1. Introduction

Carbon (C) is a vital element. Plants and animals are built with molecules containing C. Plants on ground and in the ocean feed on atmospheric or dissolved CO₂. Animals derive their carbon directly or indirectly from plants. Without CO₂ no life, as we know it, would exist.
In 2020 the concentration of CO₂ molecules in the atmosphere is ~415 ppM. (415 CO₂ molecules in one million molecules of air.) 20.000 years ago, at the end of the last glacial it was 180 ppM. Dangerously low, because with less than 150 ppM plants on land will not grow.
On the other hand a few percent of CO₂ in the air would suffocate us. During the last 150 million years the atmospheric concentration declined from ~2000 ppM (0,2 %) to 0,04 % at present.

We are living in a period with ample higher and lower safety margins. Still there is commotion about atmospheric CO₂. Currently the contents increases ~2,4 ppM/yr, according to measurements at many places around the globe notably at Mauna Loa Observatory (MLO or ML for short), on the island of Hawaii in the Pacific. The ML measurements are generally considered representative for the atmosphere as a whole, due to winds stirring the air. NASA Goddard's Space Centre composed an interesting animation of the mixing process based on its satellite measurements⁽¹⁾. The variation around ML's average data is ±10 ppM, rapidly changing locally.

There are many C-storages. Trees, books, furniture, houses, ships, plants, shells, rocks, whole mountain ranges, peat, coal, oil, gas, the ground below land and sea etc. These contain much more C than the atmosphere. And they are in constant exchange with it; some in slow, some in fast processes. Erosion of limestone rock and outgassing is a process lasting hundreds of millions of years like the storage of CO₂ by Olivine, the world's most abundant mineral. Corals are building islands of calcite skeletons within a couple of years. Flying insects supply the atmosphere with ten times more CO₂ than mankind with all its machines. They consume herbs making room for new growth, that absorbs CO₂ from the atmosphere Here we deal with cycling times from days to a few years. For trees it may take many decades before they die and decay, or are burned in wild fires or by us. Volcanoes and such fires deliver sudden big CO₂-surges at irregular intervals...

Storage and transport of carbon reserves have been investigated abundantly. F. Engelbeen⁽²⁾ reviewed and elaborated on several processes with special impact on the atmosphere, referring to earlier work by others. We apologise for not doing that again.
A special case for carbon storage and transport is the oceans. These contain 50 - 100 times more CO₂ than the atmosphere. What happens depends on the storing mechanism. There is solution in water and, depending on salinity, also dissociation and chemical storage. Each with their own typical intake and out flow process times. In chemical equilibrium with pure water CO₂ behaves like inert gasses. It follows Henry's law. In the low concentration range, we are dealing with, this would hold also for other storing mechanisms in sea water. The solubility would be proportional to the partial CO₂ pressure in the gas phase. So the ocean would capture 98% of any quantity added to the atmosphere and the fossil contribution would be insufficient to explain the ML measurements.
Solubility depends on temperature. It decreases when the temperature rises. This effect explains a rise in atmospheric CO₂ of ~100 ppM between the ice age and the present interglacial. One of us checked whether the rising temperature between 1880 and 2020 could be the cause of the present increase⁽³⁾. It isn't¹³. The temperature change would make up for some 12 ppM rise instead of ~120!
Much depends on the time it takes to reach equilibrium between the atmosphere and other CO₂ stores. If relaxation times would be in the order of one or two years, the origin of CO₂ excess would be a puzzle. With the IPCC assumption of 100+ years, we would end up with catastrophic forecasts, from a climate point of view.

The commotion around atmospheric CO₂ is not about starving plants or suffocating animals. One assumes that more CO₂ increases the Earth's temperature. It melts land ice and expands the water volume. The sea level rises and low lying lands will be flooded. There are more consequences, both annoying and favourable. Enough to justify an investigation. However, when studying the available data, we discovered, it is not so much a matter of "where does all this CO₂ come from?" But rather, "how does all that CO₂ disappear?"


2. Process times

Figure 1

Several researchers attempted to find the residence time for atmospheric CO₂ relative to many other CO₂ stores. Others made compilations thereof. The results range between 2 - 28 year. The IPCC's "Bern model" stands out with its 100+ year. Figure 1 pictures the harvest of results. It is only to show the lack of consensus as different definitions of process times were used.

