2-D Gyre Models

File last modified 23 November 1998

21.5 Another 2-D Gyre Model

How important are the non-ideal effects on the tritium-helium age? We can do "back of the envelope" type evaluations, but the concern is that the time derivative term in the equation acts as a good integrator of the effects of the non-linear terms, and hence significant biases can accumulate over time for a fluid parcel. Thiele and Sarmiento (1990) (hereafter referred to as "T&S") published a very interesting model simulation, patterned after Musgrave's model. Their goal was to evaluate the effects of non-linearity on real tracer-age systems. In this calculation, they ran simulations of the penetration of triium-³He, CFCs and an ideal tracer age. By ideal tracer age, they meant an age tracer which obeys

that is, it is a steady-state tracer age which can be advected, diffused, and increases at a rate of one second per second. This, obviously, is the answer to the question "how old is the water?". This age admits to the suggestion that water masses can be "ventilated" by mixing processes as well as by direct subduction of water from the surface. It is a generalization of a pure advective age:

(where we have put the subscript "a" to indicate it is pure advective) which is kind of a clock age felt by a float that is traveling down a streamline. The generalized ideal tracer age is a useful construct, for its correspondence with, say, dissolved oxygen concentrations, could be used to determine oxygen consumption rates. (We may talk more about that issue later). However, the most illuminating aspect of this is to compare the ideal tracer age to the other, less-ideal tracer ages.

We mentioned the tritium-helium age before. We'll introduce you to the CFC age. CFCs are anthropogenic, and have been increasing in the atmosphere since their introduction in the 1930s. In particular, two tracers are of interest: CFC-11 (CCl₃F₁) and CFC-12 (CCl₂F₂). Below is a plot of the time history of CFC concentrations in the atmosphere, and , for comparison, the surface water tritium concentration for the North Atlantic.

There are two ways of defining a CFC age. One way is to take the measured concentration in a water parcel, calculate the equilibrium atmospheric concentration (known as the P_CFC) and pick the "vintage" of the water off the above graph. This would be known as the "P_CFC Age" (how quaint!). The problem with this approach is that if ignores the issue of mixing of water masses. If the water parcel were to mix with completely CFC-free water, then the P_CFC concentrations would be lowered by some fraction, and the water would "look" older because of the lower P_CFC values. This maybe is not such a bad thing, but the mapping of P_CFC values to age is not a linear processes (the CFC values don't increase linearly with time). Another approach, which kind of ignores mixing is the CFC ratio age. The ratio of two of the CFCs (CFC11 and CFC12) has been changing in the atmosphere with time, and this ratio could be used as an age tracer. If the same water mass were to mix with CFC free water, then the CFC ratio would remain unchanged, so the age would remain the same. This is also not a desirable thing, but in combination with the P_CFC age technique, maybe something can be worked out… some day. One other problem with the CFC ratio technique is that the ratio has flattened out and remains roughly constant with time now, so that the practical resolution of the technique for shorter time scales is significantly limited. Anyway, the behavior of the latter dating technique was the target of T&S paper.

Theirs was a very clever idea. They did the simulations with the intent of comparing the non-ideal age tracer systems (tritium-helium and CFC-ratio ages) to an ideal age tracer. Running an ideal tracer in Musgrave's model, however, reveals some interesting and very general problems with running steady state tracers in models with open ocean boundary conditions. When they used the southern boundary condition of Musgrave's model, the steady-state age field became very distorted along the southern boundary of the actual circulation. What was happening was that the "sponge layer" to the south soaks up the age, and drags the water near the southern limb of the gyre to older ages. You can see this by doing a simple calculation: if the sponge layer is order 3300km wide, then the age associated with the extreme southern end of it would be ca L²/K_H or about 10¹³/1000 = 10¹⁰ seconds, which is 3000 years! This wasn't a problem with Musgrave's calculation because he was running the model only long enough (a few decades) to get the tritium into the gyre, but T&S's age simulation had to be run out to many 10's millenia to reach steady state. The big problem with doing steady-state tracers in such models is that they have an infinite amount of time to feel the distant boundary conditions that the transient tracers often don't even know or care about. Thus T&S had to compromise, and use a model with no southern boundary layer, and with no flux condition on the southern end.

