## Weather forecasting and chaos

The science of chaos has developed rapidly since the 1960s (when a meteorologist, Edward Lorenz, was one of its pioneers) along with the power of electronic computers. In this context, chaos4 is a term with a particular technical meaning (see Glossary). A chaotic system is one whose behaviour is so highly sensitive to the initial conditions from which it started that precise future prediction is not possible. Even quite simple systems can exhibit chaos under some conditions. For instance, the motion of a simple pendulum (Figure 5.7) can be 'chaotic' under some circumstances, and, because of its extreme sensitivity to small disturbances, its detailed motion is not then predictable.

A condition for chaotic behaviour is that the relationship between the quantities which govern the motion of the system be non-linear; in other words, a description of the relationship on a graph would be a curve rather than a straight line.5 Since the appropriate relationships for the atmosphere are non-linear it can be expected to show chaotic behaviour. This is illustrated in Figure 5.6, which shows the improvement in predictability that can be expected if the data describing the initial state are improved. However, even with virtually perfect initial data, the predictability in terms of days ahead only moves from about six days to about 20 days, because the atmosphere is a chaotic system.

For the simple pendulum not all situations are chaotic (Figure 5.7). Not surprisingly, therefore, in a system as complex as the atmosphere, some occasions are more predictable than others. A good illustration of an occasion with particular sensitivity to the initial data is provided by the exceptionally severe storm Lothar that crossed northern France in December 1999. It blew down hundreds of millions of trees and led to economic losses estimated at over 5 billion euros. Figure 5.8 shows an ensemble of forecasts carried out

(a) Forcing frequency w

(a) Forcing frequency w

Conical pendulum (resonant frequency wo)

Conical pendulum (resonant frequency wo)

Chaotic w =0.99766 wo

Stable w =1.00088 wo

Chaotic w =0.99766 wo

Figure 5.7 (a) A simple pendulum consisting of a bob at the end of a string of length 10 cm attached to a point of suspension which is moved with a linear oscillatory forcing motion at frequencies near the pendulum's resonance frequency f0. (b) and (c) show plots of the bob's motion on a horizontal plane, the scale being in centimetres. (b) For a forcing frequency just above f0 the motion of the bob settles down to a simple, regular pattern. (c) For a forcing frequency just below f0 the bob shows 'chaotic' motion (although contained within a given region) which varies randomly and discontinuously as a function of the initial conditions.

w by the European Centre for Medium Range Forecasting (ECMWF) starting from a set of slightly varying initial conditions 42 hours earlier.6 The best-guidance deterministic forecast only predicts a weak trough in surface pressure which is supported by a number of members of the ensemble. However, a minority of the ensemble members show an intense vortex over France similar to what actually occurred, demonstrating the value of the ensemble in its prediction of the risk of the severe event even though a precise deterministic forecast was not possible. It is interesting that more recent deterministic reforecasts with an improved model have failed to predict this storm.

Deterministic predictions

Verification

Deterministic predictions

Verification

Forecast 1 Forecast 2 Forecast 3 Forecast 4 Forecast 5 Forecast 6 Forecast 7 Forecast 8 Forecast 9 Forecast 10

Forecast 1 Forecast 2 Forecast 3 Forecast 4 Forecast 5 Forecast 6 Forecast 7 Forecast 8 Forecast 9 Forecast 10

Forecast 11 Forecast 12 Forecast 13 Forecast 14 Forecast 15 Forecast 16 Forecast 17 Forecast 18 Forecast 19 Forecast 20

Forecast 11 Forecast 12 Forecast 13 Forecast 14 Forecast 15 Forecast 16 Forecast 17 Forecast 18 Forecast 19 Forecast 20

Forecast 21 Forecast 22 Forecast 23 Forecast 24 Forecast 25 Forecast 26 Forecast 27 Forecast 28 Forecast 29 Forecast 30

Forecast 21 Forecast 22 Forecast 23 Forecast 24 Forecast 25 Forecast 26 Forecast 27 Forecast 28 Forecast 29 Forecast 30

Forecast 31 Forecast 32 Forecast 33 Forecast 34 Forecast 35 Forecast 36 Forecast 37 Forecast 38 Forecast 39 Forecast 40

Forecast 31 Forecast 32 Forecast 33 Forecast 34 Forecast 35 Forecast 36 Forecast 37 Forecast 38 Forecast 39 Forecast 40

Forecast 41 Forecast 42 Forecast 43 Forecast 44 Forecast 45 Forecast 46 Forecast 47 Forecast 48 Forecast 49 Forecast 50

Forecast 41 Forecast 42 Forecast 43 Forecast 44 Forecast 45 Forecast 46 Forecast 47 Forecast 48 Forecast 49 Forecast 50

Figure 5.8 I sopleths of surface pressure from a 51-member ensemble forecast by the European Centre for Medium Range Forecasting (ECMWF) of the storm Lothar based on initial conditions 42 hours before the storm crossed northern France on 26 December 1999. The isobars are 5 mb apart and the thicker 1000 mb isobar runs across the middle of the figures. The top left shows forecasts made from the best estimate of the initial conditions that did not indicate the presence of a severe storm. Nor did many members of the ensemble. However, some of the ensemble members show an intense vortex indicating significant risk of its occurrence. The top right shows the situation at the end of the forecast period.

Figure 5.9 Monthly values of the Southern Oscillation Index (SOI) based on normalised Tahiti minus Darwin sea level pressures. An 11-point low pass Alter effectively removes fluctuations with periods of less than eight months. The thick black line represents a decadal Alter. Negative values indicate positive sea level pressure anomalies at Darwin and thus El Niño conditions.

Around Christmas 1999, storm front Lothar raced across France, Switzerland and Germany, and 100 people died. (Also see Figure 5.8).

Figure 5.9 Monthly values of the Southern Oscillation Index (SOI) based on normalised Tahiti minus Darwin sea level pressures. An 11-point low pass Alter effectively removes fluctuations with periods of less than eight months. The thick black line represents a decadal Alter. Negative values indicate positive sea level pressure anomalies at Darwin and thus El Niño conditions.

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In the tropics, the atmosphere is particularly sensitive to sea surface temperature. This is not surprising because the largest contribution to the heat input to the atmosphere is due to evaporation of water vapour from the ocean surface and its subsequent condensation in the atmosphere, releasing its latent heat. Because the saturation water vapour pressure increases rapidly with

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