Lab 6
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Oregon State University, Corvallis *
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201
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Geology
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Feb 20, 2024
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ATS 201 Module 6 Lab:
Daisyworld
Most climate models are so complex that it's difficult (impossible) to analyze all the possible climate feedbacks in the model. Instead, we're going to use a very simple model of a hypothetical planet to study how a simple negative feedback can regulate climate. This planet is comprised of a rock planet with two species of life on it: white daisies and black daisies. Despite the simplicity of this planet, it can have a relatively constant temperature for a wide range of solar luminosities (incoming solar radiation).
Learning Objectives:
Describe how the simple Daisy World self-regulates global temperature;
Explain how positive and negative feedback processes can control system behavior.
Understand that changes in external forcing can have differing effects based on the state
of the system, occasionally leading to large changes in climate state; While this document is long, most of it is background reading. You do not need to entirely understand the model
. Just get a basic idea of the feedbacks in the model and how the different components interact. Questions for you to answer start on page 8.
About the model
This module's optional reading explains some of the history and relevance of this particular model, if you are interested.
The model is basically an energy budget model, like those we've seen before:
Solar radiation heats the planet
The planet cools by emitting terrestrial radiation (according to the Stephan-Boltzman Law)
If the absorbed solar radiation = emitted terrestrial radiation, the planet is in equilibrium.
Otherwise, planet warms or cools.
The absorbed solar radiation depends on
Luminosity of the sun, which varies in the simulation from 0.65 to 1.65 times our sun's luminosity (in the default configuration of the model)
The albedo of the planet.
The planetary albedo, in turn, depends on the surface type. There are three possible surfaces:
Bare ground, which has an albedo = 0.5 = α
g
White daisy, which has an albedo = 0.75 = α
w
(higher than bare ground)
Black daisy, which has an albedo = 0.25 = α
b
(lower than bare ground)
We also need to know what fraction of the global surface is each type:
Fraction of bare ground = f
g
Fraction covered by white daisies = f
w
Fraction covered by black daisies = f
b
1
The planetary albedo is the weighted mean (average) of these three albedos. [See http://www.mathsisfun.com/data/weighted-mean.html
for a description of weighted mean if you are not familiar with it.] To calculate the planetary albedo, we multiply the fraction of each ground type by the albedo for that ground type, and sum up:
α = α
g
f
g
+ α
w
f
w
+ α
b
f
b
You do not need to completely understand this equation, just get a feel for how the planetary albedo depends on the different amounts of daisies. As the fraction of, say, white daisies increases, the planetary albedo becomes more like the white daisy albedo (0.75).
To determine the equilibrium temperature for the planet, we use the same energy budget equation as before:
L×Q
(
1
−
α
)
=
σ T
e
4
where Q
is now multiplied by a luminosity factor L (which allows us to vary incoming solar radiation in the model).
The other thing we need to know is the fraction of the planet covered by the different daisy types. Daisies will only grow within a particular temperature range, specifically between 5 and 40 °C. They grow best in the middle of this range:
Growth rate of daisies based on Local Temp is in °C [From:
http://www3.geosc.psu.edu/~dmb53/DaveSTELLA/Daisyworld/daisyworld_model.htm]
What makes this planet interesting is that white and black daisies alter not just the planetary albedo, but their local temperature
as well. Land near white daisies is cooler
than the global average (or planetary) temperature
, while land near black daisies is warmer
than the global average temperature. Thus, black daisies can expand when the average temperature is 5 °C (because their local temperature is higher than 5 °C), but white daisies cannot. Conversely, white daisies can expand when the global average temperature is 40 °C, since their local temperature will be a little less than the average temperature. Daisies die off a constant rate, making room for other daisies to grow (conditions permitting).
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In the simulation, the solar luminosity (L) starts off low, so that no daisies can grow. As
luminosity increases, the global average temperature increases. Eventually it gets warm enough
that black daisies start to grow. The black daisies further warm the planet (due to their low
albedo), until it is warm enough for white daisies to grow. Once the white daisies start to grow,
they cool the planet. As the planet cools, black daises are more likely to grow, which leads to a
warming, which makes white daisies more likely to grow
....
