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Weather forecasts are good for a few days and then get vague, even with powerful computers. Edward Lorenz found out why by accident, while waiting for a cup of coffee. This post tells that story in his own words, shows with simple numbers how a small error can swamp a forecast, lists the mistakes students make about chaos, and ends with what you can borrow from how Lorenz worked.
Who was Edward Lorenz?
Lorenz was born in 1917 in West Hartford, Connecticut, and died on 16 April 2008 at his home in Cambridge, Massachusetts, aged 90, according to MIT News. He studied mathematics at Dartmouth and Harvard, served as a weather forecaster for the US Army Air Corps in the Second World War, and then took a doctorate in meteorology at MIT, where he spent his career.
MIT News records that he shared the 1983 Crafoord Prize and won the 1991 Kyoto Prize. The Kyoto Prize citation says he discovered "deterministic chaos" using a system of differential equations with only three degrees of freedom.
What happened when Lorenz rounded his numbers?
Lorenz told the story in his 1991 Kyoto Prize lecture, A Scientist by Choice. In the late 1950s he wanted to test a claim that a statistical method, linear regression, could forecast as well as any other. He built a model of twelve equations whose variables stood for broad features of the weather, and ran it on a desk-sized Royal McBee LGP-30 computer. It advanced in six-hour steps and printed the twelve values once per simulated day.
One day he wanted to look at a solution in more detail. He typed in twelve numbers from an earlier printout, restarted the computer and went for a cup of coffee. When he returned about an hour later, the new run no longer agreed with the old one. At first the two matched, then they differed in the last decimal place, and then the differences grew, doubling in size about every four simulated days, until after sixty days the two runs were unrecognisably different.
The cause was simple. The computer carried about six decimal places, but to fit twelve numbers on one line he had told it to print only three. The numbers he typed back were rounded. If the real atmosphere behaved like the model, Lorenz realised, long-range weather prediction would be impossible, since real weather is never measured to three decimal places.
He published the general result in 1963 as Deterministic Nonperiodic Flow020%3C0130:DNF%3E2.0.CO;2) in the Journal of the Atmospheric Sciences. Its abstract says that nonperiodic solutions are ordinarily unstable to small changes, so slightly differing initial states can evolve into considerably different states. MIT News says the name "butterfly effect" grew out of a 1972 talk titled "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?"
How fast does a small error grow? A worked example
Take Lorenz's observation that differences doubled about every four simulated days. Rounding to three decimal places can leave an error of up to 0.0005 in a number.
How many doublings until that error reaches about 1, the size of the numbers themselves? Solve 0.0005 × 2ⁿ ≥ 1, which means 2ⁿ ≥ 2,000.
2¹⁰ = 1,024, not enough
2¹¹ = 2,048, enough
So n = 11 doublings, and at four days per doubling that is 11 × 4 = 44 simulated days. After that the forecast carries no information about the original state.
Now suppose you measure ten times more precisely, with an error of 0.00005. You need 2ⁿ ≥ 20,000. Since 2¹⁴ = 16,384 and 2¹⁵ = 32,768, n = 15 and the horizon becomes 15 × 4 = 60 days. Ten times better measurement bought only 16 extra days. That is the hard lesson of exponential error growth: precision helps, but slowly. MIT News notes that Lorenz's work led to the conclusion that useful weather prediction may be impossible beyond two or three weeks.
What mistakes do students make about chaos?
"Chaos means random." A chaotic system can be fully deterministic; what fails is long-term prediction from imperfect data.
"A butterfly literally causes tornadoes." The title was a vivid question about sensitivity, not a claim about insects.
"Better computers will fix it." Faster computers do not remove the doubling. Only better starting data helps, and only by adding a few doublings.
"Chaos makes all prediction useless." Short-range forecasts work well; it is the horizon that is limited.
Chaos and fractals are closely linked. For the geometry side, read our companion post on Benoit Mandelbrot and fractals. For a simpler equation that also turns chaotic, see Pierre Verhulst and logistic growth.
What can a student learn from how Lorenz worked?
Lorenz could have blamed the computer and moved on. He first suspected a fault, which he says happened often, but then compared the two runs step by step until he found the real cause. That habit, checking line by line when two answers disagree instead of guessing, is how you find your own errors in long calculations. He also stayed with a surprise for months until he understood it.
When you meet a model with many moving parts, it helps to change one input and watch what follows. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Redo the worked example with an error that doubles every two days and compare the two forecast horizons.
