The butterfly effect — that the flapping of a butterfly’s wings in Brazil can cause a tornado in Texas — expresses the sensitivity of a chaotic system to initial conditions. In essence, because we can’t possibly track every butterfly in Brazil, we’ll never perfectly predict tornadoes in Texas, even if the equations behind our weather forecast are deterministic.
But this interpretation doesn’t fully capture the subtleties of the situation. With fluid dynamics, the small scales of a flow — like the turbulence in an individual cloud — are linked to the largest scales in the flow — for example, a hurricane. For short times, we’re actually quite good at predicting those large scales; our weather forecasts can distinguish sunny days and cloudy ones a week out. But at smaller scales, the forecast errors pile up quickly. No one can forecast that an individual cloud will form over your house three days from now. And because the small scales are linked to the larger scales, the uncertainties from the small scale cascade upward, limiting how far into the future we can reliably predict the weather.
And, unfortunately, drilling down to capture smaller and smaller scales in our models can’t fix the problem, unless our initial uncertainties are identically zero. To get around this problem, weather forecasters instead use ensemble forecasting, where they run many simulations of the weather with slightly different initial conditions. Those differences in initial conditions let the forecasters play with those initial uncertainties — how accurate is the temperature reading from that station? How reliable is the instrument reporting that humidity? How old is the satellite data coming in? Once all the forecasts are run, they can see how many predicted sunny days versus rainy ones, which ones resulted in severe weather, and so on. Often the probabilities we see in our weather app — like 30% chance of rain — depend on factors including how many of the forecasts resulted in rain.
Unfortunately, this butterfly effect permanently limits just how far into the future we can predict weather — at least until we fully understand the nature of the Navier-Stokes equations. For much more on this interesting aspect of chaos, check out this Physics Today article. (Image credit: NASA; see also T. Palmer at Physics Today)
