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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\A SIMPLE MODEL FOR CLIMATE BIFURCATION}





\markboth{A Simple Model for Climate Bifurcation}{A Simple Model for Climate Bifurcation}

\author{Mayer Humi\\
Worcester Polytechnic Institute\\
Worcester  MA 01609 USA\\
mhumi@wpi.edu
}

\makeStitlePDFLaTex


\begin{abstract}
In this paper we present (for educational purposes) simple zero and one
dimensional models for the mean temperature of the Earth. These models can 
exhibit bifurcations from the present Earth climate to an ice age or a ``Venus 
type of climate.'' The models are accompanied by Matlab programs which 
enables the user to experiment by changing the model parameters and explore 
the impact that these changes might have on the Earth's climate.
\end{abstract}



\section{SCENARIO DESCRIPTION}

The Earth's climate and its variations has always been of great interest 
to humans as it has major impact on human activities. 
In this paper we present prototype models for the mean temperature of the 
Earth which will enable us to explore possible bifurcations in the Earth's 
climate which might due in part to anthropogenic emissions and natural
forcing processes. These model are based in part on the approach presented
in \cite{WS1,GC}. More elaborate and sophisticated models are available in the 
literature \cite{MHS,WGB,GC}.

Much was written lately about the gradual change in the mean temp of  
the earth due to human intervention which is expected to range by 1-2 degrees
by the end of this century. However the real danger of these changes is that
they may lead to a rapid major change in the Earth climate (viz.~climate 
bifurcation \cite{HAD}) in the same way that a rubber band snaps suddenly 
when it is over stretched or a sudden snow avalanche occurs on the slopes of a 
mountain. 

The Earth climate system is composed of the following components: land,
biosphere, atmosphere, ocean and the cryosphere (ice and frozen water),
These components display a broad range of variability on temporal and
spatial scales such as the Dansgaard-Oeschger cycles which occur
quasi-periodically on a millennial time scale or the El Ni\~{n}o-Southern
Oscillation in the equatorial Pacific. Thus a refined model of such
a complex system requires vast amounts of data and elaborate sophistication.
In the following we stick to the basics.

\section{Some Basics}

In this section we introduce some basic concepts and data needed
for the modeling of the Earth's climate.

\subsection{The Albedo}

When radiation impinges on a (perfect) mirror all of this radiation
is reflected back and the temperature of the mirror remains unchanged.
On the other hand, if such radiation impinges on a (perfect) black body
all of the radiation is absorbed by that body. In general however
some of this radiation is absorbed by the body and some is reflected 
back. We say that in this case we are dealing with a ``grey body.''
In general the albedo  of a body is the percentage of radiation that 
is reflected back by the body into space. Thus for a perfect mirror the 
albedo is $1$ and for a black body the albedo is zero. For a grey body 
the albedo is a number between zero and one.

The Earth is a grey body. However, if albedo changes with time due to 
snow and ice cover, vegetation and human activities 
such as paving of asphalt roads and emissions from cars.
\subsection{Greenhouse Gases and Clouds}

Radiation from the Sun reaches Earth (mostly) in the visible part of 
the spectrum (viz.~wave lengths in the range of 
$380\times 10^{-9}$ - $750\times 10^{-9}$ meters).
Radiation with shorter wavelengths is absorbed (mostly) by the Van-Allen belts.
The Earth absorbs part of this radiation and emits it back in the infra
red part of the spectrum, with wave lengths in the range of 
$7.5\cdot 10^{-7}-1\cdot 10^{-4}$ meters. If the Earth had no atmosphere
this radiation will be reflected into space. However in the Earth's atmosphere
some trace gases such as carbon dioxide ($CO_2$), methane ($CH_4$) and 
water vapor can absorb this radiation and reflect it back to Earth \cite{GC}.
This leads to a warming of the Earth's surface.  Figure~\ref{fig:HumiFigure01} presents the 
absorption spectra of several trace gases. The source of this figure is unknown.



\begin{figure}[h]
\centering

\includegraphics[width=7cm]{HumiFigure01.pdf}


\caption{\label{fig:HumiFigure01} Absorption Spectra of some Trace Gases.}
\end{figure}

