Kodi Arfer / Writing / Introduction to Chaos

Okay, But What Does Any of This Have to Do with Real Life?

Many important natural processes are best modeled with dynamical systems that turn out to be chaotic. The canonical example is the atmosphere.

The weather forecasts we're familiar with today are largely the products of mathematical models of Earth's atmosphere. The models use current values of the temperature, wind velocity, humidity, and air pressure at each of many points to predict future values of the same variables. The underlying equations are mostly just applications of generic thermodynamic formulae. One such model, used as of 1992 by the European Centre for Medium Range* Weather Forecasting in Reading, England, divides Earth's surface into 45,000 regions and the space above it into 31 layers of elevation, yielding a system with five million variables. Lorenz (1992) points out "Lest a system of 5,000,000 simultaneous equations in as many variables appear extravagant, let us note that, with a horizontal grid of less than 50,000 points, each point must account for more than 10,000 square kilometers [about three Long Islands]. Such an area is large enough to hide a thunderstorm in its interior."

Of the many models utilized since the birth of numerical weather prediction fifty years ago, nearly all have a property in common: given two slightly different initial states, a model's predictions will diverge dramatically within two weeks.** Typically, the error in predicted values doubles every two days. Thus, given that measurements of current conditions are always approximate, weather forecasts at the range of two weeks perform only slightly better than chance.

How essential is chaos to the weather? Some empirical tests called the dishpan experiments suggest: less than you might expect. In these experiments, dishpans filled with water and sprinkled with aluminum powder (so one could see currents) were systematically heated and cooled as they rotated counterclockwise (in one case, by means of an old turntable). The idea was that the dishpan gave you a simulated view of the atmosphere as if you were looking down at the North Pole, Earth were flat, and the atmosphere were pure water instead of a mixture of gases. At some combinations of rotation speeds and temperatures, these dishpans exhibited surprisingly realistic meandering jet streams and vortices. At others, only symmetric, circular currents were produced. One might imagine that if Earth's day were longer, or if we were nearer to or farther from the Sun, our atmosphere would behave in a similarly predictable fashion, entirely obviating the need for elaborate weather prediction.

On the other hand, weather prediction is almost by definition a tricky business. There are many features of the climate that can be predicted reliably at long range. I know that in February 2011 in Meadville, Pennsylvania, there will be snow on the ground and temperatures will stay well below 70 °F. But this isn't the domain of weather prediction. Rather, what we want to learn from a weather forecast is exactly the things that are hard to predict—the details, like the temperature to the nearest degree, and the irregularities, like which days will be rainy. It's all marvelously quixotic.


* I should note that in weather forecasting, "range" refers to temporal range.

** In fact, a common weather-prediction technique nowadays is a Monte Carlo method called ensemble forecasting, in which you synthesize several different predictions. The component predictions may even be created with different models.