by Joachim Dengler
This post is the first of two extracts from the paper Improvements and Extension of the Linear Carbon Sink Model.
Introduction – Modelling the Carbon Cycle of the Atmosphere
When a complex system is analyzed, there are two possible approaches. The bottom-up approach investigates the individual components, studies their behavior, creates models of these components, and puts them together, in order to simulate the complex system. The top-down approach looks at the complex system as a whole and studies the way that the system responds to external signals, in the hope to find known patterns that allow conclusions to be drawn about the inner structure.
The relation between anthropogenic carbon emissions, CO2 concentration, and the carbon cycle has in the past mainly been investigated with the bottom-up approach. The focus of interest are carbon sinks, the processes that reduce the atmospheric CO2 concentration considerably below the level that would have been reached, if all CO2 remained in the atmosphere. There are three types of sinks that absorb CO2 from the atmosphere: physical oceanic absorption, the photosynthesis of land plants, and the photosynthesis of phytoplankton in the oceans. Although the mechanisms of carbon uptake are well understood in principle, there are model assumptions that cause divergent results.
The traditional bottom-up approaches are typically box-models, where the atmosphere, the top layer of the ocean (the mixed layer), the deep ocean, and land vegetation are considered to be boxes of certain sizes and carbon exchange rates between them. These models contain lots of parameters, which characterize the sizes of the boxes and the exchange rate between them. The currently favored model is the Bern box diffusion model, where the deep ocean only communicates by a diffusion process with the mixed layer, slowing down the downwelling carbon sink rate so much, that according to the model 20% of all anthropogenic emissions remain in the atmosphere for more than 1000 years.
I challenge this claim by questioning some of these assumptions. How does the Bern Model explain the high yearly absorption rate of more than 6% of the 14C from the bomb tests for 30 years after 1963? How does a downwelling diffusion model work, when the CO2 concentration in the deep ocean is higher than in the top layer? Empirically there is evidence that ocean absorption is still increasing and there is no sign of saturation.
The top-down models in contrast do not typically look at the details of the sink mechanism. On the basis of mass conservation, they measure the actual empirical sink rate as the difference between anthropogenic emissions and the CO2 concentration growth in the atmosphere, and investigate how this empirical sink rate is related to CO2 concentration. This is justified with the assumption that all contributing sink effects can be approximated with sufficient accuracy by linear functions of CO2 concentration.
The Linear Carbon Sink Model
Atmospheric concentration growth is the difference between all emissions and all absorptions – this is the carbon mass balance described by the continuity equation. Emissions are split between known anthropogenic emissions and unknown natural emissions. For simplification, the relatively unknown emissions caused by land use change are included in the unknown natural emissions. This is justified on the basis that the measurement error of land use change emissions is very large anyways, and by transferring a badly known contribution to the unknown contributions, we do not lose important information.
Obviously emissions, absorptions, and concentration must be measured using the same unit. The natural unit for evaluating mass conservation would be Pg (petagram), but atmospheric masses are usually measured as concentration, relative to the total mass of the atmosphere. For emissions and absorption, their masses translate into potential concentration change. Therefore, ppm is used here consequently, where 1 ppm (parts per million) is equivalent to 2.12 PgC (Petagram Carbon).
The difference between the unknown absorptions and the unknown natural emissions is defined as the sink effect during the time interval of typically a year, implying tacitly that the annual net absorptions are larger than the annual net natural emissions. On the other hand, the same sink effect is known from measurements of anthropogenic emissions minus the concentration growth. Together both statements are equivalent to the continuity equation, where annual concentration growth equals all (anthropogenic and natural) annual emissions minus all annual absorptions. This is displayed in Figure 1. The sink effect is modelled linearly with a constant absorption coefficient a expressing the proportionality of the absorptions to CO2 concentration and a constant n representing the annual natural emissions
Figure 1. The measured yearly sampled time series of anthropogenic emissions and yearly CO2 concentration growth. Both effects are measured in or have been converted to ppm, in order to guarantee comparability. Their difference is the growing carbon sink effect.
The estimated parameters of the least squares fit with annual data from 1959 to 2023 are a=0.0183, n=5.2 ppm. The equilibrium concentration which is reached when anthropogenic emissions are assumed to be zero, is C0= n/a = 284 ppm.
When reconstructing or predicting modelled CO2 concentrations, it is carried out by calculating the concentration growth from the mass balance equation and recursively reconstructing the concentration from emissions (equation (6) in the paper).
Figure 2 shows the comparison of the actual measured CO2 concentration with the predicted concentration data 2000–2020, using only data from 1950–1999 for the estimation of the model parameters. This shows the high quality of the prediction based on the linear model, using only data before the prediction time interval for estimating the model parameters. The 95% confidence interval of the prediction error, displayed as the grey shaded area, is extremely small, the actual deviations are much smaller still.
Figure 2. Model estimation with measured data from 1950–1999. The prediction of CO2 concentration is done by using the real emission data and the model. The grey error bar shows the estimated 95% confidence interval based on error propagation of the modelling residual error variance. Direct prediction comparison is possible due to the availability of the actual concentration data from 2000–2020.
