CO2 Residence Times, Take Two

Guest essay by Joe Born

In a recent post Christopher Monckton identified me as a proponent of the following proposition: The observed decay of bomb-generated atmospheric-carbon-14 concentration does not tell us how fast elevated atmospheric carbon-dioxide levels would subside if we discontinued the elevated emissions that are causing them. He was entirely justified in doing so; I had gone out of my way to bring that argument to his attention.

But I was merely passing along an argument to which a previous WUWT post had alerted me, and the truth is that I’m not at all sure what the answer is. Moreover, semantic issues diverted the ensuing discussion from what Lord Monckton probably intended to elicit. So, at least in my view, we failed to join issue.

In this post I will attempt to remedy that failure by explaining the weakness that afflicts the position attributed (again, understandably) to me. I hasten to add that I don’t profess to have the answer, so be forewarned that no conclusion lies at the end of this post. But I do hope to make clearer where at least this layman thinks the real questions lie.

To start off, let’s review the argument I made, which is that the atmospheric-carbon-dioxide turnover time is what determined how long the post-bomb-test excess-carbon-14 level took to decay. That argument was based on the “bathtub” model, which Fig. 1 depicts. The rate at which the quantity <i>m</i> of water the tub contains changes is equal to the difference between the respective rates <i>e</i> (emissions) and <i>u</i> (uptake) at which water enters from a faucet and leaves through a drain:

The same thing can, <i>mutatis mutandis</i>, be said of contaminants (read carbon-14) in the water. But in the case of well-mixed contaminants one of the <i>mutanda</i> is that the rate at which the contaminants leave is dictated by the rate at which water leaves:

where is the contaminant quantity and is the rate at which contaminants are added.

Consequently, if the water quantity increases for an interval during which <i>e</i> exceeds <i>u</i>, it will thereafter remain elevated if the emissions rate <i>e</i> then falls no lower than the drain rate <i>u</i>. If a dose of contaminants is added to the water, though, the resultant contaminant amount falls, even when there’s no difference between <i>u</i> and <i>e</i>, in accordance with the <i>turnover</i> rate, i.e., with the ratio of <i>u</i> to <i>m</i>. So, to the extent that this model reflects reality’s relevant aspects, we can conclude that the rate at which the carbon-14 concentration decays tells us nothing about what happens when total-CO2 emissions return to a “normal” level.

But among the foregoing model’s deficiencies is that it says nothing about a possible dependence of overall drain rate <i>u</i> on the water quantity <i>m</i>, whereas we may expect biosphere uptake (and emissions) to respond to the atmosphere’s carbon-dioxide content. Nor does it deal with the possibility that after contamination has flowed out the drain it will be recycled through the faucet. In contrast, the biosphere no doubt returns to the atmosphere at least some of the carbon-14 it has previously taken from it.

A model that takes such factors into account could support a conclusion different from the one to which the bathtub led us. Consistently with my last post’s approach, Fig. 2 uses interconnected pressure vessels to represent one such model. In this case there are only two vessels, the left one representing the atmosphere and the right one representing carbon sinks such as the ocean and the biosphere.

The vessels contain respective quantities and of an ideal gas, which represents carbon. This model is based on the assumption that those quantities’ ratio is naturally driven to a preferred equilibrium value. The vessels are assigned “volumes” and to represent that equilibrium ratio. We assume that the vessels keep their contents at a constant temperature so that the resultant pressures can represent the emission and uptake processes’ tendencies to proceed, and a volume flow rate expresses the flow rate’s proportionality to those pressures:

Those equations tell us that the carbon quantity responds as follows to a magnitude- injection of new carbon:

which the substitutions and transform into the Gösta Pettersson equation set forth in Lord Monckton’s post.

Note that in the Fig. 2 system any constituent of the gas would be exchanged between vessels in accordance with the partial-pressure difference of that constituent alone, as if it were the only component the vessels contained. By thus constraining the flow from (and to) the first, atmosphere-representing vessel, this model supports the conclusion that the overall-carbon-dioxide quantity would, contrary to my previous argument, decay just as the excess, bomb-caused quantity of atmospheric carbon-14 did.

Could providing more than one sink enable us to escape that conclusion? Not necessarily. Consider the more-complex system that Fig. 3 depicts. Just as the system that my previous post described, this one can embody the TAR Bern-model parameters. As that post indicated, describing such a system requires a fourth-order linear differential equation. So that system does have more degrees of freedom in its initial conditions and can therefore exhibit a wider range of responses.

But it still constrains the flow among its four vessels linearly in accordance with partial pressures, just as the Fig. 2 system does. From complete equilibrium, therefore, its behavior for any constituent is the same as for any other constituent as well as for the contents as a whole. In other words, this model, too, seems to support the notion that the bomb-test results do indeed tell us how long excess carbon dioxide will remain if we stop taking advantage of fossil fuels.

In a sense, though, the models of Figs. 2 and 3 beg the question; they use the same uptake- and emissions-process-representing parameters for every constituent. In contrast, although carbon’s different isotopes are often called “chemically identical,” they differ in their chemical kinetics. Whereas the inter-vessel flow in those models makes no distinction based on type of constituent, the real-world processes it represents do.

So one question is how significant that difference is in the present context. I don’t have the answer, although my guess is, not very. But readers attempting to answer that question could do worse than start by referring to a previous WUWT post by Ferdinand Engelbeen.