NOAA Earth System Research Laboratory (Global Monitoring Division)
has been monitoring the concentration of the trace gases in the 
atmosphere from an observation station on Mona Loa mountain in Hawaii USA
and globally\cite{NOA1,NOA2,NOA3} by averaging these concentrations 
as measured by observation stations throughout the globe. This data is shown in 
Figures~\ref{fig:HumiFigure02}, \ref{fig:HumiFigure03}, and \ref{fig:HumiFigure04} (courtesy of ``NOAA Global Monitoring Division''). These figures   
demonstrate the almost linear increase in the concentrations of $CO_2$ and 
$CH_4$ over many years which can be attributed human activities.
\begin{figure}[ht!]
\centering

\includegraphics[width=9cm]{HumiFigure02.pdf}

\vspace*{-.5 true cm}
\caption{\label{fig:HumiFigure02} }
\end{figure}

\begin{figure}[h!]
\centering

\includegraphics[width=9cm]{HumiFigure03.pdf}

\vspace*{-.5 true cm}
\caption{\label{fig:HumiFigure03} }
\end{figure}
\begin{figure}[h!]
\centering

\includegraphics[width=9cm]{HumiFigure04.pdf}

\vspace*{-.5 true cm}
\caption{\label{fig:HumiFigure04} }
\end{figure}


The effect of clouds in this scenario is dual. First, they block the Sun's 
radiation and reflect it into space. At the same time they also reflect
the Earth's infra red radiation back to Earth. There is still on going debate 
in the scientific community as to which of these two process is more 
prominent for the determination of the Earth's energy balance.


\subsection{\bf Stefan-Boltzmann Law}

The Stefan-Boltzmann Law states that the power radiated from a black body  as a
function of its temperature is given by
$$
P=\sigma A T^4
$$
where $A$ is the body surface area, $T$ is it temperature, and
$\sigma=5.67\times 10^{-8}$ ${\rm Watts}/(m^2 K^4)$ is the Stefan-Boltzmann constant.

\noindent For a grey body $P$ has to multiplied by the ``grayness" coefficient of
the body.


\section{A Model for the Mean Temperature of the Earth}

We enumerate here the  data that is needed to model the mean temperature 
of the Earth and then assemble these into model equations.

\begin{enumerate}

\item {\bf The Sun Forcing} 

The rate at which energy from the Sun reaches the top of Earth’s atmosphere 
is called ``total solar irradiance'' (or TSI)\cite{TSI,CLP,KL}.  
TSI fluctuates slightly 
from day to day and week to week. In addition to these rapid, short-term 
fluctuations, there is an 11-year cycle in TSI measurements related to 
``sunspots'' (a part of the Sun's surface that is temporarily cooler and darker than its neighboring regions).

The evidence shows that although fluctuations in the amount of solar energy 
reaching our atmosphere do influence our climate, the global warming trend 
of the past six decades cannot be attributed to changes in the Sun output.
Figure~\ref{fig:HumiFigure05} which was compiled by Professor G.~Kopp\cite{KL} 
(and reproduced here with his permission) is a graph of TSI for the 
last 400 years. It shows clearly 
the ``small ice age" during the 17$^{\rm th}$ century. This TSI average is referred to
as the ``solar constant,'' in power (watts) per square meter. 

\begin{figure}[h]
\centering

\includegraphics[width=13.6cm]{HumiFigure05.eps}

\vspace*{-.3 true cm}
\caption{\label{fig:HumiFigure05} }
\vspace*{-.3 true cm}
\end{figure}

From a practical point of view we can consider the Sun's rays as 
arriving in parallel to Earth. This energy flux is measured to 
be $F_s\approx 1362$ ${\rm Watt}/m^2$ \cite{TSI,CLP,KL}. However the cross-section 
of the Earth to this flux is $\pi R^2$, while it surface area is
$4\pi R^2$. Therefore we divide this flux by $4$ when we compute 
the average temperature of the Earth.

The average albedo of the Earth from the upper atmosphere, its planetary 
albedo, is $30$-$35\%$ because of cloud cover, but widely varies locally across 
the surface because of different geological and environmental features.

