Daisyworld: an Introduction to Systems

Daisyworld: An Introduction to Systems

Learning Objectives

After reading this chapter, students should be able to:

· Understand what the essentials of systems are

· Know how the components of systems interact with each other

· Describe the concept of a “feedback” both positive and negative

· Know how positive and negative feedback loops affect the equilibrium states of the system

· Know how we can learn about climate systems in general by studying the hypothetical planet Daisyworld

· Know how forcings in the daisyworld system affect albedo

· Know how albedo affects climate

Review Questions

1.) A perturbation that causes a decrease in component A leads to a decrease in component B. Is the coupling between these two components positive or negative?

Positive coupling, because decrease in one component (A) leads to change the same direction in the linked component (B).

2.) What is a feedback loop?

A feedback loop is a self-perpetuating mechanism of change and response to that change.

3.) Why do negative feedback loops tend to diminish the effect of disturbances?

For simplicity, consider a two component system linked by a positive and a negative coupling. If one component is perturbed, the sign of that perturbation is retained through the positive coupling but reversed through the negative coupling. Thus, overall the negative feedback loop reverses the sign of the initial perturbation, tending to restore the system to its initial state.

4.) What distinguishes a forcing from a perturbation?

A temporary disturbance of a system is called perturbation, while a more persistent disturbance of a system is called a forcing.

5.) Are all equilibrium states stable? Why or why not?

The equilibrium states of a system can be represented as peaks (unstable) and valleys (stable). On disturbance, the system returns to stable equilibrium states but moves away from unstable equilibrium states. A system with a single feedback loop has a stable equilibrium state if the feedback loop is negative and an unstable equilibrium if the feedback loop is positive.

6.) What is albedo? How does it influence climate?

The reflectivity of a surface (expressed as a decimal fraction or percent) is called the surface albedo. The reflectivity of an entire planet is called the planetary albedo. Increased albedo causes a decrease in temperature (a negative coupling).

7.) How are daisies on Daisyworld able to regulate the hypothetical planet’s temperature?

The more daisies, the more sunlight reflects off their white petals, the less sunlight absorbed, and finally the cooler the surface temperature. In sum, as daisy coverage increases, temperature decreases. The stable equilibrium state is below the growth optimum.

Critical Thinking Problems

1.) In the Dysfunctia family, when the children get noisy, the parents get mad. When the parents get mad, the children get noisy. Draw a systems diagram for the Dysfunctia family.

a. Is the feedback loop negative or positive?

Positive

b. Is the family stable or unstable?

Unstable

2.) Earth’s average temperature is determined in part by the amount of CO2 in the atmosphere, by way of the greenhouse effect. The atmospheric CO2 content may in turn be affected by photosynthetic activity of plants, which convert CO2 into plant tissue (organic carbon); however, the rate of photosynthesis depends on the amount of CO2 in the atmosphere and on global air temperature. The components of this system, atmospheric CO2 content, global temperature, and photosynthesis rate, are intimately interconnected. By increasing global photosynthesis rates, plants would tend to lower the atmospheric CO2 level. In doing so, however, the plants would tend to cool Earth. This cooling, together with the CO2 level, might tend to reduce the photosynthetic activity of plants.

a. On the basis of this discussion, draw a system diagram of the photosynthetic rate-CO2 –temperature system.

b. How many feedback loops are there?

Two

c. Are the feedback loops positive or negative?

Negative

d. Describe the response of the system to the following perturbations:

(i) An increase in atmospheric CO2

Increased atmospheric CO2 à Increased surface temperature à Increased photosynthetic activity à Decreased CO2

So the perturbation is damped.

(ii) A decrease in temperature

Decreased surface temperature à Decreased photosynthetic activity à Increased CO2 à Increased surface temperature

So the perturbation is damped.

e. Extra credit: How might the system respond to a continued forcing--an increase in solar luminosity through time?

Increased solar luminosity à Increased surface temperature à Increased photosynthetic activity à Decreased CO2

3.) Daisyworld has a companion planet that is similar in all ways except that the daisies are black.

a. What is the effect of an increase in black-daisy coverage on planetary temperature? Express your answer graphically.

An increase in black-daisy coverage will lead to an increase in planetary surface temperature.

b. Assuming that the effect of temperature on daisy coverage is the same on black-daisy Daisyworld as on white-daisy Daisyworld, draw a stability diagram--a diagram analogous to Figure 2-10 – for black daisy Daisyworld. Include two equilibrium states.

c. Which of the two equilibrium states in part (b) is stable?

Equilibrium state P2 is stable.

d. Is the stable equilibrium state of part (c) cooler or warmer than that of white-daisy Daisyworld?

The equilibrium state of part (c) is warmer than that of white-daisy Daisyworld, because it’s above the daisy growth maximum.

e. How would this system respond to a decrease in solar luminosity? Express your answer graphically and in terms of the feedback factor.

f = DTeq / DT0 = (smaller negative)/(larger negative), 0<f <1

f. Is f of part (e) greater than or less than 1?

DT0 and DTeq are both negative because the arrows point to the left in the diagram. |DTeq | < |DT0| , so 0<f <1.

4.) Real daises have an optimum growth temperature of 22 oC and they go extinct at temperatures 20 oC warmer or colder than this value. Let’s assume that they cover 100% of Daisyworld at their optimum growth temperature. A parabola that describes this mathematically is:

where c= percent daisy coverage and T= temperature (see Figure 2-9). Assume the temperature of (White) Daisyworld is given by T = 60 – C/2. (This describes a line with a negative slope as shown in Figure 2-7.)

a. Combine these equations and solve for the equilibrium solution, i.e., the values of c and T that satisfy both equations. How many solutions are there? Which solution is stable?

Equation of parabola:

Equation of line:

Equation of parabola = Equation of line

= à

At P1: C = 100 - 64/4 = 84%

At P2: C = 100 - 256/4 = 36%

b. Suppose that the Daisyworld Sun brightens so that temperatures increase by 4 degrees for daisy coverage of 0%, i.e., T = 60 – C/2. What are the solutions in this case?

Equation of parabola:

Equation of line:

Equation of parabola = Equation of line

= à

Resource Guide

Video/Films:

Earth Systems Science Video

Series of 12 short Quicktime videos from NASA

http://education.gsfc.nasa.gov/video/

Episodes are:

Introduction to Systems

Satellites: Looking at Earth from Space

Introduction to El Nino

El Nino

Introduction to Global Warming

Global Warming

Introduction to Drought

Drought

Introduction to Hurricanes

Hurricanes

Epilogue

Credits

Websites:

http://www.pik-potsdam.de/~bloh/

http://www.cogs.susx.ac.uk/daisyworld/daisyworld.html#ASOFC

http://www.acad.carleton.edu/curricular/GEOL/DaveSTELLA/Daisyworld/daisyworld_model.htm

http://systemerde.ipn.uni-kiel.de/Daisyworld/daisyweb.html (James Lovelock's Daisyworld)

http://earthscape.org/t1/bid01/bid01g.html

http://gingerbooth.com/courseware/daisy.html

Literature:

Watson, Andrew J. and J. E. Lovelock. "Biological homeostasis of Daisyworld," Tellus 35B, 1983, pp. 284-289.

Lovelock, James E. The Ages of Gaia: A Biography of Our Living Earth. Norton, 1988.