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