\item {\bf The  Albedo}

The albedo of the earth is constantly changing. We shall model it as having 
two values:
$$
\alpha_M=0.85,\,\,\,\, \alpha_m=0.25;
$$
where $\alpha_M$ is the albedo for the part of the Earth that is covered by snow and ice (frozen Earth),
and $\alpha_m=0.25$ is the albedo for the part of the Earth that is not 
covered by snow, ice etc. (i.e.~not frozen).
The Earth will be considered as frozen if it mean temperature is below
$T_1=240$ Kelvin and unfrozen if it mean temperature is over $T_2=275$ Kelvin.
In between $T_1$ and $T_2$ we use linear interpolation to compute the albedo
\begin{equation}
\label{4.2}
\alpha(T)=\frac{\alpha_m-\alpha_M}{T_2-T_1}(T-T_1)+\alpha_M\, .
\end{equation}
Observe that the effect of changes in the vegetation of the Earth is not 
taken (directly) into consideration in this model.

More elaborate models for the Earth's albedo are available\cite{WS2,GC} and 
satellites are being used currently to give accurate real time data on its 
value. It is estimated that the average albedo of the Earth (viz.~planetary
albedo) is $30\%$ to $35\%$ because of cloud cover and the effect of 
trace gases, but it varies widely across the surface because of 
different geological and environmental features.

\item{\bf Clouds and the Greenhouse factor}

Let $T_0$ be the ambient temperature for green house effect 
and $\kappa(T)$ the cloud cover due to the green house effect.
Without the greenhouse effect the mean temperature of the Earth will be
$\approx 240$ Kelvin and effectively there will no cloud cover.

The incoming radiation from the Sun that is absorbed by Earth
is 
$$
R_{in}=\frac{F_sA}{4}(1-\alpha(T))
$$
where $A$ is the surface area of the Earth.
The outgoing infra red radiation from Earth is
$$
R_{out}=\sigma Ag(T) T^4
$$
where $g(t)$ is a lump sum parameter that represents the impact of
the greenhouse gases and clouds. This parameter was modeled (empirically)
by Sellers\cite{WS1} as
$$
g(T)=1-\kappa \tanh[(T/T_0)^6],\,\,\,\, \kappa=0.5
$$
where $T_0\approx 275$ Kelvin.
This expression implies that as $T$ increases $g(T)$ decreases and 
the greenhouse effect becomes stronger ($R_{\rm out}$ decreases)
\end{enumerate}

\section{Zero-dimensional Climate Model}
If we denote by $C$ the heat capacity of the Earth then by the Law of 
Conservation of Energy it follows that
\begin{equation}
\label{4.0}
C\frac{dT}{dt}=R_{\rm in}-R_{\rm out}
\end{equation}
at equilibrium (viz.~Steady state) $\frac{dT}{dt}=0$ and we must have 
therefore that $R_{\rm in}-R_{\rm out}=0$.

\section{Model Predictions}

If the Earth's atmosphere was totally transparent with no greenhouse gases
(or if the Earth had no atmosphere) then $g(T)=1$.
The percentage of the Sun's energy that is reflected by the Earth into space
(viz.~albedo) is estimated to be $\alpha(T_E)=0.3$.
The equilibrium mean temperature of the Earth, $T_E$, will have to 
satisfy then 
\begin{equation}
\label{4.1}
\frac{F_sA}{4}(1-\alpha(T_E))-\sigma A g(T_E) T_E^4=0
\end{equation}
The solution of this equation yields $T_E=255$ Kelvin which is well below
the freezing point of water. Thus the Earth will be covered by snow and ice.
It follows then that the difference between this value of $T_E$ and the 
(estimated) current mean temperature of $T_0=284.9$ Kelvin is due to the 
greenhouse effect.
To find the value of $g(T)$ that leads to this equilibrium temperature 
we substitute $T_0=284.9$ in (\ref{4.1}) and solve for $g$ 
(with $\alpha(T_E)=0.3$). We find that the current value of $g$ is $0.65$.