These adjustments occur all the
time, so the swings in global average (planetary) temperature are actually pretty tiny, resulting in
a stable temperature. For most values of solar luminosity, the fractions of white and black
daisies adjust so that the planet reaches a new equilibrium temperature near the ideal growing
temperature for the daisies (22.5 °C). Here's a diagram of relationships in the Daisyworld model. Note how complex the interactions get when there are only a few model components. Imagine how ginormous a similar diagram would be for a more realistic climate model!
From: R.M. MacKay, WSU Geology
A first look at the model
This all sounds very complicated, but it's easier to just see for yourself what a simulation looks like. Please see the embedded videos
on the course webpage for this lab. These videos show all the simulations you need to answer the questions in this lab. 3
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If you can still run flash, you can play with the simulations yourself. Go to this web site and click the "Run" button:
http://www.gingerbooth.com/flash/daisyball/DaisyBall.html
Please use the Daisyball Classic version for this lab (this is the default when you open the webpage). The top plot shows the fractions of area covered by bare ground and the daisies. The bottom plot shows (in brown) the temperature the planet would have if there were no daisies and (in pink) the planet's temperature with daisies. You can mouse over the various lines to see what the values are for any given luminosity.
To learn more about this model, click the "?" button. Look at the DaisyWorld math page if you want to see all the equations.
Analysis of Feedback Loops
[Closely following assignment by R. M. MacKay, WSU Geology]
Daisyworld maintains a relatively constant temperature for so long due to negative feedback
loops. Negative feedback loops tend to stabilize the temperature (damping changes), as opposed to positive feedbacks, which tend to amplify temperature changes.
We'll start by considering what happens when we have only white or black daisies. We’ll do some basic thought experiments before actually using the model.
An analysis of white daisy coverage using feedback loops
First, consider the case where only white daisies grow:
I. White daisies reflect more sunlight than bare ground.
So as white daisies grow (the fraction of white daisies increases
), the planetary albedo increases
(indicated by the plus sign (+) in the following flow chart, which means that changes of one sign in "White Daisy Fraction" result in changes of the same sign in "Albedo"). As the planetary albedo increases
, the planetary temperature decreases
(indicated by the minus
sign (-)). The reverse is also true:
As white daisies die (the fraction of white daisies decreases
) the planetary albedo decreases
(the same sign as the daisy change) and the planetary temperature increases
(the opposite sign
of the planetary albedo change).
White
Daisies
Albedo
Temperature
4
This relationship can also be visualized with a graph of planet temperature as a function of white
daisy fraction:
The more white daisies you have, the higher the albedo, and thus the lower the planetary temperature.
Temperature
II. There is an optimum temperature for daisy growth.
To the left
of the graph’s peak (optimum temperature), increasing local
average temperature causes an increase in daisy coverage. (It was
too cold, but now it’s warmer and the daisies are more likely to grow.)
Planet Temperature
White Daisy Fraction Local Temperature
White Daisy Fraction Optimum Temperature
White
Daisies
Temperature
5
To the right
of the graph’s peak, increasing the local
average temperature causes a decrease in daisy coverage. (It becomes
too hot).
This figure also applies to black daisies in regards to the local temperature near them.
III. Combining the ideas from the first two parts.
To left of the graph’s peak (for lower temperatures)
:
An increase in temperature causes more white daisy growth, an albedo increase, and a temperature drop towards its original value.
-OR-
A decrease in temperature causes reduced white daisy coverage, an albedo decrease, and a temperature increase towards its original value.
This is a negative feedback loop. The temperature remains stable around the original value. The
red minus indicates that the feedback is negative. (The feedback is negative because there are an odd number of negative links. If you find this diagram confusing, remember that the signs by the arrows indicate whether the two variables have the same sign of change, not what the sign of the change actually is
. For the arrow from Temperature to White Daisies, the plus sign
indicates that White Daisies increase with temperature. But temperature decreases with increasing albedo. So white daisies decrease with increasing albedo.)
You can think of each link as multiplying by either +1 or -1. If you have an increase
in white daisies, multiply by +1 to get an increase
in the albedo. Then, you multiply the increase
in the White
Daisies
Temperature
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albedo by -1 to get a decrease
in temperature. You multiply the decrease
in temperature by +1 to get a decrease
in white daisies. When you go around the loop, you multiply +1 by -1 by +1 = -
1. To right of graph’s peak
: The effect of temperature on white daisies is negative, unlike for the previous case. An increase
in temperature causes reduced
white daisy coverage, an albedo decrease
, and a temperature increase,
which is added to the original temperature increase, so that the temperature moves even further away
from its original value. OR
A decrease in temperature causes more white daisy coverage, an albedo increase, and a temperature decrease further away from its original value. This is a positive feedback loop, which makes the temperature unstable. (Two negative arrows combine to make a positive arrow.)
