a jump when the Earth enters glaciated or exits non-glaciated conditions? What is the physical effect responsible for this?
- 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 = 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):

*R**i** *= *A *+ *BT**i*

* *

- 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 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*T*0–10,*T*10–20, and*T*mean as a

function of the value of *A *in the one diagram, and *T*70–80 and *T*80–90 in another. Put *A *on the

*x*-axis and the temperatures on the *y*-axis.

- Explain the change in global mean temperature as
*A*What physical mechanism does lowering*A*represent in the real world? - 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):

*F**i** *= *C*(*T**i** *– *T *)

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.

- Increased greenhouse
- Melting sea
- Weakening ocean
- Injecting aerosols into the stratosphere to reduce shortwave radiation (geoengineering).
- Placing mirrors on major deserts (geoengineering).

## References

** **

McGuffie, K., and A. Henderson-Sellers (2005), *A Climate Modelling Primer*, John Wiley & Sons, Ltd.

## 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

*S**i** *(1- *a**i** *) = *R**i** *+ *F**i** *,

where

*S**i** *is the mean annual shortwave radiation,

*a**i** *is the albedo,

*R**i** *is the longwave

radiation and

*F**i** *is the loss by heat transport. The subscript *i *denotes the latitude zone.

*S**i** *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

*S**i** *.

*R**i** *and

*F**i** *are given by the formulae,

*R**i** *= *A *+ *BT**i** *,

*F**i** *= *C*(*T**i** *– *T *),

where *A*, *B *and *C *are positive-valued parameters that can be changed, and

*T**i** *and

*T *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 ( *T**i** *< *T *),

the heat loss

*F**i** *is negative, implying that heat is transported to that zone.

The albedo

*a**i** *can take two values. If

*T**i** *is less than the critical temperature (default is

−10°C), the zone is considered ice-covered and

*a**i** *takes the ice albedo (default is 0.62). If *T**i*

is higher than the critical temperature, the zone is considered ice-free and

defined value which can be changed.

*a**i** *takes a pre-

The energy balance equation can be solved for

*T**i** *,

*T**i** *=

*S**i** *(1- *a**i** *) + *CT* – *A *.

*B *+ *C*

* *This equation, together with the calculation of computed iteratively until the solutions converge.