To investigate further the insights that can obtained from this model
we wrote a Matlab program $mean03.m$ that implements it.
At first we used this program, with the model parameters that are quoted 
above\cite{WS1,GC}. Using this program with an initial (mean) temperature of 
$285$ Kelvin we obtained Figure~\ref{fig:HumiFigure06} ($20$ years simulation). 
This initial mean temperature is based on data which was collected by NASA 
which shows that the Earth mean surface temperature in $2017$ was $14.9 C$.
Figure~\ref{fig:HumiFigure06} shows a steady increase in the average temperature of the 
Earth for the first few years which then stabilizes around $305$ Kelvin.
(This program is available for download from the author's web page \cite{MH}).


\begin{figure}[t]
\centering
\vspace*{-4.7 true cm}
\includegraphics[width=12cm]{HumiFigure06.pdf}

\vspace*{-5 true cm}
\caption{\label{fig:HumiFigure06} }
\end{figure}

Many of the original parameters\cite{WS1,GC} used in this program are estimates
(at best) and  change with time. Therefore one can use this program to 
probe also for the dependence (or sensitivity) of the results to the values 
of the various parameters of this model. As an example, we varied the value of 
$T_2$ in (\ref{4.2}) from $275$ (original value) to $296.11$ with no
substantial changes in the results. However, if we increase $T_2$
to over $296.12$ we obtain Figure~\ref{fig:HumiFigure07}  where the Earth's mean temperature
plunges to an ice age. Thus the model exhibits a climate bifurcation
if $T_2$ is in the interval $[296.11,296.12]$.


\begin{figure}[t]
\centering
\vspace*{-4.7 true cm}
\includegraphics[width=12cm]{HumiFigure07.pdf}

\vspace*{-5 true cm}
\caption{\label{fig:HumiFigure07} }
\end{figure}

\section{One-dimensional Climate Model}

In this model the Earth's temperature is considered to be a function of time and 
latitude $\theta$. Using this one dimensional model it is possible 
to study the oscillation of the Earth's climate between ice ages in the past.

To present this model we need to develop first, two mathematical formulae.

\subsection{Area of a Strip on a Circle}

We want to find a formula for the area of a strip on a circle of radius $R$
that is enclosed between the equator and {\it south} of latitude line $\theta$ 
(see Figure~\ref{fig:HumiFigure08}).  This area equals the sum of the area of the rectangle 
$2R\cos\theta\sin\theta$ and the area enclosed by equator and the two arcs 
of the circle. Hence, the total area of the strip is
\begin{equation}
\label{6.1}
A(\theta)=2R^2\cos\theta\sin\theta+
2\displaystyle\int_{Rcos\theta}^R \sqrt{R^2-x^2} dx=
R^2(\cos\theta\sin\theta+\theta)=R^2\left(\frac{\sin(2\theta)}{2}+\theta\right).
\end{equation}
It follows that the area of a strip on the circle between $\theta$ and
$\theta+d\theta$ is
\begin{equation}
\label{6.2}
dA=\frac{dA(\theta)}{d\theta}d\theta=R^2(\cos(2\theta)+1)d\theta.
\end{equation}

\begin{figure}[h]
\centering
\vspace*{-4.75 true cm}
\includegraphics[width=12cm]{HumiFigure08.pdf}

\vspace*{-5 true cm}

\caption{\label{fig:HumiFigure08} }
\end{figure}


\subsection{Area of a ``polar cap'' on a Sphere}

The formula for the area {\it north} of a line of latitude $\theta$
can be computed directly as in the previous subsection. It is given by,
\begin{equation}
\label{6.3}
S(\theta) = 2\pi R^2(1-\sin(\theta))
\end{equation}
The area enclosed by a strip on the sphere between $\theta$ and 
$\theta+d\theta$ is
\begin{eqnarray}
\label{6.4}
\Delta S &=& 2\pi R^2(1-\sin(\theta))- 2\pi R^2(1-\sin(\theta+\Delta\theta)
=2\pi R^2(\sin(\theta+\Delta\theta)-\sin(\theta))\\ \nonumber
&=& 2\pi R^2\frac{\sin(\theta+\Delta\theta)-\sin(\theta)}{\Delta\theta}\Delta\theta
\end{eqnarray}
which in the limit as $\Delta\theta \rightarrow 0$ yields
\begin{equation}
\label{6.5}
dS=2\pi R^2\cos(\theta)d\theta\, .
\end{equation}

