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O.R. Come Rain Or Shine, It Is All Chaos Really!

by Nigel Cummings

Edward Norton Lorenz

This article is in part a tribute to the distinguished mathematician, Edward Norton Lorenz, who died in April 2008. (This article first published June 2008)

On a typical summer’s day – torrential rain, and despite a fine weather forecast, I sit indoors in front of my word processor, and ponder the accuracy of weather forecasting. Will there ever be a time that weather can be predicted with 100% certainty?

I know that O.R. techniques have been employed in weather modelling. Indeed, O.R. has contributed to various weather related scientific fields, one of these being meteorology, an interdisciplinary science that focuses on weather processes and forecasting. Weather it seems, can be modelled, predicted and defined in mathematical terms, as weather related events appear to be bound by the variables that exist in Earth's atmosphere. (Or do they?)

So how many variables are needed to accurately calculate weather, how long is a piece of string? The list of variables to be taken into account when considering weather forecasting can, upon close observation, seem almost endless.

Modellers could consider variables such as temperature, pressure, water vapour, planetary movement, and the gradients and interactions of each variable, and how they change in time. The Earth’s atmosphere is a fluid on a rotating sphere. Evaporation of water from the surface, and its precipitation from the atmosphere, makes our air a mixed phase system. (In a simple model, quite predictable.)

Other variables may concern the reduction of density in the vertical which makes the system stratified and how wind velocity is calculated using the laws of motion- conservation of mass, momentum and the equation of state. (The major forcing for atmospheric motion is radiative heating from our sun.)

But other factors can be considered too, factors that change the temperature are heating due to phase change (from vapour to liquid) and heating of air at the surface (sensible heating). The changes in temperature cause changes in pressure which in turn affects the flow.

We know that evaporation increases the concentration of moisture in the atmosphere while precipitation reduces its concentration. The concentration can also change due to advection of moisture. These changes are accounted for by a species conservation equation.

Some of the more sophisticated weather models also have species conservation equations for other tracers such as nitrogen dioxide, sulphur dioxide, aerosol hydrocarbons, etc. (Though it appears that aerosols have only recently been incorporated into atmospheric models,)

In the space of a few short paragraphs I have attempted to define some aspects of the weather that lend themselves to modelling. We know that weather modelling is by no means new, neither is it an exact science, some scientists have shown that the unpredictable may have to be taken into account in such modelling too. This is where chaos theory, well known in O.R. for its use in accommodating exogenous random factors particularly in econometrics and supply chain analysis may be of use.

Despite its name, chaos theory does not imply randomness: chaotic systems are both deterministic and nonlinear, but show evidence of what mathematicians call sensitive dependence. One peculiar feature of chaotic systems though, is their sensitivity to initial conditions, which are responsible for the unpredictability we experience in such phenomena. One of the most popular quantities that measure this property is the maximum Lyapunov characteristic exponent (MLCE). In supply chain analysis for example, degrees of chaos can be quantified using the Lyapunov exponent across levels of the supply chain.

Returning to the weather and chaos theory though, some of the most significant contributions to how weather can be modelled may be attributed to Edward Norton Lorenz (May 23, 1917– April 16, 2008), an American mathematician, meteorologist, and pioneer of chaos theory.

Lorenz was a mathematician and MIT meteorologist who tried to explain why it is so hard to make good weather forecasts, and wound up unleashing a scientific revolution by becoming one of the early exponents of chaos theory and in particular of the “butterfly effect”: the notion that a tiny event, such as the movement of a butterfly's wings somewhere in Brazil could have enormous effects such as the setting off of a tornado in Texas. Lorenz said that chaos theory could be used to improve our knowledge of apparently unstable systems such as the weather, and thus describe and analyse them, and improve our forecasts.

LaPlace Pierre-Simon, Marquis de Laplace (March 23, 1749 - March 5, 1827), French mathematician and astronomer.

