2-D Gyre Models

File last modified 23 November 1998

21.4 Tracer Ages

Implicit within some tracer systems is the concept of the "age" of the water. For example, in looking at radiocarbon distributions in the deep ocean, the first attempts involved essentially determining the "radiocarbon age" of the water, just like you were doing some archaeology in the oceans. A similar concept can be developed for tritium and ³He. The model is really rather simple: if you have a fluid parcel at the sea surface with tritium in it, and the tritium is decaying, the resultant ³He is lost to the atmosphere. Once the parcel leaves the surface, however, the ³He accumulates, and you can estimate the time elapsed since the parcel was at the sea surface from:

which can be easily derived from the radioactive decay equation and the fact that ³He is just dead tritium. Times as short as a few months, or as long as a few decades are measurable this way. These timescales are particularly useful for shallow water circulation and water mass formation, as well as oxygen consumption rates.

Now this age tracer is potentially very useful, because, in sense, it ratios out some of the history and dilution of the water (who cares how much tritium there is, as long as there's enough to make a difference?). In some respects, it embodies the time information inherent in the tracer fields. However, there is a little problem here: water parcels don't retain their identity in the ocean, and will tend to mix with other water parcels. This presents some interesting problems for this age tool, since the age of mixture of two water parcels will be an average weighted by the tritium concentrations of the original water parcels. Further, the age equation is in itself a non-linear (logarithmic) function. So the age may respond in a funny way to mixing. How do we assess this?

First we can write the two dimensional advection-diffusion equations for tritium and ³He in the following form:

where is tritium and is ³He. Here also, we have assumed that the diffusivity is spatially constant. As we mentioned in our previous discussions, this is not such a significant restriction. Note that the equations are telling us that these tracer are transient (the time derivatives are non-zero), and are advected and diffused. Tritium decays, and ³He grows in (note the theta in the second equation!). Interestingly, we can add these equations together, and define a new tracer called "stable tritium" which is the sum of tritium and ³He, and obeys:

which behaves like a simple, conservative dye tracer, and where the tritium-helium age reduces to

We can combine these equations with the definition of the tritium-³He age to derive the advection-diffusion equation for the age:

which looks a little gnarly, but isn't really as bad as it looks. Let's take it apart. First, you have the time derivative, which means that the observed age distribution may change with time. This isn't surprising, because the tritium and ³He distributions are changing with time. Next, you can advect and diffuse the age, just like any other tracer. The third term on the right hand side is just the tendency for the age to increase at a rate of one second per second. Now this has one additional aspect that wasn't immediately obvious: the equation is non-dimensional. Yes, that's right, the number "1" you see there is really a number 1. The other terms are demonstrably non-dimensional: for example the time derivative is time/time, the velocity divergence term is distanceXtime/time/distance, etc. Check out the other terms to see what we mean. The final term, which appears like a kind of velocity (since it multiplies the age gradient), is the effect of non-linear mixing on the age. That is, you could rewrite the equation in the form:

where is a kind of "augmented velocity" which consists of the true velocity with an added effect of mixing. The sense of the gradients in and actually tell you what the age does: picture a "shock wave" of tritium (and zeta) propagating into the ocean. Their downstream gradients will be negative (i.e. they will decrease downstream) so that sense of the second term in the equation above will be positive. That is, the age will look like it's moving faster than it should. This is consistent with the qualitative feeling you get from simple experiments mixing two water masses, one with tritium and one without. The resultant mixture will represent only the age of the tritium-loaded component. Thus this equation quantifies this effect in a continuum case.

Before we go on, we can do some "back of the envelope" types of calculations on that last term. We can measure the size of the gradients in tritium and stable tritium, as well as the computed age, and if we have some idea what the diffusivity is, we can calculate the size of these terms. This has been done in the subtropical gyre (Jenkins, 1987) and seems to be relatively small (order 10% or smaller) in the shallow part of the water column. But not completely negligible. We will now turn to some numerical models to further evaluate the effect.

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