The "Butterfly effect"

There's recently been a bit of a flap on some blogs concerning the "butterfly effect". As that wikipedia page indicates, this is the poetically-named phenomenon whereby chaotic systems (such as the atmosphere) exhibit sensitive dependence on initial conditions. That is, if you had two identical copies of the same system, and perturb one ever so slightly, the perturbation will grow over time and eventually the states of the two systems will be completely uncorrelated. Its discovery and significance in the context of atmospheric science was first noted by Lorenz in his famous 1963 paper:This picture (snaffled from this page) shows some results from Lorenz's original paper.

Less widely-known than Lorenz's work is the fact that the chaotic dynamics of the 3-body problem had been shown much earlier by Poincare. Following the links on wikipedia (and googling) should cover the historical and technical background adequately.

The significance of chaos to weather prediction is sometimes overstated - the existence of a Lyapunov (watch the spelling, wmc!) exponent of > 1 does not in itself preclude accurate long-term predictions, since it relates only to infinitesimal perturbations integrated over infinite time (note that we are staying in the realm of mathematics and classical Newtonian physics here - I'm not going to start talking about molecular and quantized effects). In realistic applications, it is the current (and near-future) growth rate of finite perturbations that is more important, and this may well be negligible or negative over significant periods of time. And even that ignores the effect of model error, which means that even a perfectly initialised model state will inevitably diverge from truth over coming days. Some analysis of weather prediction I've seen indicates that model error is a substantial source of forecast error, with chaos relatively unimportant. But this is not my current field of interest, and this is also probably getting a bit obscure...

Anyway, the point is that a random perturbation will have a non-zero projection onto Lyapunov vectors with exponents greater than 1 (ie, growing vectors). In the atmosphere and many other chaotic systems, most of the perturbation will project onto decaying vectors, so overall it will initially shrink. However, ultimately the growing component will come to dominate and the pertubation will grow (with probability 1).

This is the sort of thing that is easy to show with numerical models of the atmosphere. Simply perturb a reference run, and see what happens. There are a couple of "gotchas" that could confuse the results. Firstly, limited numerical precision means that a small pertubation could fall below the level of numerical precision and be rounded out of the system. This is demonstrated by running the model at diffferent precisions. A perhaps more subtle problem is that many codes contain max and min operators, to prevent nonphysical solutions (eg turbulent energy is not allowed to fall below a threshold, velocities may be limited for computational reasons). If a variable crosses such a threshold and is reset to the limit, then a slightly perturbed version will be reset to the same point and again the perturbation is lost. But these are nit-picks in a result that is otherwise robust across all adequately realistic models. It is possible in theory to devise a perturbation that shrinks even without relying on these factors (align it precisely with only the shrinking LV), but a random pertubation will not have this property. Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small pertubation to grow to a significant size - if it affects the weather 100 years from now, this will be obvious in the atmospheric state 99 years and 51 weeks hence, and so will be accounted for in the forecast.

I might as well point out again here that the chaotic sensitivity of a system has no direct bearing on forecasting the climate (ie the long-term statistics) of the system. To copy from my comment on realclimate:

Not only is the climate of the Lorenz model easy to understand, it is also simple to predict how it will respond to a variety of "external forcings", in the form of either a parameter perturbation or direct forcing term in the dynamical equations. Eg see here and here. However the detailed trajectory is unpredictable except in the very short term.

Anyway, this is all pretty well-known stuff - or so I thought, before reading Roger Pielke Sr's blog. He seems to deny that this "butterfly effect" exists at all. Myself, Gavin and William have all pulled him up on it, so it will be interesting to see how he justifies this in his promised follow-up post.