\subsection{Steady State 1-dimensional Model}

To derive a one dimensional steady state model for the Earth's
temperature as a function of $\theta$ we consider a strip on the Earth 
between $\theta$ and $\theta+d\theta$. The cross section of this strip 
to the sun radiation is given by () . 
The incoming radiation from the Sun that is absorbed
by this strip on Earth is
\begin{equation}
\label{6.6}
R_{in}={F_s}(1-\alpha(T(\theta))dA=
{F_s}(1-\alpha(T(\theta))R^2(\cos(2\theta)+1)d\theta\, .
\end{equation}
The outgoing infra red radiation from Earth from this strip is
\begin{equation}
\label{6.7}
R_{out}=\sigma g(T(\theta) T(\theta)^4 dS=
\sigma g(T(\theta)) T(\theta)^4(2\pi R^2\cos(\theta)d\theta)\, .
\end{equation}
Hence when the system is out of equilibrium and with $T=T(t,\theta)$ we have
by the Conservation of Energy Law: 
\begin{equation}
\label{6.0}
C\frac{\partial T}{\partial t}=k\nabla T+ R_{\rm in}(\theta)-R_{\rm out}(\theta)\, ,
\end{equation}
where $R_{\rm in}$ and $R_{\rm out}$ are given by (\ref{6.6}) and (\ref{6.7})
respectively. The term $k\nabla T$ represents the heat transport (or
diffusion) due to temperature difference (viz.~gradient) in the
different latitudes (due to atmospheric and ocean currents). 
The coefficient $k$ is called the diffusion coefficient. Without this 
meridional heat transport the equator will become increasingly warmer 
while the poles increasingly colder. A crude approximation of this term 
is to replace it by a term proportional to the temperature difference 
${\bar k}(T(t,\theta)-T_{\rm ave})$ where $T_{\rm ave}$ is an estimate for the 
global temperature average.

We observe also that this model equation does not take into account the
change in the Earth tilt with respect to the Sun at different seasons.  

Neglecting the diffusion term in (\ref{6.0}) we must then have at the
steady state $R_{\rm in}=R_{\rm out}$ and the following expression for $T(\theta)$
\begin{equation}
\label{6.8}
{F_s}(1-\alpha(T(\theta))\cos(\theta)=
\pi\sigma g(T(\theta)) T(\theta)^4
\end{equation}
(where we have used the trigonometric formula 
$2\cos^2(\theta)=1+\cos(2\theta)$). 

Equation  (\ref{6.8})  has to be solved either graphically or iteratively since the 
values of $\alpha(T)$ and $g(T)$ depend on the final equilibrium temperature.

The results of this model are rather unusual. If one starts the simulation
from above the freezing temperature (e.g., 280 Kelvin) one obtains Figure~\ref{fig:HumiFigure08},
where at least part of Earth is well above the freezing point. On the other 
hand if one starts the simulation from $240$ Kelvin one finds that the Earth
is a frozen ball (see Figure~\ref{fig:HumiFigure09}. This demonstrates that this model exhibits 
the existence of two stable climates, one is ``warm" and the other is ``frozen.''
The first corresponds to the present climate while the second corresponds
to an ice age. Matlab programs mod03.m and mod04.m which implement these 
results can be downloaded from \cite{MH}.

\begin{figure}[h]
\centering
\vspace*{-4.75 true cm}
\includegraphics[width=12cm]{HumiFigure09.pdf}

\vspace*{-5 true cm}

\caption{\label{fig:HumiFigure09} }
\end{figure}


If the parameters of the system change (e.g., if $F_S$ changes) ``slightly''
and the Earth is at the warm stable climate it will stay at warm
temperatures. On the other hand if it is in the cold stable climate
it will remain at an ice age. The transition from one stable climate
to another occurs due to variations in the solar output and the $CO_2$
concentration cycle in the atmosphere. 

