From Science Matters
By Ron Clutz
The latest rebuttal of IPCC CO2 hysteria comes from Peter Stallinga in his 2023 publication Residence Time vs. Adjustment Time of Carbon Dioxide in the Atmosphere. Excerpts in italics with my bolds and added comments and images.
One of the major points in discussion of the anthropogenic global warming (AGW) scenario is the time the added carbon dioxide (CO2) stays in the atmosphere. In an extensive study, Solomon concluded that the residence time of carbon atoms in the atmosphere is of the order of 10 years , see Table 1. Such a short time would undermine the prime tenet of AGW, since a molecule of CO2 will not have time to contribute to any greenhouse effect before it disappears to sinks where it cannot do any thermal harm.
However, some claim that the residence time (the amount of time a molecule on average spends in the atmosphere before it disappears from it) is not relevant for this discussion; what matters is the adjustment time (or relaxation time or (re)-equilibration time), the time it takes for a new equilibrium to establish, the time constant seen in the observed transient, and, allegedly, these two are different. In a recent work, Cawley explains it as 
… natural fluxes into and out of the atmosphere are closely balanced and, hence, comparatively small anthropogenic fluxes can have a substantial effect on atmospheric concentrations.
In the current work, we use these exact two concepts, with turnover time called residence time. We also focus on first-order systems mentioned here by the IPCC. We discuss the difference between residence time on the one hand, and adjustment time on the other hand, and test the hypothesis that the adjustment time can be longer than the residence time by mathematical methods. After having addressed this core point, we perform a calculation based on the available data to see how they fit.
2. Residence Time and Adjustment Time (Methods, Data, Results and Analysis)
Figure 1. Two-box model of the carbon dioxide cycle. The top box represents the atmosphere, with a total carbon dioxide mass of 3403 Gt. Humans add 38 Gt per year to the system. Nature adds Fn+ and takes away Fn− to a sink represented by the bottom box. That sink has a total CO2 mass equal to S. The residence time in the atmosphere, τa is well known and estimated to be 5 years, the residence time in the sink τs is not well known.
In what follows, we will use a simple two-box first-order model, see Figure 1. The atmosphere has a mass of carbon dioxide equal to A. CO2 molecules can be captured into a sink and this occurs at a certain rate, a fraction of the molecules being trapped per time unit. Each individual molecule has a certain probability to be captured over time. In other words, a molecule has a residence time τa in the atmosphere (also sometimes called the ’turnover time’), which is the reciprocal of the rate, ka. Likewise, in the sink, there is a carbon dioxide mass equal to S, where molecules have a residence time τs; an individual molecule has a certain probability over time to be released by the sink into the atmosphere, or a rate ks.
Humans add an extra flux into the atmosphere labeled Fh. On the basis of this, we can determine the adjustment time τ of the atmosphere in terms of the residence times. This requires solving a simple mathematical differential equation; we do not have to worry at this moment about the thermodynamics and explain why the reaction constants are what they are. The questions we ask are, if we add an amount of carbon dioxide ΔA to it:
- What are the new equilibrium values of A and S?
- How long does it take to establish this new equilibrium?
Before injection of 100 into the atmosphere, the atmosphere-sink system was in equilibrium at 100 each, with the residence ‘times’ in both atmosphere and sink 1000 iterations. At each iteration A/τa is moved from atmosphere to sink and S/τs moved from sink to atmosphere. As can be seen, the observed adjustment time (relaxation time) of the system is 500 iterations, as predicted by Equation (9). After 500 iterations, the surplus quantity in the atmosphere relative to the new equilibrium has been reduced to 1/e, a level indicated by a horizontal dashed line. Further, a half-life can be defined, a time at which half of the transient amplitude has passed, t1/2=τln(2)= 347. This is indicated by a dotted line. (b) The adjustment time τ, as a function of the sink residence time τs, normalized by the atmospheric residence time τa. The dot indicates the value of the plot in (a), τs=τa, resulting in τ=τa/2 .
As can be seen, the adjustment time is shorter than the atmospheric residence time
for all values of the sink residence time, with, for large τs,
the adjustment time τ approaching the atmospheric residence time τa.
