Climate change modelling in the classroom Teach article

Why not get your students to make their own predictions of climate change – with the help of Dudley Shallcross and Tim Harrison from Bristol University, UK?

Climate change and global warming are ‘hot’ topics and deserve an important place in the school science curriculum. But how do we predict how our climate is going to change? It seems timely to introduce students to simple climate modelling. In this article, we demonstrate that students can use a straightforward spreadsheet to investigate the major factors that affect Earth’s climate.

A first attempt to model the climate

The simplest model of the climate is one in which incoming solar energy is equal to outgoing terrestrial energy emitted from the planet, i.e. an ‘energy in equals energy out’ model. In this article, the term ‘energy’ really means energy flux, i.e. energy per second. We know from measurements that the energy from the Sun reaching the top of the atmosphere (per second), termed the solar constant S, is 1370 Wm^-2.

We start by calculating the average surface temperature of Earth, T_E. If we take the radius of a perfectly spherical Earth to be R_E, in this very simple model we can see that Earth absorbs solar radiation over an area πR_E² (i.e. a flat disc of atmosphere) but emits energy from an area 4πR_E² (i.e. from the entire surface).

(a) Energy in = energy out

and using the Stefan-Boltzmann Law (see box)

(b) Energy per unit area per unit time x total area_(disc) = energy per unit area per unit time x total area_(sphere)

(c) 1370 x πR_E² = σT_E⁴ x 4πR_E²

Rearranging the equations gives:

σT_E⁴ = 1370/4 = S/4

T_E⁴ = 1370 / (4 x 5.67 x 10^-8)

T_E = 279 K (6°C)

where σ is the Stefan-Boltzmann constant (5.67 x 10^-8 Wm^-2K^-4).

At first glance, this looks like a sensible figure for such a crude model, although of course the actual average surface temperature of Earth is known to be 16 °C. (There are several ways to work it out, such as by splitting Earth into latitude bands, working out the average temperature per band and summing and averaging all of these.) The problem with this very simple model, though, is that some solar energy is not absorbed by Earth, but is instead reflected back out to space by clouds and ice.

Approximately 24% of the incoming energy is reflected by clouds and another 6% is reflected by the surface, e.g. by ice. This gives a total reflectivity of Earth – known as the albedo (A) – of 30% or 0.3. Therefore, the left-hand side of equation (c) must now be re-written as 0.7 x 1370 x πR_E² and the calculation of T_E becomes:

T_E⁴ = (1370 x 0.7) / (4 x 5.67 x 10^-8)

T_E = 255 K (-18°C)

This value is obviously far too low, and leads naturally to the question: why is Earth so warm? In order to answer this question we need a slightly more complex model.

The one-layer atmosphere model of Earth

If we assume that the atmosphere is made up of a single layer of miscible gases, we can create a more accurate model that students can use with a spreadsheet. In this model, allowances are made for absorption by the atmosphere of the incoming visible light from the Sun and absorption of the outgoing infra-red light emitted from Earth.

The figure below summarises the elements of the model. F_S is the solar constant divided by 4 (S/4), which arises from the difference between incoming energy spread over a disc, πr², and outgoing energy radiated from the surface of a sphere, 4πr² (the assumption made in the first model). The incoming energy from the Sun is then F_S(1-A), where A is the albedo – the portion reflected back to space. This incoming energy is in the UV and visible regions. τ_VIS is the proportion of this incoming energy that is not absorbed by the atmosphere: if the atmosphere absorbs it all, τ_VIS is zero; if the atmosphere absorbs none of it, τ_VIS is one. Hence the energy reaching the surface of the planet is F_S(1-A) x τ_VIS.

Earth will behave similarly to a black body and will emit the energy denoted as F_g from its surface. This terrestrial radiation is centred in the infra-red region of the spectrum. Certain gases in the atmosphere absorb the infra-red energy (greenhouse gases). Similar to τ_VIS, τ_IR is the proportion of infra-red energy not absorbed by these gases in the atmosphere, and so the outgoing energy is F_g x τ_IR.

Assuming that the energy from the atmosphere is denoted as F_a and that energy in and out at the surface of Earth and the top of the atmosphere are both balanced, then:

at the surface of Earth:

F_S(1-A) x τ_VIS + F_a = F_g                                                         (Equation 2)

and at the top of the atmosphere:

F_g x τ_IR + F_a = F_S(1-A)                                              (Equation 3)

Combining Equations 2 and 3 gives:

F_g = (F_S(1-A)(1 + τ_VIS)) / (1 + τ_IR)

Finally,

F_g = σT_E⁴ = (F_S(1-A)(1 + τ_VIS)) / (1 + t_IR)

T_E = [(F_S(1-A)(1 + τ_VIS)) / ( σ(1 + τIR))]^0.25           (Equation 4)         

If we take the following values:

F_S = 1370/4 = 342.5 Wm^-2 (solar constant divided by 4)

A = 0.3

τ_VIS = 0.8

τ_IR = 0.1

Equation 4 gives us:

T_E = 288.5 K (15.5°C),

which is a close approximation to the current average surface temperature of Earth.

