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

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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

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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."

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• 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.

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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)

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• 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

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• 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.

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• 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.

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• 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.

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• 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.

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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

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Lorenzs 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."

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• 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.

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Butterfly Effect

• Lorenzs 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

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• 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 t30.
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Chaos and Climate

• Lorenzs 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.

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• Lorenzs 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.

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1980s

• 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.

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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

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Germanys Super Computer
Super Computer Predictions
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Sources

• Edward N. Lorenz (1963). "Deterministic
  Nonperiodic Flow". Journal of the Atmospheric
  Sciences 20 130141. doi10.1175/1520-0469(1963)0
  20lt0130DNFgt2.0.CO2
• Edward N. Lorenz (1969). "Atmospheric
  predictability as revealed by naturally occurring
  analogues". Journal of the Atmospheric Sciences
  26 636646. doi10.1175/1520-0469(1969)26lt636APA
  RBNgt2.0.CO2
• Edward N. Lorenz (1969). "Three approaches to
  atmospheric predictability". Bulletin of the
  American Meteorological Society 50 345349.
• 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". Nonlinear
  ity 10 1289.doi10.1088/0951-7715/10/5/014
• Kolmogorov, A. N. (1954). "Preservation of
  conditionally periodic movements with small
  change in the Hamiltonian function". Doklady
  Akademii Nauk SSSR 98 527530.
• Michael Berry, "Quantum Chaology," pp 1045
  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 517532.