Atomic atmospheric bomb tests provided researchers with a carbon tracer. This enabled them to calculate the process time of atmospheric refreshment. Each year about 1/7th of the atmospheric CO₂ is exchanged with numerous sinks and sources. So, at first sight, one would expect a half life of refreshment of about 10 year. And that is what, according to several authors, this tracer study showed. Refreshment, however is different from a process of reaching equilibrium after a pressure or concentration excess. Refreshment occurs independently of excess pressure or concentration. The refresh exchange between atmosphere and other sinks is limited to a rather thin boundary layer or a small in-and-out taker, whereas the store of CO₂ is within the whole bulk. In many media the internal mixing or transport of CO₂ is much slower than in the atmosphere. Disregarding this difference caused much confusion about decay time of added CO₂ as well as the cause of rising atmospheric content.

The previous article⁽⁴⁾, in which one of us tried to answer the question "How does all that CO₂ disappear?", was based on sloppy, noisy data. Results from scattering data depend strongly on the way of averaging. It is only during the last fifty years or so, that the 'signal' exceeds the noise, i.e. the natural fluctuations. Hence, here we used the best 'state of the art' available data, and we changed the computation from the integral to the differential processing. This also offers a wider range for conclusions about related consequences.
The presumption is simple. The atmosphere is a reservoir. It accumulates by many sources and dissipates CO₂ into many sinks. We know how much fossil CO₂ we put into it. And we measure that its increase is less than the fossil fuel supplies. So for many years all the other sources and sinks together act as a net joint sink. We check whether an accepted physics theory describes this joint behaviour.
Others did the same, as we learned from comments we received after appearance of the earlier article^(2,5). Roy Spencer⁽⁶⁾ and Peter Dietze⁽⁷⁾ recently used a similar approach as we did. The results are mutually confirming within the margins of scattered data they used.

3. Data and method

Reliable data we used are
1. ML annual average CO₂ concentrations (ppM) 1960 - 2019.
2. CDIAC (US Dep. of Energy DOE) (https://cdiac.ess-dive.lbl.gov/). Amount of fossil fuel tons of oil equivalent (toe).
3. Conversion of atmospheric 1 ppM CO₂ = 2,12 GtC = 7,77 GtCO₂ (GtCO₂ = 1000 million ton of CO₂). Following CDIAC, IPCC a.o..
4. Data and the algorithms we used can be found and downloaded in the Excel spreadsheet

The CO₂ concentration in the atmosphere at time t is N(t). It changes due to many sources and sinks. One of them is the annual anthropogenic fossil CO₂ supply B(t), which is fairly well known. We are interested in the behaviour of all other sources and sinks together. Looking at ML annual increment data from 1960 onward, and comparing with the fossil CO₂ input, it is obvious that CO₂ disappeared. All other sources and sinks together act as a fluctuating but on the average, steadily growing sink. If the system would be in equilibrium, there should be equilibrium concentrations in all subsystems. We assume such an equilibrium existed at some time t = 0. The atmospheric concentration at the time being N(0). In nature there are many such systems in which equilibrium conditions are approached at a rate proportional to a driving force. In this case that would be the concentration difference between N(t) and N(0)⁽⁹⁾. This leads to a differential equation

Herein the constant proportionality factor is 1/T. T is the characteristic residence time of excess CO₂ in the atmosphere. (In the previous article⁽⁴⁾ the integrated formula was used with k as flow coefficient; k = 1/T.)
Even when the other sinks and sources would also change with a rate proportional to N(t) - N(0) but at a different T_i, formula (1) still holds with 1/T = ∑ 1/T _i. This approach even holds for slow changing conditions in the various processes like e.g. the pH balance of the oceans, which will shift the equilibrium concentration for that CO₂ absorption process. The same happens with variation in temperature.
Irregular CO₂ sources like volcanism, extreme wildfires or influences from phenomena like El Nino/La Nina in the past are part of the Mauna Loa data. They seem to have had virtually no impact on the trend. That is fortunate, because when in future some unforeseen big event happens and we know how much CO₂ it involves, we shall be able to predict from that moment on, how N(t) will evolve further. That is, if T and N(0) are firmly established.
Another - fortunately unlikely - disturbing factor would be the appearance of an unexpected new trend source or sink. We don't know, what we know not.

Please note. This paper focusses on long-term trends and will only use eq. (1) averaged over the annual cycle to take out any season related effects that are otherwise very well visible in atmospheric CO₂ monitoring data.
Furthermore, eq. (1) is a specific form of the general relation between the rate of change of the atmospheric concentration N(t) and the sum of all sources B_i(t) and sinks S_i(t)

It is tempting to take into account all individual sources and sinks. This is, what e.g. the Carbon Project does⁽⁸⁾. Next to the fossil fuel emission source B(t) in eq.(1), the project provides best guesses about all other contributing processes like for instance deforestation, change of land use, degassing of rock etc.. It also considers the various sinks like the ocean, soils and biosphere with their specific parameters for these different CO₂ absorption processes. Recently this was extensively reported by P.Friedlingstein & some 70+ colleagues⁽⁸⁾.

