Chaos

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Chaos
Dr Mark Cresswell

• 69EG6517 Impacts Models of Climate Change

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Lecture Topics

• Introduction
• Lorenz and chaos
• The attractor
• Bifurcation
• Mandelbrot and Helge von Koch
• Summary

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Introduction

• If atmospheric processes were constant, or
  strictly periodic, describing them mathematically
  would be easy..weather forecasting would also be
  easy.and meteorology would be boring!
• Instead, the atmosphere exhibits variability and
  fluctuations that are irregular
• In order to deal quantitatively with uncertainty,
  it is necessary to use the tools of probability
  the mathematical language of uncertainty

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Introduction

• Computer models generate deterministic solutions
  of the future state of the atmosphere
• Such model solutions do not provide an estimate
  of uncertainty if re-run with the same starting
  conditions, they generate the same solution
• These models could NEVER generate forecasts with
  zero uncertainty because of their limited model
  physics and the fact some processes operate at
  very small spatial scales far smaller than the
  finest resolution of the model

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Lorenz and Chaos

• Even if all relevant physics could be included in
  atmospheric models we would still not escape
  uncertainty because of dynamical chaos
• This was a problem discovered by Ed Lorenz in
  1963 - and effectively rules out any hope of an
  uncertainty-free forecast
• Lorenz began to formulate his ideas on weather
  predictability and chaos from an accidental
  discovery in his lab

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The effect of 0.506 instead of 0.506127 !!
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Lorenz and Chaos

• What Lorenz observed was to become known as the
  butterfly effect
• The flapping of a single butterfly's wing today
  produces a tiny change in the state of the
  atmosphere. Over a period of time, what the
  atmosphere actually does diverges from what it
  would have done. So, in a month's time, a tornado
  that would have devastated the Indonesian coast
  doesn't happen. Or maybe one that wasn't going to
  happen, does.
• (Ian Stewart, Does God Play Dice? The Mathematics
  of Chaos, pg. 141)

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Lorenz and Chaos

• Lorenz found that the time evolution of a
  non-linear dynamical system (like the atmosphere)
  is very sensitive to the initial conditions of
  that system
• If 2 realisations of the system are started from
  2 very slightly different initial conditions, the
  two solutions will eventually diverge markedly

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Lorenz and Chaos

• Since the atmosphere is always incompletely
  observed, even a model with perfect physics will
  never start exactly like the real world but
  will have gaps where initial conditions are
  unknown or slightly inaccurate
• These small errors and unknown values will
  accumulate and be magnified over time such that
  model and reality diverge over time

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Lorenz and Chaos
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The Attractor

• Lorenz decided to produce an experimental set of
  equations to investigate chaos further
• He took the basic equations for convection and
  stripped them down to an unrealistically small
  equation that retained a sensitive dependence on
  initial conditions
• Later, it was discovered that his equations
  exactly described the motion of a simple water
  wheel

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The Attractor

• The simple equations gave rise to entirely random
  behaviour BUT when the results were graphed
  they showed something extraordinary
• The output always stayed on a curve a double
  spiral. Previously, only a steady state or
  perfectly periodic state was known about
• Lorenzs new state was ordered but never
  repeated themselves nor followed a steady state

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The Attractor

• When output from Lorenzs equations are
  visualised, we see a strange attractor.
• Now known as the Lorenz Attractor

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Bifurcation

• A similar problem occurs in ecology, and the
  prediction of biological populations
• The equation would be simple if population just
  rises indefinitely, but the effect of predators
  and a limited food supply make this equation
  incorrectso the simplest equation taking this
  into account is

next year's population r this year's
population (1 - this year's population)
r growth rate
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Bifurcation

• A biologist, Brian May decided to change the
  growth rate and plot the results graphically
• As the growth rate increased, the final
  population would increase as well
• Butstrangely.. As the rate passed a threshold
  the line broke into 2 and became 2 different
  populations
• As the rate increased, the populations would
  break again but unpredictably chaotically!

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Bifurcation
Bifurcation Upon closer inspection, it is
possible to see white strips. Looking closer at
these strips reveals little windows of order,
where the equation goes through the bifurcations
again before returning to chaos
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Mandelbrot

• An employee of IBM, Benoit Mandelbrot was a
  mathematician studying the self-similarity of May
  and Lorenzs work
• One of the areas he was studying was cotton price
  fluctuations
• No matter how the data on cotton prices was
  analysed, the results did not fit the normal
  distribution. Mandelbrot eventually obtained all
  of the available data on cotton prices, dating
  back to 1900

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Mandelbrot

• The numbers that produced aberrations from the
  point of view of normal distribution produced
  symmetry from the point of view of scaling
• Each particular price change was random and
  unpredictable. But the sequence of changes was
  independent on scale curves for daily price
  changes and monthly price changes matched
  perfectly

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Mandelbrot

• At one point, he was wondering about the length
  of a coastline. A map of a coastline will show
  many bays. However, measuring the length of a
  coastline off a map will miss minor bays that
  were too small to show on the map
• walking along the coastline misses microscopic
  bays in between grains of sand. No matter how
  much a coastline is magnified, there will be more
  bays visible if it is magnified more

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Mandelbrot
One mathematician, Helge von Koch, captured the
ideas of Mandelbrot in a mathematical
construction called the Koch curve. The Koch
curve brings up an interesting paradox. Each time
new triangles are added to the figure, the length
of the line gets longer. However, the inner area
of the Koch curve remains less than the area of a
circle drawn around the original triangle.
Essentially, it is a line of infinite length
surrounding a finite area
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Summary

• Climate does repeat itself (consider the 4
  seasons of Spring, Summer, Autumn and Winter)
  but NEVER EXACTLY
• Models cannot replicate a system if it is missing
  information about initial conditions
• Uncertainty degrades predictability
• Randomness and unknown forcings generate
  seemingly chaotic turbulence but such
  randomness actually has order

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Summary

• We will NEVER be able to generate accurate
  weather forecasts or climate predictions because
  chaos arises from uncertainty
• Improvements in model physics will have a limited
  effect (garbage in garbage out)
• Parallel improvements in observing systems will
  help to produce model simulations that see the
  influence of chaos occurring later in the
  integration and hence providing longer
  lead-time forecasts