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Non-linear systems: Chaos and Complexity in Meteorology and Climatology

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### Non-linear systems: Chaos and Complexity in Meteorology and Climatology

Brad Gutnik

Chaos Theory

- Chaos Theory Defined: a field of study in mathematics, physics, economics, and philosophy studying the behavior of dynamic systems that are highly sensitive to initial conditions.
- Meteorology and Climatology

Early Chaos Theory

- An early proponent of chaos theory was Henri Poincaré
- In the 1880s, while studying the three-body problem:
- 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.
- He found that:
- There can be orbits which are nonperiodic, and
- Yet not forever increasing nor approaching a fixed point
- In 1893, Nikola Tesla noted:
- "A single ray of light from a distant star falling upon the eye of a tyrant in bygone times may have altered the course of his life, may have changed the destiny of nations, may have transformed the surface of the globe, so intricate, so inconceivably complex are the processes in Nature."

In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature

- In the system studied, "Hadamard's billiards", Hadamard was able to show that all trajectories are unstable in that all particle trajectories diverge exponentially from one another.
- Much of the earlier theory was developed almost entirely by mathematicians, under the name of Ergodic theory.

Climate Cycles

- Things from tides to rabbit populations go through regular cycles, and it was easy to suppose that climate also was cyclical
- If you could detect a regular cycle in climate:
- You could develop a scientific explanation for climate change
- Use it to calculate predictions of economic value (and perhaps make a killing on the wheat futures exchange, etc)
- Given enough different bodies of data, people could also turn up correlations between a weather cycle and some other natural ebb and flow, notably the eleven-year cycle of sunspots
- A 1941 U.S. Weather Bureau publication noted that some 50 climate cycles had been reported, ranging from days to centuries (not to mention the ice ages, which come and go regularly over hundreds of thousands of years)

By the middle of the 20th century, opinion among meteorologists was divided about the same way as at the start of the century:

- Some expected that a few cycles would eventually be pinned down
- Others believed that no cycles existed — the variations of climate were purely random
- Progress was stalled unless clues could be found in some new approach.
- A big clue came in the 1950s, when a few scientists decided to build actual physical models of climate.
- These "dishpan" studies turned out to be surprisingly effective in modeling features of the atmosphere like weather fronts.
- What was most thought-provoking was the way the circulation of a fluid in the laboratory could show different patterns even when the external conditions remained the same.
- The choice of pattern depended in some arbitrary, unpredictable way on the system's past history

The intellectual basis of a new viewpoint was well expressed in 1961 by R.C. Sutcliffe at an international climate conference:

- Using the popular new language of cybernetics, he described climate as a complex nonlinear feedback system.
- Unceasing variation might be "built-in," an intrinsic feature of the climate system.
- Thus it might be pointless to look for external causes of climate change, such as solar variations or volcanic eruptions. Every season the pattern of the general circulation of the atmosphere was newly created, perhaps in a quite arbitrary way.
- The "sudden jumps" seen in the climate record, Sutcliffe concluded, are "suggestive of a system controlling its own evolution.“

The father of cybernetics himself, mathematician Norbert Wiener, insisted that attempts to model the weather by crunching physics equations with computers, as if meteorology were an exact science like astronomy, were doomed to fail.

- Quoting the old nursery rhyme that told how a kingdom was lost "for want of a nail" (which caused the loss of a horseshoe that kept a knight out of a crucial battle)
- Wiener warned that "the self-amplification of small details" would foil any attempt to predict weather.
- One pioneer in computer prediction recalled that Wiener went so far as to say privately that leaders of the work were "misleading the public by pretending that the atmosphere was predictable”.

Despite initial insights in the first half of the twentieth century:

- Chaos theory became formalized only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments.
- The main catalyst for the development of chaos theory was the electronic computer.
- Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand.

The more people worked with computers, the more examples they found of oddly unstable results.

- If you start two computations with exactly the same initial conditions, and they must always come to precisely the same conclusion.
- But, make the slightest change in the fifth decimal place of some initial number, and as the machine cycles through thousands of arithmetic operations the difference might grow and grow, in the end giving a seriously different result.

Edward Lorenz

- May 23, 1917 - April 16, 2008
- Earned two degrees in meteorology from MIT
- Was a professor there until his death
- During the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology, as most atmospheric phenomena involved in weather forecasting are non-linear

Lorenz’s “Mistake”

- In 1961, Lorenz was using a numerical computer model to rerun a weather prediction, when, as a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127 the computer would hold.
- After a simulated month or so, the weather pattern diverged from the original result. A difference in the fourth decimal place was amplified in the thousands of arithmetic operations, spreading through the computation to bring a totally new outcome.
- "It was possible to plug the uncertainty into an actual equation," Lorenz later recalled, "and watch the things grow, step by step."

Lorenz was astonished. While the problem of sensitivity to initial numbers was well known in abstract mathematics, and computer experts were familiar with the dangers of truncating numbers, he had expected his system to behave like real weather.

