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\title{Laplace equation and using Sturm-Liouville theory on a sphere:
  the energy balance of the atmosphere}

\author{Eli, APM105b}
\maketitle

\section{Derivation of equation}

Consider a simple model for the earth atmospheric surface temperature.
Because most of the atmosphere is concentrated in the first 10 km
above the surface, while the earth radius is 6300km, we consider the
atmosphere to be a very thin layer of fluid on a sphere.  We ignore
variations in temperature in both height and longitude and assume the
temperature is only a function of latitude.  The physical processes
affecting the atmospheric temperature are now diffusion in the
north-south direction, heating by the sun, and heat radiating away
from the earth back to outer space.

We need to work with spherical coordinates $(\rho,\phi,\theta)$, where
$\theta$ is longitude, and $\phi$ is latitude, with $\phi=0$ at the
equator.  There is input of radiation from the sun which varies with
latitude, which may be written to a very good approximation as
\begin{equation*}
  S(x)=S_0(1+S_2P_2(x)),
\end{equation*}
where $S_0=\frac{\sigma}{4}(1-\alpha)$, and $\sigma/4$ is the
radiation from the sun averaged over the earth area and over a full
day.  $\alpha$ is the fraction of solar radiation reflected back to
space by clouds and ice, also known as the ``albedo''.  It is
convenient to use $x\equiv\sin\phi$ as explained below,
$P_2(x)=\frac12(3x^2-1)$ is the second Legendre polynomial.  There is
also output of energy that is radiated away from the earth depending
on the temperature at the surface.  This energy output may be modeled
as black body radiation, which may be linearized in the surface
temperature $T(\phi)$ as follows
\begin{equation*}
  I(T)=\epsilon\sigma T^4\approx A+BT.
\end{equation*}
Including in the diffusion only terms due to derivatives of the
temperature with respect to latitude and dropping derivatives with
respect to $\rho$ and $\theta$ we have
\begin{equation*}
  0=\frac{D}{\rho^2}
    \frac{d}{dx}\left((1-x^2)\frac{dT}{dx}\right)
  +S_0(1+S_2P_2(x))-(A+BT)
\end{equation*}
where $\rho$ is now the radius of the earth.  This we may rearranged
into
\begin{eqnarray}
  \frac{d}{dx}\left((1-x^2)\frac{dT}{dx}\right)
  -\hat{B}T&=&\hat{A}-\hat{S}(1+S_2P_2(x)) 
  \nonumber \\
  &=&(\hat{A}-\hat{S})P_0-\hat{S}S_2P_2(x)
  \label{eq:final-heat-balance}
\end{eqnarray}
where $\hat{B}\equiv\frac{B\rho^2}{D}$,
$\hat{A}\equiv\frac{A\rho^2}{D}$ and
$\hat{S}\equiv\frac{S_0\rho^2}{D}$; in addition, $P_0(x)=1$ and
$P_2(x)=\frac12(3x^2-1)$.  This is a Legendre-like equation, so that
it makes sense to expand the solution in terms of Legendre
polynomials.  Note that the rhs is already written in terms of
Legendre polynomials, which should make the solution simpler.  This is
the reason for approximating the solar radiation as function of
latitude in terms of the second Legendre polynomial $P_2(x)$. 

The numerical values of constants here may be taken as follows
\citep{North-Cahalan-Moeng-1981:energy}, $A=203.3Wm^{-2}$,
$B=2.09Wm^{-2}{}^{\circ}C^{-1}$, $S_2=-0.477$, $\sigma=1340Wm^{-2}$,
$\alpha=0.3$, $D=0.649Wm^{-2}{}^{\circ}C^{-1}$ and
$\rho=6.3\cdot10^6m$, so that $\hat{A}/\hat{B}=97.3$,
$\hat{S}/\hat{B}=112.2$.  Note that $S_0$ is the solar constant
divided by 4 and corrected for reflection by clouds.


\section{Solution via a Sturm-Liouville expansion}

Remember that the Legendre polynomials satisfy the Legendre
equation, 
\begin{equation*}
  \frac{d}{dx}\left((1-x^2)\frac{dy(x)}{dx}\right)+\lambda y(x)=0.
\end{equation*}
This equation is singular at the poles, $x=\pm1$.  The appropriate
Sturm-Liouville boundary conditions there are therefore that the
solution is bounded at $x=\pm1$.  This is obtained when the eigenvalue
is $\lambda=n(n+1)$, when the power series terminates leading to finite
polynomial solutions.  These polynomials, $P_n(x)$, are therefore the
eigenfunctions of the above diffusion operator on the sphere and
satisfy
\begin{equation*}
  \frac{d}{dx}\left((1-x^2)\frac{dP_n(x)}{dx}\right)=-n(n+1)P_n(x).
\end{equation*}
Their orthogonality condition is (weight function is 1, see Greenberg
p 910, section 17.8),
\begin{equation*}
  \int_{-1}^1 P_n(x)P_m(x)dx=\delta_{mn}\frac2{2m+1}.
\end{equation*}
\textit{The fact that the Legendre equation is a Sturm-Liouville
  problem guarantees that the Legendre polynomials are a complete set,
  so that we know that any solution for the temperature may be
  expanded in terms of these polynomials.}  Substituting a solution of
the form
\begin{equation*}
  T(x)=\sum_n a_nP_n(x),
\end{equation*}
into \eqref{eq:final-heat-balance} gives, 
\begin{equation*}
  \sum_{n=0}^\infty a_n(-n(n+1)-\hat{B}) P_n= 
  (\hat{A}-\hat{S})P_0 -\hat{S}S_2P_2.
\end{equation*}
We now multiply by $P_m(x)$ and integrate from $-1$ to $1$ to find
\begin{eqnarray*}
  a_m(-m(m+1)-\hat{B})\frac{2}{2m+1}
  &=&(\hat{A}-\hat{S})\frac{2}{2\times0+1}~\delta_{m0}
  -\hat{S}S_2\frac{2}{2\times2+1}~\delta_{m2} \\
  &=&2(\hat{A}-\hat{S})~\delta_{m0}
  -\frac{2}{5}\hat{S}S_2~\delta_{m2}
\end{eqnarray*}
which leads to the solution for the coefficients,
\begin{equation*}
  a_m=
\begin{cases}
  \frac{\hat{S}-\hat{A}}{\hat{B}} & \hbox{if $m=0$}\\
  \frac{2}{5}\frac{\hat{S}S_2}{6+\hat{B}} & \hbox{if $m=2$}\\
  0 & \hbox{else} \\
\end{cases}
\end{equation*}
The solution 
\begin{equation}
  T=\frac{S-A}{B}P_0+\frac{2}{5}\frac{\hat{S}S_2}{6-\hat{B}}P_2(x)
\end{equation}
is shown in the figure.

\begin{figure}[!bh]
\begin{center}
  \includegraphics[width=6in]{Figures/energy-balance-solution.pdf}
\end{center}
\end{figure}


This solution is not too bad given our simplifying assumption.  The
most obvious problem with this solution is that the temperature is too
warm at the pole.  This is because we have neglected the reflection of
solar radiation by polar ice (e.g. Arctic sea ice).  Incorporating ice
involves making $\alpha$ a function of temperature and makes things
very interesting, we may get to that at a later part of the course.

\bibliographystyle{apalike}
\bibliography{/home/eli/tex/Bibliography/all,/home/eli/tex/Bibliography/mypapers,/home/eli/tex/Bibliography/myproceedings,/home/eli/tex/Bibliography/mychapters,/home/eli/tex/Bibliography/mymisc}
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