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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.
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