Lyapunov time for the atmosphere is roughly two weeks. That's a hard wall. Not a limit of current computers, or current satellites, or current models. A property of the system. Every forecast in your weather app past about day ten isn't a bad forecast, exactly. It's a different kind of object wearing a forecast's clothes.

Lorenz and the rounding error

The story is famous enough that I'll keep it short. In 1961 Edward Lorenz was running a simple atmospheric model on a computer at MIT, twelve equations, nothing like a real forecast. He wanted to re-examine a run, so he restarted it partway through by typing in the numbers from a printout. The printout showed three decimal places. The machine had been using six. He typed 0.506 where the machine had 0.506127.

The rerun tracked the original for a while and then diverged completely. Within a couple of simulated months the two runs had nothing in common. A difference in the fourth decimal place, one part in ten thousand, had grown until it was the whole state of the weather.

What Lorenz had found, and spent the next decade formalizing, was sensitive dependence on initial conditions. In a chaotic system, nearby trajectories separate exponentially. The rate of separation is the Lyapunov exponent, and its inverse, the time it takes for a small error to grow by a factor of e, is the Lyapunov time. For the large-scale atmosphere, the error-doubling time is about two days. Give it two weeks and a tiny error is as big as the difference between any two random days in the same season.

What the wall is made of

Here's why this is a wall and not a challenge.

Suppose you wanted a good forecast at day twenty. Errors double every two days, so at day twenty your initial error has grown by a factor of about a thousand. To have day-twenty error as small as today's day-ten error, you'd need your initial measurement to be a thousand times more precise. Every temperature, every pressure, every wind vector, everywhere, including the ocean surface and the upper stratosphere and the places where there are no instruments.

Then, having done that, you get ten more days. Then you need a thousand times better again for the next ten. The precision required grows exponentially with lead time, and there is a floor: at some point the required precision is smaller than the size of the turbulent eddies you can't model, or smaller than the measurement noise of any instrument, or smaller than the effect of a thing you were never going to measure, like a wildfire in Siberia or a ship's exhaust.

Lorenz made this argument in 1969 and estimated the limit at about two weeks. Fifty years of progress have refined that estimate and not moved it much. Some recent work suggests you might squeeze it to fifteen days for large-scale patterns with perfect models and near-perfect data. You won't get to thirty. The system doesn't permit it.

Progress happened, up to the wall

None of this means forecasting has stalled. Forecast skill has improved at a steady rate of about one day per decade: a five-day forecast today is roughly as accurate as a four-day forecast was ten years ago and a three-day forecast twenty years ago. Better satellites, better models, more compute, the assimilation of hundreds of millions of observations a day.

The gains have come from pushing closer to the wall, not through it. Skill at day three is now excellent. Day five is good. Day seven is useful. Day ten is marginally better than just guessing the climatological average for the date. And then it flattens, because there is nothing on the other side.

Ensembles: forecasting the uncertainty instead

The modern response to chaos was not to fight it but to measure it. Instead of one forecast from one set of initial conditions, the major centers run ensembles: the European Centre runs fifty-one versions of its model, each started from a slightly perturbed initial state within the measurement uncertainty.

For the first few days the fifty-one runs agree, and the agreement is the confidence. Then they start to spread. The spread is the forecast's own estimate of its ignorance. When forty-eight of fifty-one members have rain on Thursday, that's a 94% chance. When they're split down the middle at day eight, the honest forecast is "we do not know," and the probability you see is the ensemble reporting that honestly.

This is the actual content of "40% chance of rain." Nobody is hedging. That number is the fraction of plausible atmospheres, consistent with what we measured, that produced rain. Chaos didn't make forecasting impossible. It made forecasting probabilistic, and the probability is the product.

The category error

So what is the fourteen-day forecast in your app?

Past day ten, the ensemble members have spread so far that their average is close to the climatological average for that date. The app shows you that average, with an icon. It's a forecast in the sense that it has a date and a temperature. It's a horoscope in the sense that the temperature is what late November is usually like, and would have been displayed regardless of what the atmosphere was doing today. The app isn't lying. It is answering a question, "what is a typical day like then," and presenting it in the format of a different question, "what will that day be like."

That's the category error. A trajectory question is being answered with a distribution answer, and the interface hides the swap.

Climate is the other category

Which is also, and this is the part worth carrying around, why climate prediction isn't undermined by weather chaos. People confuse these constantly, in both directions.

Weather is a trajectory: where will the system be on a given day. Chaos destroys that past two weeks. Climate is a distribution: what is the average and variance of the system over decades. Chaos does not touch that at all. You can't predict a single coin flip. You can predict the fraction of heads in ten thousand flips very precisely, and if someone bends the coin you can predict how the fraction changes.

Climate models aren't weather forecasts run for a hundred years. They are statistical descriptions of a system under a changed energy budget. They don't claim to know what July 14, 2087 will be like. They claim to know what Julys will be like, on average, and they're answering a question chaos has nothing to say about.

Living at the wall

I check the ten-day forecast anyway. Everyone does. There's something in the brain that wants the trajectory and will take a distribution dressed as one rather than nothing.

But knowing where the wall is changes what the check means. Days one through three: plan around it. Four through seven: probably, bring a jacket. Eight through ten: it might rain sometime that week. Past ten: it is late November. That was always going to be the forecast. Lorenz could have told you in 1969, and did.