Another way in which carbon-14 differs from the other two carbon isotopes is that it’s unstable. So, if the Fig. 3 model is adequate for carbon-12 or -13, a different model, which Fig. 4 depicts, would have to be used for carbon-14 if its radioactive decay is significant. That diagram differs from Fig. 3 in that it includes a flow to represent beta-decay “leakage” from the deep oceans.

To the extent that those different models produce different responses, using bomb-test data to predict the total carbon content’s behavior is problematic. But the Engelbeen post mentioned above seems to say that even deep-ocean residence times tend to be only a minor fraction of carbon-14’s half-life: this factor’s impact may be small.

A possibly more-significant factor is that the carbon cycle is undoubtedly non-linear, whereas the conclusions we tentatively drew from the models above depend greatly on their linearity. Before I reach that issue, though, I should point out an aspect of the Bern model that was not relevant to my previous post. The Bern equations I set forth in my last post were indeed linear. But that does not mean that their authors meant to say that the carbon cycle itself is. Although for the sake of simplicity I’ve discussed the models’ physical quantities as though they represented, e.g., the entire mass of carbon in a reservoir, their authors no doubt intended their (linear) models’ quantities to represent only the differences from some base, pre-industrial values. Presumably the purpose was to limit their range enough that the corresponding real-world behavior would approximate linearity.

But such linearization compromises the conclusions to which the models of Figs. 2 and 3 led us. A linear system is distinguished by the fact that its response to a composite stimulus always equals the sum of its individual responses to the stimulus’s various constituents; if the stimulus equals the sum of a step and a sine wave, for example, the system’s response to that stimulus will equal the sum of what its respective responses would have been to separate applications of the step and the sine wave. And this “superposition” property was central to drawing the conclusions we did from those models: the response to a large stimulus is proportionately the same as the response to a small one.

To appreciate this, consider Fig. 5, which depicts scaled values of the Fig. 2 model’s left-vessel total-carbon and carbon-14 contents. Initially, the system is at equilibrium, with zero outside emissions , and with balanced emissions and uptake recycling contents fast enough to turn that vessel’s contents over in seven years.

At time t = 5, a bolus of carbon-14 appears in the (atmosphere-representing) left vessel. Compared with the total carbon content, the added quantity is tiny, but it is large enough to double the small existing carbon-14 content. As the distance between the red dotted vertical lines shows, the resultant increase in carbon-14 content decays toward its new equilibrium value with a time constant of just about seven years. (I’ve assumed that the processes greatly favor the sink-representing right vessel—i.e., that its “volume” is much greater than the left vessel’s—so that the new equilibrium value is not much greater than the original.)

Now consider what happens at t = 45, when the left vessel’s total-carbon quantity suddenly increases. Although the two quantities are scaled to their respective initial values, this total-carbon increase is orders of magnitude greater than the t = 5 carbon-14 increase. Yet, as the black dotted vertical lines show, the decay of the left vessel’s total-carbon content proceeds just as fast proportionately as the much-smaller carbon-14 content did. As was observed above, this could tempt one to conclude that the carbon-14 decay we’ve observed in the real world tells us how fast the atmosphere would respond to our discontinuing fossil-fuel use.

But now consider what can happen if we relax the linearity assumption. Specifically, let’s say that the Fig. 2 model’s proportionality “constant” varies with the system’s carbon content. Maybe various sinks saturate or become less efficient with increased concentration. Whatever the reason, the resultant non-linearity could cause behavior like that shown in Fig. 6.

In that plot, the red lines show that the carbon-14 decay occurs just as fast as in the previous plot, the carbon-14 content falling to exp(-1) above its new equilibrium value in around seven years. But the much-larger total-carbon increase brings the system into a lower-efficiency range, so that quantity subsides at a more-leisurely pace, taking over forty years. If such results are any indication, bomb-test results are a poor predictor of how long total-carbon content will settle.

Now permit me a digression in which I attempt to forestall pointless discussion of precisely what the quantities are that the graphs show. I believe the exposition is clearest if it is directed, as in Figs. 5 and 6, to ratios that carbon 14 and total carbon bear to their own initial values. But it appears customary to express the carbon-14 content instead in terms of its ratio to total carbon content. This means that, since total carbon has been increasing, the numbers commonly used in carbon-14 discussions could fall below the pre-bomb values, even though total carbon-14 has in fact increased.

For the sake of those to whom that issue looms large, I have attached Fig. 7 to illustrate how the values for carbon-14 itself could differ from those of its ratio to total carbon in a situation in which new (carbon-14-depleted) carbon is continually injected into the atmosphere.

But that’s a detail. More important is the issue that Fig. 6 raises.

Now, I “cooked” Fig. 6’s numbers to emphasize the point that nonlinearity can undermine conclusions based on linear models. Specifically, Fig. 6 depicts the results of making the flows proportional only to the fifth root of the carbon content.

But non-linearity must have some effect. How much? I don’t know. Together with the differences in behavior between carbon-14 and its stable siblings, though, it is among the considerations to take into account in assessing how informative the bomb-test data are.

As I warned at the top of the post, this post draws no conclusions from these considerations. But maybe the foregoing analysis will prompt knowledgeable readers’ comments that help narrow the issues.