The Earth's climate is a very complex system. It involves the atmosphere, the oceans, the Sun, vegetation and ice, not to mention human activity. Predicting what the climate will do over the next hundred years or so requires all the power of science and mathematics. We've explored this in the article Climate change: Does it all add up?

However, it turns out that sometimes even a simple bit of mathematics can go a long way. One of the most useful examples of a simple, but powerful, mathematical model used by climate scientists is the energy balance model — it only uses ideas from A level mathematics.

In the energy balance model we assume that the Earth is heated by the radiation from the Sun and that it has an average (absolute) temperature $T.$ Some of this heat energy is absorbed and the rest is radiated back into space. We then reach an equilibrium when these two balance. Now the heat energy from the Sun is given by $$(1-a)S,$$ where $S$ is the incoming power from the Sun (which is around $342 W m^{-2}$ on average), and $a$ is the albedo of the Earth: it measures how much of this energy is reflected back. The current value is $a =0.31.$ (The albedo would be higher if the Earth were covered in ice, since ice reflects the energy from the Sun.) The heat energy radiated back into space is given by $$\sigma e T^4,$$

Here $$\sigma = 5.67 \times 10^{-8} W m^{-2} K^{-4}$$ is Stefan-Boltzmann constant (with temperature measured in Kelvin, denoted by $K$), and $ e$ is the emissivity, which is a measure of how transparent the atmosphere is. On the Moon, with almost no atmosphere, we have $e = 1$. Currently on the Earth we have $e =0.605.$

To find the Earth's temperature we balance these two expressions so that $$\sigma e T^4 = (1-a)S,$$ and then we solve this for $T $ to give $$T = \left(\frac{(1-a)S}{\sigma e}\right)^{1/4},$$

which you can evaluate on a calculator. Isn't that nice! Try it with the values above to find the current mean temperature of the Earth. Now take $e = 1$ to find an estimate for the average temperature of the Moon (all the other parameters stay the same). To check if you got the right answer, see here.

The power of this expression is that we can perform what if experiments to see what can happen to the climate in the future. For example, if the ice melts then the albedo $a$ decreases which means that $(1- a)$ and hence $T$ increases. Similarly if the emissivity $e$ decreases then the temperature $T$ increases. This is a worrying prediction as it is well known that increasing the amount of greenhouse gases, such as carbon dioxide, in the atmosphere leads to a decrease in $e.$ Thus there is a direct cause and effect link between an increase in carbon dioxide (which is of course what we are seeing) and a rise in the predicted mean temperature of the Earth.

Thus, even a simple model can provide us with useful, if worrying, insights into the future of our planet. Along side basic models such as this one, climate scientists use hugely complex models that take into account all the factors that influence the Earth's climate. To find out more, see Climate change: Does it all add up?

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About this article

This article first appeared in the March/April 2016 edition of MEI's M⁴ Magazine.

Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the Institute of Mathematics and its Applications, Chair of Mathematics for the Royal Institution and an honorary fellow of the British Science Association. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.

He has co-written the popular mathematics book Mathematics Galore!, published by Oxford University Press, with C. Sangwin and features in the book 50 Visions of Mathematics ed. Sam Parc.