Butterflies, tornadoes and climate modelling

The notion of the butterfly effect itself was drawn from a simple but astute observation of the way the solutions of certain nonlinear equations behave when they are solved using a computer. Start with a greatly simplified representation of thermal convection, first formulated by Barry Saltzman using a technique called “low order modelling.” If you run a simulation using these equations and then try and replicate it using starting values that only differ in the last decimal place, you will find that the simulations quickly diverge from one another – and by quickly, it means that the differences grow exponentially fast. Lorenz found this phenomenon by accident, but quickly recognised the profound implications. If the real weather system displayed the same behaviour, it meant that since however well one knew the initial conditions of the atmosphere, there would always be some uncertainty, that uncertainty would be quickly magnified, rendering weather forecasts useless after a few exponential doubling times. The practical implication is that – even if you had a perfect model – for every halving of the error in the initial conditions you only get one extra time period of useful forecast. Given this time period is only a few hours in many cases, the practicality of true weather forecasts for periods longer than two weeks or so, is vanishingly small.

The mathematically inclined reader who takes a look at Ed’s early papers on what is now called the “Lorenz Attractor” will be astonished at the depth and modernity of his ideas about chaos. This line of work was no mere remark on a numerical exercise. Lorenz actually teased out the geometry of chaos — the many-leaved structure of the attractor — realizing that it was no simple geometric entity like a sphere or a folded sheet of paper. It was indeed “strange” in a sense which he made geometrically precise. This is why the work had such lasting impact on the area of pure mathematics known as dynamical systems theory. He went beyond that to develop or apply many fundamental concepts in chaotic systems, quantitatively formulating various measures of predictability and connecting the Lyapunov exponent — a certain precise mathematical characterization of chaos — with the structure of strange attractors. But that’s for the mathematicians. What makes Lorenz’s work interesting to the entity on the Clapham omnibus is the notion of sensitive dependence on initial conditions. Some have even seen in this deterministic chaos the resolution to the problem of free will!

The idea that small causes can have large and disproportionate effects is not at all new of course. The poem: “For the want of a nail, the battle was lost” (medieval in origin) encapsulates that well, and popular culture is full of such examples, “It’s a wonderful life” (1946) and Ray Bradbury’s “A Sound of Thunder” (1952) for instance. Curiously, Bradbury’s story also involves a butterfly, and since it predates Lorenz’s coining of the phrase by a decade or so, people have speculated that there was a connection to Lorenz’s choice of metaphor (he started off with a seagull in his 1963 Trans. N.Y. Ac. Sci. paper). But that doesn’t appear to be the case (see here for a history). It’s worth adding that all of Lorenz’s papers were exceptional in their clarity and are well worth tracking down as an example of science writing at its best.

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