Daisyworld

Model Description

Daisyworld is a simple planetary model designed to show the effects of coupled climate-vegetation dynamics. In this simple model, Daisyworld is a cloudless planet with a negligible atmospheric greenhouse, its ground is grey and it is inhabited by two species of different colors daisies. 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 as a flat planet, orbitting around a star.

The growth of daisies responds to the equations of population growth. The evolution of area fractions of white (${\alpha }_w$) and black (${\alpha }_b$) daisies is given by the following differential equations:

$$\begin{array}{c}\frac{\partial {\alpha }_w}{\partial t}={\alpha }_w\left[{\alpha }_g\beta \left(T_w\right)-\gamma \right]\hfill \\ \frac{\partial {\alpha }_b}{\partial t}={\alpha }_b\left[{\alpha }_g\beta \left(T_b\right)-\gamma \right]\hfill \end{array}$$

(1)

where ${\alpha }_g$ is the area fraction of bare ground, $\beta \left(T\right)$ is the birth rate for a given temperature $T$ and $\gamma$ is the death rate. In this case, $\gamma$ is kept fixed and has the same value for both black and white daisies ($\gamma$ = 0.3). Here ${\alpha }_g\equiv p-{\alpha }_b-{\alpha }_w$, with $p$ refering to the proportion of fertile ground in the system (we take $p=$ 1). $T_w$ and $T_b$ are local temperatures felt by each daisies species. $\beta \left(T\right)$ is defined as follow:

$$\beta \left(T\right)=\left\{\begin{array}{cc}1-k{\left(T-T_{\text{ opt}}\right)}^2\hfill & \text{ if }\left|T-T_{\text{ opt}}\right|<k^{-\frac{1}{2}}\hfill \\ 0\hfill & \text{ otherwise}\hfill \end{array}\right.$$

(2)

where $T_{\text{ opt}}=295.5\text{ K}$ is the optimal temperature. The parabolic width $k$ is chosen so that the daisies grow for temperature between 5˚ C and 40˚ C, i.e. $k=17.5^{-2}$.

Fixed albedos are prescribed for the white daisies ($a_w$), for the black daisies ($a_b$) and for the bare ground ($a_g$). The planetary albedo $A$ is therefore given by:

$$A={\alpha }_w{a}_w+{\alpha }_b{a}_b+{\alpha }_g{a}_g$$

(3)

where $a_g=\frac{1}{2}$ by convention and $a_b<a_g<a_w$.

The local temperatures $T_w$ and $T_b$ and the bare ground temperature $T_g$ are obtained through a simplification of the heat tranfer equation. Local temperatures are defined as:

$$\begin{array}{c}{T}_w^4=q\left(A-a_w\right)+T^4\hfill \\ T_b^4=q\left(A-a_b\right)+T^4\hfill \\ T_g^4=q\left(A-a_g\right)+T^4\hfill \end{array}$$

(4)

where $T$ is the planetary temperature and $q=$ 2.06 $\times$ 10${}^9$ K${}^4$ is the heat transfer coefficient. This planetary temperature responds the Stefan-Boltzmann law for black body radiation:

$$SL\left(1-A\right)=\sigma T^4$$

(5)

where $SL$ is the average solar energy flux incident on the Daisyworld. $S=$ 917 W m${}^{\text{ -2}}$, $L$ is an adjustable parameter representing the luminosity of the star and $\sigma$ is the Stefan-Boltzmann constant ($\sigma =$ 5.67 $\times$ 10${}^{\text{ -8}}$ W m${}^{\text{ -2}}$ K${}^{\text{ -4}}$).

Exercises

After clicking on the button ”Launch Daisyworld” below, an applet showing several graphs should appear. The evolution of daisyworld variables is shown through increasing and decreasing values of luminosity. The model equations are integrated using the finite difference method. The methodology used 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 ${\alpha }_w$ and ${\alpha }_b$ of the new simulation are set to the steady state value of the previous simulation, or 0.01 if these equal 0. The procedure is repeated for increasing values of $L$. The same procedure is also applied to decreasing values of $L$.

The system variables present hysteresis, i.e. their behaviour for increasing luminosity is not the same as the one for decreasing luminosity. This particular property can be seen on the graphs of the applet.

Several parameters of the daisyworld model can be modified to see their effects on the system variables. This should help you to answer the questions in the following quiz. After answering each question, please check it using the box on the left before going to the next question.

Note: If you are using Internet Explorer or Firefox, you may need to install Java in your browser to run the applet. Here are installation instructions for Internet Explorer and for Firefox.

Useful References

$\left[1\right]$ Bonan G., 2008, Ecological Climatology – Concepts and Applications, Cambridge University Press, 2nd. Ed., 550 pp.
$\left[2\right]$ Claussen M., C. Kubatzi, V. Brovkin and A. Ganopolski, 1999, Simulation of an abrupt change in Saharan vegetation in the mid-Holocene, Geophysical Research Letters, vol. 26, No. 14, 2037-2040.
$\left[3\right]$ 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.
$\left[4\right]$ Watson A. J. and J. E. Lovelock, 1983, Biological homeostasis of the global environment – The parable of Daisyworld, Tellus, Ser. B, 35, 284-289.
$\left[5\right]$ 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.

Acknowledgements

Components of the applet are based on the Java Components for Mathematics developed at Hobart and William Smith Colleges.