Daisyworld

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Daisyworld

An Introduction to Systems

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Key Questions

• What are systems?• What are feedback loops?• What are equilibrium states?• Does viewing Earth as a system allow for deeper insight

into the interrelationships among the physical and biological worlds?

• Can Earth’s climate be self-regulating?

• Chapter focus – the fundamentals of systems theory needed to study Earth

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Systems Approach• Human physiology– Systems interrelated, function together to maintain

the body in a healthy state

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System Essentials

• System – composed of diverse but interrelated components that function as a complex whole– Components can be:• reservoir of matter• reservoir of energy• system attribute• subsystem

• Introduction to Systems

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State of a System• Set of important attributes that characterize the

system at a particular time– Components interact so that a change in state is sensed

by the whole system• This linkage allows for the control of important attributes

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Couplings• Links between systems that allow the flow of information

from one component to the next– Electric blanket example

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Systems Diagrams• Keep track of couplings within a system

Positive Coupling – a change ( or ) in one component leads to a change in the same direction in the linked component - represented by

Negative Coupling - a change in one component leads to a change in the opposite direction in the linked component - represented by

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Feedback Loops

• Feedback is a self-perpetuating mechanism of change and response to that change

Positive Feedback Loops – amplify the effects of the disturbance

Negative Feedback Loops – diminish the effects of the disturbance

“Sign” of the Loop –• odd number of negative couplings – negative• even number of negative or all positive couplings- positive

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Equilibrium State

• Condition will not change unless the system is disturbed– Stable• Created by negative feedback loops• Modest disturbances will be followed by return

to equilibrium state– Unstable• Slight disturbance carry the system further and

further away from the state

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Stable State

• Small disturbances followed by return to equilibrium state• Large disturbances can lead to a new different equilibrium state• There are limits to the stability of stable equilibrium states

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Unstable States

• No region of stability•Will not return to original state on its own• Slightest disturbance pushes system to a new stable equilibrium

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Equilibrium State

• For natural systems with a single feedback loop it is usually true that – Stable systems result from negative feedback

loops– Unstable systems result from positive feedback

loops

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Perturbations and Forcings• Perturbation – temporary disturbance of a system

– Volcanic eruption example – average climatic response to the 5 largest eruptions in the last 100 years

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Perturbations and Forcings• Forcing – more persistent disturbance of a system– Solar Luminosity example

• Increase in temperature countered by decrease in CO2

LLGHG = Long Lived Greenhouse Gases

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Daisyworld

• Hypothetical planet with a simpler climate system– Only life forms are daisies– Creation of Lovelock & Watson– Demonstrates that natural systems can be self-

regulating on a global scale without the need for intelligent intervention

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Daisyworld Climate System

• A two component system– Area of white daisy coverage– Average surface temperature

• Daisy coverage affects temperature• Temperature affects daisy coverage

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Daisyworld Couplings• Albedo – reflectivity of a surface– Expressed as a decimal fraction of the total

incoming energy reflected from the surface

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Surfa

ce Te

mpe

ratu

re v

s Dai

sy C

over

age

• Negative Coupling• Negative slope

• Daisy Coverage - Surface Temp

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Coupling:Daisy Coverage – Albedo - Temperature

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Daisy Coverage in response to Temp

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Equilibrium States

• Put the two together• Intersection shows• Effect of daisies on

temperature AND• Effect of temp on

daisies• EQUILIBRIUM STATES•Two Feedback loops• One above optimum• One below

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Feedback Loops

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THE

DAIS

Y W

ORL

D CL

IMAT

E SY

TEM

Response of Daisy world to perturbations depends on the temperature

-Below optimum has negative feedback loop and is stable-Above optimum has positive feedback loop and is unstable

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Response to External Forcing

• Increased solar luminosity• Daisies would increase (immediately) and

albedo increase and warming would be slowed• Persistent increasing solar luminosity would

eventually lead to a new higher equilibrium temperature, but it would happen at a much slower rate (daisies & environment feedback loops)

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More Accurate Response to External Forcing

• Assumption – daisies respond to temperature change only

• So – no change to

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• However, change in surface temperature and daisy coverage expected

• Temperature will be higher for any amount of daisy coverage

• So – change to

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Combine the two Graphs

• P1' stable

• P2' unstable• Both temperature and

daisy coverage higher at new stable equilibrium

• Stability limit for P1' is lower

• New equilibrium state less resistant to perturbations

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Mathematically Speaking . . .

• Comparing equilibrium temperature with and without feedback

∆Teq = ∆T0 + ∆Tf

• The overall temperature change resulting from increase solar luminosity is the sum of the temperature change with no feedback and the temperature change due to feedback

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For Any Stable Equilibrium in a System Involving Feedback Loops

• The change in state of a system as it moves from one equilibrium to the next is the sum of the state change that would result without feedback and the effect of the feedback itself

• To qualify the strength of the feedback effect . . .

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The Feedback Factor• The ratio of the equilibrium response to forcing (the

response with feedback) to the response without feedback

= temperature change with feedback = ∆Teq temperature change w/out feedback ∆T0

Negative feedback loop if 0 < < 1Positive feedback loop if 1 < Feedback factor defined only for stable systems

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Daisyworld Climate History

History of Daisy Coverage

Temperature History-Initially temp rises quickly-Once min temp for daisies met, daisies increase-Growth of daisies cools planet-Eventually, when optimal temp is met, daisy coverage at max-Increasing solar luminosity not countered by daisy growth and daisies die, causing increasing temp-Feedback loop positive & unstable-Surface temp rises, daisies extinct

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The Lessons of Daisyworld• A planetary climate system is not passive in the face of internal or

external forces– Negative feedback loops counter external forcings

• Non-human systems that self-regulate seem intelligent, yet no foresight or planning is involved

• In a natural self-regulating system, there is no preset state that the system is programmed to “seek-out”

• Thresholds often exist in systems that when surpassed can lead to rapid changes in system state– Abrupt changes can have no forewarning

• Earth is like Daisyworld– Strong negative feedback loops lead to long-term stability

• Are we approaching a climate threshold that will result in a much warmer state?

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