On CO2 residence times: The chicken or the egg?

While some model based claims say that CO2 residence times may be thousands of years, a global experiment in measurable CO2 residence time seems to have already been done for us.

By Christopher Monckton of Brenchley

Is the ~10-year airborne half-life of ¹⁴CO2 demonstrated by the bomb-test curve (Fig. 1, and see Professor Gösta Pettersson’s post) the same variable as the IPCC’s residence time of 50-200 years? If so, does its value make any difference over time to the atmospheric concentration of CO2 and hence to any consequent global warming?

Figure 1. The decay curve of atmospheric ¹⁴C following the ending of nuclear bomb tests in 1963, assembled from European records by Gösta Pettersson.

The program of nuclear bomb tests that ended in 1963 doubled the atmospheric concentration of ¹⁴CO2 compared with its cosmogenic baseline. However, when the tests stopped half the ¹⁴C left the atmosphere in ten years. Almost all had gone after 50 years. Why should not the other isotopes of CO2 disappear just as rapidly?

Mr. Born, in comments on my last posting, says the residence time of CO2 has no bearing on its atmospheric concentration: “It’s not an issue of which carbon isotopes we’re talking about. The issue is the difference between CO2 concentration and residence time in the atmosphere of a typical CO2 molecule, of whatever isotope. The bomb tests, which tagged some CO2 molecules, showed us the latter, and I have no reason to believe that the residence time of any other isotope would be much different.”

He goes on to assert that CO2 concentration is independent of the residence time, thus:

The total mass m of airborne CO2 equals the combined mass m₁₂ of ^12,13CO2 plus the mass m₁₄ of ¹⁴CO2 (1):

(1) .

Let CO2 be emitted to the atmosphere from all sources at a rate e = e₁₂ + e₁₄ and removed by uptake at a rate u. Then the rate of change in CO2 mass over time is given by

(2) ,

which says the total mass m of CO2, and thus its concentration, varies as the net emission, which is the difference between source e and sink u rates.

For example, if e = u, the total mass m remains unchanged even if few individual molecules remain airborne for long. Also, where e > u, m will rise unless and until u = e. Also, unless thereafter u > e, he thinks the mass m will remain elevated indefinitely. By contrast, he says, the rate of change in ¹⁴CO2 mass is given by

(3) ,

which, he says, tells us that, even if e were to remain equal to u, so that total CO2 concentration remained constant, the excess ¹⁴CO2 concentration

(4) ,

which is the difference between the (initially elevated) ¹⁴CO2 concentration and the prior cosmogenic baseline ¹⁴CO2 concentration, would still decay with a time constant m/u, which, therefore, tells us nothing about how long total CO2 concentration would remain at some higher level to which previously-elevated emissions might have raised it. In this scenario, for example, the concentration remains elevated forever even though x decays. Mr. Born concludes that the decay rate of x tells us the turnover rate of CO2 in the air but does not tell us how fast the uptake rate u will adjust to increased emissions.

On the other hand, summarizing Professor Pettersson, reversible reactions tend towards an equilibrium defined by a constant k. Emission into a reservoir perturbs the equilibrium, whereupon relaxation drains the excess x from the reservoir, re-establishing equilibrium over time. Where µ is the rate-constant of decay, which is the reciprocal of the relaxation time, (5) gives the fraction f_t of x that remains in the reservoir at any time t, where e, here uniquely, is exp(1):

(5) .

The IPCC’s current estimates (fig. 2) of the pre-industrial baseline contents of the carbon reservoirs are 600 PgC in the atmosphere, 2000 PgC in the biosphere, and 38,000 PgC in the hydrosphere. Accordingly the equilibrium constant k, equivalent to the baseline pre-industrial ratio of atmospheric to biosphere and hydrosphere carbon reservoirs, is 600 / (2000 + 38,000), or 0.015, so that 1.5% of any excess x that Man or Nature adds to the atmosphere will remain airborne indefinitely.

Empirically, Petterson finds the value of the rate-constant of decay µ to be ~0.07, giving a relaxation time µ^–1 of ~14 years and yielding the red curve fitted to the data in Fig. 1. Annual values of the remaining airborne fraction f_t of the excess x, determined by me by way of (5), are at Table 1.

Figure 2. The global carbon cycle. Numbers represent reservoir sizes in PgC, and carbon exchange fluxes in PgC yr^–1. Dark blue numbers and arrows indicate estimated pre-industrial reservoir sizes and natural fluxes. Red arrows and numbers indicate fluxes averaged over 2000–2009 arising from CO2 emissions from fossil fuel combustion, cement production and land-use change. Red numbers in the reservoirs denote cumulative industrial-era changes from 1750–2011. Source: IPCC (2013), Fig. 6.1.

Table 1. Annual fractions f_t of the excess x of ¹⁴CO2 remaining airborne in a given year t following the bomb-test curve determined via (5), showing the residential half-life of airborne ¹⁴C to be ~10 years. As expected, the annual fractions decay after 100 years to a minimum 1.5% above the pre-existing cosmogenic baseline.

