Daisy World an introduction to systems and equilibria.
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Transcript of Daisy World an introduction to systems and equilibria.
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Daisy World
an introduction to systems and equilibria
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Geological Example: Albedo Feedback
• high albedo: most energy is reflected back into space
• low albedo: most energy is absorbed and heats the surface
sunlightreceived
sunlightreflectedAlbedo
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Some Albedo Values(from Table 2-1, Kump et al., 1999)
Sand: 0.20 – 0.30
Grass: 0.20 – 0.25
Forest: 0.05 – 0.10
Water (overhead sun): 0.03 – 0.05
Water (sun near horizon): 0.50 – 0.80
Fresh snow: 0.80 – 0.85
Thick cloud: 0.70 – 0.80
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Image source: NASA Visible Earth, http://visibleearth.nasa.gov/view_detail.php?id=4701
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Albedo Feedback cont.
Surface Temperature
SnowCoverage
Albedo
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Albedo Feedback cont.
Surface Temperature
SnowCoverage
Albedo
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A variation of the Albedo Feedback
• how do plants respond to climate ?
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possible outcomes of feedback loops• Possible choices:
– Arrived at stable end state– Never settles down in a stable end state but
oscillates– Never reaches any stable end state
• What kind of feedback loops would produce these options?
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Equilibrium States
• If system is in equilibrium it will not change unless it is disturbed
• two types: stable, unstable
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Stable Equilibrium
• (within limits) system will return to old equilibrium state
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Unstable Equilibrium
• small perturbations will dramatically affect the state of the system
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What kind of feedback loops tend to produce stable equilibria ?
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Long-term temperature fluctuations
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Daisy World
• Gaia hypothesis
• Earth a self-regulating system
• How can it do that without a brain
• Daisy World: local processes affect global state of climate
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Influence of Daisy Growth on Surface Temperature
Daisy Coverage Avg. Surface Temp.
negative or positive feedback ?why ?
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Daisy (Albedo) Feedback
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Influence of Daisy Growth on Surface Temperature
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Influence of Daisy Growth on Surface Temperature• two modes:
– below optimum temperature → positive feedback
– above optimum temperature → negative feedback
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Two Feedback Mechanisms
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Some Mathematical Conventions
• y = f(x) means that y is a function of an independent variable x
• mathematicians plot the independent variable on the abscissa (horizontal axis) and the dependent variable on the ordinate (vertical axis)
• physicists and other scientists are not always that rigid!
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Two Feedback Mechanisms
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How to Solve for Both:
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Two Equilibrium Points
• Both conditions have to be fulfilled
• Daisy coverage affects surface temperature
• Surface temperature affects daisy coverage
• under which conditions is the system in equilibrium?
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How Does the System Reach Equilibrium?
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Are both Equilibria Created Equal?
• try out a few permutations yourself
• is the system always reaching an equilibrium?
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Two Types of Equilibria
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External Forcing of the Daisy World System• Forcing: outside influence on the system
(e.g., insolation, chemical composition of the climate system etc.)
• How does Daisy World respond to external forcings ?
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How will Insolation changes affect the Daisy World ?
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Effect of Luminosity on Temperature
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How does Feedback Affect new Equilibrium Temperature?
Temperature change withoutDaisy feedback ?(number of daisies is constant)
Temperature change withDaisy feedback ?(Daisies will spread)
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How to Quantify Importance of Feedback?
Temperature change withoutDaisy feedback ?(number of daisies is constant)
Temperature change withDaisy feedback ?(Daisies will spread)
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How to Quantify Importance of Feedback?
• compare ΔT0 and ΔTeq
• feedback factor:
0T
T
feedbackwithoutchangeetemperatur
feedbackwithchangeetemperaturf eq
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How will Stability of Equilibria be Affected?
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Stability of Equilibrium Positions
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Effects of Daisy Feedback
• biological albedo feedback
• positive and negative feedback loop
• two equilibrium positions (one stable, one unstable)
• resulting equilibrium is generally not optimum growth temperature for daisies
• presence of daisies lowers planetary temperatures
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Effects of Daisy Feedback (cont.)
• Dampens effects of external forcings
• feedback factor f quantifies feedback effect
• Stability of equilibria changes