Exercises Ch 21

Chapter 21. Climate Control on Daisyworld -- 9 Exercises

1. Check your Answer to the 3rd Exercise in the book:

The model of daisyworld includes a 1st order lag between the change in the local temperatures and the subsequent change in the flower growth rates. The length of the lag time is set at 4 years, somewhat longer than the life of the daisies. Now, to match the original model by Watson & Lovelock, you would eliminate this lag by setting the binary variable (lag time?)to zero. To test the effect of longer lag times, leave lag time? at 1 and experiment with changes in the length of the lag time. If you complete this exercise correctly, you should see the results below

All three runs were generated with the the 5th variety of daisies (the W&L variety) and solar scenario #2 (the"heat shock" scenario).

2. Zeng's Criticism of Daisyworld as Chaotic

Zeng (1990) criticizes daisyworld on the basis that stable climatic conditions are not always maintained, despite the presence of daisies which supply the required feedback. He argues in his article that he used "both qualitative and quantitative methods from modern chaos theory ... to verify the presence of chaos in daisyworld." The key step in the analysis is the introduction of a time lag to allow the daisies to adjust to temperature variations. Page 310 of his article leaves the impression that the time lag corresponds to the time lag tested in the previous exercise. But when Zeng implements the time lag mathematically, he obtains surprisingly complex behavior (including chaos). Review Zeng's use of finite difference equations to implement a time lag in the flowers' adjustment to temperature changes. How does his time lag compare with the time lag in the previous exercise?

3. Jascourt's Response to Zeng

Jascourt's (1992) commentary helps to clear up the confusion left by Zeng's misleading article. Jascourt emphasizes the important differences between a collection of differential equations used by Watson and Lovelock)and a set of finite difference equations used by Zeng. Jascourt observes correctly that Zeng's approach introduces a one generation lag time in both the population feedback and the environmental temperature feedback. Review Jascourt's comments to learn if he finds Zeng's model descriptive of natural systems? If Jascourt were willing to accept Zeng's model, does he believe that it contradicts the original conclusions by Watson and Lovelock?

4. Experiments with a Discrete Daisyworld

Review the advice from Chapter 11 to "avoid thinking of DT as a real world parameter" and that "DT has NO counterpart in the real world." Now, for purposes of experimentation similar to Zeng (1990), let's ignore this advice. That is, assume that a larger value of DT may be used to represent a time lag in the system. Simulate daisyworld (variety #5, solar scenario #2) with larger and larger values of DT. How large must DT become before you observe curious behavior? Do you agree with Zeng that these experiments "raise questions about the validity of the Gaia hypothesis" (Zeng 1990, p. 309). Or do they simply reveal a "DT problem" as explained in Chapter 11?

5. Reflective Permafrost on Daisyworld

Two of the exercies in chapter 21 deal with Kirchner's critique of the Gaia Hypothesis. Kirchner (1989, p. 230) is quite correct when he observes that daisyworld was designed as a mathematical model "to illuminate the implications of the Gaia hypothesis, but not to establish its validity." He criticizes the work by Lovelock and Margulis because their research appears slanted toward the stabilizing forces of negative feedback: "[they] proposed a wide range of biological feedback mechanisms that might control the climate. Conspicuously absent from the bulk of the ensuing research, however, is any mention of processes that might destabilize the climate." This is an important criticism that certainly seems fair in light of the positive feedback loops shown in Figure 8.6. One of the destabilizing loops involves the loss of reflective properties of the permafrost with a warmer climate. Consequently, a revealing exercise would be the addition of reflective permafrost to daisyworld.

Begin by assigning a separate stock to the area covered by permafrost. Assume that the permafrost albedo is 0.85, a typical value for fresh snow. Initialize the area of permafrost at 100 acres. Include the permafrost area in the calculation of the planet's average albedo. Next, assume that the permafrost expands or shrinks depending on its local temperature. (When the local temperature falls below zero degrees, the permafrost expands. When it rises above zero degrees, it shrinks.) You may assume that the temperature in the vicinity of the permafrost is 25 degrees below the planet's average temperature. Should the average temperature rise above 25 degrees, the permafrost temperature will rise above freezing. This will trigger shrinkage in the permafrost and a reduction in its reflective power. The world would then become warmer causing further shrinkage in the permafrost.

Build this new model and compare the results with Watson and Lovelock with the solar luminosity constant at 1.0

6. Permafrost and the Span of Control

Use the model from the previous exercise to study the the span of control. Compare the span of control with the findings from the 1st exercise in chapter 21. Does the addition of permafrost expand or shrink the span of control?

7. Methane Trapped in the Permafrost

In the same spirit of the previous two exercises, expand the permafrost model to include methane trapped in the permafrost. Assign a stock to methane in the permafrost and a separate stock to methane in the atmosphere. Assume that 100 pounds of methane are trapped in each acre of permafrost. Since the initial area of permafrost is 100 acres, you would initialize the stock of methane in the permafrost at 10,000 pounds. Set the initial value of atmospheric methane at zero. Assume that methane enters the atmosphere with shrinkage in the permafrost. (A shrinkage of 1 acre per year releases 100 pounds of methane per year.) Assume that atmospheric methane acts as a greenhouse gas to increase the fraction of solar luminosity absorbed on daisyworld. If there is no atmospheric methane, the planet's average albedo is the same as the previous model. If 1,000 pounds of methane end up in the atmosphere, the planet's average albedo would be lowered by 8% from the values calculated in the previous exercise. If the entire 10,000 pounds escapes to the atmosphere, the albedo would be lowered by 80%. Build this new model and compare its results with Watson and Lovelock with the solar luminosity constant at 1.0

8. Trapped Methane and the Span of Control

Use the model from the previous exercise to study the span of control. Compare the span of control with the findings from exercise #6. Does the addition of trapped methane expand or shrink the span of control?

9. Lovelock's Variations in Daisyworld

Lovelock (1991, p. 70-72) has designed several versions of daisyworld. Instead of just two, there are many species of daises with varying pigments in the new models. Also, there are models in which the daisies evolve and change color. There are models in which rabbits eat the daisies and foxes eat the rabbits. And there are examples with major catastrophes which wipe out 30% of the daisies at regular intervals. According to Capra (1996, p. 110), Lovelock "finds that daisyworld is remarkably resilient under these severe disturbances." Review these variations in light of Kirchner's (1989) criticism that Lovelock's point of view is slanted in favor of the stabilizing feedbacks in nature. Do you believe his many variations are slanted in favor of negative feedback?