Questions to answer Start by repeating the feedback analysis above for the case of black daisies. Complete all missing parts of the following text. Your answers should use similar terminology as those in the white daisy example above. {0.5 pt per blank}
1.
Black daisies absorb ____
more
_____________ sunlight than the bare ground. 2.
So as the black daisies grow, the planetary albedo ______d
ecreases
_______ and the planetary (global average) temperature _______
increases
___________. 3.
As black daisies die the planetary albedo _______
increases
____________ and the planetary temperature _____d
ecreases
____________.
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4.
Which of the following plots (
a or b) represents the effect of black daisies on temperature?
5.
Is there a + or a - for the first question mark? The second?
first is – second is +
The black daisies have the same optimal growth diagram as the white daisies.
6.
To the left of the graph’s peak, increasing the local average temperature causes a(n) ______
increase
___________ in daisy coverage. 7.
What sign should go in the following diagram in this case:
Planet Temperature
Black Daisy Fraction Planet Temperature
Black Daisy Fraction a)
b)
Black
Daisies
Albedo
Temperature
?
?
Local Temperature
Black Daisy Fraction Optimum Temperature
Black
Daisies
Temperature
?
8
a +
8.
To the right of the graph’s peak, increasing local average temperature causes a(n) _______
decrease
__________ in daisy coverage.
9.
What sign should go in the following diagram in this case:
-
Combining the ideas:
10. To left of graph’s peak: An increase in temperature causes ___
increase
_____ black daisy growth, an albedo ____
decrease
_________, and a temperature _____
increase away from
______ its original value. OR
A decrease in temperature causes ____
decrease
_______ black daisy coverage, an albedo ____
increase
________, and a temperature _________
decrease
__________ its original value. 11. This is a ____
negative
______ feedback loop. 12. The temperature is ____
not stable
_______ {stable or not stable}.
13. Indicate whether the signs at 1
(neg)
, 2
(pos)
, 3(positive)
, and 4
(negative) are positive or
negative.
Black
Daisies
Temperature
?
9
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14. To right of graph’s peak: An increase in temperature causes ____
increase
_______ black daisy growth, an albedo ____
increase
_________, and a temperature _______
decreases toward
________ its original value. OR
A decrease in temperature causes ___
less
________ black daisy coverage, an albedo _____
decrease
_______, and a temperature _________
increase
__________ its original value. 15. This is a ___
positive _______ feedback loop, which makes the temperature ____
unstable
______.
16. Indicate whether the signs at 1, 2, 3, and 4 are positive or negative.
1-neg, 2-pos,3-neg,4-pos
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Now consider both the white and black daisy feedbacks together. 17. When temperatures are below
the optimal temperature for daisy growth, which type of daisy is likely to expand quickly
with a temperature increase? {1}
black
18. When temperatures are above
the optimal temperature, which type of daisy is likely to die back quickly
with a temperature increase? {1}
white
19. In the simulation, we are increasing the luminosity, which tends to increase the temperature.
Therefore, which type of daisy will increase faster at the start of the simulation? {1}
Black Daisy
OK. It's time to watch the simulation! (if you have not already done so)
Watch the simulation with both black and white daisies. 20. Describe [5-6 sentences] what happens as the luminosity increases. What happens to temperature? To % area covered by white daisies? To % area covered by black daisies? {4}
As the luminosity increases, temperature increases. White increases as black decreases
21. Which type of daisy has a faster increase at the beginning? Which has a faster decrease at the end? Use the earlier discussion of feedback loops for the two daisies to explain these results. [5-6 sentences] {4}
black fast beginning and white fast at the end bc positive and negative feedback loops. 22. Is the local temperature in the black daisy fields warmer or cooler than that in the white daisy fields? {0.5}
Local temperature for black daisies fields are warmer than white daisy fields.
Because the local
temperature differs from the planetary (global average)
temperature, the curves for optimal temperature for white and black daisies are effectively offset from each other. Thus, the white and black daisies will be able to survive for different ranges of 11
planetary temperature.