Another unknown, which Musgrave struggled with, was the portion of the gyre which was outcropped. The way to look at this is that if you keep the circulation (streamlines) constant, but vary the portion of the gyre which is "exposed" to the surface, and hence forced to the surface boundary conditions. You might regard this outcrop region as an area of the gyre which is subject to deep, wintertime convection (deep winter mixed layers) which bring the water into equilibrium (or close to it) with the atmosphere. This has a huge effect on the penetration of tracer into the circulation. What happens is that as you increase the outcrop portion, you increase the number of streamlines which intersect the outcrop region increases, and the volume of tracer swept into the gyre increases. You probably experienced this effect indirectly when you dialed the diffusivity in your gyre model runs in assignment 10. That is not quite the same thing as was done here, but you get the idea. The figure below shows their tritium penetration simulations for the same circulation (advection and mixing) but different outcrop portions:

They settled on case (d) (the 3500 km outcrop) as the one which seems to carry in the right amount of tritium into the model. Having established that, they then ran the model to a point in time (1977) for a variety of circulation and mixing strengths. Like Musgrave, they got characteristic distributions that were diagnostic of the circulation and mixing, but because they had settled on a specific northern boundary condition, their solutions were less ambiguous (but not necessarily more "correct") than Musgrave's. From those simulations, they were then able to compute the tritium-helium age in the model, and the CFC ages, and compare them to the ideal tracer age. Below is a composite plot showing the model results for their "middle of the road" case (50 Sv, 1000 m²/s) showing the penetration of tritium, the generation of ³He and the computed tritium-helium age for this case:

You can see in the left-most panel, advected into the gyre, an embedded tritium maximum, much like the maximum you encountered in the pipe model: not very strong, but definitely there. The corresponding ³He distribution looks monotonic, largely because the pipe "kind of wraps around onto itself" in this gyre. Another way of saying this is to say that the gyre recirculation time is less than the time since the tritium transient, and the tritium is starting to become homogenized in the gyre. The resultant tritium-helium age distribution is shown on the right. You see the young age, or we should say "youth" being pulled into the gyre circulation, much as you'd expect.

Now they looked at the ideal tracer age, the ratio of tritium-helium age to ideal age, and the ratio of CFC age to ideal age for a variety of experiments, of which we show the most likely candidate model:

The age patterns are interesting, and characteristic of what you might expect for the gyre. The ideal age looks a lot like the tritium-helium age, at least at first glance, but it's the deviations that are interesting. The ratios, shown in the middle and right panel for the tritium-helium age and the CFC ratio age show a characteristic pattern. Inspection of these figures indicate that the tritium-helium age seems to be better behaved than the CFC ages. Both are close to the ideal age (ratio=1) in the eastern part of the gyre, where the streamlines enter the circulation directly, but they begin to deviate from ideality in the western part of the gyre. The CFCs seem to be worse, but there does appear to be two characteristic regions: good agreement in the east and bad agreement in the west.

Why is this so? Below we combine the tritium-helium age and age-ratio patterns with the streamlines to make a point (note that the detailed numbers disagree, since this is from an entirely different numerical experiment, but it does make the point):

In the upper plot is the tritium-helium age (solid contours) and the streamlines (dashed lines). Note how there are two types of streamlines in the gyre circulation: open streamlines which connect directly to the outcrop region (they appear predominantly in the east, to the right), and closed streamlines which form closed orbits (they appear in the west, to the left), and do not intersect the outcrop region. Now the age fields have a rather different character on the open streamlines: the age contours tend to be at right angles to the streamlines (but not perfectly) while on the closed streamlines, the age gradients tend to be parallel to the streamlines. It is in these latter regions that the tritium-helium (and CFC) ages diverge most strongly from ideality. One way of looking at this is that in the region of open streamlines, the so-called "directly ventilated" part of the gyre, the advection-diffusion balance is dominated largely by advective processes, with a smattering of "laplacian diffusion", and a very small amount of pseudovelocity nonlinearity. That is, if you order the terms in the tritium-helium age advection diffusion equation, you have for the eastern region

Now notice that the dominant term (aside from the "1") is the scalar product of the velocity with the age gradient, hence the two will tend to line up, so that the lines of constant age, which are at right angles to the gradient, will be at right angles to the streamlines. That is what we see.

Now in the western region of closed streamlines, where you are reliant on cross-streamline diffusion to carry tracer (and age) into the center of the gyre, you have something like

or something equally nasty like that. This seems a little difficult to deal with, since the dominant terms are no longer advective. Actually, this might be turned to your advantage, because if you play your cards right in this area, you can use the age equations to determine the diffusivity, since it is such a dominant force (see Jenkins, 1991).

Well, T&S's conclusion was that it was safe to use the tritium-helium age in the directly ventilated region to do things like measure oxygen consumption rates, but you'd better be careful in other regions. But the take home lesson on this is that there is a use for such simple numerical models: validating and evaluating other more "direct" measurement tools. We'd like to take you just one small step further in the next section.

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