Three Decimal Places

In 1961 Edward Lorenz was running a simple weather simulation on a machine at MIT. Wanting to re-examine part of a run, he restarted it from the middle rather than from the beginning, typing in the values from a printout.

The new run diverged from the old one. Slowly at first, then completely: within a simulated couple of months the two weather patterns had nothing to do with each other.

He assumed a hardware fault, and the cause was that the printout gave three decimal places while the machine held six. He had restarted from a state differing from the original by about one part in ten thousand.

That difference had doubled, and doubled again, until it was the whole forecast.

This is sensitive dependence on initial conditions, and the important thing to be clear about is that Lorenz's system was entirely deterministic. There is nothing random in it. Given perfect knowledge of the starting state, the trajectory follows exactly.

The trouble is that measurement always has finite precision, and in a chaotic system the difference between two nearly identical states grows exponentially. Whatever error you start with, however small, it doubles at a characteristic rate until it dominates.

A chaotic system is not a random one, and the distinction is worth holding. A chaotic system has structure: it stays on an attractor, its statistics are stable, its behaviour is bounded. Lorenz's equations produce the famous butterfly-shaped figure, and the trajectory never leaves it and never repeats. You can know a great deal about where a chaotic system will be and very little about when.

The practical consequence is a forecast horizon, and it is a property of the system rather than of the technology.

For the atmosphere it is around two weeks. Beyond that, the error in the initial state (which is unavoidable, since we do not have a thermometer on every cubic metre) has grown enough to swamp the signal.

The cruel part is how improvement scales. Halving the initial error does not double the forecast range; it adds a fixed increment, because the growth is exponential. So a hundredfold improvement in observation buys a few extra days, and this is why weather forecasting has become enormously more accurate within its horizon over fifty years while barely extending how far ahead it reaches. A modern five-day forecast is about as good as a one-day forecast was in 1980. There is no version of this technology that tells you about next month.

What forecasters do instead is give up on the single trajectory.

An ensemble forecast runs the model fifty or more times from slightly different starting states, all consistent with what was actually measured. If the runs stay close together, the atmosphere is in a predictable state and confidence is high. If they diverge rapidly, it is in a sensitive state and it is not.

This is where a probability of precipitation comes from: it is the fraction of the ensemble that produced rain. The spread is itself information, and a forecast that reports only the most likely outcome has discarded the most useful part of the calculation.

Which leads to the standard misunderstanding. A forecast said 30% chance of rain, it rained, and the forecast was wrong.

A single outcome cannot falsify a probability. The relevant test is calibration: across all the occasions a forecaster said 30%, did it rain on about 30% of them? Weather services measure this routinely and are strikingly well calibrated, considerably better than doctors, lawyers or economists on comparable measures, largely because they get immediate feedback on every prediction, every day, for decades.

Finally, the confusion that matters most in public argument.

Chaos limits weather forecasting. It does not limit climate projection in the same way, and the reason is that they are different kinds of question.

Weather is the system's specific trajectory: what happens on Tuesday. Climate is its statistics: the distribution of temperatures over decades. Nobody can tell you whether it will rain in Manchester on a particular day next year, and a great deal can be said about how many wet days Manchester will have.

The analogy is a die. You cannot predict a single roll. You can predict the average of ten thousand rolls with considerable precision, and if somebody weights the die, you can predict how the distribution shifts without knowing any individual result.

Climate projection is a boundary-condition problem (what happens to the distribution when you change the energy balance) rather than an initial-condition problem. Chaos is about initial conditions, which is precisely why it does not transfer.