Exercise 11 -- Daisyworld

For years students have asked me if there is anything available for extra credit. In general, the answer has been no. In part, because the question was usually raised around the time of the final evaluation when the cause of the motivation was rather apparent. This exercise is worth 20 points and will be added to the total for home work exercises.

The following paragraphs are from the COVIS discussion of Daisyworld.

Predictions of climate change depend on feedback loops. One type of a feedback loop is called a vicious circle. The most famous example of a vicious circle is blood feuds: A member of one tribe or family is killed by another tribe or family, so that injured party strikes back, often killing more than one; this in turn leads to a response, often more severe, which leads to more and more killing. Similarly, some people worry the increase in global temperature could begin a cycle of increases where the earth will get warmer and warmer until it boils off all the water on the planet and the atmosphere becomes pure carbon-dioxide. This condition is called 'runaway greenhouse effect' and accurately describes what happened on Venus. The cycle would proceed as follows: the carbon-dioxide that people have added to the atmosphere leads to increased temperatures, this temperature increase leads to more water being evaporated, which raises temperatures more and (since over 99% of the world's carbon is stored in the ocean) the evaporation leads to more carbon being released into the atmosphere -- which returns us to the beginning of the cycle.

Vicious circles can be broken when there is some type of negative feedback or force that brings a system to balance instead of continuing the change. For example, feuds can be broken when the level of grief overwhelms the survivors and stops the cycle. Similarly, the Earth might be able to stop the vicious circle of climate warming by having a negative feedback. One source of negative feedback could come from the adaptation of life systems (or the biosphere). This possibility has been explored by James Lovelock who calls his theory the Gaia hypothesis. This hypothesis claims, for example, that if the level of CO₂ in the atmosphere increases, then plant life will increase in order to absorb the CO₂ and convert it back into oxygen through photosynthesis.

Imagine a world populated only by black and white daisies. As with life on earth, the daisies are at the mercy of the climate and can only survive moderate temperatures. However, just like the rainforests, even the modest little daisies can effect the environment by absorbing varying amount of sunlight, altering the temperature. On Daisyworld there are black daisys and white daisys. When the temperature is high white daisys fluorish as they reflect heat (increased albedo) and the temperature is lowered. At lower temperatures black daisys fluorish as they absorb heat (lowered albedo) and the temperature. Lovelock hypothesized that such a simple world could (under certain conditions) reach a steady state such that the temperature is stabilized at a value which allows both kinds of daisys to grow. At temperatures lower than 5^oC and higher than about 40^oC daisys cannot survive.

What happens is based on the assumption that daisies have a set temperature where they grow best. If the earth is colder than this optimum black daisies will grow more than white daisies, since black objects absorb more of the sun's energy than white objects do leading the black daisies to be warmer than their white daisies. Having more black daisies causes the earth to absorb more energy since black objects absorb more sunlight than white objects. This increase in energy absorption leads to higher temperatures on earth. Now the feedback occurs -- since the earth's temperature has increased it is now closer to the optimum temperature for growing daisies. Alternatively, if the temperature were higher than the optimum daisy temperature white daisies will grow better since when they reflect more of the sun's energy than the black daisies do. Having more white daisies means the earth will absorb less energy (since white reflects better than black) which will in turn lower the earth's temperature. This lower temperature again serves to bring the earth's temperature closer to the temperature daisies prefer. This feedback cycle shows how living systems on earth can create feedback loops that regulate the earth's temperature and cause it to gravitate towards temperatures which are best for their growth. You can experiment with the balance between temperature and the amount of the surface of daisyworld populated by black and white daisys using a Mathematical Model. You can set the fraction of the surface originally covered by black and white daisys and the amount of sunlight. In working with a model of this type it is easy to plug in values, look at the results and plug in different values without getting a feeling for what is happening. Following a systematic investigation in which you hold at least one of the variables constant is a good way to begin.

We will assume that the original surface contains 0.7 (70%) white daisys and 0.1 (10%) black daisys. The amount of sunlight is one unit. Click the RUN button. After a brief delay two graphs are plotted along with a summary of the experiment. The two graphs are reproduced below. Time is plotted across the bottom and the fraction of white daisys (shown in red) and black daisys (shown in that awful green which is harder to see) is plotted on the Y-axis. Graphs for this first experiment are reproduced below.

Diasy Populations

Note that the original population of white daisys initially drops. It must be too cold for them to survive. As the white daisy population drops less of the incident radiation is reflected from the surface and the temperature drops. Black daisys begin to increase as the temperature drops. These black daisys absorb heat and the surface temperature rises. Eventually the fractions of white and black daisys remain constant at a constant temperature and a steady state system is realized.

Temperature

Note that the white daisys are associated with a lower average temperature since they reflect a higher percentage of the incident radiation whereas the black daisys are associated with a lower temperature.

Beneath the plots is a summary of the "equilibrium" conditions:

1. 40.9% White Daisys
2. 26.7% Black Daisys
3. 21^oC is the average surface temperature.

Scroll back to the start of the model and reduce the fractions to 0.0 and rerun the simulation. This is deadworld and the temperature of deadworld is what the temperature of daisyworld would be in the absence of daisys. You should get an average surface temperature of about 27^oC. Note that the initial temperature of the first simulation was nearly 0^oC. The high percentage of white daisys, which reflect a high percentage of the incident radiation, reduces the temperature.

Also note that "life" on daisyworld reduces the average surface temperature by about 6^oC.

1. Perform two addition experiments at the same level of incident radiation.

   White Start	White Final	Black Start	Black Final	Temp Start	Temp Final
   0.70		0.1
   0.35		0.35
   0.10		0.70

2. Fill out the above table. What relationship do you see between the initial temperature (of each simulation) and the initial composition of the daisys? Does this relationship make sense?

   For the conditions of the simulation (sunlight = 1.0) and the total fraction of the surface covered by daisys (0.70) the different initial compositions lead to the same equilibrium composition. Thus, we could view the initial compositions as disturbances to an equilibrium state -- the system should shift to reduce the disturbance. If there are more black daisys to start with, the system should shift in favor of the white daisys; that is, the temperature will decrease as white daisys replace black daisys.

   <3>Increase the sunlight constant to 1.25 and fill in the spaces in the table that follows. First, however, what do you think will happen to the temperature of deadworld? What will happen to the initial temperature of each simulation? What will happen to the equilibrium composition (if there is one)?

   White Start	White Final	Black Start	Black Final	Temp Start	Temp Final
   0.70		0.1
   0.35		0.35
   0.10		0.70

   Note that for these conditions, equilibrium is not attained by all initial compositions. Explain what you think is happening. What relationship do you see between the initial temperature (of each simulation) and the initial composition of the daisys? Does this relationship make sense? Run a few more initial compositions to try and pin down the initial composition where the daisys rapidly go to zero. How did your predictions compare with the results of the simulations? [Be Honest]

3. Yet another simulation is called for. Change the sunlight constant to a value of 0.75 (that is, the incident radiation declines by 25%). Repeat with the values shown in the table below. First, however, what do you think will happen? Which group of daisys will most likely dominate the equilibrium population (if one exists)? Why?

   White Start	White Final	Black Start	Black Final	Temp Start	Temp Final
   0.70		0.1
   0.35		0.35
   0.10		0.70

4. Simulations are only as good as the equations which are written and the values assigned to the different variables. For example, you don't know how rapidly white daisys "die off".

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