predictability-chaos-lorenz attractor mea719 lec4 jan28 2009

7/28/2019 Predictability-Chaos-Lorenz Attractor MEA719 Lec4 Jan28 2009

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Climate Modeling

MEA 719

Lecture Set 4Predictability-Chaos-Lorenz Attractor


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Main Topics

GENERAL BACKGROUND

TO THE CONCEPT OF CHAOS

CONTRIBUTION FROM ASTRONOMY

APPLICATIONS TO METEOROLOGY

LORENZS MATHEMATICAL MODEL

ENSEMBLE CLIMATE PREDICTION (set-6)

AN EXERCISE IN ENSEMBLE MODELING (set-

6)


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GENERAL BACKGROUND

TO THE CONCEPT OF CHAOS


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"What is Chaos?"

In physics, chaos is a word with a specializedmeaning, one that differs from the everyday useof the word

To a physicist, the phrase "chaotic motion" reallyhas nothing to with whether or not the motion ofa physical system is frenzied or wild inappearance.

In fact, a chaotic system can actually evolve in away which appears smooth and ordered.


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"What is Chaos?"

Rather, chaos refers to the issue of whether or not it ispossible to make accurate long-term predictions aboutthe behavior of the system

For four centuries in physics, the laws of physics havereflected the complete connection between cause andeffect in nature

Thus until recently, it was assumed that it was alwayspossible to make accurate long-term predictions of anyphysical system so long as one knows the startingconditions well enough.


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"What is Chaos?"

The discovery of chaotic systems in natureabout 100 years ago has all but destroyedthat notion.

Before that scientists and mathematiciansbelieved in the Philosophy of Determinism

This is the belief that says that everycause has a unique effect, and vice versa.


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"What is Chaos?"

The Philosophy of Determinism imply that using

the assumed link between cause and effect, the

initial conditions are used to make predictions at

later and earlier times

To the contrary it is now believed, based on the

theory of chaos, that no measurement can be

made with infinite accuracy and ultimately the

forecast becomes useless


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"What is Chaos?"

Uncertainty of measurements give rise toDynamical Instabilities, which to most physicistsis a term synonymous with Chaos.

Newton's laws are completely deterministicbecause they imply that anything that happensat any future time is completely determined by

what happens now, and moreover thateverything now was completely determined bywhat happened at any time in the past.


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CONTRIBUTION FROM

ASTRONOMY


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"What is Chaos?"

The equations of motion for planets are anapplication of Newton's laws, and thereforecompletely deterministic.

That these mathematical orbit equations are

deterministic means, of course, that by knowingthe initial conditions---in this case, the positions

and velocities of the planets at a given startingtime---you find out the positions and speeds ofthe planets at any time in the future or past


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Limits of Predictability

It is impossible to actually measure the initial positionsand speeds of the planets to infinite precision, evenusing perfect measuring instruments, since it isimpossible to record any measurement to infiniteprecision. Thus there always exists an imprecision,however small, in all astronomical predictions made bythe equation forms of Newton's laws

Up until the time ofPoincar, the lack ofinfinite precisionin astronomical predictions was considered a minorproblem, however, because of an incorrect assumptionmade by almost all physicists at that time.


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Infinite Precision

In practical terms infinite precision may be

interpreted to mean that the accuracy

required to define the initial conditions may

be much greater (by orders of magnitude)

than the one required to observe and

monitor the physical phenomena ofinterest


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The assumption was that if you could shrink theuncertainty in the initial conditions---perhaps by usingfiner measuring instruments---then any imprecision in the

prediction would shrink in the same way.

In other words, by putting more precise information intoNewton's laws, you got more precise output for any later

or earlier time. Thus it was assumed that it wastheoretically possible to obtain nearly-perfect predictionsfor the behavior of any physical system.


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But Poincar noticed that certain astronomicalsystems did not seem to obey the rule thatshrinking the initial conditions always shrank thefinal prediction in a corresponding way.

By examining the mathematical equations, hefound that although certain simple astronomical

systems did indeed obey the "shrink-shrink" rulefor initial conditions and final predictions, othersystems did not.


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The astronomical systems which did not obeythe rule typically consisted of three or moreastronomical bodies with interaction between allthree. For these types of systems, Poincar

showed that a very tiny imprecision in the initialconditions would grow in time at an enormousrate.

Thus two nearly-indistinguishable sets of initialconditions for the same system would result intwo final predictions which differed vastly fromeach other.


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Poincar mathematically proved that this"blowing up" of tiny uncertainties in the initialconditions into enormous uncertainties in thefinal predictions remained even if the initial

uncertainties were shrunk to smallest imaginablesize.

That is, for these systems, even if you could

specify the initial measurements to a hundredtimes or a million times the precision, etc., theuncertainty for later or earlier times would notshrink, but remain huge.


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The extreme "sensitivity to initial conditions"mathematically present in the systems studiedby Poincar has come to be called dynamicalinstability, or simply CHAOS.

