Devlin's Angle

November 2010

The Other Thing Fourier Did

Quick, what first comes to mind when I mention the name Joseph Fourier?

Almost certainly, you answered "Fourier analysis" or "Fourier transform." If not, you probably found yourself on this website by mistake. For those are what the famous French mathematician (1768 - 1830) is best known for to mathematicians.

Well, I suppose if your interests are the history of mathematics, you might have answered "He is one of the first mathematicians to carry out a detailed mathematical analysis of heat flow." But what his name probably did not evoke is the response "Greenhouse effect." Yet he is the first person to observe a phenomenon that has become a hot (sic) topic of political discussion in the United States.

In October 1824, Fourier published a scientific paper titled "Remarques generales sur les Temperatures du globe terrestre et des espaces planetaires" in the journal Annales de Chimie et de Physique, Tome XXVII (pp.136-167), in which he presented his results from a mathematical analysis, that climate-change experts today (the ones who actually are experts) generally regard as the start of climate-change science.

Fourier stumbled across the greenhouse effect when he puzzled over this particularly tantalizing question: Every day the Sun's rays strike the Earth's surface and warm it up, so why doesn't the planet keep heating up until it is as hot as the Sun itself?

His answer was that the heated surface must emit invisible infrared radiation, which carries the heat energy away into space. But when he calculated the effect mathematically, he got a temperature well below freezing, much colder than the actual Earth. The difference, he suggested, must be due to the Earth's atmosphere. Somehow it kept part of the heat radiation in. He tried to explain this by comparing way Earth's atmosphere holds in heat from the Sun to the way the glass of a greenhouse keeps in the heat. He actually wrote about a glass box rather than a greenhouse, but the name "greenhouse effect" for the effect he described was coined soon afterwards.

Fourier did not set out to think about climate change as such. Rather he was investigating the purely scientific question of what determines the average temperature of a planet like the Earth? This was the sort of question that physicists were just beginning to learn how to attack back in the early 19th century. To understand heat transfer, Fourier invented the powerful mathematical techniques he is best known for to mathematicians today - techniques that turned out to have many applications besides heat flow, in particular, forming the basis of modern music synthesizers and MP3 players.

In fact, Fourier's glass box example was far too simple. It's quite different physics that keeps heat inside a greenhouse. The main effect of the glass is to keep the air heated by contact with sun-warmed surfaces from wafting away, although the glass does also keep heat radiation from escaping. In 1862, the Irish physicist John Tyndall gave the correct explanation for how the atmosphere retains heat. He discovered in his laboratory that certain gases, including water vapor and carbon dioxide, do not transmit heat rays. Such gases high in the air help keep our planet warm by interfering with escaping radiation. He wrote: "As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial rays, produces a local heightening of the temperature at the Earth's surface."

So we knew the basic scientific principle behind global warming back in 1862. But it was not until the mid-20th century that scientists would fully understand how the effect works, and with the aid of computers could calculate it with some precision. Today, we know that the actual physics is fairly complex, but a rough explanation goes like this.

Visible sunlight penetrates easily through the air and warms the Earth's surface. When the surface emits invisible infrared heat radiation, this radiation also easily penetrates the main gases of the air. But as Tyndall found, even a trace of CO2, (a single bottle-full in his laboratory) is almost opaque to heat radiation. Thus a good part of the radiation that rises from the surface is absorbed by CO2 in the middle levels of the atmosphere. Its energy transfers into the air itself rather than escaping directly into space. Not only is the air thus warmed, but also some of the energy trapped there is radiated back to the surface, warming it further. The more CO2 there is in the atmosphere, the greater the warming.

As I noted, the actual mechanism is much more complicated tan the above description. The American Institute of Physics has an excellent description at www.aip.org/history/climate/simple.htm.

In addition to being a brilliant mathematician who became a professor at the prestigious Ecole Polytechnique in Paris, Fourier was an interesting person. He was a promoter of the French Revolution. He accompanied Napoleon on his Egyptian expedition in 1798, whereupon the emperor appointed him governor of Lower Egypt and secretary of the Institut d'Egypte. When they were cut off from France by the English fleet, he organized workshops to supply the French army with munitions. After the British victories and the capitulation of the French in 1801, Fourier returned to France, and was made prefect of Is¸re. It was while there that he carried out his investigations of the propagation of heat.

So now you know.


Devlin's Angle is updated at the beginning of each month. Find more columns here. Follow Keith Devlin on Twitter at @nprmathguy.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.