2-D Gyre Models
File last modified 23 November 1998
21.3 Tritium and 3He Invasion Into A Gyre
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 idea is to ask questions like "how does the total amount of tritium entering the gyre depend on mixing rates and velocities?" and "how does the tritium-3He relationship respond to these factors?".
The physical model involves a circulation on a surface between two isopycnals (surfaces of constant density, along which water very much prefers to move) in the subtropical gyre of the North Atlantic, much like that pictured below. The circulation consists of a Stommel Gyre (with boundary current) spinning in this isopycnal layer . It outcrops at some latitude, where some of the streamlines intersect the surface.
The flow field, we mentioned is a Stommel Gyre, which is simply a gyre satisfying the coriolis force, and having a frictional boundary layer on the western side. This friction is necessary to prevent the gyre from spinning up to some enormous speed under wind stress, and controls the scale over which the western boundary current exists. The general formula for the streamfunction is given by
where we have
and 
is just an arbitrary factor to have
part of the gyre outcrop at the northern edge, and 
is the fractional width of the boundary current (for example,
a very tight boundary current would have this value as .01, whereas
a broad current would be more like 0.1).Now we won't go into the
derivation of this equation, except to say that the general form
should begin to look a little familiar (in particular the forms
of c1 and c2, they arise from satisfaction
of the boundary conditions to the problem).
Also, we'd like to comment on the general strategy of using the streamfunction to define the velocity field. The streamfunction is something that you may have seen before, since lines (contours) of constant streamfunction are called "streamlines" and trace the flow field. They have the generic characteristic that the fluid flow is along lines of constant streamfunction, and that where the streamlines are bunched together, the flow is fast, and where they are far apart, the flow is slow. The precise mathematical linkage between streamfunction and velocities is given by
where u and v are the x and y components of the velocity. The general description we made of the character of streamlines follows from these definitions, but one very important thing can also be noted. If we define and compute our velocity fields from the streamfunction, we can guarantee that the velocity fields are non-divergent. This is a very important quality, as it assures that flow conserves (water) mass, and that the fluid is incompressible. (Yes, we know that water actually is compressible, but from the viewpoint of the large scale flow fields, this is not important). Here's the proof:
where we have sneakily swapped the order of differentiation in the last step (honest, it's legal!). Thus the divergence of a velocity field derived from a streamfunction is guaranteed to be zero.
Boundary conditions are crucial in this and any model. Remember that! When you describe a model, it is not enough to describe the physical shape, and the velocity field. You need to think very carefully about the boundary conditions. The outcrop BCs are very straightforward: you just set the water to the surface water tritium and 3He values. The left- and right- edge boundary conditions are equally simple: since the gyre is bounded by continents, we specify zero flux across the boundary. How do you do this? We're glad you asked. The simplest way of doing this is to specify slightly different weights for your FD algorithm, such that there is no diffusive loss out of the gyre on the east/west sides. You don't have to worry about flow out of the box, because your velocity field/streamline setup doesn't have any by definition. OK, that's pretty simple.
The really tough problem is the southern boundary. The gyre is open-ended. This is a classic difficulty in ocean models which have an open end to them: what do you do to the open boundary? If you specify zero flux like the continental boundaries, then things will go awry, because you tend to trap tracer along the southern boundary. This is not what happens in the real ocean. What Musgrave did was to put in a diffusive box (one with diffusion, but no advection) which passively "filled up" in some slow way during the simulation. This sounds a little grotesque, but actually is not such a bad thing, in that water to the south of the southern limb of the subtropical gyre have lower tritium, and more slowly accumulate tracer than a closed box. In actual fact, the water to south is advected by from the tropics and the southern hemisphere, which responds much more slowly and weakly to the tritium transient. It turns out that this crude approximation is actually good enough for Musgrave's purposes. That is, it approximates the behavior of the southern boundary well enough to mimic the real ocean on these time scales.
Musgrave performed a number of gyre simulations, with varying mixing rates and transports. There appeared to be a relatively small range of these parameters whereby one could get enough tracer into the gyre to match observed penetration, and where the tritium-3He relationship (the scatter plot) could be replicated. However, he also came to the conclusion that it is difficult to separate the effects of interior mixing from processes near the ventilation region (i.e. how the tracers are introduced into the gyre) in setting tracer relationships. This, unfortunately, is so awfully characteristic of these kinds of model studies: you often can get the "right" answer for a variety of "wrong" reasons. One needs to "lift the degeneracy" by measuring and modeling other, independent properties.
This seems a little depressing, but there are other ways of extracting further quantitative information from the tracer fields.
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