Weather: Catastrophe, Chaos, Complexity and Crisis

Developments in mathematics over the past 30 years offer different ways of explaining anomalous phenomena such as sudden changes in the weather. The maths may not lead to better forecasting, but it offers a good explanation of why the forecasters sometimes get it wrong.

We have heard a good deal in recent years about the susceptibility of the weather to the mathematics of Chaos. Edward Lorenz started it all in 1975 with the memorable image of the flap of a Brazilian butterfly's wings causing a typhoon in Texas. People rapidly got the wrong message, however, and started viewing butterflies with intense suspicion wherever they dared flap their wings. You couldn't even avert a typhoon by swatting them, because the flap of your newspaper might lead to a hurricane somewhere.

Yet the mathematics of Chaos, as Lorenz developed it, showed a far more precise picture. The existence of Chaos in a dynamic system depends on the starting conditions. For most such systems, if you measure the initial conditions reasonably accurately, then you can predict what will happen with an equal degree of accuracy. The Chaos theorists showed, however, that in certain systems, under certain special conditions, the initial measurements are infinitely sensitive. The slightest discrepancy in measurement could throw out the results by a vast amount. Most of the time, a butterfly's flap will make no real difference. Only at moments of extreme sensitivity of the initial conditions will its wretched wing-flap throw the whole of Texas into Chaos.

Another theory, however, suggests that we may always be on the brink of Chaos. According to Crisis Theory - a branch of mathematics developed only in the Nineties, complex systems tend naturally to settle not, as you might think, into the most stable state they can reach, but into the most unstable state that does not collapse completely. Look what happens when you make a pile of dry sand: it will always settle into a neat cone, but when you add more sand at the top, it sets off a series of small avalanches down the sides before it settles again. The settling points are perfectly balanced, yet inherently unstable. Just as in the cases when Chaos Theory operates, the smallest perturbation of the initial conditions causes major disruption.

Chaos Theory and Crisis Theory come together in Complexity Theory, which suggests that complexity of structure arises quite naturally out of any simple system, and the complex structures that emerge then become subject to sets of simple rules of their own. Calm seas, mild breezes and Newton's laws of motion may produce hurricanes, floods and whirlwinds, which then obey their own physical laws as simple as, but quite different from, those we started with.

If you want a description of weather disasters, however, you can hardly do better than the Catastrophe Theory developed in the Seventies. This was designed to deal with the mathematics of sudden change and was used to bring the precision of maths to such inexact sciences as psychology and economics. When animals (or people) change from cowering fear to aggression, when stock-exchanges suddenly collapse, or when a dark cloud suddenly unleashes a torrent of rain, Catastrophe Theory explains what drives a previously stable system over the edge.

Thanks to all of these theories, we may soon be able to state one thing with complete confidence: that the weather is inherently unpredictable.