In this model the average Earth temperature as a 
function of the system variables follow a hysteresis diagram and the 
transition between these two stable climates can happen within a very 
short period of time. This is illustrated in Figure~\ref{fig:HumiFigure10}. In this plot
the red line denote a ``warm climate'' while the blue line ``ice age.''
In the interval $[-0.3849,0.3849]$ both states are stable i.e.~if the 
Earth is in the ``warm climate" it will remain on the same branch
if the system parameters change (and vice versa if the Earth is an ``ice age''). 
However if the Earth's system is in the ``warm climate' and reachs the 
hypothetical point $-0.3849$ the ``warm climate'' destabilizes and the Earth 
system `jumps'' (or transition rapidly) to the other stable branch which 
is an ``ice age.'' Similary if the Earth is in an ice age and reaches the 
point $0.3849$ the ice age becames unstable and the Earth's system transitions 
to the warm climate. 

\begin{figure}
\centering
\vspace*{-4.5 true cm}
\includegraphics[width=12cm]{HumiFigure10.pdf}

\vspace*{-5 true cm}
\caption{\label{fig:HumiFigure10} }
\end{figure}

In one dimension (and spherical coordinates) the diffusion term for a
strip between $\theta$ and $\theta+d\theta$ in (\ref{6.0}) has the form
$\frac{k}\nabla T=\frac{k}{R}\frac{dT}{d\theta}\,d\theta$.
Using (\ref{6.6}) and (\ref{6.7}), the steady state equation
(\ref{6.0}) becomes
\begin{equation}
\label{6.9}
{K}\frac{dT}{d\theta}={F_s}(1-\alpha(T(\theta))(1+\cos(2\theta))-
\sigma g(T(\theta)) T(\theta)^4(2\pi\cos(\theta) 
\end{equation}
where $K=\frac{k}{R^3}$. We simulated this equation with $K=1,1.5,2.$
and initial condition $T=280$ Kelvin at the equator ($\theta=0$).
The solution(s) are depicted in Figure~\ref{fig:HumiFigure11}.  Matlab program diffusion.m
that was used to obtain these results can be downloaded from \cite{MH}
\begin{figure}
\centering
\includegraphics[width=9cm]{HumiFigure11.png}

\caption{\label{fig:HumiFigure11}}
\end{figure}


\section{Geoengineering}

Several countries signed the Paris Agreement \cite{UN} to curb the concentration of
greenhouse gases in the atmosphere. However, this agreement is 
``not totally binding'' and there is little political will to abide by this 
agreement by some countries.

Several ideas have been considered in the last few years to reverse
the possible effects of climate change. One of these is to
release a compound into the stratosphere that would reflect some of the Sun’s
energy back into space.

To experiment and gauge the impacts of such a program Harvard University researchers
\cite{HGE,LMR} sent a balloon into the stratosphere, where it
released about 100 grams of calcium carbonate
This compound was chosen since it could stay in the upper
atmosphere for a long period of time and reflect sunlight back into space.

It is expected that this small scale experiment will provide data about
the risks and rewards of a large scale geoengineering programs.

Carbon sequestration is another scheme that was considered as a way to reduce 
greenhouse gas emissions into the atmosphere. Carbon capture and storage 
(CCS) (or carbon capture and sequestration) is the process of capturing 
waste carbon dioxide, transporting it to a storage site, and depositing 
it where it will not enter the atmosphere, normally into underground geological 
formation \cite{BC}. In some cases the captured carbon dioxide is pumped 
underground 
to force more oil out of oil wells. This set-up allows companies to actually 
make money from capturing carbon rather than it just being a financial burden.
Carbon capture has seen the most success in the United States, where so far 
projects have stored nearly 160 million metric tons of carbon dioxide in 
underground geological formations. 


\section{Conclusions}

The models presented in this paper are prototype models for the steady state 
temperature of the Earth. They ignore many features that control the Earth's 
climate. In spite of this they provide a ``window'' to the most 
important factors that influence the Earth climate viz.~Albedo and 
greenhouse effects. They provide a warning signal about the impact of human 
activities on global warming and climate. 