However, we only use their data of fossil CO₂ supply as we are on a different track. Our aim is to single out the role of fossil CO₂ 'vis à vis' the other sources and sinks together. We wanted to see if the data, we consider reliable, show some joint behavior, allowing to calculate from eq. (1) stable values for T and N(0). Otherwise we cannot use these thus obtained values again with eq. (1) in forecasting the development of CO₂ concentration in the atmosphere depending on various fossil fuel consumption scenarios.
The data listed and computed in the Excel sheet to be downloaded are depicted in figure 2.

Figure 2

N(t): Data of Mauna Loa annual CO₂ averages with accompanying 3^rd degree trend line;

B(t): Data of Carbon Project fossil fuel and cement emissions with 2^nd degree trend line;

dN(t)/dt: time derivative of the N(t) trend line.

The results are shown in figure 3

Figure 3

Variance of decay time T over the years with different assumed equilibrium concentrations.

Although indicative, the outcome of this analytical approach with curve-fitted polynomials for determining N(0) and T is not sufficient as the variations in T over the years seems to be too big to justify the method. But then we should realize that the approach uses the data with equal weight and that due to the long residence time we apparently have to deal with, small errors in the early years of this period have a far to "heavy" weight in the overall outcome. Using these results for forecasting of the effect of future emission scenarios would therefore lead to questionable results.

4. Optimizing T & N(0)

To obtain more accurate and stable results, we apply eq. (1) to reconstruct the development of the atmospheric concentration from 1960 to 2020 purely on basis of the original fossil fuel emission data. For that purpose, we rewrite eq. (1) into a recurrent relation between the CO₂ concentration C(t) in any year to the concentration a year earlier plus the fossil emission B(t-1) in that year minus the amount of CO₂ that has been absorbed in the sinks of the environment, i.e. {C(t-1)-N(0)}/T:

It is obvious that for this reconstruction we need input values for N_i(0) and T_j as well as a value for the concentration at the starting point from which we will build up this C(t) sequence.
By comparing the thus obtained sequence C(t), N_i(0) and T_j with the measured data N(t) for every year, we can apply a straightforward "minimum average least square" approach in an iterative process to determine the optimum N(0) and T values.
Those values also depend on the starting point we have chosen to build up that particular sequence. Although it seems logical to take the concentration in the year 1960, N(t=1960) and build the sequence C(t) from there on, there is no guarantee that any particular starting concentration isn't biased by a non-predictable emission-event as discussed earlier in the introduction. Such a bias would propagate through eq. (4) and heavily influence the end-result like in the analytical approach. So, as any choice for a starting point would be subject to such an unknown bias, we should do in principle this optimization process for all starting points in this period 1960 to 2019 and take an average of thus obtained ensemble of 60 different N(0) and T combinations. Realizing the amount of work we started with the most simple approach of taking the two most extreme starting concentrations N(1959)and N(2019) respectively, and reconstructed their corresponding time sequences of "up" and "down" concentrations C(t), took the average of both and determined N(0) and T as described. We checked this approach by applying a more extensive calculation using 7 different starting points 1959, 1969, 1979,..., 2019, but found no significantly different N(0) and T values as from the "2-way reconstruction".

Figure 4a shows the outcome of this exercise, in the form of the standard deviation σ between these sequences for a 3x3 set of N(0) and T values, centered around their optimum values. We have chosen the values of this matrix in such a way that the non-centric combinations show at least a value of 2σ, indicating a rather steep minimum of the "sum of squares" surface in the [N(0),T] space.

Through this we arrive at the most reliable values of N(0) & T:

In figure 4b we show to which value of C(T) these 9 cases would lead in the year 2080 if fossil fuel use keeps growing with 1,5%/yr. The most likely result is 647 ppM in 2080 with a limited spread of less than 1%.

We speculated on the 7 ppM difference between the most quoted 1880 CO₂ concentration (280 ppM) and the one found this way (287 ppM). In a previous paper⁽³⁾ we looked at the change in equilibrium partial pressure with changing temperature⁽¹⁰⁾. In pure water the change would be about 12 ppM/°C, while for the salty oceans we have seen some 16 ppM/°C. The latter thanks to differences between glacial and interglacial.
One is tempted to ascribe the 7 ppM difference to the matrix iteration result to half way between 1880 and present temperature of the oceans.

