Some helpful references are:
1.) Two-dimensional gyre This is a problem about modeling a two dimensional gyre. Because our goal is to have you develop skills interpreting the results, we'll give you a "starter m-file" called dogyre_new.m with a skeleton outline of code for you to do the model calculations. You may surmise from the name of this file that this is a "new" version of this code; it is. In particular this version has the handy property that it conserves mass.
We want you to study it, and modify it to suit your purposes in the following exercise. You can run this code to do the first part of the problem. The velocity field is a symmetric gyre whose streamlines and velocity field are shown in the diagram below
The direction of flow is counter-clockwise, and the velocities have been generated using the formulation of the Stommel gyre given in the class notes (see the m-file). The model simulation starts with zero concentration of a tracer within the model domain (which is 2500 km x 2500 km ). At the beginning of the simulation, the bottom most row is fixed at a concentration of ten units, and the model is run for 10 years.
A contour of concentrations (contour interval of one unit) is plotted every year, and the final one saved. Also plotted (in figure 2) are several diagnostics of the model tracer distributions. They are, respectively, the total integrated tracer content ("mass") of the model domain, the concentration at the approximate center of the gyre, and two cross-sections (meridional and zonal) through the middle of the gyre. The legend in subplot(222) indicates which color goes with what diffusivity.
Run the model, which takes about 1 minute to run on a 1.2 GHz pentium M laptop, and observe the evolution of the tracer field for the various diffusivities. Explain the dependence of the overall behavior of the tracer field on diffusivity in qualitative terms. Discuss the dependence of the various diagnostics of the distributions shown in figure 2 in as quantitative way as possible. For example, why and how does the total mass depend on diffusivity? How do the boundary layer widths depend on diffusivity? Explain your results as thoroughly as possible. Think about what is going on in the model!
2. Numerical diffusion in higher dimensions Now we want you to modify the code in the model to explore the numerical diffusivity of the model. (You might want to make a second copy of the m-file) We want you to do four model runs, all with zero explicit diffusivity. (That is, with km=0). Start with interior concentrations being zero, and leave the boundaries fixed at zero. Now create a delta function (i.e. a single node with concentration set to 1) and use this as a probe like you did in an earlier problem set to estimate the effective diffusivity at the following four locations within the gyre: c(5,5), c(15,15), c(25,25), c(25,45). Do the simulations for each "spike" separately, so that you can easily determine the maximum concentration. For each year of a ten years simulation, contour the concentration of the spike, and plot the inverse of the maximum concentration as a function of time in seconds. Bear in mind that the relation we derived in problem set 7 does not hold in two dimensions, but rather the geometric mean of the two diffusivities is given by
Where the slope is the slope of 1/Cmax as a function of time. Obtain this slope by plotting 1/Cmax vs time for the 10 year simulations. Comment on the apparent spatial/temporal dependence of the diffusivities thus obtained, and explain what you see. Specifically, compare the observed evolution over the first 2-3 years, with what is observed over the full 10 year simulation. Can you predict what you'd expect? Explain the temporal evolution in terms of some of the things we discussed in Lecture 15. Considering the results, explain why you actually see a difference between the four simulations in question number 1.