Daisyworld
Daisyworld is a simple planetary model designed to illustrate the influence of a coupled climate-vegetation feedback related to the different albedo of different vegetation types. In this simple model, Daisyworld is a cloudless planet with a negligible concentration of greenhouse gases in its atmosphere. Its ground is grey and it is inhabited by two species of daisies with different colors. One
species is black and has a low albedo while the other one is white and has a high albedo. The black
and white daisies are dinstinct species and there is therefore no possibility of mixed replication of
the types. For simplicity, Daisyworld is considered to be a flat planet, orbitting around a star.
Fractions of daisies and temperatures as function of the luminosity
Changing the parameters may lead to a small time delay before the figures adjust to the new values.
Model description
The fraction of daisies responds to the equations of population growth and the evolution of area fractions of white ($a_w$)
and black ($a_b$)
daisies is given by the following differential equations:
|
$$\table ∂a_w/∂t=a_w[a_gβ(T_w)-γ];
∂a_b/∂t=a_b[a_gβ(T_b)-γ]$$
|
(1) |
where $a_g$ is the area fraction
of bare ground, $β(T)$ is the birth
rate for a given temperature $T$
and $γ$ is the death
rate.
As a consequence, $a_g≡p-a_b-a_w$, with $p$
refering to the proportion of fertile ground in the system (we take
$p=1$). $t$ is a unitless time as we only look for steady state solutions (see below).
Here, $γ$
is kept fixed and has the same value for both black and white daisies
($γ=0.3$).
$T_w$ and $T_b$ are local temperatures felt by each daisies species. $β(T)$
is defined as follows:
|
$$β(T)=\{\,\cl"ma-join-align"{\table {1-k(T-T_{\text"opt"})}, \, ,{\text"if " ∣T-T_{\text"opt"}∣<k^{-1/2}};
{0}, \, , {\text"otherwise"}$$
| (2) |
where $T_{\text"opt"}$=295.5 K (22.5°C) is the optimal
temperature. The parabolic width $k$
is chosen so that the daisies grow for temperature between 278 K and 313 K (5 °C and 40 °C), i.e.
$k=17.5^{-2}$ K$^{-2}$.
Fixed albedos are prescribed for the white daisies
($α_w$), for the black
daisies ($α_b$) and for the
bare ground ($α_g$). The
planetary albedo $α_p$
is therefore given by:
|
$$α_p=a_wα_w+a_bα_b+a_gα_g$$ | (3) |
where $α_g=1/2$ by
convention and $α_b<α_g<α_w$.
The local temperatures $T_w$ and $T_b$ and the bare
ground temperature $T_g$
are obtained through a simple heat balance including a heat transfert between the different regions. Local temperatures are obtained
as:
|
$$\table
T_w^4=q(α_p-α_w)+T^4;
T_b^4=q(α_p-α_b)+T^4;
T_g^4=q(α_p-α_g)+T^4;
$$
| (4) |
where $T$ is the planetary
temperature and $q=
2.06 × 10^9$ K$^4$ is
a heat transfert coefficient. The planetary temperature is derived from the global heat balance of Daisyworld:
where $S_oL$
is the average solar energy flux incident on the Daisyworld.
$S_o=917$ Wm$^{-2}$.
$L$ is an adjustable parameter representing the luminosity of the star and
$σ$ is the Stefan-Boltzmann
constant ($σ=5.67×10^{-8}$Wm$^{-2}$K$^{-4}$).
The evolution of the model variables is shown for increasing and decreasing values of
luminosity. The model equations are integrated using the finite difference method. The
methodology applied here is the one introduced by Watson and Lovelock (1983). For a given value of
$L$,
the model equations are integrated until a steady state is reached. The value of
$L$ is then incremented and
the initial conditions for $a_w$ and $a_b$ of
the new simulation are set to the steady state value of the previous simulation,
or 0.01 if these equal 0.
The same procedure is applied to decreasing values of
$L$. The system variables display an hysteresis, i.e. their behaviour for increasing luminosity is not the
same as the one for decreasing luminosity.
In order to answer the questions in the following quiz, you can modify several parameters of the daisyworld model and identify their effect on the state variables. After
answering each question, please check it using the box on the left before going to the next
question.
Bonan
G., 2008, Ecological Climatology – Concepts and Applications, Cambridge University Press, 2nd.
Ed., 550 pp.
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2037-2040.
Liu Z.,
Y. Wang, R. Gallimore, M. Notaro and I. C. Prentice, 2006, On the cause of abrupt vegetation
collapse in North Africa during the Holocene: Climate variability vs. vegetation feedback,
Geophysical Research Letters, vol. 33, L22709, doi:10.1029/2006GL028062.
Watson A. J. and J. E. Lovelock, 1983, Biological homeostasis of the global environment – The
parable of Daisyworld, Tellus, Ser. B, 35, 284-289.
Wood
A. J., G. J. Ackland, J. G. Dyke, H. T. P. Williams, T. M. Lenton, 2008, Daisyworld: A review,
Rev. Geophys., 46, RG1001, doi:10.1029/2006RG000217.
