Dear APM115 friends,

So, the first projects were very nicely done, and we'll give you some
feedback next class.  

Here is a list of possible subjects for your second project.  These
projects are less straightforward than the first projects, and we
suggest you start right away rather than wait for the last minute.
Because the projects require additional explanation and discussion,
you will also need to come and consult with either Eli or Drew
depending on which project you choose *before* Tuesday March 13 and
*after* you spent some time thinking about this.  Office hours for
this purpose will be announced shortly.  Each group of three of you
needs to choose your first, second and third choices for a project
from this list.

Please email these **three** selections to both TFs by Thursday March
8: Kurt House <khouse@fas.harvard.edu>; Michael Weidman
<mweidman@fas.harvard.edu>; and **get their confirmation on what your
project is before you start working on the project**.

The number of groups working on each of the choices below will be
limited, so you may only get your second choice (allocation will be on
a first come first serve basis).

The list follows,

1) (Eli) Global warming: build an energy balance model representing
   the greenhouse effect and predict the effect of various scenarios
   of increased atmospheric CO2.  Model should be composed of two
   atmospheric layers and one ground level temperature (that is, three
   different model temperatures).  No ice is needed in this case.
   Background: a given layer of the atmosphere does not absorb any of
   the solar radiation passing through it.  It does absorb a fraction
   epsilon (called absorptivity or emissivity) of the long wave
   radiation passing through it.  It will also emit epsilon*sigma*T^4
   radiation up and the same quantity down.  epsilon (0<epsilon<1) is
   about 0.6+/-0.3 and increases with concentration of CO2.  The
   surface absorbs all solar radiation and long wave radiation it
   receives, and emits sigma*T^4 upward.  If the temperatures are Tg,
   T1, T2, The equation for T2 is (I think...):

   [Cp*rho*V]*d T2/dt=(-2*epsilon_2*sigma*T2^4
                      +epsilon_2*epsilon_1*sigma*T1^4+
                      +epsilon_2*(1-epsilon_1)*sigma*Tg^4)*A
   where A=area, Cp=Specific heat capacity, rho=density, V=volume. 

2) (Eli) Glacial cycles: Earth went through an ice age every 100,000
   years, last one being 20,000 years ago.  Can this be due to the
   ocean circulation?  Add to the two box Stommel model both
   stochastic noise (representing random weather events) plus periodic
   fresh water forcing (representing periodic changes to the solar
   radiation with a period of 100,000 years due to periodic changes in
   the earth orbit around the sun (look up "Milankovitch").  Analyze
   the interaction between the periodic and stochastic forcing as
   function of the amplitude of both (look up "stochastic
   resonance"...).

3) (Eli) thermohaline circulation models:

   variation (a): a four box Stommel-like model of the THC with a
   single flow through all boxes.  Note that diffusion will cause salt
   exchanges between every two adjacent boxes and find a way to
   represent this in the equations.  As part of the analysis, consider
   the effects of different possible melting scenarios of Greenland on
   the circulation.  4box Configuration could be something like:

    --------------------  surface
    |        |         |
    |   ^   -|->       |
    |   |    |    |    |
    --------------------  2km
    |   |    |    |    |
    |      <-|-   v    |
    |        |         |
    --------------------  4km
    Equator  45N      90N

   variation (b): couple two Stommel models to each other to represent
   the interaction between the Atlantic and Pacific.  Note that total
   evaporation needs to be equal total rain, but this doesn't need to
   hold in each ocean basin separately.

4) (Eli) add temperature equations to the two box Stommel model and
   analyze the resulting system.  Calculate the temperature by
   including energy balance (incoming solar radiation, outgoing long
   wave black body radiation) plus transport between boxes in the
   temperature equation.  Allow the flow in the model to be a function
   of both water density difference and of the wind, which in turn is
   a function of the temperatures (e.g. the larger the temperature
   difference the stronger the wind).

5) (Drew) Develop and analyze a dynamic model of an open-access
   fishery.  To do this, assume that fisherman are myopic and react
   only to current profit.  For example, you might suppose that fishing
   effort E simply equals the stock of fishing boats, and specify how
   the stock of boats adjusts. Or you could let the hours that each
   boat is used vary as well.  Characterize the steady states of the
   system and their stability.  If you have time, develop a model
   where fishermen react to some function of recent prices, and not
   just the current one, or to forecasts of future prices. What would
   it mean for the fisherman to have correct expectations here?

6) (Drew) Develop and solve a model of optimal harvesting when there
   is a second, species that is not fished but which interacts with
   the stock of the commercially harvested fish- this could either be
   a predator species or a species that competes for nutrients with
   the commercial fish. How does this change the solution we derived
   in class?

7) (Drew) Develop and solve a model of the optimal extraction of an
   exhaustible resource when there is a backstop technology that is
   available at constant cost B and is a perfect substitute for the
   exhaustible resource. Then develop a model where the availability
   of the backstop is endogenous. What might this depend on and how?

8) We are open to other related suggestions.  (Eventually you will
   need to come up with your own idea for a project for the third and
   final projects.)

Any of these subjects should be fun to work on, we hope you'll enjoy
this and are happy to help you along the way.  The final product of
your project is a 10 minutes presentation which will be given on March
20.  We will split the class on that day, such that half the
presentations will be given during regular class hours and half during
the Tuesday workshop time.  You only need to come to the half during
which you will be presenting, but are expected to sit through all
presentations during that half (and participate with questions to the
presenters!).  You will need to email your Matlab code used for the
project and the presentation in pdf format to the TFs.

best, Drew, Eli, Kurt, Michael.