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EVERY DAY, meteorologists try to predict next week’s weather using immensely
complicated mathematical descriptions of how the atmosphere behaves. Research
is also under way to develop models that will help them make predictions
on an even longer timescale, seasonal forecasts of monsoon rains, for example.
Meteorologists would even like to be able to estimate changes in climate
resulting from human activities, such as the greenhouse effect.

And yet we know that the atmosphere is a chaotic system. As such, it
is inherently unpredictable . So, are these attempts at longer-range weather
andclimate prediction a waste of time? Should we content ourselves with
the television forecast oftomorrow’s weather, and leave the rest to chance?
Although the weather can change every day as individual systems progress
eastwards, depressions and their associated weather fronts, for example,
certain spells of weather can last for weeks, months or even whole seasons.
These spells are not characterised by individual weather systems, but by
the position of the so-called jet streams, regions of strong wind in the
upper atmosphere. They determine whether we will have a wet or dry summer,
mild or severe winter.

Figure 1 shows the track of the jet stream over the Atlantic and Europe,
which is associated with two different weather ‘regimes’. In Figure 1a,
the jet stream is more or less oriented along a line of latitude. Individual
weather systems tend to be steered along the jet stream. Over the British
Isles, this configuration would probably give a rather wet and unsettled
spell of weather, as rain bands pass by with monotonous regularity. Meteorologists
call this a ‘zonal’ regime. In Figure 1b, the jet stream splits into two
branches over the mid-Atlantic, with one branch positioned north of the
British Isles, the other to the south. In the summer, this configuration
brings about a warm fine spell of weather over Britain; inthe winter, it
produces dull, overcast, and possibly verycold weather. Meteorologists usually
refer to this as a ‘blocked’ regime.

We can define weather regimes quantitatively from historical records
of data of weather in the northern hemisphere, using what are called ‘cluster-analysis’
techniques. In practice, about 10 different regimes characterise most of
the large-scale variability of the atmosphere in the northern hemisphere.

Meteorologists have been interested for a long time in the predictability
of these weather regimes. Can we forecast how they evolve up to a month
ahead, even though we can predict what happens to individual weather systems
for only a few days ahead? These sort of problems motivated the meteorologist
Edward Lorenz, whose work at the Massachusetts Institute of Technology in
the early 1960s spawned much ofthe activity in chaos theory today. Theatmosphere
behaves like a turbulent fluid, and Lorenz was only too aware that it was
governed by a set of mathematical equations that were nonlinear and were
extremely sensitive to small changes – in other words, showed modes of instability.
He had an intuitive feeling that this would make weather prediction a tough
problem. To confirm his hunch, he sought a way of simplifying these equations
so that he could study them mathematically, while retaining their essential
nonlinearity and instability.

The most drastically simplified version of the full fluid-dynamical
equations led to a ‘model climate’ with just three variables, x, y and z.
A state of instantaneous ‘weather’ in Lorenz’s model can, therefore, be
represented by a point in a three-dimensional ‘phase space’ and the evolution
of the weather with time can be represented by a line, or trajectory, in
this space (as described in Ian Stewart’s article ‘Portraits of chaos’,
New ÐÓ°ÉÔ­´´, 4 November). The climate of the model, the set of all possible
model weather states, is known as the Lorenz attractor (see Figure 2).

This attractor has no volume in this three-dimensional space, yet is
neither a simple one-dimensional line, nor a smooth two-dimensional surface.
As Ian Stewart’s article explains, the attractor has a fractional dimension
(2.06), and therefore, not surprisingly, carries the epithet ‘strange’.
It represents one of a generic class of strange attractors whose topology
characterises the chaotic, unpredictable properties of the basic equations.

Although the three-component equations that Lorenz proposed do not realistically
describe the evolution ofweather regimes, they have similar chaotic properties
tomore realistic models. So we can use the Lorenz model to describe in a
qualitative way the chaotic behaviour of the evolution of weather regimes
in the atmosphere.

First, notice from Figure 2 that the Lorenz attractor has two separate
branches, sometimes called butterfly wings. We can think of these wings
as representing, in our abstract state space, the two weather regimes shown
in Figure 1 in real space. For the sake of argument, suppose the left-hand
wing corresponds to the zonal regime of Figure 1a, and the right-hand wing
corresponds to the blocked regime of Figure 1b. In other words, any two
points on the left-hand wing relate to different instantaneous weather,
but the large-scale flow would be the same.

