Butterflies, tornadoes and climate modelling

Ed was not a user of general circulation models. His essential approach was to crystallize profound phenomena into very small sets of equations for how a handful of variables change with time. He left behind him a dozen or so such models, each of which would repay many lifetimes of study. He was indeed a master of “seeing the world in a grain of sand.” You can read about some of these models in the talk raypierre gave at the 1987 Lorenz ‘retirement’ symposium — not that this slowed him down!

Now let’s take a closer look at that butterfly effect. Despite the fact that there are no butterflies or tornadoes in climate models, Lorenz’s discoveries and their implications played a central role in climate modelling efforts and in the most recent IPCC report.

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!

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