Tomorrow's Weather
or, the trouble with three decimal places
In the winter of 1961, an MIT meteorologist named Edward Lorenz was running a small weather model on a Royal McBee LGP-30, a vacuum-tube computer the size of a large desk and slower than your phone by a factor of around a hundred million. The model had twelve equations and a handful of variables. He had been running it for some time, watching long sequences of simulated weather scroll out on a printout, looking for patterns. One afternoon he wanted to look more closely at a particular stretch of weather. Rather than starting the simulation over from the beginning, which would have taken hours, he typed in numbers from the middle of the printout as initial conditions, set the program running, and went to get a cup of coffee.
When he came back, the new weather had nothing to do with the old. The same program, given what looked like the same starting state, was producing an entirely different sequence of storms and lulls. Lorenz first thought the computer had broken. It hadn’t. The printout he had typed from showed three decimal places. The computer’s memory had been running the simulation to six. He had typed 0.506 where the machine’s state at that moment had been 0.506127. The difference was about one part in a thousand. Tiny. Tiny was enough.
The behavior wasn’t a bug. It was a property of the math. The equations of school physics are well-mannered: a small error in the input gives a small error in the output, and the error stays small as the equation runs forward. But the kind of equations that describe a fluid — air, water, blood, plasma — have feedback. A wobble at one timestep is amplified by the next, and amplified again, and after enough doublings the original wobble is the size of a continent. Time, in these systems, is an amplifier of imprecision. Lorenz worked out the rate of amplification for his model and arrived at a hard ceiling for forecasting. Today’s models, fed by every satellite and weather balloon humans can put in the air, are reliable about a week out, ten days in good cases, and improving slowly; but the theoretical horizon sits somewhere around two to three weeks, and it does not move no matter how much computer or how many sensors you throw at it. Past that line the atmosphere has erased the difference between what was true and what is now possible. A perfect computer fed with the position of every molecule in the atmosphere except for one would still get next month’s weather wrong.
He published the result in 1963, in the Journal of the Atmospheric Sciences, where it sat largely unread for a decade. In 1972, asked to give a talk at the AAAS meeting in Washington, he and his session chair settled on a title: “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” The metaphor stuck because it landed on something true. A small flap, somewhere upstream in time, can be the difference between two completely different futures. The atmosphere does not have to be magical or random for this to be true. It only has to be deterministic and chaotic, which it is.
The deeper move here, the one that quietly broke a long-standing assumption in physics, is the separation of determinism from predictability. Pre-Lorenz, the unspoken faith of the science of motion was Laplace’s demon: a sufficiently brilliant intelligence, given the position and velocity of every particle in the universe at one instant, could compute the entire future. Quantum mechanics had bruised that dream earlier in the century, but for systems above the atomic scale, classical determinism was supposed to hold. Lorenz did not break determinism. The atmosphere obeys its rules. He broke the bridge between determinism and prediction. The rules can be exact. The future, computed forward from any actually-measurable present, is still uncomputable. This combination — deterministic in principle, unpredictable in practice — is called deterministic chaos.
Chaos has structure, though. It is not noise. Weather is unpredictable but it has seasons. Heart rhythms wander but stay alive. Turbulent fluids never repeat themselves but they fall into recognizable patterns — cells, vortices, cascades. Inside the unpredictability there are attractors: regions of state space the system keeps coming back to, even when you cannot say which corner of the region it will be in next Tuesday. The chaotic dance has rooms it doesn’t leave. This is good news for life. The universe is full of systems that are unpredictable in detail and yet livable in aggregate. You don’t know what next year’s spring will look like; you do know there will be a spring.
The Experiment
Below is a small version of Lorenz’s puzzle. A row of cells — sixty of them — each holding a few weather variables nudged forward by a simple rule. The rule has the kind of feedback that produces fluid motion in actual fluids, and that produces something like fluid motion in this toy. The cells are colored by “temperature”: blue for cold, red for hot, green for in between.
The simulation runs two copies of the same row, top and bottom. The top strip is “reality” — the actual weather, evolving by the rule. The bottom strip is “forecast” — an identical model started from initial conditions that differ from reality by 0.001 in a single cell. Between them is a difference strip, which lights up wherever the two have diverged. Initially the strips look identical and the difference strip is dark. After enough time, they have nothing to do with each other, and the difference strip is a glowing band.
Things to try:
Hit Reset and just watch. The two strips look identical and the difference strip is dark. Both strips show colored bands writhing slowly across the row in lockstep. This is the first part of Lorenz’s afternoon, before he came back from coffee.
Wait. Within a few seconds of simulated time, the difference strip will start to show a faint glow, somewhere near where the original perturbation was placed. The glow grows. It spreads to neighboring cells. After a minute or so the difference strip is a bright band and the two weather strips look like different planets. This is the moment Lorenz noticed the second printout.
Hit Reset and immediately click somewhere on either strip. You’ve added a butterfly to the forecast. Watch the spot you clicked light up briefly in the difference strip, then smear into a region, then become indistinguishable from the rest of the divergence. The flap is visible at first and unrecoverable later. Same trick Lorenz spotted, demonstrated in seconds instead of days.
Slow the speed slider to 1× and watch the strips evolve in slow motion. Each time you see a band of red move across the row, you are watching local “convection” — the equation’s analog of warm air rising. It is not random. It is coupled to the bands beside it in a specific way. This is the structure inside chaos.
Crank the speed slider up and watch the “mean diff” number. It starts near zero, holds there for a while, then climbs — slowly at first, then faster — and finally levels off around 5 to 8. The plateau is the attractor. The two strips have become statistically independent and the difference cannot grow any larger than that. The chaos has rooms it doesn’t leave.
The rule is a few lines. The starting conditions were the same to within a thousandth. Nothing was added to either strip. Nothing was random. And after a few minutes of clock time, the two strips have nothing to do with each other. This is what determinism actually looks like when you let it run on the kind of equation that runs the world.
It is also why long-range weather forecasts fail, why the heart’s beats vary breath to breath, why two identical cars driven the same way down the same highway by twins do not arrive in the same lane, why nobody alive has ever predicted the next big swing of an economy, and why your plans for next year, examined closely, turn out to be plans for some neighboring year that did not arrive. It is also why those plans were not wasted. Chaos has structure. The seasons come. Spring comes. The universe writes weather every day without writing any particular weather, and much of what looks designed in the world was made by processes built out of exactly this kind of mathematics. Predictable in the rooms they live in. Unpredictable in any specific corner of any specific room.