daisy world an introduction to systems and equilibria
TRANSCRIPT
Daisy World
an introduction to systems and equilibria
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
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
Image source: NASA Visible Earth, http://visibleearth.nasa.gov/view_detail.php?id=4701
Albedo Feedback cont.
Surface Temperature
SnowCoverage
Albedo
Albedo Feedback cont.
Surface Temperature
SnowCoverage
Albedo
A variation of the Albedo Feedback
• how do plants respond to climate ?
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?
Equilibrium States
• If system is in equilibrium it will not change unless it is disturbed
• two types: stable, unstable
Stable Equilibrium
• (within limits) system will return to old equilibrium state
Unstable Equilibrium
• small perturbations will dramatically affect the state of the system
What kind of feedback loops tend to produce stable equilibria ?
Long-term temperature fluctuations
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
Influence of Daisy Growth on Surface Temperature
Daisy Coverage Avg. Surface Temp.
negative or positive feedback ?why ?
Daisy (Albedo) Feedback
Influence of Daisy Growth on Surface Temperature
Influence of Daisy Growth on Surface Temperature• two modes:
– below optimum temperature → positive feedback
– above optimum temperature → negative feedback
Two Feedback Mechanisms
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!
Two Feedback Mechanisms
How to Solve for Both:
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?
How Does the System Reach Equilibrium?
Are both Equilibria Created Equal?
• try out a few permutations yourself
• is the system always reaching an equilibrium?
Two Types of Equilibria
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 ?
How will Insolation changes affect the Daisy World ?
Effect of Luminosity on Temperature
How does Feedback Affect new Equilibrium Temperature?
Temperature change withoutDaisy feedback ?(number of daisies is constant)
Temperature change withDaisy feedback ?(Daisies will spread)
How to Quantify Importance of Feedback?
Temperature change withoutDaisy feedback ?(number of daisies is constant)
Temperature change withDaisy feedback ?(Daisies will spread)
How to Quantify Importance of Feedback?
• compare ΔT0 and ΔTeq
• feedback factor:
0T
T
feedbackwithoutchangeetemperatur
feedbackwithchangeetemperaturf eq
How will Stability of Equilibria be Affected?
Stability of Equilibrium Positions
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
Effects of Daisy Feedback (cont.)
• Dampens effects of external forcings
• feedback factor f quantifies feedback effect
• Stability of equilibria changes