2-D Gyre Models
File last modified 23 November 1998
21.6 Going the Whole Eleven Yards
Well what T&S did was a benchmark thing: it brought tracer age dating into a quantitative framework, but we'd like to suggest taking their approach one step further. This, we feel is an excellent opportunity (while we have you in our clutches, you poor miserable fools!) to demonstrate yet another use for numerical models: the ability to analyze what is going on. Think of the term "analysis". It stems from two Greek words, the second of which is "lysis" meaning "to take apart". T&S were very intent on testing the validity of an age tracer technique, but they lost sight of an opportunity (or perhaps the editor of the journal or the reviewers wouldn't let them) to take apart the model run fields to examine what processes dominated where and when in the experiments.
It's not as hard as it sounds, and we won't bore you with a lot of details, but we'll do a couple of things. We'll take a model run (50 Sv, 1000m2/s diffusivity) and look at the results in 1981. The year is chosen because that is the year that we had a lot of major cruises (TTO, etc). The first trick is to take to consecutive years, say 1980 and 1981 and look at the age distributions in those years. The difference between the fields divided by the time lapse give the rate of change in the fields (dtau/dt). Just load the matrices into MATLAB from the model runs and subtract 'em. Hard, isn't it? This is what you get:
Sorry about the garish color scheme, but that's MATLAB for you. Anyway, the point is that the age field is changing less than 0.1 years per year in the ventilated region, but begins to become very unsteady in the closed streamline region. The strongest region, surprisingly, is not in the center of the gyre, but on the outer eastern edge of the core region. Note also the strong changes along the southern boundary and in the western boundary current. We could go on ad nauseum with this stuff, but we hope you get the point that you can begin to take the model apart and look at the innards to see what's going on. Here's another example. This time, we'll show you the MATLAB code that we use to calculate the pseudoadvective term (remember the nasty little term on the far right of the equation?
well, here's what we would do in MATLAB, assuming that we have loaded the data for tritium, helium and age in to matrices "T", "H" and "A":
Z=T+H; % the stable tritium "zeta"
dx=5e4; % the delta X for node spacing in matrix
dt=3.15e7; % needed to convert from years to seconds
k=1000; % diffusivity in m2/s
nlt=k*dt*abs((gradient(Z)./Z + gradient(T)./T).*gradient(A))/dx/dx;
Now a few technical notes about the last line. Note the "./" and ".*" operations are to make them array operations. Note also that the "gradient" function returns a complex number to represent the gradients of the arrays (a gradient, recall is a vector quantity) so when we do a product of two complex numbers like that, they behave like scalar products of two vectors, which yields another complex number that we need to take the "length" of (hence the "abs") operator. Anyway, this gives us the size of the non-linear term (the pseudovelocity term) and it looks like this:
where we have an approximately logarithmic contour interval, given by [0 .1 .5 1 5 10 50]. Now note where the "action" is in this figure. There is the usual non-linear nasties going on at the southern boundary (and now you are beginning to see why Musgrave and co. worried about it!), but you also see a rather striking thing up in the top left corner. This is the region where you are blowing Gulf Stream water up against the outcrop: you are creating very strong gradients by having advection working against a hard boundary condition, and creating strong non-linear effects in the age. You can see this feature advected out into the gyre. Fascinating! … well we think so, anyway. Each to their own.
Do you think you can do more analysis in this direction? How about the age laplacian. How do these terms begin to evolve with time? We've made available a few example data files for you to play with. They're called d80.mat, d81.mat, d90.mat , and d91.mat. Don't forget to "shift-click" when you download them. You can explore some of the features that we've alluded to. Enjoy.
Well, that brings us to the end of our exciting little adventures in 2-D land. We hope you learned something about numerical techniques here. Yes, you can run this model in MATLAB: you've done something like this in one of the problem sets. However we hope you've learned something even more important in this last section: how to begin to analyze your model results. You see it's not just pretty pictures, it's a powerful tool to begin to take things apart in a fundamental way to understand both how the models are working, and perhaps something about how things are working in the real world too.
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