2-D Gyre Models
File last modified 23 November 1998
21.2 Boundary Currents and Mixing
After the initial bomb tests of the 50s and 60s (which created the bulk of tritium in the world today) the distributions of tritium and its daughter 3He in the oceanic thermocline quickly evolved into a characteristic pattern. The tritium was more-or-less homogeneous in the upper part of the thermocline, and decreased in an approximately exponential fashion through the thermocline to low values in the deeper waters. This can be readily explained by the fact that after the initial "bomb-spike" of tritium, surface concentrations have decreased only slightly faster than radioactive decay rates. The result is that within a decade or two of the bomb tests, those water masses that were going to become "tritiated" had already done so. The exponential decrease in the thermocline is characteristic of rapidly decreasing ventilation rates with depth, coupled with some weak in situ vertical exchange mechanisms. The 3He distributions are equally easily explained: they decrease from near zero values at the surface, to a maximum situated somewhere below the depth where tritium starts to drop off, and then decreases just like the tritium into the deep water. The co-behavior of these two tracers is to be expected, since they are linked in a fundamental way: 3He is just "dead tritium".
The problem arises when we try to obtain more quantitative information from their distributions. One approach, which you have already seen, is to construct a "pipe model" and to run simulations of tritium and 3He in the model. The physical situation is simple: you have a long pipe which into one end flows surface waters which have some characteristic time history of tritium and zero 3He. At some point in time, we can look at the distribution of properties along the pipe. This seems a stupid thing to do, since we all know that the ocean isn't a one-dimensional pipe, but maybe if we think of the ocean as an ensemble of these one-dimensional pipes, where flow and exchange tends to occur along streamlines, then this is not such a bad thing. By looking at such simple model, we are hoping to catch the essence of the importance of direct flow in the ocean.
If we do this experiment for a variety of velocities and diffusivities, and we find that many of the experiments look identical, except for the details of the length scale along the pipe. The reason for this is that the experiments scale with a characteristic Peclet Number (you remember, Lecture 15!). That is, the experiments are similar to within a constant, given by
where U is the velocity, 
is the diffusivity,
and 
is the tritium decay constant. What
kind of value would we expect in the ocean? Well, for typical
upper thermocline velocities of about .01 m/s, and horizontal
mixing rates of 1000 m2/s, and 
=1.764x10-9s-1,
we get Pe=50-60. Fine. This says the thermocline is
basically dominated by advection over diffusion (recall the peclet
number is the ratio of advective to diffusive strengths).
Rather than looking at the actual distributions down the pipe, and trying to relate them to actual locations within the real ocean - this would seem like a kind of feckless enterprise - it is tempting to look at a tritium-3He scatter plot to compare model and data. Since the two tracers are inextricably linked, their covariation in both model and data may be of interest in a position independent manner. Below is a plot showing the model curves as a function of Peclet number, and the observed data for the subtropical North Atlantic.
The surprising thing is that the optimum Peclet number is more like 1, not 50-60! This is quite a robust result. The data, which was accumulated from many stations all over the subtropical North Atlantic (not just one tiny corner) unambiguously point toward the low Peclet number curve. Does this mean that the oceans are behaving rather differently than we thought? Or is there something fundamentally two dimensional about the transport of tracers in the ocean that is not captured by our simple model? Well the answer to the second question is: what precise process might this be?
One candidate process is shear dispersion in the Gulf Stream. Think of it in the following way: the effect of diffusion is to move tracer and materials across stream lines (see your numerical experiments in problem set #10). Now out in the main (eastern) part of the gyre, the streamlines are very spread out, so tracer has to move a long distance horizontally to get from one streamline to another. But if you look in the western boundary current region of the gyre, streamlines are bunched together. Below is a plan view of streamlines in an idealized ocean gyre with a western boundary current:
thus in that region, tracer need only diffuse a very short distance to make it to the next streamline. Looks obvious, doesn't it? Except for one thing, fluid spends less time in the boundary current, because speeds are greater there. So what's the trade-off?
This problem was discussed by Bill Young (Young, 1984) who approached the problem analytically, and attempted to demonstrate the conditions under which this might be the case. He used simplified flow models with highly idealized geometries and computed analytic solutions. However, the difficulty was that he could only treat cases for very large (>100) or very small ( <1) Peclet numbers, and the most interesting region was in between. The only way of looking at this region was with numerical models.
To this end, Dave Musgrave published two papers in 1985 and 1990 where he looked at these problems. The first paper was a more general treatment of passive tracer dispersion within a gyre, and the role of the western boundary current in the process. To do this study realistically, however, you need to be very careful about making sure that your simulations are not affected by numeric diffusion. If you'll recall, most practical finite difference schemes smear out property distributions because they have truncation errors (no, that's not a mistake, it's the limits of the formal precision of the algorithm). This becomes a big problem in these kind of gyre studies because the numeric diffusivity depends on the velocity, and the velocity is high in the boundary current, precisely where you want to see the effects of "real" diffusivity. How can you separate out the effects of explicit diffusion when your finite difference code is inherently (implicitly) diffusive?
Now you'll recall that going to higher order schemes isn't necessarily the best bet, because not only are these higher order schemes not always truly accurate (in a mass conserving sense), they tend to be less stable and harder to control. There is a definite trade-off, and it can be demonstrated that the numeric diffusion (often referred to as "implicit") actually stabilizes the behavior of the algorithm. High order, low diffusion algorithms often generate negative tracer concentrations due to instabilities and dispersion.
Well, one way around this was an algorithm developed in the early '80s which was extended to multidimensional problems by Smolarkiewicz (1984) . We won't get into the details of this algorithm, for we think it unlikely that you will have a need to do these kind of numeric experiments, but we wanted to let you know that such things exist, and where to look for them if you need them. The basic scheme is to use "antidiffusivity velocities" to correct for the dispersion induced by the simple finite difference scheme. You simply take one time step forward, then apply a correction step (or a number of correction steps) that approximately compensate for the mixing. This correction takes place in the form of advection rather than "negative diffusion" for rather important fundamental reasons: for the same reason that diffusion stabilizes the algorithm, negative diffusion will destabilize the code. Advection serves to be a more tame correction process which has not such damaging effects. Provided one pays attention to the sums (i.e. you derive the size of the antidiffusive velocities from the formal truncation errors) you can get away with it, at least to a good enough approximation that you can represent high peclet number flows.
Musgrave's calculations in 1985 provided some interesting insight into the role of boundary current mixing in tracer dispersion, but it didn't tell us anything about the tirtium-3He paradox: why does the gyre look so diffusive? So Dave went one step further in 1990, by trying to simulate the invasion of tritium and 3He into a gyre. The physical model involves a circulation much like that pictured above,
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