23. At what luminosity do black daisies appear (above 0.1% area
1
)? [You can find this out by moving the cursor along the % area graph until you find the luminosity where the area goes above zero.] {0.5}
.73% luminosity
24. At what luminosity do white daisies appear (above 0.1% area)? {0.5}
.73% luminosity
25. At what luminosity do black daisies disappear (0.1% area or less)? {0.5}
1.33% luminosity
26. At what luminosity do white daisies disappear (0.1% area or less)? {0.5}
1.58 luminosity
27. If there were only black daisies (no white), at what luminosities do you think black daisies will appear and disappear? [This is just a guess
, not based on any equations. It's OK if you guess the wrong numbers here. I just want you to think about it a bit before you run the simulation.] {1}
I think the black daisies will appear at the 0.7 luminosity and disappear around the 1.2 luminosity
28. Check your guess by running the simulation with "1-black only" in the drop-down menu. What are the actual "appear" and "disappear" luminosities? {1}
appear .73 disappear 1.08
29. Compare the black daisy “appear” and “disappear” luminosities to those in the "2 - black and
white" [default] case. {1} 30. Which of these two cases ("1-black only" or "2 - black and white") has a larger range
of luminosities where the temperature is near the optimal temperature (a stable climate), regardless of whether or not black daisies are present? {1}
Now do the same for the white case:
31. If there were only white daisies (no black), at what luminosities would you expect white daisies to appear and to disappear? [It's OK if you guess the wrong numbers here. As before, I just want you to think about it a bit before you run the simulation.] {1}
32. Check this by running the simulation with "1-white only" in the drop-down menu. What are the actual "appear" and "disappear" luminosities? {1}
33. Compare the white daisy “appear” and “disappear” luminosities to those in the "2 - black and
white" [default] case. {1}
The appear rate is a bigger difference than the disappear luminosity is. The appear difference is .08 while the disappear difference is .01
34. How does the range
of luminosities corresponding to a stable climate for the white case compare to the black case? Do you find it surprising that they differ? (I did.) This is an example of why you can't just think of the white and black daisies as "opposites" of each other. {1}
1
In order for there to always be "seeds" for daisies to spread, the model periodically adds a small fraction of daisies if there are none. I think this is why you get 0.1% coverage outside the normal range sometimes.
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Return to the white and black case:
35. Do you think this range would be larger or smaller if you increased the albedo of the white daisies? {1}
36. Try it. Increase the albedo to 0.8 for white daisies in the "Advanced" settings. What is the corresponding maximum luminosity that has daisies? [You may need to increase the max luminosity under settings as well.] {1}
37. Does having a bigger range of possible albedos (that is, the white and black albedos are further apart) increase or decrease the stability of Daisyworld (i.e., increased range of luminosities with daises)? {1}
Now watch the second video with simulation using 16 daisy colors. Or if you have
a working version of flash and want to run the simulation yourself, go to the "Advanced" setting again, and "reset" the values. Now increase the number of colors in the "Daisies" drop-down menu to 16. This simulation adds more daisy types with albedos at various values between the black and white albedo values. 38. Compare this run to the default run of 2 colors. [It's helpful to open another browser window to run the default case.] Which simulation shows more temperature variation
in the "stable" luminosities? {1}
39. Explain why adding more daisy types changes the temperature variability within the stable range
. [2-3 sentences] {2.5}
[I think the fact that the stable range is smaller when there are more daisies is due to the fact that there are those "random" 0.1% coverages due to the model limitation mentioned in the footnote earlier. In any case, ignore the apparent decrease in the max luminosity with daisies.]
40. Finally, in all of these simulations, where are the largest temperatures changes, relative to the region of stable luminosities? {1.5}
While the daisies moderate temperature changes for the stable range, due to negative feedbacks, positive
feedbacks are dominating during these transition periods with fast temperature changes. (Remember that feedbacks are not constant, and in fact do change with temperature, as the feedback analysis above showed). For these luminosities, a small change in forcing (i.e., increase in luminosity) has a large change on temperature. This is called abrupt climate change
.
41. Explain how positive feedbacks can result in abrupt climate change, if there are no (or insufficient) countering negative feedbacks. [2-3 sentences] {2.5}
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