Now, it is at once evident that Professor Pettersson’s analysis differs from that of the IPCC, and from that of Mr. Born, in several respects. Who is right?

Mr. Born offers an elegantly-expressed analogy:

“Consider a source emitting 1 L min^–1 of a fluid F₁ into a reservoir that already contains 15.53 L of F₁, while a sink is simultaneously taking up 1 L min^–1 of the reservoir’s contents. The contents remain at a steady 15.53 L.

“Now change the source to a different fluid F₂, still supplied at 1 L min^–1 and miscible ideally with F₁ as well as sharing its density and flow characteristics. After 50 minutes, 96% of F₁ will have left the reservoir, but the reservoir will still contain 15.53 L.

“Next, instantaneously inject an additional 1 L bolus of F₂, raising the reservoir’s contents to 16.53 L. What does that 96% drop in 50 minutes that was previously observed reveal about how rapidly the volume of fluid in the reservoir will change thereafter from 16.53 L? I don’t think it tells us anything. It is the difference between source and sink rates that tells us how fast the volume of fluid in the reservoir will change. The rate, observed above, at which the contents turn over does not tell us that.

“The conceptual problem may arise from the fact that the ¹⁴C injection sounds as though it parallels the second operation above: it was, I guess, adding a slug of CO2 over and above pre-existing sources. But – correct me if I’m wrong – that added amount was essentially infinitesimal: it made no detectable change in the CO2 concentration, so in essence it merely changed the isotopic composition of that concentration, not the concentration itself. Therefore, the ¹⁴C injection parallels the first step above, while Man’s recent CO2 emissions parallel the second step.”

However, like all analogies, by definition this one breaks down at some point.

Figure 3. Comparison between the decay curves of the remaining airborne fraction f_t of the excess x of CO2 across the interval t on [1, 100] years.

As Fig. 3 shows, the equilibrium constant k, the fraction of total excess concentration x that remains airborne indefinitely, has – if it is large enough – a major influence on the rate of decay. At the k = 0.15 determined by Professor Pettersson as the baseline pre-industrial ratio of the contents of the atmospheric to the combined biosphere and hydrosphere carbon reservoirs, the decay curve is close to a standard exponential-decay curve, such that, in (5), k = 0. However, at the 0.217 that is assumed in the Bern climate model, on which all other models rely, the course of the decay curve is markedly altered by the unjustifiably elevated equilibrium constant.

On this ground alone, one would expect CO2 to linger more briefly in the atmosphere than the Bern model and the models dependent upon it assume. To use Mr. Born’s own analogy, if any given quantum of fluid poured into a container remains there for less time than it otherwise would have done (in short, if it finds its way more quickly out of the container than the fixed rate of exit that his analogy implausibly assumes), then, ceteris paribus, there will be less fluid in the container.

Unlike the behavior of the contents of the reservoir described in Mr. Born’s analogy, the fraction of the excess remaining airborne at the end of the decay curve will be independent of the emission rate e and the uptake rate u.

Since the analogy breaks down at the end of the process and, therefore, to some degree throughout it, does it also break down on the question whether the rate of change in the contents of the reservoir is, as Mr. Born maintains in opposition to what Pettersson shows in (5), absolutely described by e – u?

Let us cite Skeptical Science as what the sociologists call a “negative reference group” – an outfit that is trustworthy only in that it is usually wrong about just about everything. The schoolboys at the University of Queensland, which ought really to be ashamed of them, feared Professor Murry Salby’s assertion that temperature change, not Man, is the prime determinant of CO2 concentration change.

They sought to dismiss his idea in their customarily malevolent fashion by sneering that the change in CO2 concentration was equal to the sum of anthropogenic and natural emissions and uptakes. Since there is no anthropogenic uptake to speak of, they contrived the following rinky-dink equationette:

(6) .

The kiddiwinks say CO2 concentration change is equal to the sum of anthropogenic and natural emissions less the natural uptake. They add that we can measure CO2 concentration growth (equal to net emission) each year, and we can reliably deduce the anthropogenic emission from the global annual fossil-fuel consumption inventories. Rearranging (6):

(7) .

They say that, since observed e_a ≈ 2ΔCO2, the natural world on the left-hand side of (7) is perforce a net CO2 sink, not a net source as they thought Professor Salby had concluded. Yet his case, here as elsewhere, was subtler than they would comprehend.

Professor Salby, having shown by careful cross-correlations on all timescales, even short ones (Fig. 4, left), that CO2 concentration change lags temperature change, demonstrated that in the Mauna Loa record, if one examines it at a higher resolution than what is usually displayed (Fig. 4, right), there is a variation of up to 3 µatm from year to year in the annual CO2 concentration increment (which equals net emission).

Figure 4. Left: CO2 change lags and may be caused by temperature change. Right: The mean annual CO2 increment is 1.5 µatm, but the year-on-year variability is twice that.

