Predicting chaos.

Recently, I was caught out by a sudden downpour that was absolutely not predicted by the BBC, and it got me thinking about how weather-persons still so often get it hilariously wrong. The human race has achieved some spectacular things: we've sent cameras to Pluto, we've decoded DNA, we're untangling quantum mechanics, and we've invented the iPad ... and yet, whether or not it will rain 4 Tuesdays from now is almost completely unknowable (alright alright, it's knowable if you're in India mid-Monsoon season, smartypants). The reason it's (often) unpredictable is chaos, and as an Englishman I feel particularly suited to talk about unpredictable weather. This story begins with a classic problem in mathematics/physics, something unimaginatively known as the three body problem ...

After Isaac Newton - arguably England's finest contribution to physics - reluctantly published his now famous equations of motion and gravity, mathematicians started to wonder about what our solar system might look like in a few million years. They were also very interested in detailed equations of the motion of the planets and moon so they could better use the position of the moon to navigate by, even on cloudy days.

When they plugged just two bodies (such as the Earth and the Sun) into Newton's equations, simple paths through time popped out. It worked beautifully. They could arbitrarily add mass (or take it away, if they wanted to) or speed to the Earth, and pick an arbitrary time in the future (say, 5 billion years from now, assuming the Sun hasn't gobbled it up by then) and predict exactly where it would be. Everything worked like clockwork. So far so straightforward.

This may as well have been what their Solar system predictions looked like.

This may as well have been what their Solar system predictions looked like.

There's a slight problem though ... What about the moon? The moon has its own gravity that affects the Earth and even slightly affects the Sun, so we can't disregard it. And, what about the other planets? Unfortunately for the mathematicians, our Solar system isn't just made of two bodies; there's a star (taking up 99.8% of all the mass), 4 solid planets, 4 gassy planets (or maybe 5?), dozens of moons (some bigger than Mercury) and thousands of miscellaneous planetoids, comets and asteroids - and they're all pulling on each other. When the mathematicians factored in the 8 known planets, and fast-forwarded a few million years to see what the system would look like, no one could agree on, well, anything! It was a right mess.

For a little context, at this point in history it was believed that basically every law of physics had been written down and solved, and all the remaining mysteries were just because we didn't have the tools to know all the "hidden parameters". For example, we had very good understanding of heat flow/transfer, but no one claimed to be able to predict a volcanic eruption, because no one had the tools to make even crude estimations of what was going on underground. If all the "hidden parameters" could be measured, they argued, then surely everything can be predicted with absolute certainty.

But with the planets, the results seemed to vary depending on who was asking, and no equations of the paths of all the planets were popping out. It looked ... random. Physicists don't like randomness, it implies that the universe doesn't obey laws, that it instead just does what it wants. No matter how precise they made the inputs into the maths (such as the mass of Jupiter), the outcomes still just eventually descended into randomness. This wasn't a complex system - there was no chemistry, no solar flares, no evolution, no turbulent atmospheres, no collisions, no radiation, etc. there were just balls gravitationally falling around each other in a vacuum. Physicists are supposed to like spheres in a vacuum! The problem was such an embarrassment that Oscar II, the King of the Sweden, announced a sizeable cash prize for whomever could solve the problem and bring the solar system back into order.

No one got very far, despite the promise of cold hard cash, not until a man called Poincaré came along. Ironically, Poincaré didn't actually solve the problem, he instead proved that it can't solved. The reason is subtle, but in a nutshell, when three objects are all pulling on each other (in the case of the Sun, Earth and Moon they're pulling via gravity), if you nudge one, you affect the pull on the other two, which will then have a feedback effect on the one you nudged, which will then again affect the motion of the other two, and so on and so on. What Poincaré proved is that if you want to know the exact future of three bodies, you have to know the exact details of all their starting conditions, to an infinite degree of accuracy.

Although they had no problems with the equations of gravity, it was (and still is) impossible to know the masses and trajectories of the planets and moons to an infinite degree of accuracy. They were all working with (very good) approximations, and in a chaotic system tiny changes make all the difference. Because the Solar system is made of lots of parts that are all interacting with each other, any small change or nudge to one tiny part will eventually affect everything, and any small uncertainty in one factor will eventually lead to total uncertainty in all factors. Quite how long it takes for these affects to become noticeable is complicated, but take my word for this: if the solar system were to last forever, an infinite amount of spins and orbits, then the action of me striking my keyboard right now be enough to eventually alter the shape of Saturn's rings. So would your eyes popping out of your head at that thought.

And so it goes for the Butterfly Effect, which states, in simplistic terms, that the flap of a butterfly wing can be enough to set off a cascade of air flows that could eventually lead to a hurricane on the other side of the world. The film of the same name starring Ashton Kutcher touched on this, and if you watched it you probably thought "how can a butterfly possibly cause a hurricane?" (or, more likely, you gave it zero thought and instead spent 2 hours wondered how Ashton ever made it as an actor).

Well, in reality, a butterfly probably won't cause a hurricane any more than a bout of flatulence will. But, whether a butterfly did or didn't flap its wing one day will eventually lead to two totally different predictions of the global weather patterns (assuming everyone else stayed perfectly still), because all gusts of wind interact with each other in a feedback process similar to nudging one of those three gravitating bodies. This is partly why fluid mechanics is still a very active area of research. The concepts are simple, the outcomes are a soup of uncertainty. Eventually, all wind patterns look random, so it becomes impossible to make long-term predictions about whether there will be a hurricane or not. If there is a hurricane, does that mean the butterfly caused it? Almost definitely no. But the butterfly did perturb a chaotic system, and that eventually leads to noticeable changes and complete unpredictability everywhere.

In 1987, British weather reporter Michael Fish uttered the famous last words "Earlier on today,... a woman rang the BBC and said she heard there was a hurricane on the way... well, if you're watching, don't worry, there isn't!", and lo and behold, there was a monster of storm that killed 19 people (1 year old me was happily sunbathing in Athens at the time). It is definitely going too far to let chaos theory excuse dear Michael, but it is the reason why weather forecasts can't make reliable predictions further than a couple of weeks away. Even with our numerous and constantly watchful weather monitors, we just don't know. And in fact, even when we drive flying cars, live to be 1000, and take 5 hours a day out to worship the great and noble deity Elon Musk, we probably still won't have a reliable weather forecast.

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