Another way of getting a sense for the quality of a model is to compare the model reconstruction with the original data within the range, from which the model was built. Figure 3 displays the comparison of the actual CO2 concentration data with their model reconstruction based on the linear model over the whole time range from 1959 to 2023. Surprisingly, the actual concentration is a bit smaller than the one predicted by the model. This suggests that in the near future no saturation of the sink effect is to be expected.
Figure 3. The measured CO2 concentration (in ppm) is compared with the concentration reconstruction based on the linear model. The parameters of the model are estimated from emission and concentration data of the whole time range from 1959–2023.
Identifying the Inflection Point in the CO2 Concentration
An important consequence of the linear sink model needs to be mentioned. When we look again at Figure 1 we clearly see that the large short term variability of the concentration growth is reflected in the sink effect. This variability is removed in the sink model, without changing the trend of the data. Therefore, the reconstructed concentration growth also does not exhibit its original short-term variability; its only “noise” is caused by the anthropogenic emission data.
From a recent publication of the CarbonBrief Project, we know that global emission data have been constant for more than 10 years. For constant emissions the linear sink model implies declining concentration growth, due to increasing sink effect while concentration increases. Figure 4 shows that the measured yearly concentration growth data have an absolute maximum in 2016 and a declining trend afterward. But the concentration growth data, when stripped of short-term effects by means on the linear sink model, have their maximum already in 2013 and are declining since then. This means that the concentration graph has an inflection point in 2013, turning from concave to convex behavior. The effect appears even clearer when emission data are also smoothed. This is a remarkable validation of a model prediction — the fact that atmospheric carbon concentration growth is declining since 2013 has not been published before.
Figure 4. Comparison of the measured atmospheric CO2 concentration growth (in ppm) with the reconstruction of concentration growth by means of the linear sink model from both the original anthropogenic emission data as well as the smoothed anthropogenic emission data.
Figure 4 also explains why the significant Covid-19 related drop in anthropogenic emissions in 2020 did not have any visible effect on concentration growth. The reconstructed “noise-free” concentration growth clearly reflects the drop in emissions. But it so happened that this coincided with a positive spike in the “random” component of the concentration growth originating from natural emissions.
Making Land Use Change Emissions Consistent
When comparing the ex-post prediction in Figure 5 of 2000–2020 concentrations using the linear model with data from 1950–1999, where emissions caused by land use change were included (copied from the previous post), with the new prediction in Figure 2 without explicit land use change emissions, it is obvious that the predictive quality has become considerably better when discarding explicit land use change emission data for estimating the model parameters.
Figure 5. Prediction of 2000–2020 concentration with data from 1950–1999 from previous post. Emissions caused by land use change had been included as anthropogenic emissions. This graph is included for comparison with the better prediction in Figure 2, which does not explicitly include emissions from land use change. The grey area represents the 95% confidence interval of the predicted data.
This does not mean that there are no land use change emissions; it rather means that the best assumption is that they have been constant between 1950 and 2000 and beyond. It is a direct consequence that constant annual land use change emissions are interchangeable with natural emissions, and we are free to interpret a part of the measured natural emissions as land use change emissions.
The most likely annual value of the land use change emissions during this specific time interval can be inferred from the assumption about the equilibrium CO2 concentration by postulating an equilibrium concentration value without land use change emissions, and let land use change account for the difference to the actually measured equilibrium concentration. This obviously assumes that the ocean and land sink mechanisms have remained rather stable over the time of observation.
Let’s postulate that the “real” equilibrium CO2 concentration value should be the same as the preindustrial assumed value of 280 ppm. The estimate of the equilibrium based on anthropogenic emissions is 𝐶0=284 ppm. Therefore, we can infer that between 1950 and 2020 the most likely annual value of the Land Use Change emissions LUC is
𝐿𝑈𝐶=((284ppm−280ppm)*0.0183)*2.12 PgC/ppm = 0.15 PgC
The measured data constrain the possible range of the land use change emissions. Increasing their assumed value implies lowering the equilibrium CO2 concentration. With the most likely equilibrium concentration of 280 ppm the best estimate for land use change emissions, they are 0.15 PgC per annum. Obviously changing the assumption of the “real” natural equilibrium value to e.g. 270 ppm, would consequently change the inferred Land Use Change emissions to 0.54 PgC.
The postulated value of land use change emissions may contradict the state-of-the-art literature, depending on the value of the assumed equilibrium concentration. I see, however, no other possibility to reconcile the four constraints of anthropogenic emission measurements, concentration growth measurements, consistent sink coefficient, and equilibrium concentration consistent with preindustrial value of 280 ppm. The satisfied consistency of these constraints is reflected in the quality of prediction, as shown with the ex-post prediction of the 2000–2020 concentration data in Figure 2.
Future Emission Scenarios
To make predictions, assumptions about future CO2 emissions have to be made. Obviously, the standard scenarios of IPCC AR6 are a possible first choice. They have, however, severe handicaps. Originating from the time of exponential emission growth, at least 2 IPCC scenarios (SSP5-8.5 named “Avoid at all costs” and SSP3-7 named “Dangerous”) are so far from reality and even from the availability of fossil fuel resources that it is not meaningful to discuss them. For more than the last 10 years, global emissions have been constant within the range of measurement error. This knowledge is not yet reflected adequately in official emission statistics of the International Energy Agency, but also in these statistics there are no significant global emission changes since 2018.
