Chaos and Climate

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“.

The x-axis is time in days: 0-5 (top); then 0-15 (mid); then 0-89 (bottom). By day about 25, and definitely by day 60, the difference between the runs has saturated: their weather is totally different, so no further growth in RMS occurs. The 5-10 day oscillations in the last month are what you’d expect to see from weather evolution.

The y-axis is in Pascals. 100 Pascals is an hPa, ie 1 mb. Standard weather charts tend to plot pressure in contours of 4 hPa, so in real weather terms the diffs are sort-of negligible out to about day 15 (although this is a global value, so locally there will probably be bigger values).

The top pic is about day 4 (in the not-much-happening phase). The middle, day 15 (in the exp growth). The bottom, day 31 (saturated). Note that the pics have a different contour interval. By days 15/31 we’re into “real meteorology” and hence the MSLP field is most different in extratropics, as it always is (its tropical dynamics, folks).

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.

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