According to James Gleick’s book ‘Chaos’. Lorenz proposal of chaos theory came about in a rather unusual way. In the 1820’s Pierre Laplace had suggested a deterministic universe in which prediction would be possible if one knew exact details of all the ‘laws of nature’ and had both the ability to plot the position of ‘all’ physical elements, and an intellect which could submit this data to analysis. This theory became known as “ Laplace's demon”, for many years it was used to explain why “noise” (or unknown background factors) made it difficult to establish “true” scientific values in complex systems.

Lorenz’s academic life became affected by “ Laplace’s Demon” in 1961, when he re-entered data from a weather simulation he had previously run into his computer (a Royal McBee LGP-30). But having retyped the numbers from the printout of the first experiment, he found that it produced wildly different results.

The computer program was the same, so the weather patterns of the second run should have exactly followed those of the first. Instead, the two weather trajectories quickly diverged on separate paths. At first, Lorenz thought the computer had a malfunction. Then he realised that he had not entered the initial conditions exactly. The computer stored numbers to accuracy of six decimal places, but to save space, the printout of results shortened the numbers to just three decimal places.

At the time he was typing in the computer printout, Lorenz had not realised the numbers had been rounded-off, and even this small discrepancy, of less than 0.1 per cent, completely changed the end result. Even though his model was vastly simplified, Lorenz realised this made perfect weather prediction a fantasy.

He realised that the reason for the differing results provided by the computer program, was that the original computer had entered numbers to six decimal points, but the printout provided only the first three. Entering 0.506, rather than 0.506127 for example - though a margin of error of less than 0.1 per cent over the experiment, regarded then as utterly trivial - resulted in huge changes and made prediction virtually impossible.

A perfect forecast would require not only a perfect model, but also perfect knowledge of wind, temperature, humidity and other conditions ‘everywhere’ around the world at any given moment. Even a small discrepancy would lead to a drastically different weather forecast.

In January 1963 the Journal of the Atmospheric Sciences published Lorenz’s findings in a paper entitled Deterministic Nonperiodic Flow, which found that “slightly differing initial states can evolve into considerably different states” and examined the feasibility of very long range weather forecasting in the light of the results.

Also in 1963, Lorenz revealed an image which is known as the ‘Lorenz Attractor’, this image is reproduced here, and is characterised by its distinctive butterfly shape. The Lorenz Attractor image is derived from the simplified equations of convection rolls arising in his equations of the atmosphere. From a technical standpoint, the system portrayed, is nonlinear, three-dimensional and deterministic. In 2001 a proof was given by Warwick Tucker, that for a certain set of parameters the system exhibits chaotic behaviour and displays what is today called a ‘strange attractor’.

The Lorenz Attractor
Warwick Tucker. Cornell University. 'The Lorenz Attractor exists', a paper published in 2001

The following year Lorenz published another paper that described how only small adjustment of parameters in a model could produce enormously different behaviour, transforming regular, periodic events into a seemingly random, chaotic pattern.

At a meeting of the American Association for the Advancement of Science in 1972, Lorenz gave a talk with a title that captured the essence of his ideas: Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?

HPoincareLorenz's early insights marked the beginning of a new field of study that impacted not just the field of mathematics but virtually every branch of science-biological, physical and social. In meteorology, it led to the conclusion that it may be fundamentally impossible to predict weather beyond two or three weeks with a reasonable degree of accuracy.

Jules Henri Poincaré, (April 29, 1854 – July 17, 1912), French mathematician and theoretical physicist.

Lorenz was not the first to stumble on chaos. At the end of the 19th century, the mathematician Henri Poincaré showed that the gravitational dance of as few as three heavenly bodies was practically impossibly complex to calculate, even though the underlying equations of motion seemed simple. But Poincaré's findings were forgotten through the first three-quarters of the 20th century. Interestingly Lorenz's papers on chaos also attracted little notice until the mid-1970s.

Some scientists have since asserted that the 20th century will be remembered for three scientific revolutions--relativity, quantum mechanics and chaos.

First published to members of the Operational Research Society in Inside O.R. June 2008