5. Results

The proof of the pudding... In order to check the decay theory, we plotted the calculated C(t) using measured B(t) against the measured N(t) of Mauna Loa. We used the optimized values of T and N(0). See figure 5.

Figure 5
We think there is a striking result. The theory allows for a prediction of the Moana Loa measurements within ± 1,2 ppM virtually over the entire 60 year period. This difference between reconstructed and measured data is extremely small, also in the early years of the period. Add this to the knowledge that back from 1960 the increments in fossil CO₂ are further declining, making the fossil CO₂ influence less and less, we feel pretty confident that our formula (1) describes the atmospheric CO₂ concentration over the entire industrial period of about 140 years with an accuracy within ± 0,5%. Or rather it describes the CO₂ loss each year with appreciable precision.
Although it goes beyond the scope of this paper, the pattern of variations between calculated and measured data can probably be linked to certain events like volcanic activity or large wildfires, that had major (order of 1 ppM) impact on the CO₂ concentration.

Another test for the model is the comparison between the fraction of B(t) each year that stays in the air, the 'airbound part', and that fraction according to our calculations. It is shown in figure 6 .

1. N(t) 'business as usual' growth of fossil CO₂ emissions 1,5 %/yr.
2. N(t) with the same fossil emission as present, i.e. 5 ppM/yr.
3. N(t) as II but now with a reduction of 1,5% annually.
4. N(t) for immediate ban on all fossil fuel use.

The result of the exercise is depicted in figure 7, drawn lines. We added a tail of the historical process. There, on this scale, is no difference to be seen between the actual data and the computed results.

Figure.7

If the fossil CO₂ emissions in the following years remain as present, scenario (II), we 'll see a rise until we approach a level where the natural sink approaches the input of 5 ppM/yr at 552 ppM. It won't rise above that level, which is less than two times the equilibrium of 287 ppM. That 2x the pre-industrial level will never be reached.
With a mall annual reduction of -1,5 %/yr, (scenario (III), the maximum will be reached around the year 2050 at a level of about 445 ppM. After that there will be a lowering, first accelerating and then slowing when approaching 287 ppM.
In scenario (I), the exponential growth case of fossil fuel use, 1,5%/yr, the atmospheric concentration will be double the equilibrium value before the year 2070. The growth will continue till all fossil fuel is depleted.


6. Side step for policy makers

In fig. 7 we have shown graphically the development of the atmospheric CO₂ concentration for a number of future fossil fuel emission scenarios using the recurrent expression of eq. (4). All these scenarios that can be more generally described as B(t) = B(0). e^(a.t) )(a being a reciprocal time constant, with a = 0 for the case B = constant) can also be solved analytically using eq. (1). However, policy makers are not as much interested in the development of the CO₂ concentration for certain scenarios as such, but more in the maximum value it will eventually reach. In case there is indeed for a certain scenario a maximum CO₂ concentration N_max, this value can be easily expressed by using eq. (1) under the condition dN(t)/dt = 0:

which can only be true if the development of B(t) levels off at a certain time at a maximum of B_max.
The determination of N(0) and T in this paper makes the expression in eq. 5 powerful, as it quantifies the effect of measures on fossil fuel emission for the expected atmospheric CO₂ concentration. With current assumptions in climatology (AGW hypothesis), this is a determining factor for the change in the Earth's temperature. Although the importance of CO₂ for the temperature is contested, it might be interesting for policy making to quantify the consequence of our findings in case they think the AGW-hypothesis is right.
If so, eq. (5) can simply be used to calculate the maximum temperature increase in relation to B_max by applying the climate sensitivity expressed as the Transient Climate Response, TCR i.e. the temperature change in degrees centigrade when the CO₂ concentration doubles:

If we would take the Paris agreement of a maximum acceptable temperature raise relative to pre-industrial conditions of 2 degrees centigrade, and a TCR of 2, eq. (6) leads directly to a B_max = N(0)/T. With the values for N(0) = 287 ppM and T = 53,5 years, B_max = 5,4 ppM, about the current level of fossil fuel emissions annually. For the four scenarios depicted in fig. 6 we have also added the corresponding temperature developments (dashed lines) applying a TCR = 2℃/2CO₂. Mind that the temperature developments as shown in fig. 7 are calculated according to ΔTemp = TCR.ln(C(t)/N(2020))/ln 2, i.e. relative to the temperature in 2020 and not relative to the pre-industrial level. The idea of some, that we have to go back to temperatures in those days is not realistic, if not even completely wrong for our wellbeing.