- The truncation errors in the fourth decimal place were tiny compared with any of a hundred minor factors that might nudge the temperature or wind speed from minute to minute.
- Lorenz had assumed that such variations could lead only to slightly different solutions for the equations, "recognizable as the same solution a month or a year afterwards... and it turned out to be quite different from this.“
- Storms appeared or disappeared from the weather forecasts as if by chance.
- His work on the topic culminated in the publication of his 1963 paper Deterministic Nonperiodic Flow in Journal of the Atmospheric Sciences, and with it, the foundation of Chaos theory.

Butterfly Effect

- Lorenz’s description of the Butterfly effect followed in 1969:
- Butterfly Effect-
- Sensitive dependence on initial conditions
- The flapping butterfly wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events
- Concept seems first to have appeared in a 1952 short story by Ray Bradbury about time travel

These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ only by 10−5 in the x-coordinate.

The final position of the cones indicates that the two trajectories are no longer coincident at t=30.

Chaos and Climate

- Lorenz’s work on chaos theory did not necessarily apply to the climate system, which averaged over many states of weather.
- So Lorenz next constructed a simulation of climate in a simple mathematical model with some feedbacks, and ran it repeatedly through a computer with minor changes in the initial conditions.
- His initial plan was simply to compile statistics for the various ways his model climate diverged from its normal state. He wanted to check the validity of the procedures some meteorologists were promoting for long-range "statistical forecasting," along the lines of the traditional idea that climate was an average over temporary variations.
- He could not find any valid way to statistically combine the different computer results to predict a future state. It was impossible to prove that a "climate" existed at all, in the traditional sense of a stable long-term average. Like the fluid circulation in some of the dishpan experiments, it seemed that climate could shift in a completely arbitrary way.

Lorenz’s ideas spread among climate scientists, especially at a landmark conference on "Causes of Climate Change" held in Boulder, Colorado in August 1965.

- Lorenz, invited to give the opening address, explained that the slightest change of initial conditions might randomly bring a huge change in the future climate.
- "Climate may or may not be deterministic," he concluded. "We shall probably never know for sure."
- Other meteorologists at the conference pored over new evidence that almost trivial astronomical shifts of the Earths orbit might have "triggered" past ice ages.. Summing up a consensus at the end of the conference, leaders of the field agreed that minor and transitory changes in the past "may have sufficed to 'flip' the atmospheric circulation from one state to another.

1980’s

- The meteorological questions that had launched chaos theory remained among the hardest to answer
- Some scientists now insisted that the climate system's intrinsic fluctuations would utterly defeat any attempt to calculate its changes.
- Thus the 1980 edition of one classic textbook said that predictions of greenhouse effect warming were dubious because of chaotic "autovariations."
- Lorenz and others argued that the recently observed global warming might be no evidence of a greenhouse effect or any other external influence, but only a chance excursion in the drunkard's random walk.

Future of Predicating Weather/Climate

- “To be able to identify and analyze long-term climatic trends and changes, it is important to have access to near-continuous data of our planet over long periods of time, which is made possible by Earth-observation (EO) satellites” -IPCC
- In 2009 German scientists inaugurated what they describe as the world's most powerful climate-predicting super-computer
- The custom-built machine at the German Climate Computing Centre (DKRZ) in the port-city of Hamburg can take any region and forecast how that place's climate will alter with global warming.
- Scientists said the supercomputer enables much more detailed climate calculations than were possible until now.
- "It's the biggest computer in the world to be dedicated solely to climate research," said German Science Minister Annette
- Even with the fastest computers and most sophisticated software imaginable climate models will always be skewed because
- There is no current way to continuously obtain the enormous amount of weather information from every square inch on (and above) earth at the very same instant.
- Unless every atom on earth can be observed continuously and completely the uncertainty that is Chaos Theory will make analyzing results and make climate and weather predictions forever flawed

Super Computer Predictions

Sources

Edward N. Lorenz (1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences20: 130–141. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

Edward N. Lorenz (1969). "Atmospheric predictability as revealed by naturally occurring analogues". Journal of the Atmospheric Sciences26: 636–646. doi:10.1175/1520-0469(1969)26<636:APARBN>2.0.CO;2

Edward N. Lorenz (1969). "Three approaches to atmospheric predictability". Bulletin of the American Meteorological Society50: 345–349.

Kenneth Chang (2008-04-17). "Edward N. Lorenz, a Meteorologist and a Father of Chaos Theory, Dies at 90". New York Times. Retrieved 2010-05-01.

Saber N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, 1999, page 117, ISBN 1-58488-002-3.

Fu, Z.; Heidel, J. (1997). "Non-chaotic behaviour in three-dimensional quadratic systems". Nonlinearity10: 1289.doi:10.1088/0951-7715/10/5/014

Kolmogorov, A. N. (1954). "Preservation of conditionally periodic movements with small change in the Hamiltonian function". Doklady Akademii Nauk SSSR98: 527–530.

Michael Berry, "Quantum Chaology," pp 104–5 ofQuantum: a guide for the perplexed by Jim Al-Khalili (Weidenfeld and Nicolson 2003).

Robert G. Watts, Global Warming and the Future of the Earth, Morgan & Claypool, 2007, page 17.

Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background",Environmetrics, vol. 8, no. 5, pages 517–532.

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