It is Equation 4 that can be put into a programme such as Microsoft Excel so that students can see the changes in global temperature when the equation parameters are changed. This has been done and an interactive version can be used online or downloaded from the Bristol University website^w1.

Table 1 shows examples of the output if the variables A, τ_VIS and τ_IR are altered. In Example 1 we assume that the atmosphere does not absorb any of the incoming or outgoing energy fluxes (i.e. both τ_VIS and τ_IR are equal to 1.0) and the albedo is 0.3, giving a temperature of 255 K. In Example 2, we assume that there are no clouds or ice (i.e. A = 0.0), which raises the temperature of Earth to 279 K, showing the importance of the albedo. In Example 3, there are also no clouds or ice (A = 0.0) but the atmosphere now absorbs all the outgoing infra-red radiation (i.e. τ_IR = 0.0) and Earth warms to 331 K. If we now reintroduce clouds and ice in Example 4 (A = 0.3), the temperature drops to 303 K.

Possible questions

Typical questions that could be asked, and which would require students to use this model, are:

1. Which of the variables has the greatest effect on average global temperature?
2. If the average distance from Earth to the Sun was increased by 1% of the current value, the solar constant would be reduced by a factor of 1.0201, i.e. it would become 1343 Wm^-2. What would the temperature be? Assume an albedo of 0.3, τ_IR of 0.3 and τ_VIS of 0.6.

(The solar constant will scale with distance squared, so moving Earth 10% closer to the Sun will mean S = 1370/(0.9)² = 1691 Wm^-2 and moving Earth 1% further away will mean S = 1370/(1.01)² = 1343 Wm^-2)

Of course, far more challenging questions could be asked.

3. If the sand in the Sahara desert could be made into a glass mirror (sand can be melted into glass),

1. How big would the mirror have to be in order to cool the planet by 1°C?
2. What fraction of the Sahara desert would that be?

A more sophisticated model

How does the simple model compare with more sophisticated models such as the Hadley Centre climate model used by the UK’s Meteorological Office (see images below)? In fact, the two models are very similar, except that the Hadley Centre model does not consider the atmosphere as one layer, but splits it into a number of boxes based on altitude, latitude and longitude. For each box, the model directly calculates the amount of incoming UV/visible radiation transmitted and scattered within that box, and the amount of outgoing infra-red radiation transmitted by that box, based on the concentrations of key greenhouse gases and the surface area of cloud and ice. The most sophisticated versions of the Hadley Centre model also consider the heat flux into and out of the ocean and the uptake of CO₂ by vegetation. But if you understand the principles of the simple model, you are well on your way to understanding the more complicated real climate models.

Taking part in a real climate simulation

Climateprediction.net^w2 is the largest experiment to try and produce a forecast of the climate for the 21st century. It does this by recruiting help from people around the world who can offer time on their computers – such as when the computers are switched on, but are not being used to their full capacity. The full climate model has many parameters that can be adjusted; to explore all of these parameters, a phenomenal number of simulations must be performed. Even with the computer resources available to the climateprediction.net team, such an ensemble of simulations would take a very long time. The idea behind climateprediction.net is that anyone can download a version of the model that will explore one of these particular parameters (preset). The model will take about three months to run in the background while you work, without compromising the speed of the computer.

Calculations are performed in three parts. The first part runs calculations using data from the years 1850 to 1900, checking the resulting predictions against temperature records: this is known as the calibration run. The second part runs a simulation from 1901 to the present day. The third stage then runs a simulation of the future climate (2000-2100) with one parameter changed, for example the sensitivity of climate to uncertainties in the sulphur cycle. Once the calculations are complete, the data is automatically uploaded to the UK’s Meteorology Office the next time that the computer is online. The interface software provided (free of charge) with the simulation gives the computer user a graph of the changes to the climate, as they are calculated. Temperature variations by season with latitude, longitude and altitude are just some of the variables that can be visualised. Such features would enable a cross-curricular school project linking geography and the physical sciences.

In the next issue of Science in School, the authors suggest some chemistry experiments relevant to climate change.

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