Because long-term mathematical predictionsmade for chaotic systems are no more accurate

than random chance, the equations of motioncan yield only short-term predictions with anydegree of accuracy.


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Examples in Real World

Occurs in a large number of RETHYMIC SYSTEMS

Applied in e.g., arrhythmic pacemakers,

fluid dynamics, etc

the stock market provides trends which

exhibit behavior of strange attractors

a dripping faucet seems random to theuntrained ear, but when plotted exhibits

behavior of strange attractor


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APPLICATIONS TO METEOROLOGY


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Edward Lorenzs Pioneering

Theory

One of the most important discoveries was

made in 1963, by the meteorologist

Edward Lorenz, who wrote a basic

mathematical computer model to study asimplified model of the weather.

Specifically Lorenz studied a primitive

model of how an air current would rise and

fall while being heated by the sun.


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Lorenz Model

Lorenz's computer code contained the mathematicalequations which governed the flow the air currents.

Since computer code is truly deterministic, Lorenz

expected that by inputing the same initial values, hewould get exactly the same result when he ran theprogram.

Lorenz was surprised to find, however, that when heinput what he believed were the same initial values, hegot a drastically different result each time.


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Implications of Lorenzs Theory to

Meteorology Lorenz had not realized that the initial values for

each run were different because the differencewas incredibly small, so small as to beconsidered microscopic and insignificant byusual standards.

Gradually it came to be known that even the

smallest imaginable discrepancy between twosets of initial conditions would always result in ahuge discrepancy at later.


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Application of Chaos to

Meteorology

Scientists now believe that like Lorenz's simple computermodel of air currents, the weather as a whole is a chaoticsystem. This means that in order to make long-termweather or climate forecasts with any degree of accuracyat all, it would be necessary to take an infinite number ofmeasurements.

Even if it were possible to fill the entire atmosphere of

the earth with an enormous array of measuringinstruments---in this case thermometers, wind gauges,and barometers---uncertainty in the initial conditionswould arise from the minute variations in measuredvalues between each set of instruments in the array.


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Butterfly Effect in the Atmosphere

Because the atmosphere is chaotic, theseuncertainties, no matter how small, wouldeventually overwhelm any calculations anddefeat the accuracy of the forecast.

This principle is sometimes called the "ButterflyEffect." In terms of weather forecasts, the"Butterfly Effect" refers to the idea that

whether/climate or not a butterfly flaps its wingsin a certain part of the world can make thedifference in whether or not a storm arises oneyear later on the other side of the world.


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Butterfly Effect in the atmosphere

Because of the "Butterfly Effect," it is now

accepted that weather/climate forecasts

can be accurate only in the short-term,

and that long-term forecasts, even made

with the most sophisticated computer

methods imaginable, will always be nobetter than guesses.


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MATHEMATICAL MODELING

OF LORENZS THEORY


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What is Lorenz Attractor?

The so called "lorenz attractor" was first studied by Ed N. Lorenz, ameterologist, around 1963. It was derived from "Navier-Stokes"equations.

The Lorenz model is defined by three nonlinear differential equations givingthe time evolution of the variablesX(t), Y(t), Z(t)

dx / dt = a (y - x)

dy / dt = x (b - z) - y

dz / dt = xy - c z

One commonly used set of constants is a = 10, b = 28, c = 8 / 3. Another is

a = 28, b = 46.92, c = 4. "a" is sometimes known as the Prandtl number and"b" the Rayleigh number. The never reaches a steady state. Instead it is an example of deterministic

chaos. As with other chaotic systems the Lorenz system is sensitive to theinitial conditions, two initial states no matter how close will diverge.


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Lorenz Attractor Independent Variables

In the context of the atmosphere, a is

proportional to the temperature difference

across the layer responsible for driving the

fluid motion at a rate given by the variableX.

Yand Ztell us about the changes in the

temperature distribution in the layer due tothe heat carried by the moving fluid


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Lorenz Attractor

Solution


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Lorenz Attractor


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Lorenz Attractor


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Lorenz's Attractorand Avogadro's number

The attractor represents the behavior of gas at any given time, andits condition at any given time depends upon its condition at aprevious time

If the initial conditions are changed by even a tiny amount, say as

tiny as the inverse ofAvogadro's number (a heinously small numberwith an order of 1E-24), checking the attractor at a later time mayyield numbers totally different.

This is because small differences will propagate themselvesrecursively until numbers are entirely dissimilar to the original

system with the original initial conditions.
http://www.zeuscat.com/andrew/chaos/lorenz.html


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Implications of chaos for climate

variability & ensemble prediction

If you make a forecast based on model integration thereis a chance that the error in the initial conditions is justenough to put the solution into the wrong attractor

By performing an ensemble is forecasts starting fromslightly different initial conditions within the bounds oferror increases the chance to predict the correctbutterfly wing of the strange attactor


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Lorenz Attractor

Slight change in

ICs ends up in a

different wing

Lorenz Attractor


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Lorenz Attractor

Slight change in

ICs ends up in a

same wing

predictability-chaos-lorenz attractor mea719 lec4 jan28 2009

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