Chaos Theory

The first true experimenter on the impact of chaos on natural systems was meteorologist, Edward Lorenz. In 1960, he was working on the problem of weather prediction. He progammed a computer with a set of equations designed to predict how pressures would vary at a point. Once the model predicted the pressures at each point in the future, a weather map showing isobars for this future date was drawn.

The program required entry of initial (or observed) pressures at each location. But this original entry included the pressures measured to six decimal places (e.g., 1012.234231 mb). The resulting predicted weather map, incidentally, did not look very much like the map that actually occurred. In essence, that part of the experiment had failed.

In 1961, Lorenz attempted to duplicate the same results using the same computer model. . He entered the same numbers, but rounded off to three decimal places (e.g., 1012.234 mb), and left the computer to run the program to its conclusion. When he came back an hour later, the the computer program had produced a sequence that had evolved differently. Instead of producing the same pattern as it did in 1960, the computer produced a much different pattern. Figure 1 shows the pressure traces (sort of a metgram) at one location for the two experiements. Note that the pressure traces initially were identical, but began to diverge from one another about 1/3 of the time through the expermient. Eventually the pressure traces did not resemble each other (See figure 1.)

Eventually Lorenz figured out what happened. The problem was not with the computer model itself, but with the initial data given to the computer. The computer stored the numbers to six decimal places in its memory. To save time, in 1961 he had provided the computer with input data only to three decimal places.

Fig. 1: Example of Two Curves of Pressure Variation at a Location. The second curve was obtained by varying the initial conditions by a percentage of only .0127%.

By all conventional ideas of the time, the model should have correctly predicted the evolution of the pressures. And the model should have worked exactly the same both times; Lorenz should have gotten a sequence of the pressure change very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Lorenz had assumed that inputting pressure observations to the fourth and fifth places, impossible to measure using reasonable methods, would not have a huge effect on the outcome of the experiment.

Lorenz proved that accurate prediction of the evolution of a weather variable, like pressure, depends sensitively on the accuracy of the initial observations. Note in Fig. 1 above, that the two traces overlap initially, but diverge widely by the end of the sequence. In essence, using the same computer model, and the same initial conditions, two dramatically different results occurred.

Lorenz realized that forecasting future pressures was compromised by the fact that even the most sophisticated barometer at the time had an intrinsic error of around 0.1 mb. Since the model suggested that the atmosphere might be sensitive to pressure variations on the order of 0.0001 mb, he concluded that the accuracy of weather forecasts based upon computer models would be compromised by the quality of the initial weather observations. He also concluded that weather forecasts of this sort were not possible because the end results would be so different each time a prediction was made with basically the same starting data, the final results would appear to be chaotic. Thus was born CHAOS THEORY.

This effect also came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings.

"...The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does..." (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

Just a small change in the initial conditions can drastically change the long-term behavior of a system.

Ensemble Forecasting

At the time it was thought that chaos theory implied that weather forecasting was not possible, because to make an accurate forecast, the initial conditions would need to be known to an accuracy far surpassing that of our best instruments. In reality, weather forecasting IS possible, even taking into account chaos theory, but only if one carefully limits the results one expects to get (in class discussion). While it is not possible to forecast exact conditions at an exact time too far into the future, it is possible to forecast general conditions accurately across a reasonable time interval (say that the temperature will be in the low 70s in the morning five days from now as opposed to predicting accurately the exact temperature at 7:42AM).

It turns out that all predictions made of a specific weather element cluster around an "average" value called the strange attractor. While that average value is almost never perfectly accurate in predicting what exactly will be observed at an exact instant in time, it is very accurate in predicting the general "flavor" of the weather in advance.

As you might imagine, the "strange attractor" gets less and less accurate the farther out from present time (the starting point of the experiment) one attempts to make a forecast. Scientists in general and meterologists in particular have gotten quite ingenious in using the concept of the "strange attractor" to increase the accuracy of long range forecasting.

One way of approaching this is by putting a purposeful error into the observations to see what will happen. In fact, one can do this several, to many, times, to produce a cluster of possible future forecasts, called the "ensemble." Ensemble forecasting is a way of evaluating the "believability" of a forecast. The basis for this approach from Lorenz's original experiments on the impact of chaos on weather prediction.