Imagine two points arbitrarily close to each other, on the left-hand
wing of the attractor. Using our conventions, these two points represent
almost identical weather states in a regime characterised by rather unsettled
weather conditions over the British Isles. We now follow the initial evolution
of these two weather states. There are three possibilities:both trajectories
remain on the left-hand wing (see Figure 3a);both trajectories evolve towards
the right-hand wing, as inFigure 3b; or one trajectory remains on the left-hand
wing,while the other moves to the right-hand wing, as in Figure 3c. Note
that in all three cases, the two trajectories have diverged, implying quite
different forecasts of instantaneousweather. On the other hand, in the first
two scenarios(see Figures 3a and 3b), the two trajectories evolve to the
sameweather regime (remaining unsettled in the first case; becoming more
settled in the second case).

You can see, therefore, that although the atmosphere is fundamentally
chaotic, you can predict the weather regime from certain initial conditions
in the atmosphere. To find out what these initial conditions are, we need
to make a series of weather forecasts from (a sufficiently large number
of) similar but not identical initial states. Figure 4 shows the evolution
in phase space of two ensembles of realistic weather forecasts over a certain
period. For the first set of initial conditions, in Figure 4a, the forecasts
start to diverge only a little, indicating that the predictability for that
set of initial conditions in the atmosphere is pretty good, so we can confidently
forecast the evolution of weather regime over that period.

On the other hand, in Figure 4b, the paths of evolving weather patterns
from similar initial conditions disperse considerably, indicating that the
atmosphere is in a particularly chaotic state during the period of the forecasts,
so we cannot make any meaningful predictions.

Now, the Lorenz model has only three variables, or three degrees of
freedom. It is much too simple to describe accurately the evolution of the
real atmosphere. In fact, increasing the number of degrees of freedom in
models has improved the quality of weather prediction for the first few
days ahead. Today’s weather-prediction models have about a million degrees
of freedom.

Figures 5 and 6 show an example of an ensemble of eight forecasts from
the weather prediction model we use at the European Centre for Medium-Range
Weather Forecasts (ECMWF). The pictures show contours of the height of a
pressure surface (500 hectopascals) in the middle of the atmosphere. The
wind blows parallel to these contours in the direction shown by the arrows
– the strength of the wind is proportional to the contour gradient. Figure
5 shows the initial conditions for the eight forecasts. Note that they look
very similar. Differences between individual members of the ensemble correspond
to uncertainties associated with the weather observing network. The initial
flow in Figure 5 corresponds approximately to the pattern shown in Figure
1; over Britain, the flow is very weak, with the region of strong winds
splitting over the east Atlantic towards the north-east and south-east.

We now run the computer model eight times from each of these initial
conditions and see what the model predicts for the weather one week later.
This is shown in Figure 6. You can see that the forecasts are now far from
identical; their trajectories in phase space (not shown) have begun to diverge
significantly. However, over some parts of Europe, you can see a weather
pattern common to many of the members of the ensemble. Over Italy, for example,
the initial state showed an air flow from the south; after a week, the winds
swing round so that they have a northerly component. So, an Italian forecaster
could confidently predict that Italy should expect much colder weather a
week ahead. Using the ensemble, the forecaster could estimate probabilities
that temperature will fall within certain ranges.

By contrast, the British weather forecaster would have a more difficult
task – the forecasts over Britain diverge much more (confirming the saying
about the unreliability of British weather). Nevertheless, it would not
be unreasonable to predict a trend to more westerly flow and unsettled weather.
This example is one that is neither particularly chaotic, nor exceptionally
predictable, and highlights the practical consequences of chaos for the
weather forecaster. There are many research centres around the world exploring
the possibility of predicting the evolution of weather regimes up to a month
ahead. It is early days in this business, and meteorologists still need
to refine their description of various important physical processes in the
computer models. If they are to forecast weather regimes this far ahead,
they will have to run ensemble forecasts of the sort just described, so
as to be able to estimate how reliable the prediction is. At present, raw
computing power is still insufficient to do this routinely. But with advances
in computer technology, this kind of forecasting may not be many years away.

