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\centerline{Workshop \#3, outline for TF, not students}
\centerline{APM 115: mathematical modeling}

\bigskip
Climate, hysteresis, multiple equilibria

Model the Thermohaline circulation using a 3 box model, similar to the
Stommel two-box model discussed in class.  The boxes represent the
equatorial ocean (box 1, 0-30N in the North Atlantic), the
mid-latitude ocean (box 2, 30-60N) and the polar ocean (box 3,
60-90N).  Students may start from the Matlab program Stommel2box.m for
the Stommel box model.  The program Stommel3box.m contains the
solution.
\begin{enumerate}
\item Write the equations for the salinity for all three boxes,
  including a transport term between boxes 1 and 2 and between boxes 2
  and 3.
\item Integrate the equations using ode45, slowly varying the fresh
  water forcing to find a hysteresis behavior.  Plot it using some
  appropriate variables (q12, q23, S2, etc) as function of the fresh
  water forcing.
\item Starting from the hysteresis plot of, say the THC between boxes
  2 and 3 as function of the fresh water forcing, sketch an
  approximate bifurcation diagram for this model that includes a guess
  of where the unstable solution branch is.  Do this first for the 2
  box model.
\item Explain which bifurcations are seen and how they result in the
  observed hysteresis behavior (saddle nodes). 
\item \textbf{Optional, time permitting:} Solve for the steady state
  (fixed point) solutions using Matlab's fsolve.  Draw the bifurcation
  diagram for q12 and q23 based on this solution.  Also, integrate the
  equations using ode45 for a few values of the FW forcing to find
  which of the calculated equilibria states are stable and which
  unstable.
\end{enumerate}
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