Chaos and Climate

By James Annan and William Connolley

In this post, we will try to explain a little about chaos theory, and its relevance to our attempts to understand and forecast the climate system. The chaotic nature of atmospheric solutions of the Navier-Stokes equations for fluid flow has great impact on weather forecasting (which we discuss first), but the evidence suggests that it has much less importance for climate prediction.

Chaos is usually associated with the sensitivity of a deterministic system to infinitesimal pertubations in initial conditions (the full definition is a bit more difficult: see technical bit at the end). The identification of chaos in atmospheric systems is due to an accidental discovery by Lorenz in 1961. Using a greatly simplified model of the atmosphere, he restarted a computation from part-way though a previously-completed run. However, for the initial conditions, he used a printout that only had 3 figures of precision, compared to the 6 used internally by the computer. The outputs of the two runs initially appeared indistinguishable, but then diverged and became wholly decorrellated. So it was an atmospheric model which provided some of the first insight into the ‘chaos effect’, thus teaching us something quite profound about nature

In fact, this type of behaviour had already been identified and studied more than 60 years earlier by Poincare, in the form of the “3-body problem” of celestial dynamics. 2 stars (or planets etc) in orbit around each other will each follow a regular ellipsoidal trajectory around their joint centre of mass. However, when a 3rd (or more) body is thown into the mix, their future trajectories may be highly sensitive to the precise initial conditions. One extremely useful result of chaos theory is the design of complex orbits that enable spacecraft to travel great distances in a fuel-efficient manner, by analysing the Earth, Sun and spacecraft as a 3-body system (eg see the articles here and here).

Back in atmospheric physics, chaotic behaviour is a highly-studied and well-understood phenomenon of all realistic global models, arising directly from the nonlinearity of the Navier-Stokes equations for fluid flow. So any uncertainty in the current atmospheric state, however, small, will ultimately grow and prevent accurate weather forecasts in the long term. 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. So long as the perturbation is not rounded out by the limited numerical precision of the model, it will invariably grow.

Here is an example using the HADAM3 model. One standard run was performed, and then another run was started where the pressure in a single grid box was changed by 10-10 (one part in 1015 of the model value) For a bit more about this experiment, see here. The first graph shows how the RMS difference in sea level pressure increases over time, and the second graph shows the evolution of the spatial pattern of differences. The perturbation rapidly saturates the highly local convective mode in the tropics, before more slowly spreading to the much larger mesoscale differences that matter to weather forecasters (note the scale changes). If this model resolved hurricanes, then their appearance and paths would be completely uncorrelated in the two runs – a classic example of the “Butterfly effect“.

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