The annual changes in anthropogenic CO2 emission are nothing like 3 µatm (Fig. 5, left). However, Professor Salby has detected – and, I think, may have been the first to observe – that the annual fluctuations in the CO2 concentration increment are very closely correlated with annual fluctuations in surface conditions (Fig. 5, right).

Figure 5. Left: global annual anthropogenic CO2 emissions rise near-monotonically and the annual differences are small. Right: an index of surface conditions (blue: 80% temperature change, 20% soil-moisture content) is closely correlated with fluctuations in CO2 concentration (green).

Annual fluctuations of anthropogenic CO2 emissions are small, but those of atmospheric CO2 concentration are very much larger, from which Professor Salby infers that their major cause is not Man but Nature, via changes in temperature. For instance, Henry’s Law holds that a cooler ocean can take up more CO2.

In that thought, perhaps, lies the reconciliation of the Born and Pettersson viewpoints. For the sources and sinks of CO2 are not static, as Mr. Born’s equations (1-4) and analogy assume, but dynamic. Increase the CO2 concentration and the biosphere responds with an observed global increase in net plant productivity. The planet gets greener as trees and plants gobble up the plant food we emit for them.

Similarly, if the weather gets a great deal warmer, as it briefly did during the Great el Niño of 1997/8, outgassing from the ocean will briefly double the annual net CO2 emission. But if it gets a great deal cooler, as it did in 1991/2 following the eruption of Pinatubo, net annual accumulation of CO2 in the atmosphere falls to little more than zero notwithstanding our emissions. It is possible, then, that as the world cools in response to the continuing decline in solar activity the ocean sink may take up more CO2 than we emit, even if we do not reduce our emissions.

Interestingly, several groups are working on demonstrating that, just as Professor Salby can explain recent fluctuations in Co2 concentration as a function of the time-integral of temperature change, in turn temperature change can be explained as a function of the time-integral of variations in solar activity. It’s the Sun, stupid!

It is trivially true that we are adding newly-liberated CO2 to the atmosphere every year, in contrast to the ¹⁴C pulse that ended in 1963 with the bomb tests. However, the bomb-test curve does show that just about all CO2 molecules conveniently marked with one or two extra neutrons in their nuclei will nearly all have come out of the atmosphere within 50 years.

To look at it another way, if we stopped adding CO2 to the atmosphere today, the excess remaining in the atmosphere after 100 years would be 1.5% of whatever we have added, and that is all. What is more, that value is not only theoretically derivable as the ratio of the contents of the atmospheric carbon reservoir to those of the combined active reservoirs of the hydrosphere and biosphere but also empirically consistent with the observed bomb-test curve (Fig. 1).

If the IPCC were right, though, the 50-200yr residence time of CO2 that it imagines would imply much-elevated concentrations for another century or two, for otherwise, it would not bother to make such an issue of the residence time. For the residence time of CO2 in the atmosphere does make a difference to future concentration levels.

To do a reductio ad absurdum in the opposite direction, suppose every molecule of CO2 we emitted persisted in the atmosphere only for a fraction of a second, then the influence of anthropogenic CO2 on global temperature would be negligible, and changes in CO2 concentration would be near-entirely dependent upon natural influences.

Atmospheric CO2 concentration is already accumulating in the atmosphere at less than half the rate at which we emit it. Half of all the CO2 we emit does indeed appear to vanish instantly from the atmosphere. This still-unexplained discrepancy, which the IPCC in its less dishonest days used to call the “missing sink”, is more or less exactly accounted for where, as Professor Pettersson suggests, CO2’s atmospheric residence time is indeed as short as the bomb-test curve suggests it is and not as long as the 50-200 years imagined by the IPCC.

And what does IPeCaC have to say about the bomb-test curve? Not a lot:

“Because fossil fuel CO2 is devoid of radiocarbon (¹⁴C), reconstructions of the ¹⁴C/C isotopic ratio of atmospheric CO2 from tree rings show a declining trend (Levin et al., 2010; Stuiver and Quay, 1981) prior to the massive addition of ¹⁴C in the atmosphere by nuclear weapon tests which has been offsetting that declining trend signal.”

And that is just about all They have to say about it.

Has Professor Pettersson provided the mechanism that explains why Professor Salby is right? If the work of these two seekers after truth proves meritorious, then that is the end of the global warming scare.

As Professor Lindzen commented when Professor Salby first told him of his results three years ago, since a given CO2 excess causes only a third of the warming the IPCC imagines, if not much more than half of that excess of CO2 is anthropogenic, and if it spends significantly less time in the atmosphere than the models imagine, there is nowhere for the climate extremists to go. Every component of their contrived theory will have been smashed.

It is because the consequences of this research are so potentially important that I have set out an account of the issue here at some length. It is not for a fumblesome layman such as me to say whether Professor Pettersson and Professor Salby (the latter supported by Professor Lindzen) are right. Or is Mr. Born right?

Quid vobis videtur?

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