Therefore, approximately constant emissions are to be considered as the worst-case scenario in the real world. This is slightly above the IPCC scenario SSP2-4.5 named “Middle of the road” during the second half of this century.
At the other end of the scale, the IPCC scenario SSP1-1.9 named “Most optimistic” is equally in denial of reality, because it assumes global emissions will to be cut to zero by 2050. None of the large countries that dominate global emissions has any plans to reduce emissions to zero. Also, SSP1-2.6 named “Next best”, with zero emissions after 2050, ignores industrial transition times, even if there was the political will. Both these scenarios also ignore the stabilizing effect of natural carbon sinks on CO2 concentration, which is the key message of this post.
Therefore, I want to focus on four scenarios, displayed in Figure 6, which are less restrictive than SSP1-2.6. First, the mentioned worst-case reference scenario with constant future emissions, extrapolating the recent 5 years.
Figure 6. Historical CO2 emissions until 2022 and from 2023 emission scenarios 0%, 0.3%, 1%, and 2% annual emission reductions.
Then, the IEA “Stated Policies” scenario, which is the most likely future emission scenario according to extensive research about existing policy decisions, approximately reducing worldwide carbon emission by 0.3% per annum, strictly speaking after 2040. This, in fact, corresponds closely to the IPCC SSP2-4.5 emission scenario.
A more severe emission reduction scenario would be 1% per annum, reducing worldwide emissions by 50% every 70 years, and finally the most aggressive reduction scenario with 2% reduction per annum, reducing emissions by 50% every 35 years. This comes close to the SSP1-2.6 “Next best” scenario without reducing to zero completely.
The predictions based on the discussed linear carbon sink model for all four scenarios are shown in Figure 7. With the linear carbon sink model, all four emission scenarios will not raise the CO2 concentration beyond 520 ppm, and the three emission reduction scenarios will reach the peak concentration within this century. I do not draw conclusions about consequences for global temperature here, because the difficult question of climate sensitivity is clearly beyond the scope of this post.
Figure 7. Historical CO2 concentration time series until 2022 and from 2023 concentration prediction scenario from linear carbon sink model with 0%, 0.3%, 1%, and 2% annual emission reductions.
For future historians, I include Figure 8, in order to be able to compare the same scenarios with the CO2 concentration predictions after 2022 from the Bern model of the 2013 publication. The IPCC predictions are based on similar models to the Bern model, with comparable outcomes.
Figure 8. Historical CO2 concentration time series, measured in ppm, until 2022 and from 2023 concentration prediction scenario from Bern model with 0%, 0.3%, 1%, and 2% annual emission reductions.
The prediction result of the 1% per annum reduction scenario from the Bern model corresponds to the constant emissions prediction scenario result from the linear sink model, and the 2% per annum reduction scenario from the Bern model corresponds to the 0.3% reduction scenario from the linear sink model. Therefore, the question of which model is correct may greatly affect future policy decisions.
Within the next 10 to 20 years, it will be easier to see which model will come closest to reality.
Conclusions
The linear carbon sink model is primarily a consequence of mass conservation resp. the continuity equation. From measurements we see an increasing sink effect which has been a strict linear function of CO2 concentration for the last 65 years. When this statistically highly significant model is accepted for the past – where its validity is obvious –, important implications follow.
By removing the “noise” from the CO2 concentration growth while keeping the trend, the modelled concentration growth data exhibit a clear maximum in 2013 and a declining trend since then. This is fully consistent with the fact that since more than 10 years anthropogenic emissions have been approximately constant. The fact that concentration growth declines when emissions are constant, is a nice validation of the linear carbon sink model.
The linear carbon sink model introduces a strict relation between the measured data, and the model parameter of equilibrium concentration. When the traditional natural equilibrium without anthropogenic emissions of 280 ppm is accepted, then the assumed constant rate of land use change emissions of the last 65 years is restricted to 0.15 PgC per annum.
The linear carbon sink model has proven to be of high predictive value. The concentrations of the years 2000-2020 have been predicted with high accuracy from the 2000-2020 anthropogenic emissions and the model built with the 1950-1999 data.
There is one potential weakness in the linear carbon sink model. The oceanic and photosynthesis sink systems are of finite size, but the model assumes no saturation effect. This contrasts with the box and diffusion models used by other researchers. The Bern model, in particular, claims that the foreseeable capacity of the natural sink systems is effectively only 4 times larger than the atmosphere, with the result, that 20% of all emissions remain in the atmosphere for at least 1000 years. Up to now, not the slightest sign of saturation of the natural sink systems can be detected. We can assume, therefore, that there will be no drastic change of this in the near future.
The simple fact that both models can explain the emission and concentration data of the past very well, makes it necessary to check the deviations in the future. For this purpose, 4 possible emission scenarios have been evaluated by both models, and future researchers and historians will find out which model will have made the better predictions of CO2 concentrations.
