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Introduction to Chaos

by

Tim Palmer

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Introduction to chaos for:

Weather prediction

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Wilhelm Bjerknes (1862-1951)

Proposed weather forecasting as a deterministic initial value problem based on the laws of physics

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Lewis Fry Richardson (1881-1953)

The first numerical weather forecast

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“Why have meteorologists such difficulty in predicting the weather with any certainty? Why is it that showers and even storms seem to come by chance ... a tenth of a degree (C) more or less at any given point, and the cyclone will burst here and not there, and extend its ravages over districts that it would otherwise have spared. If (the meteorologists) had been aware of this tenth of a degree, they could have known (about the cyclone) beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it all seems due to the intervention of chance”

Poincaré, 1909

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Edward Lorenz (1917 – )

bZXYZ

YrXXZY

YXX

“… one flap of a sea-gull’s wing may forever change the future course of the weather” (Lorenz, 1963)

The Lorenz (1963) attractor, the prototype chaotic model…..

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is a nonlinear system

Since is a nonlinear function of

( )

dXF X

dtd X dF

X J Xdt dXF X

J J X

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Isopleth of initial pdf

Isopleth of forecast pdf

0, ttM

0 0 0

0 00 0

0 0 00 0 0 0 0 0

,, ,max max max

, , ,x t x t x t

x t M M x tx t x t M x t M x t

x t x t x t x t x t x t

0 0M M x t x t

x t

0x t

Nb M*M symmetric operator therefore eigenvectors form a complete orthogonal basis

Tangent propagator

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Because M is not a normal operator (M*MMM *) , M’s eigenmodes are not orthogonal

Substantial perturbation growth is possible over finite time, even if M has no growing eigenmodes

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Lothar: 08Z, 26 Dec. 1999

100 Fatalities400 million trees blown down3.5 million electricity users

affected for 20 days3 million people without water

Lothar +Martin

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Ensemble Initial Conditions 24 December 1999

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Lothar (T+42 hours)

ECMWFThanks to Rob Hine

Headlines: Oct 28 2002

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Frost free Frosty

Charlie loses L if concrete freezes.

•But Charlie has fixed (eg staff) costs

•There may be a penalty for late completion of this job.

•By delaying completion of this job, he will miss out on other jobs.

•These cost C

Is Lp>C? If p > C/L don’t lay concrete!

Charlie is planning to lay concrete tomorrow.

Consensus forecast: frost free YES

Should he?

NO

NO

?

Let p denote the probability of frost

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Value of EPS over high-res deterministic forecast for financial weather-derivative trading based on Heathrow temperature (Roulston and Smith, London School of Economics, 2003)

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Introduction to chaos for:

Seasonal climate prediction

Atmospheric predictability arises from slow variations in lower-boundary forcing

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Edward Lorenz (1917 – )

X X Y f

Y XZ rX Y f

Z XY bZ

What is the impact of f on the attractor?

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The influence of f on the state vector probability function is itself predictable.

f=0 f=2

f=3 f=4

Add external steady forcing f to the Lorenz (1963) equations

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The tropical Pacific ocean/atmosphere

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Introduction to chaos for:

Stochastic parametrisation

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3221313

3123

2122

3221311

49.043.01363.0

14.049.049.07.262

57.049.02.63.2

aaaaaaa

aaaaaa

aaaaaaa

Eg 2) Lorenz(1963) in an EOF basis

3rd EOF only explains 4% of variance (Selten, 1995) .

Parametrise it?

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Lorenz(1963) in a truncated EOF basis with parametrisation of a3

Good as a short-range forecast model (using L63 as truth), but exhibits major systematic errors compared with L63, as, by Poincaré-Bendixon theorem, the system cannot exhibit chaotic variability – system collapses onto a point attractor.

),(

14.049.049.07.262

57.049.02.63.2

213

3123

2122

3221311

aafa

aaaaaa

aaaaaaa

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Stochastic-Lorenz(1963) in a truncated EOF basis

Stochastic noise

3

3123

2122

3221311

14.049.049.07.262

57.049.02.63.2

a

aaaaaa

aaaaaaa

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Lorenz attractor

Truncated Stochastic-Lorenz attractor –weak noise

Truncated Stochastic-Lorenz attractor

Error in mean and variance

Palmer, 2001 (acknowledgment to Frank Selten)

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“He believed in the primacy of doubt; not as a blemish upon our ability to know, but as the essence of knowing”Gleick (1992) on Richard Feynman’s philosophy of science.