This post is a continuation from this previous post.

The weather is a classic example of a chaotic system. Try as we might, we just cannot predict the weather well. Even with computers the size of villages we haven’t been able to produce a model that mimics the real weather beyond a few days’ time. Why is that?

For the answer, we have to turn to Edward Lorenz; mathematician, meteorologist and Father of Chaos (theory). Lorenz tried to predict the weather by creating a computer model that could calculate how meteorological variables such as temperature, wind-speed, air-pressure, etc. would interact over time. The idea was that if he could represent each of those variables mathematically, and calculate the effect each would have on the others, there was no reason why he couldn’t calculate the future behavior of the weather. Just as long as he could get his model right. To this end, Lorenz took extraordinarily accurate measurements of the weather, so that at the starting point his model and the real weather were perfectly matched. Once everything was in place, Lorenz hit ‘Enter’, let the model run, and compared the model’s predictions to the real weather conditions.

Initially, the two systems mirrored each other quite well. However, within a few days the model started behaving erratically, and would no longer give him a reliable prediction. Confused, Lorenz went back and checked his data. The variables he put into his model were exactly the same as the weather measurements, and his model matched the real climate perfectly at the point at which he pressed ‘Go’. So how could the two systems possibly diverge if their starting conditions were the same?

It dawned on Lorenz that his model was only a perfect match to a certain level of accuracy. The measurement he had taken of temperature, for example, was accurate only to a limited number of decimal places. So whereas in his model the starting temperature might be set to, say, 22.00000000000°C and not 21.99999999999°C, in reality the actual temperature could lie somewhere between the two. This is what’s known as a rounding error, and Lorenz’s model was full of them.

Like the hairs on our snooker table, these tiny rounding errors have very little effect over short periods of time, but after a while they add up to produce drastically different behaviours. This is the famous ‘Butterfly effect’, where a tiny change in one variable (a butterfly flaps its wings in Hong Kong) can lead to huge differences (a hurricane in New York instead of sunshine) given enough time. Because these tiny errors are catastrophically additive, a digital model of a chaotic system will only behave exactly like the original if its starting parameters are exactly the same as the original, totally free from rounding errors.

Which brings us back to the digital brain…

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