So far, we have talked about weather in the middle latitudes. In the
tropics, the dynamics of the atmosphere is somewhat different, mainly because
the Coriolis effect of the Earth’s rotation is less dominant. In particular,
although there are weather systems in the tropics, such as hurricanes, monsoon
depressions and so on, that arise from instabilities of the larger-scale
flow, these weather systems do not feed back into the larger-scale flow
to the same extent as with weather systems in the middle latitudes. In fact,
the behaviour of the large-scale tropical atmosphere is very strongly linked
to the temperature of the ocean surface, which evolves over months, rather
than days. One well-studied phenomenon that intimately involves the coupled
dynamics of the ocean-atmosphere system is the so-called El Nino event,
during which ocean temperatures in the tropical east Pacific can rise up
to 4 Degree C above normal during a season. Meteorologists believe that
El Nino can influence weather patterns over a substantial fraction of the
globe.

It may be feasible to predict flow on a planetary scale in the tropical
atmosphere for a season, using models that take into account the dynamics
of both the atmosphere and the oceans. Certainly there have been some encouraging
successes in forecasting El Nino and its consequences a season ahead. In
a few years’ time, weather forecasters may be able to predict seasonal rain
over Africa, India and other tropical countries from such weather prediction
models. However, nonlinearity and instability, the hallmarks of chaos, are
not totally absent in these tropical predictions, and it is likely that
the ‘multiple-ensemble integrations’ which people have already applied to
weather at mid-latitudes will still be a necessary tool for the tropical
forecaster.

Finally, does chaos theory prevent us from predicting possible climate
change in the next century? The answer here is no. The type of prediction
is quite different from that outlined above. Here, the goal is not predicting
an individual trajectory on the climate attractor; the goal is to determine
the shape and position of the whole climate attractor itself when, for example,
greenhouse gases increase.

The critical question that climatologists are trying to answer is whether
the climate attractor will suffer a minor perturbation (for example, small
shift of the whole attractor along one of the axes of phase space), or whether
there will be a substantial change in the whole shape and position of the
attractor, leading to some possibly devastating weather states not experienced
in today’s climate. Chaos theory certainly does not forbid the possibility
of some substantial change to the atmosphere’s climate attractor as a result
of modest increases in the amount of carbon dioxide. At the moment, we cannot
be sure of the answer. The same sort of models employed in predicting the
weather are also being used to try to find out what the greenhouse effect
has in store for us. As with attempts at predicting weather regimes, uncertainty
in climate forecasting is mitigated by the fact that several of the climate
centres around the world now have sophisticated models. Again, researchers
can evaluate their confidence in predicting greenhouse warming in terms
of the dispersion of the ensemble of predictions from these different models.

Nevertheless, chaos dynamics should caution us from making too premature
a judgment about climatic changes. Returning to the Lorenz attractor, there
is no pre-ordained number of times that a given trajectory must circle around
one of the butterfly wings; it could be once, 10 times, or 100 times, depending
on the precise position of the trajectory on the attractor. If we have a
mild winter, a warm summer, then another mild winter, we might not necessarily
be in the throes of man-made climate change; the system might just be revolving
happily around one small part of phase space, and on that hundred and first
revolution, the system might, quite unexpectedly, and for no apparent reason,
evolve towards another part of phase space, which is associated with cold
winters and miserable summers. For this reason, many meteorologists are
quite guarded about whether global warming due to the greenhouse effect
really has arrived.

To the lay person, the unpredictability of the weather may be a curse;
to the meteorologist, it is what makes the subject fascinating, and fun
to study. Above all, chaos does not mean that we must throw in the towel,
and leave all to chance. This blend of fundamental science with state-of-the-art
computer technology is leading to unparalleled insights into the workings
of the fragile gaseous envelope that surrounds and sustains us.

Tim Palmer is head of the predictability and diagnostics section of
the European Centre for Medium-Range Weather Forecasts in Reading, Berkshire.

Further reading: Predictability in Science and Society, John Mason,
Peter Mathews and J. H. Westcott (editors), Cambridge University Press,
1984; Topics in Geophysical Fluid Dynamics: atmosphere dynamics, dynamo
theory, and climate dynamics, H. Ghil and S. Childress, Applied Mathematical
Sciences, volume 64, Springer Verlag, 1987.

Next week: Robert May explains how chaos arises in population dynamics,
natural selection and physiology.