We, thus, refute the claim of the climate-skeptics-skeptics [skepticalscience.com] that:
“individual carbon dioxide molecules have a short life time of around 5 years in the atmosphere. However, when they leave the atmosphere, they are simply swapping places with carbon dioxide in the ocean. The final amount of extra CO2 that remains in the atmosphere stays there on a time scale of centuries.”
Their flawed reasoning is that the adjustment time (relaxation time) is the mass perturbation in the atmosphere divided by the flux balance, and, so goes the reasoning, while fluxes can be great (and the residence time short), the balance is close to zero and the relaxation time can then approach infinity.
We can now do a more detailed analysis based on the available data.
Table 2. Carbon dioxide facts, with the natural outflux Fn− derived from the mass in the atmosphere and the residence time. Other important parameters, influx Fn+, sink mass S, and sink residence time τs are less well known and should be considered adjustable.
The residence time in the atmosphere can be estimated quite well from the above-ground atomic bomb tests , which makes us happy that these at least served the purpose of advancing atmospheric science, if nothing else. The best estimate is about τa= 5 years . Other references mention different times, with the IPCC mentioning the shortest (4 years) in their 5th Assessment Report (p. 1457 of Ref. ), showing that this value is not settled yet; we will use 5 years in this work.
The equilibrium amount of carbon dioxide in the atmosphere is open for debate, but, for this purpose, we might use the consensus value of 280 ppm (A∗= 2250 Gt). To estimate the amount of CO2 in the sink is very difficult. However, there seems to be a general view that it is fifty times more than in the atmosphere, S=50A=113,400 Gt (relatively unchanged since pre-industrial times). Using the combination of these values does not allow for consistent bookkeeping, as the reader can easily verify. Something has to yield. In what follows, we will try out some scenarios based on specific assumptions.
3.1. Scenario: Pre-Industrial Atmosphere Was at Equilibrium
First we assume that the pre-industrial level of 280 ppm was indeed an equilibrium value with influx equal to outflux in the absence of human flux, as we are wont to believe, but that the mass in the sink S and the residence time τs in the sink are unknown.
Figure 4. Above-ground atomic-bomb explosions produced a lot of 14 C that stopped in the 1960s. From a fit (dashed line) of data from 1965, we find an adjustment time of τ = 14.0 a, and an amplitude of ΔA = 740, with a final value of A′∞ = 30. This enables stating that the sink must be at least 24 times larger than the atmosphere. Data from Enting (blue) found in a work of Perruchoud  and Nydal et al.  (green), extracted with WebPlotDigitizer .
It seems that the idea of the pre-industrial level stable at 280 ppm (with Fn+=Fn−at 280 ppm) is untenable. It seems very likely that the sink was already off-balance and emitting amounts of carbon dioxide at the beginning of the industrial era and the increase in the atmospheric CO2 at any time in human history is not solely due to human activity.
This would also explain the large pre-Mauna-Loa values found with chemical methods summarized by Beck  and Slocum . For instance, values of 500 ppm have been observed around 1940. Ignoring these facts, on the other hand, would be equivalent to throwing entire generations of scientists under the bus.
[Comment: CO2 higher concentrations prior to 20th century are also indicated by use of plant stomata as paleo proxies for CO2 estimations.]
3.2. Scenario: The Sink Is Fifty Times Larger Than the Atmosphere
Next, we adopt the assumption that the sink at this moment really has 50 times more carbon than the atmosphere, in other words, S=50A= 170,000 Gt, and release the restriction that the atmosphere was stable at 280 ppm; in pre-industrial times there can have been a flux imbalance.
We see indeed a tremendous outgassing from the sink in pre-industrial times. The system was far from equilibrium, with an imbalance being a net influx of F∗n+−F∗n−= 207 Gt/a. Where, at the moment, there is a net natural flux of 18 Gt/a out of the atmosphere, in pre-industrial times, in this two-box first-order model with a sink 50 times larger than the atmosphere, there was a net natural influx of 207 Gt/a.
Somewhere, we must have passed the equilibrium value and,
considering the above numbers, this value must be
rather close to today’s concentration of 420 ppm.