\section{Student Activities}

In the following we present a sample of projects to accompany the module 
on ``Earth Climate.'' In these projects students will use a zero-dimensional
and one dimensional (spatial) models to simulate the mean, and spatial 
distribution, of the Earth temperature. This includes possible climate 
bifurcations in the near future. These projects emphasize the role of 
trace gases such as $CO_2$ or $CH_4$ on the energy balance of the Earth 
and its climate.

\begin{enumerate}
\item Choose the initial conditions for the concentrations of the trace 
gases at the pre-industrial revolution and use the zero-dimensional model 
to compute the Earth's mean temperature. To make a proper choice of these 
conditions, you will have to search for this data on the web.

\item Derive a model for the future rates of $CO_2$ and $CH_4$ increase 
in the atmosphere based on actual data about these concentrations of in 
the atmosphere. (Using, e.g., the global or Mona-Loa $CO_2$ data for the last 
20 years.)

\item Run the models of the previous project to to obtain a ``reference 
projection'' for the concentrations $CO_2$ and $CH_4$ in the next 50 years.

\item Simulate the climate model with different rates of increase for the 
concentrations of the trace gases to study the sensitivity of the results 
to these changes (which might be due, in part, to different global energy 
policies).

\item Use the results from the previous steps to study the mean temperature 
of the earth using a zero-dimensional model. In particular study the 
sensitivity of these results to the Sun output.

\item Compute the change in the mean temperature of the Earth
if the Sun output $F_s$ changed from $-15\%$ to $+15\%$ in steps of $1\%$.

\item Use (\ref{4.1}) to find the mean temperature of the Earth if
$\alpha=0.3$ and $g(T)=1$.

\item You are given that the current mean temperature of the
Earth is $T_0=284.9$ Kelvin and the albedo $\alpha$ is between
$30\%-35\%$. Use this data to estimate the value of $g$ in (\ref{4.1}).

\item Study possible bifurcations in the Earth climate i.e possible 
``snow-ball'' effects that will change the Earth climate drastically
over a short period of time. (open ended question).

\item Simulate equation (\ref{6.0}) by replacing $k\nabla T$ by
${\bar k}(T(t,\theta)-T_{ave})$ and choosing appropriate values for $C$ 
and $k$ 
(see eg. https://scholarsandrogues.com/2013/05/09/csfe-heat-capacity-air-ocean/)

\item Study how different global energy policies might affect  
predictions about the future climate of the Earth (open ended question)

\item Suppose you were in charge of the global energy policy of the Earth
(using different energy sources to satisfy humanity's needs). What you will
do to avoid a climate bifurcation?
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}

\centerline{\bf Hints for Student Exercises}

\begin{itemize}
\item[1.] For $CO_2$ concentration Tree rings from that era can provide a clue OR
one can use the data in Figure~\ref{fig:HumiFigure02}  (for 1960) as an approximation. The Sun
forcing is available in Figure~\ref{fig:HumiFigure05}.

\item[2-3.] Use first order least squares to extrapolate the graphs in
Figures \ref{fig:HumiFigure02}, \ref{fig:HumiFigure03}, and \ref{fig:HumiFigure04}. 

\item[4.] Assume a margin of error of $2-3\%$ in the values of $ g(T)$ (and/or albedo)
and calculate the impact of these change on the mean temperature.

\item[5.] The sun output varies according to Figure~\ref{fig:HumiFigure05}. Some other estimates put its
present value as 1370.

\item[6-8.] Use equation (5.3) for this calculations

\item[9.] Simulate the program mean03.m by changing $T_2$ in the range
of [296.11, 296.12].

\item[11-12.]
 As the world ``Energy Czar'' one has to:
\begin{itemize}
\item[a.] make sure that there is enough energy produced to satisfy the demand
of Humanity.
\item[b.] make sure that energy is at affordable price
\item[c.] Energy sources are sustainable for future generations
\item[d.] Have minimum or no impact on the environment.
\end{itemize}
\end{itemize}
}

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\end{document}

