12.747 Problem Set #6

Issued: 23 Oct 2008
Due: 30 Oct 2008

In addition to our notes on the web, here are some helpful references for this problem set:

Broecker and Peng (1982) Tracers in the Sea (A source for some numbers)

1. A Five-box P model. Let's extend what we've learned about phosphorus box models into a five-box model of the global distribution of phosphorus. In the figure below we show the world ocean divided into five boxes: surface Atlantic (SAT), Antarctic (ANT), surface Indo-Pacific (SIP), deep Indo-Pacific (DIP), and deep Atlantic (DAT). The volumes of the reservoirs (boxes) are in m³ and follow: V_SAT = 3 x 10¹⁶, V_ANT = 1.5 x 10¹⁷, V_SIP = 9 x 10¹⁶, V_DIP = 8.1 x 10¹⁷, V_DAT = 2.7 x 10¹⁷.

As you can see, it is only a little bit more complicated than the two box model we used in class lecture. The blue arrows represent fluxes of water in Sverdrups (1 Sv = 10⁶ m³ s^-1). The red arrows represent the flux of phosphous due to biological conversion of phosphorus into sinking particles. We'll use an average river phosphorus concentration of 1.3 mmol m^-3 in this model, so let's build a model of phosphorus in units of mmol-P m^-3. As a starting point, use a value of 10 years for the "tau" and 0.99 for the remineralization efficiency.

a). Code up the model and a main (or driver) program in MatLab. Prior to running the model evaluate the model's "stiffness" and choose an integrator from Table 8.2 in the ODE class notes, be sure to tell us what your decision was and your reasoning. Now run the model out to steady state (run the model long enough to achive steady state, but less than the life of the universe) starting from an initial condition of zero everywhere, provide an EPS plot file of the time evolution of the five model "state variables", and report the following quantities: approximately how long it took the model to achive steady state, the steady state concentrations of each box, the mean concentration of phosphorus in the global (model) ocean, and the ratio of the deep minus surface concentration between the Pacific and Atlantic (i.e. Pacific over the Atlantic). This last quantity is Broecker's so-called "horizontal segregation", given as (deepConcPac-surfConcPac)/(deepConcAtl-surfConcAtl), in this case for phosphorus.
b). Now do a sensitivity analysis of the diagnostics we identified in part (a) by making contour plots of the mean phosphorus concentration and horizontal segregation as a function of tau and remineralization efficiency (choose what you think is a good representative range of these parameter's values). Include with your answer these contour plots as EPS plot files.
c). Broecker and Peng (1982) report the global mean phosphorus concentration is approximately 2.3 mmol-P m^-3 AND the horizontal segregation is approximately 2. Is there a region in your parameter space where both of these diagnostic conditions are true? Using subplot make a 2 by 2 plot of the surface Atlantic and Pacific and deep Atlantic and Pacific concentrations in your parameter space (provide this as an EPS plot file). What can you say about the way the model is constructed that would help explain your diagnostic and state variable results?