Supplementary notes on Daisyworld
Effect of daisy coverage on surface temperature
The first of these relationships is that the larger the fraction of the surface of the planet that is covered by white daisies, the higher the albedo (albedo = fraction of sunlight which is reflected by the surface), and consequently the cooler the planet. You can see this relationship more clearly if you rotate the diagram counterclockwise 90 degrees so that the DAISY COVERAGE axis is along the bottom. (The DAISY COVERAGE scale will go backwards but that doesn't matter.) The family of diagonal lines describes the equilibrium temperature of the planet for various fractions of daisy coverage ranging from 0% on the right to 100% on the left. Each blue line corresponds to a different luminosity of Daisyworld's sun. The more luminous the sun, the warmer the temperature. If the sun is weak, the temperature of Daisyworld will be cold, even if there are no daisies present to whiten the surface. If the surface is 100% covered by daisies, the planet will reflect all the solar radiation impinging on it back to space without absorbing any of it, so its equilibrium temperature will be absolute zero, regardless of how luminous the sun is.
Effect of surface temperature on
daisy coverage
The second relationship
involves the influence of the temperature of the planet upon the daisy
coverage (red curve in the above figure). Here we try to simulate
the biosphere on earth. We assume that the daisies can survive only
within some limited temperature range, and have an optimum temperature
at which they cover the maximum fraction of the planet's surface.
This relationship is seen most clearly when the figure is viewed as presented,
with the TEMPERATURE axis at the bottom.
These two relationships were then used
to examine the climate history of Daisyworld in response to the increasing
luminosity of this planet's sun.
Response of Daisyworld to increasing
solar luminosity
Now suppose that we
start out with a very faint sun in Daisyworld's early history and let the
sun's luminosity gradually increase with time, just as it has in the Earth's
history. At first the surface of the planet will be too cold to support
daisies so the daisy coverage will be 0%. The state of the system will
move from left to right (point O). As the luminosity increases further,
daisies begin to grow. As daisies grow, the albedo of the planet
increases so that it reflects a larger fraction of the incoming solar radiation
without absorbing it. Hence the planet doesn't warm nearly
as rapidly in response to the increasing luminosity as it did when the
temperature of the planet was too cold to support daisies (negative
feedback between daisy coverage and surface temperature). Instead
of following along the x-axis, the state of the planet goes up the hump
past point P.
As the state approaches the top of the hump Q, the daisy coverage begins to level off and the temperature of the planet begins to increase more rapidly in response to increasing solar luminosity until it reaches point Q, which marks the optimal temperature for the daisies. Any further increase in luminosity will cause daisies to begin to die off, lowering the albedo, making the planet warmer, causing more daisies to die off, etc., etc. until the daisies are all gone (positive feedback between daisy coverage and surface temperature). This catastrophe will happen very rapidly and once the daisies are gone the temperature of the planet will be much hotter than it was when they were present (i.e., the state of the system would jump from Q directly to S in just the time it takes the daisies to wilt). Hence, the daisies are able to maintain the temperature of the planet in the range between their lower limit for survival and their optimal temperature for a remarkably long time just by 'doing their thing', but once the planet reaches their optimal temperature, their population abruptly 'crashes'.
Stable equilibrium
Points like P that
lie along the left side of the hump represent stable equilibrium states
of the system. Suppose invaders come in and pick a few daisies and
then leave. The reduction in daisy coverage will increase the albedo
of the planet, causing it to warm. A warmer planet will support more daisies.
Hence the daisies that the invaders picked will grow back. If the
invaders plant daisies, the planet will cool to a level less favorable
for daisies and the extra daisies will die off.
Unstable equilibrium
Points like P' that
lie along the right side of the hump represent unstable equilibrium states
of the system. Suppose the planet is in one of these states and invaders
come in and pick a few daisies and then go home. The reduction in
daisy coverage will increase the albedo of the planet, causing it to warm.
A warmer planet will support fewer daisies, so daisies will start dying,
causing the planet to warm more until all the daisies are gone. If
the invaders plant daisies, the planet will cool, and more daisies will
start growing, cooling the planet, enabling more daisies to grow until
daisy coverage reaches the value at the top if the hump. In either
case the system reacts in a way that causes the state to depart from the
equilibrium state, rather than return to it.
So if we ran the experiment
in reverse and started with a strong sun with a luminosity so strong that
Daisyworld was too hot to support daisies, and if we gradually reduced
the luminosity what would happen? In this case the state of the system
would start near S move from right to left along the x-axis until it encountered
the base of the hump R. Then it would jump abruptly to the top of
the hump Q and
then descend slowly down the left side.
It would be the exact same sequence in reverse, but for one difference:
in this case, the points between points R and S wouldn't be bypassed.
Daisyworld simulation
You can run your own simulation of Daisyworld using the JAVA module developed
by Ginger Booth and her colleagues at Yale: click here.
Last
Updated:
03/01/2001