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\title{Gaia's daisy word}
\author{great papers, Eli}
\maketitle


\section{Model formulation}

Following \citet{Lenton-Lovelock-2001:daisyworld}, and
\citet{Watson-Lovelock-1983:biological}.

Let area fraction occupied by black/ white daisies be $a_b$, $a_w$.
Corresponding temperatures are $T_b$, $T_w$.  The temperature of bare
ground is $T_g$.  Let also $a_g$ be the fraction of the area not
occupied by daisies (that is, bare \textbf{g}round, denoted $x$ in the
above papers),
\begin{align*}
  a_g=1-a_w-a_b.
\end{align*}
Growth equation for the areas are basically the logistic equation,
\begin{align*}
  \frac{da_w}{dt}&=a_w(a_g\beta-\gamma)=a_w((p-a_w-a_b)\beta-\gamma)\\
  \frac{da_b}{dt}&=a_b(a_g\beta-\gamma)=a_b((p-a_w-a_b)\beta-\gamma).
\end{align*}
The growth rate is temperature dependent, with a maximum at the
optimal temperature,
\begin{align*}
  \beta(T)=\max(0,1-(22.5-T)^2/17.5^2).
\end{align*}
Let the planetary temperature, $T_p$, be defined as the area-weighted
average,
\begin{align*}
  T_p=a_gT_g+a_wT_w+a_bT_b.
\end{align*}
The energy balance of the areas covered by black and white daisies,
and that of bare ground, includes the solar input, the albedo of each
of the daisy types, and a transport (mixing) term between each of the
two areas and the planetary temperature, represented by a transport
coefficient $q$,
\begin{align*}
  \pd{T_w}{t}&=0=S(1-\alpha_w)+q(T_p-T_w)-\sigma T_w^4\\
  \pd{T_b}{t}&=0=S(1-\alpha_b)+q(T_p-T_b)-\sigma T_b^4\\
  \pd{T_g}{t}&=0=S(1-\alpha_g)+q(T_p-T_g)-\sigma T_g^4.
\end{align*}
This way if
one adds the two equations to get one for the total heat content
\begin{align*}
  \frac{d}{dt}(a_bT_b+a_wT_w+a_gT_g)=\ldots,
\end{align*}
the transport term drops out, as it should because it only moves heat
within the earth and does not affect to total heat content.


\section{Solving the model equations}

To solve these equations, assume that the temperature adjustment time
is rapid so that we can treat the temperature equations as being in a
steady state to solve for the temperatures as function of the areas.
But first, need to linearize the long wave radiation around a
reference temperature, $T_0=273$, e.g.,
\begin{align*}
  \sigma T_b^4=\sigma T_0^4+4\sigma T_0^3(T_b-T_0)=A+B(T_b-T_0).
\end{align*}
The temperature equations therefore become,
\begin{align*}
  0&=S(1-\alpha_w)+q(T_p-T_w)-(A+B(T_w-T_0))\\
  0&=S(1-\alpha_b)+q(T_p-T_b)-(A+B(T_b-T_0))\\
  0&=S(1-\alpha_g)+q(T_p-T_w)-(A+B(T_g-T_0)).
\end{align*}
Collect terms, remembering that $T_p$ is a function of the other
temperatures,
\begin{align*}
  T_w\Big[q(a_w-1)-B\Big]+
  T_b\Big[qa_b\Big]+
  T_g\Big[qa_g\Big]
  &=\Big[-S(1-\alpha_w)+(A-BT_0)\Big]\\
  T_w\Big[qa_w\Big]+
  T_b\Big[q(a_b-1)-B\Big]+
  T_g\Big[qa_g\Big]
  &=\Big[-S(1-\alpha_b)+(A-BT_0)\Big]\\
  T_w\Big[qa_w\Big]+
  T_b\Big[qa_b\Big]+
  T_g\Big[q(a_g-1)-B\Big]
  &=\Big[-S(1-\alpha_g)+(A-BT_0)\Big].
\end{align*}
This may be written in matrix form,
\begin{align*}
  \mathsf{A}\left(\!\!\begin{array}{c}T_w\\T_b\\T_g\end{array}\!\!\right)=\Gamma,
\end{align*}
and solved to find the temperatures.  If the bare ground area, $a_g$,
approaches zero, we do not need the ground temperature for the
planetary temperature average and the third temperature equation is
not needed (it also becomes singular).  Given the temperatures we can
advance the equations for the areas in time.  For the transport term
to be a significant player but not too dominant, we want
$q\lesssim{}B=4\sigma{}T_0^3$.

\begin{figure}[!htbp]
  \begin{center}
    \includegraphics[width=6in]{Figures/dasyworld-model-results.pdf}
  \end{center}
  \caption{Results of \texttt{gaia.m}, solving the equations in these
    notes for areas and temperatures as function of a nondimensional
    factor multiplying the solar luminosity.  Note the regulation of
    the temperature for luminosity in the range of about 0.7 to 1.}
  \label{fig:model-results}  
\end{figure}



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