Open Ocean 1-D Advection-Diffusion Models

File last modified 9 November 1998

17.5 Radioactive Non-conservative Tracers: Solving for w

Now comes the challenging part. We need to solve for the most complicated case:

Now the solution to this equation is

which should look vaguely familiar: it looks like the stable, conservative tracer solution. But how can this be? We have a much more complicated equation, and we get a simpler solution than the stable, non-conservative case. Well, there is a good deal of complexity now, since the form of "alpha" must be set up to match the boundary conditions. When we express the equation in the boundary matched form, we have

where we have

and recall that the definition of the hyperbolic sine is given by

which is getting a little gnarly, but is actually functionally very simple (recall the general functional form). The complexity comes in when you have to match the functional form (by solving for the a₁ and a₂ ) to the end point concentrations. This becomes analytically rather more complicated because you have competing effects of decay and consumption to deal with when matching things up.

OK, now that the gory mathematical details are seen to, we need to do two more things before getting down to business. The first is we need to estimate J^*(CO₂) from what we know about J^*(O₂). Well, we use the redfield stoichiometry, as modified by Takahashi, Broecker and coworkers, that AOU:CO₂ is 172:106, so we use J^*(CO₂) = -0.62 J^*(O₂). Note the negative sign, for when we consume oxygen, we create CO₂. Thus the value of J^*(O₂) = -0.018, giving J^*(CO₂)=0.011 .

The second thing has to do with the mistake that Munk made in dealing with radiocarbon profiles in the Pacific. He used , which is an isotope ratio anomaly and not the actual radiocarbon concentration! This is a common error in geochemical modeling, which must be avoided at all costs. The tendency is to model the anomalies and not the actual concentrations, but nature advects/diffuses/produces/consumes concentration not anomaly. Sometimes it's OK to deal in anomalies, for the equations sometimes work out, but often it doesn't. So be careful, and deal with concentrations, not anomalies. In Munk's case, the problem was that although isotope ratio anomaly decreases with depth, the absolute activity may not, since the total CO₂ tends to increase with depth.

We start with the and profiles from a nearby GEOSECS station (the WOCE data are not available yet).

Note the total CO₂ maximum, and that it has some noise in it. We then need to compute the equivalent radiocarbon concentration, which is given by

where R is the mole ratio of ¹⁴C:¹²C (a very small number). It turns out that we can ignore this number, since it is a constant, and cancels out of the equation. We'll be lazy and leave it out. However, there is one final complication: this arises from the fact that surface water radiocarbon is generally about 5% depleted relative to the atmospheric isotopic ratio (and what we define as the reference level for the isotope ratio anomaly). The reason it is depleted has to do with the fact that radiocarbon exchanges very slowly, (see Broecker and Peng for an explanation). The net result is that we have

not a big deal considering the other uncertainties in our calculations, but we may as well be as precise as we can. Anyway, let's plot up the equivalent radiocarbon profile (the actual radiocarbon concentration divided by the R value) and crank through the gnarly equation we have above. We use the value of z^* that we obtained from T and S, and we get:

Now the variable that is plotted is , i.e. a scale height associated with decay its in units of meters… check it out!). Here the results become more uncertain, because of larger errors, but we can settle on a value of 60,000 (uncertain by about 20,000). Knowing the half life of radiocarbon, we now have that

which is remarkably similar to what we estimated from basic principles.

17.6 Denouement: Computing the Other Numbers

Now comes the real fun. Getting the numbers that we were actually after. Using our J^* values for oxygen, and knowing w now, we can compute

or

And we can also calculate the vertical diffusivity, by using the fact that

which is how the canonical "1 cm²/s" comes about for the old thermocline theories.

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