We are well aware that the outcome of the possible temperature increase as calculated by applying eq. (6) depends not only on the values for N(0) and T which are determined in this study, but depends as much on the choice of the value for the TCR, or even more fundamentally, on the choice of applying TCR or ECS, the Equilibrium Climate Sensitivity. Both TCR and ECS concepts are still subject to many debates in which we don't want to get involved.
We only want to illustrate how our findings on the development of atmospheric CO₂ concentration, can be "translated" in a pragmatic way. Thus providing policy makers who want to follow the AGW-hypothesis and plan to take measures to limit CO₂ emissions accordingly, with simple and transparent models. At present, all policy making is based on the output of very complex Global Circulation Models (GCM's); calculations that nobody outside the small, inner circle of climate experts can verify. Eq.(5) and (6) provide such a simple model and people are free to put in their own "beliefs" regarding the climate sensitivity to CO₂ levels. Our choice for TCR = 2 is just to illustrate the strength of eq. (5) and (6).
A recent study of Schurer et al⁽¹²⁾ based on a thorough analysis of existing temperature data yields an estimate TCR = 1,7 ℃/2CO₂ within a 90% certainty range of 1,2-2,4. While Witteman⁽¹¹⁾, using a 'first principles' method of assessing the CO₂-temperature relation, concluded 1/10th of the IPCC assumption. Which means TCR ≈ 0,2 ℃/2CO₂.


7. Conclusions

In the preceding paragraphs we have included our comments about the precision of our computation and the data we used. This article supports the statement

8. Acknowledgment & additional comment

We gratefully acknowledge the following colleagues for valuable comments, criticism and help; André Bijkerk, Wim Witteman, Ferdinand Engelbeen, Frans Schrijver, Peter Dietze, Chris Schoneveld, Guus Berkhout, Brendan Godwin.
We did not include all remarks we received and all remaining flaws are entirely the responsibility of the authors.

Additional: P. Dietze⁽⁷⁾ found t_h = 38 yr, almost the same as we did ( t_h = 37 yr ). He also extrapolated this for a scenario with constant fossil input. However he introduced a fast 33% enlarged buffer to cope with "light biomass, surface water and soil moisture". That led him to predict a maximum atmospheric concentration of 487 ppM, while we calculated a maximum of 555 ppM in that scenario without an extra buffer.

2020 06 08
Revisions 2020 07 14,
2021 06 20,
2021 06 06 & 2022 08 23.

9. Notes

1. NASA Goddard Space Centre: Animation of the CO₂ distribution variation during one year over the atmosphere (2006).
2. Ferdinand Engelbeen: Origin of the recent CO2 increase in the atmosphere.
3. C.(Kees) le Pair: Evenwicht CO₂ opgelost in water en lucht.
4. C.(Kees) le Pair: De toename van atmosferisch CO₂
5. Frans Schrijver reviews some similar approaches on his website.
6. Roy Spencer (2020 02 05) used an approach similar to ours.
7. Peter Dietze, 2020 03 07: Decarbonisation
8. P. Friedlingstein et.al.: Global Carbon Budget 2019; Earth Syst.Sci.Data, 11, 1783-1836, 2019.
9. It is generally assumed that a "natural equilibrium" existed around 280 ppM in 1880, at the beginning of the industrial revolution. This equilibrium shifts with temperature. It increases 16 ppM/°C⁽³⁾.
   Our data allowed for computation of a more likely equilibrium concentration of 287 ppM.
10. The IPCC climate panel assumes that the CO₂ temperature rise will be re-enforced by subsequent increase in water vapour. Which would bring the rise to ~4 °C/2CO₂.
11. W.J. Wittman: The Absorption Of Thermal Emitted Infrared Radiation by CO₂. (Also as PDF free of charge).
12. Schurer, A., G. Hegerl, A. Ribes, D. Polson, C. Morice, and S. Tett, 2018: Estimating the Transient Climate Response from Observed Warming. J. Climate, 31, 8645–8663,
13. Note added 2021 06 20. The rise in atmospheric CO₂ concentration with temperature would be about 12 ppM/°C if the oceans consisted of fresh water. Salinity enhances the solubility by dissociation. This explains the rise in concentration between glacials and interglacials of about 18 ppM/°C. This was demonstrated in a little known interesting study by Dr. Jarl Ahlbeck.