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“.
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 perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat overstated. Chaos is defined with respect to infinitesimal perturbations and infinite integration times, but our uncertainties in the current atmospheric state are far too large to be treated as infinitesimal, and furthermore, all of our models have errors which mean that they will inevitably fail to track reality within a few days irrespective of how well they are initialised. Nevertheless, chaos theory continues to play a major role in the research and development of ensemble weather prediction methods.
Although ultimately chaos will kill a weather forecast, this does not necessarily prevent long-term prediction of the climate. By climate, we mean the statistics of weather, averaged over suitable time and perhaps space scales (more on this below). We cannot hope to accurately predict the temperature in Swindon at 9am on the 23rd July 2050, but we can be highly confident that the average temperature in the UK in that year will be substantially higher in July than in January. Of course, we don’t need a model to work that out – historical observations already give strong evidence for this prediction. But models based on physical principles also reproduce the response to seasonal and spatial changes in radiative forcing fairly well, which is one of the many lines of evidence that supports their use in their prediction of the response to anthropogenic forcing.
Fortunately, the calculation of climatic variables (i.e., long-term averages) is much easier than weather forecasting, since weather is ruled by the vagaries of stochastic fluctuations, while climate is not. Imagine a pot of boiling water. A weather forecast is like the attempt to predict where the next bubble is going to rise (physically this is an initial value problem). A climate statement would be that the average temperature of the boiling water is 100ºC at normal pressure, while it is only 90ºC at 2,500 meters altitude in the mountains, due to the lower pressure (that is a boundary value problem).
We can demonstrate this sort of climate response clearly in the Lorenz model, or any more complex climate model. Perturbing the initial conditions gives a completely different trajectory (weather), but this averages out over time, and the statistics of different long-term runs are indistinguishable. However, a steady perturbation to the system can generate a significant change to the long-term statistics. Here is some output from a run of the Lorenz model in which a change was applied half way through. At time t=0, the parameter “r” (which relates to an idealised thermal forcing) is changed from 26 to 28. When viewed in close-up detail, the trajectory looks qualitatively similar before and after the change, but in fact the long-term statistics such as the mean value of z, and its 95% range, are changed. In this simple model, the steady perturbation changes the climate in a highly linear manner – increasing r again to 30 would add the same change on top of that shown for 26 to 28, and r=27 would sit half-way between the cases shown. Of course, these results cannot be directly extrapolated to the real climate system, but they do disprove the common but misguided claim that chaotic weather necessarily prevents meaningful climate prediction. In fact, all climate models do predict that the change in globally-averaged steady state temperature, at least, is almost exactly proportional to the change in net radiative forcing, indicating a near-linear response of the climate, at least on the broadest scales. The uncertainty is in the steepness of the slope, which is what “climate sensitivity” describes.
It was thought until relatively recently that chaos provided a substantial practical challenge to so-called “optimal” model tuning and climate prediction with state-of-the-art climate models, since it generally prevents the use of one of the most powerful and widely-used optimisation and estimation procedures (an adjoint – eg Lea et al 2002). The obvious alternative method of exhaustively searching parameter space requires a huge number of model simulations, and the Climateprediction.net project is pursuing this approach. However, one of us recently showed how another efficient estimation method known as the ensemble Kalman filter can be applied to this type of problem (various applications here). This is certain to remain an active area of research for some time to come.
The climate of a model can be easily defined in terms of the limit of the statistics of the model output as the integration time tends to infinity, under prescribed boundary conditions. This limit is well-defined for all climate models. However, the real world is slightly messier to deal with. The real climate system varies on all time scales, from daily weather, through annual, multi-year and decadal (ENSO), Milankovitch, glacial-interglacial cycles, plate tectonics and continental configurations, right up to the ultimate death of the Sun. The average temperature, and all other details of the climate system, will vary substantially depending on the time scale used. So how can we talk meaningfully about “the climate” and “climate change”? Well, although there are interesting scientific questions to ask across all the different time scales, the directly policy-relevant portion is on the multi-decadal and centennial time scale. It is quite clear that the perturbation that we are currently imposing is already large, and will be substantially larger, by up to an order of magnitude, than any plausible natural variability over this time scale. So for the policy-relevant issues, we generally focus on the physical atmosphere-ocean system, sometimes with coupled carbon-vegetation system, and treat the major ice sheets, orbital parameters and planetary topography as fixed boundary conditions. It’s an approximation, but a pretty good one.
[Technical para. The phenomenon of chaos can be formalised through the use of "Lyapunov Exponents" (LE) and their associated Lyapunov Vectors (LV). The leading LE can be defined as the limit of the maximal time-averaged logarithmic growth rate of the distance between two nearby model states, as the integration interval increases without bound. The leading LV gives the direction in phase space (which depends on the specific initial state, in contrast to the LE which is a fundamental constant of the system) for which this maximum growth rate is attained. Chaos is indicated by a leading LE greater than 1, indicating that initially similar model states diverge over time. See also Chaos theory]