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\centerline{MONT 100N -- Modeling the Environment}
\centerline{Chapter 1 Project}
\centerline{September 15, 2017}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
The ice sheet covering the island of Greenland is currently one of the major
concentrations of frozen fresh water situated over a land mass on the Earth.  It is
estimated to contain about $\frac{1}{9}$ of the total freshwater ice on the planet.
Because this water is presently on land, if it should melt, that water would
add to the depth of the oceans.  This is different from the situation
for the Arctic sea ice, or much of the ice in the shelves surrounding
Antarctica, which rest on sea water.  It is thought that melting in those 
cases would not appreciably change sea levels because that ice is floating 
on and supported by water already.\footnote{This is good since as of July 12, 2017 
the Larsen C ice shelf, with an area of about $44,200$ square kilometers--about the 
same as the land area of the U.S. state of Delaware--has 
broken off from the Antarctic coast and become what is probably the largest
iceberg ever observed by humans.}   There are important freshwater 
ice sheets covering the land mass of Antarctica as well as these sea ice formations,
though, and their water \emph{would} add to the depth of the oceans.   
\begin{figure}
\centering
\includegraphics[height=7in,width=4.0in]{GreenlandIce.png}
\caption{The Greenland Ice Sheet, as of about 2013, source
{\tt https://en.wikipedia.org/wiki/Greenland$\_$ice$\_$sheet}}
\label{fig:Greenland}
\end{figure}

As we will see, there is a large volume of water contained in the
Greenland ice sheet.  For this reason, its fate is of more
than passing interest for all humans--many of our major cities
and settled coastal areas occur in regions that are close enough to the current
sea level that any significant increase will cause major disruptions. 
For instance, the \emph{highest} (natural) elevation of any point
on the island of Manhattan in New York City is only 81 meters ($\doteq 265$ feet)
above sea level and most of the island lies much lower than that.
The effects of the relatively small increase in average sea level
that have already occurred were evident, for instance, in the
flooding of Manhattan that happened during ``super-storm Sandy''
in October, 2012.  

The news on this front is not encouraging because
major melting has been observed on Greenland in recent summers.  This is
thought to be a
result of a combination of higher air temperatures
and decreases in the reflectivity of the ice due to
deposition of soot particles from sources on other
continents (this increases the amount of energy absorbed from solar
radiation during the summer months and increases the rate of melting). 

Refer to the map of Greenland showing the depth of the ice sheet covering
most of its land area in Figure~\ref{fig:Greenland}.
We want to use the information here to estimate the volume of the
water contained in this ice sheet and understand the possible (now
maybe even probable!) effects if it melts completely. 

Our methods will yield rough estimates or approximations of
the ice volume.  They will be based on the following simplifying
assumptions:
\begin{itemize}
\item Even though it appears as a large area on the familiar Mercator
projection maps you may have seen, that map projection distorts areas
of regions near the poles and makes them look much larger than
they actually are.  Greenland does not make up a very large
a portion of the surface area of the (roughly spherical) Earth. 
As a result we will not lose too much if we simply estimate areas as though
they corresponded to areas on the flat map (i.e. without trying to take the
curvature of the Earth into account).  The distance scale marked in
the legend of the map can be used, together with a ruler, to approximate linear
dimensions of regions. 
\item The ice depths (in meters) of the various portions of the ice sheet are encoded
in the map by the 6 different colors.  For the purposes of this
exercise, let's take the ice depth in a region to be \emph{constant}
at the \emph{lower limit} of the range of depths shown in the legend
of the map.  For example, this will mean that all points colored
with the darkest shade of blue will have ice depths equal to 3000 m
(even though the actual depths can be larger).  Note that this will
produce estimates on the small side of the actual depth.
\end{itemize} 

\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
The chapter project will involve investigating the following questions
and writing up your results as directed below.
\begin{enumerate}
\item[(A)] Estimate the areas of each of the five different ice sheet
regions identified by the colors.  Each group will get transparencies I traced
by hand showing the \emph{outside boundaries} of the regions in the map 
where the ice depth is at least 10 meters (solid black), at least
1000 meters (dotted black), at least 1500 meters (green), at least 2000 meters
(blue) and at least 2500 meters (red).   Explain how you are doing
this in a clearly-written paragraph.   Note:  There are \emph{many ways}
to do this in a reasonable fashion and there is not a single correct answer! 
One suggestion:  Use the rectangular grid marked on the second transparencies that your
group will receive.  {\it The grid squares marked are $200$km on a side according 
to the scale of the map.}   You don't need to get super-detailed or picky, but
be as accurate as is reasonably possible. 
\item[(B)] Multiply each of your area estimates by the depth estimate
to get a volume estimate.  Add the ice volume estimates to get a
total volume and express in units of \emph{cubic kilometers}.  (As a
``reality check'' for your method, the total volume of the Greenland
ice sheet is often estimated to be about $3,000,000$ cubic kilometers.  How
close did you come to that?)  Recall that the Greenland ice sheet accounts
for about $\frac{1}{9}$ of the total freshwater ice on Earth.  What is your
estimate of the total volume of freshwater ice on Earth?
\item[(C)]  Now imagine that an amount of water equal to your total volume estimate
is added to the oceans all at once.  How much would sea levels rise as a result?  One
way to estimate that is to use the same idea as in part (3) of
Example 1.5 in the text.  What is the total surface area of the oceans
on the Earth? (You should look this up online if you don't know what proportion 
of the Earth's surface area is accounted for by the oceans. If you find different estimates,
how will you choose which one to use?  Explain your thinking.) 
If the water from the melted Greenland ice sheet was spread evenly over that
area, how deep would it be, in meters?  Would the actual change in sea level
be less than or greater than this estimated height?  Explain.\footnote{A technical note: 
In case you are worried about the fact that we
are ignoring the spherical shape of the Earth, this method is actually
sufficient (i.e. accurate enough) for our purposes because of the fact
that the depth of the water would be much smaller than the radius of the Earth. 
Here's one way to think about it:  If there is no land area, then
adding the water from the melted ice sheet is
equivalent in mathematical terms to changing the radius of the spherical Earth from $r$
to $r + \Delta r$ (with $\Delta r$ representing the depth of the new water, much smaller than $r$ itself).
The volume of the added water is the difference between the volume of the larger sphere of
radius $r + \Delta r$ and the volume of the original sphere of radius $r$:
$$\frac{4\pi (r + \Delta r)^3}{3} - \frac{4\pi r^3}{3} = 4\pi r^2\times \Delta r + 4\pi r\times (\Delta r)^2 +
\frac{4\pi (\Delta r)^3}{3}.$$
Since we assume $\Delta r$ is much smaller than $r$, then the last two terms on the right are
negligible in size compared to the first term
and we obtain an estimate
$${\rm volume\ of\ added\ water} \doteq 4\pi r^2 \times \Delta r = {\rm surface\ area\ of\ sphere} \times \Delta r.$$
The change in sea level is then approximated by
$$\Delta r \doteq \frac{\rm volume\ of\ added\ water}{\rm surface\ area\ of\ sphere}.$$
The same idea works even if the water covers only a portion of the surface area of the sphere.}
\item[(D)] Find an elevation contour map of Manhattan.  Use that information
to estimate what portions of that island would be under water if all the
freshwater ice sheets melted.
\end{enumerate}
Write up your solutions for these questions as a project report. Include
all of your calculations, the version of the map you used to estimate
the areas of the different regions of the ice sheet, and your answers
to all the ``explain'' portions of the questions above.  
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