3.3. Scenario: Residence Time in the Sink Is Much Larger Than in the Atmosphere
If we only assume that the residence time in the sink is much larger than in the atmosphere, τs≫τa, then we can get a good idea of what has happened to our anthropogenic contribution to the carbon in the atmosphere, Fh, based on the two-box model.
Figure 3. (a) Yearly global CO 2 emissions from fossil fuels. (b) Cumulative emissions (integral of left plot). The yellow curve is the remainder of the anthropogenic CO 2 in the atmosphere if we assume a residence time in the sink much longer than the 5-year residence time in the atmosphere; in this case τs=50τa was used. (Source of data: Our World In Data ).
Figure 3 shows the yearly carbon dioxide emissions into the atmosphere (left panel; data source: Our World In Data ). The total amount so far emitted is 1696.5 Gt. The right panel shows the cumulative emissions, ∑yeariFh(i). If at every year we apply the fluxes according to Equation (1), then we can see at each year how much of the anthropogenic CO2 is still in the atmosphere. The right panel of Figure 3 shows this for τs=50τa.
We see that only 202.3 Gt of the total injected 1696.5 Gt is still in the atmosphere.
In these years, the amount of CO2 in the atmosphere has risen from 280 ppm (2268 Gt) to 420 ppm (3403 Gt), an increment of 1135 Gt. Of these, 202.3 Gt (17.8%) would be attributable to humans and the rest, 932.7 Gt (82.2%), must be from natural sources.
In view of this, curbing carbon emissions seems rather fruitless;
even if we destroy the fossil-fuel-based economy (and human wealth with it),
we would only delay the inevitable natural scenario by a couple of years.
3.4. Scenario: Abandoning Constant Residence Times
We have seen here how the first-order-kinetics two-box model results in conclusions contrary to data. We could, of course, change our model. We could abandon the idea of first-order kinetics (where flux is proportional to mass), but that would be problematic to justify with physics.
We could also add more boxes to the system, distinguishing the sinks, or differentiating between deep ocean and shallow ocean, dissolved carbon dioxide gas, CO2 (aq), and dissolved organic carbon (sea-shells), or between CO2 disappearing in the oceans and being sequestered in biological matter on land, etc.
However, we expect the most likely improvement to the model to come from
abandoning the idea that the residence times τa and τs are constant.
They, in fact, are very much dependent on temperature. As an example, the ratio between the two that tells us the concentrations (and, thus, the masses) between carbon dioxide in the atmosphere and in the sink, if we assume this sink to be the oceans, is governed by Henry’s Law, and this concentration ratio is then dependent on temperature.
When including such effects, we might even conclude that the entire concentration of carbon dioxide in the atmosphere is fully governed by such environmental parameters and fully independent of human injections into the system. A is simply a function of many parameters, including the temperature T, but not Fh. It is as if the relaxation time is extremely short and any disturbances introduced by humans, or by other means, rapidly disappear, rapidly reaching the equilibrium determined by nature.
This fits very nicely with the recent finding that the stalling of the economy and the accompanying severe reduction in carbon emissions during the Covid pandemic had no visible impact on the dynamics of the atmosphere whatsoever . The result of that research, the hypothesis that the carbon dioxide increments in the atmosphere were fully due to natural causes and not humans, fits the experimental data very well, and the hypothesis that humans are fully responsible for the increments can equally be rejected scientifically. This then also agrees with the conclusions of Segalstad that “The rising atmospheric CO2 is the outcome of rising temperature rather than vice versa” .
The pre-industrial atmosphere might indeed have been in equilibrium,
and we are currently also in, or close to, equilibrium.
That seems to us to be the most likely scenario.
Once we admit the possibility of non-anthropogenic sources of carbon dioxide, we can start finding out what they might be. Examples such as volcanic sources, planetary and solar cycles spring to mind. It might well be that the climate puzzle is solved in such areas as the link between solar activity and seismic activity and climate .
This is, however, not the focus of this work. We conclude here by summarizing the major findings of this analysis using a first-order-kinetics two-box model:
(1) The adjustment time is never larger than the residence time and is less than 5 years.
(2) The idea of the atmosphere being stable at 280 ppm in pre-industrial times is untenable.
(3) Nearly 90% of all anthropogenic carbon dioxide has already been removed from the atmosphere.
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