# Refined Reservoir Routing (RRR) and Its Application to Atmospheric Carbon Dioxide Balance

## Abstract

**:**

What is more I loved, and still do love, mathematics for itself as not allowing room for hypocrisy or vagueness, my two pet aversions.Stendhal [1] (p. 111).

## 1. Introduction

_{2}and H

_{2}O, producing organic matter. The water availability drives the uptake of CO

_{2}through stomata, creating an interconnected cycle of gas exchange. In addition, both plants and animals respire, emitting CO

_{2}and H

_{2}O, while plants also transpire. These processes determine the inflow of both CO

_{2}and H

_{2}O to the atmosphere. Furthermore, decomposers break down organic material, releasing CO

_{2}, while water facilitates the breakdown of organic compounds, influencing the decomposition rates and thus CO

_{2}emissions. The hydrological cycle influences plant growth by providing the water needed for photosynthesis, thereby driving CO

_{2}absorption. Furthermore, both CO

_{2}and H

_{2}O affect the climate, as both are greenhouse gases, with water being the determinant one, as, in addition to its much larger absorption of longwave radiation, it is also responsible for clouds, which also absorb radiation [15].

_{2}balance for three reasons:

- Its “lumping” in a systems approach is direct, because its concentration varies slowly, while that of atmospheric water varies dramatically with time, geographic location, and altitude.
- As we will see below, there is controversy about the atmospheric CO
_{2}budget, reflecting incomplete understanding and quantification of the processes, which the simple RRR framework may shed light on. - Exporting a methodological framework developed in hydrology to the study of climate may be beneficial to both hydrology and climatology and may demonstrate the potential and usefulness of hydrology in climate research.

## 2. Theoretical Analysis

#### 2.1. System Components and Determination of Their Temporal Evolution

#### 2.2. Residence Time

**Definition 1.**

- Linear reservoir (in which $q\left(\tau \right)/s\left(\tau \right)=1$), any inflow:$${F}_{\underset{\_}{w}}\left(w\right)=1-{e}^{-w}$$
- Superlinear benchmark reservoir, $q\left(t\right)={\left(s\left(t\right)\right)}^{2}$, constant inflow:$${F}_{\underset{\_}{w}}\left(w\right)=1-{\displaystyle \frac{2{e}^{\frac{w}{\sqrt{{q}_{0}}}}}{\left(2+\left({e}^{\frac{2w}{\sqrt{{q}_{0}}}}-1\right)\left(1+\sqrt{{q}_{0}}\right)\right)}}$$
- Sublinear benchmark reservoir, $q\left(t\right)=\sqrt{s\left(t\right)}$, constant inflow:$${F}_{\underset{\_}{w}}\left(w\right)=1-{\displaystyle \frac{{\mathrm{W}\left(\left({q}_{0}-1\right){e}^{-\frac{{q}_{0}w}{2}+{q}_{0}-1}\right)}^{2}}{{({q}_{0}-1)}^{2}}}$$

#### 2.3. Response Time

**Definition 2.**

**Definition 3.**

**Proposition 1.**

**Corollary 1.**

**Corollary 2.**

**Remark 1.**

**Remark 2.**

**Remark 3.**

#### 2.4. Parameters and Their Estimation

## 3. Carbon Cycle: A Summary of the Established Approach

#### 3.1. Concepts and Terminology

Lifetime is a general term used for various time scales characterizing the rate of processes affecting the concentration of trace gases. The following lifetimes may be distinguished:[…] Response time or adjustment time (T_{a}) is the time scale characterizing the decay of an instantaneous pulse input into the reservoir. The term adjustment time is also used to characterize the adjustment of the mass of a reservoir following a step change in the source strength. Half-life or decay constant is used to quantify a first-order exponential decay process. […]The term lifetime is sometimes used, for simplicity, as a surrogate for adjustment time.In simple cases, where the global removal of the compound is directly proportional to the total mass of the reservoir, the adjustment time equals the turnover time: T = T_{a}.[…]Turnover time (T) (also called global atmospheric lifetime) is the ratio of the mass M of a reservoir (e.g., a gaseous compound in the atmosphere) and the total rate of removal S from the reservoir: T = M/S.[…]Response time or adjustment time In the context of climate variations, the response time or adjustment time is the time needed for the climate system or its components to re-equilibrate to a new state, following a forcing resulting from external processes. It is very different for various components of the climate system. The response time of the troposphere is relatively short, from days to weeks, whereas the stratosphere reaches equilibrium on a time scale of typically a few months. […] In the context of lifetimes, response time or adjustment time (T_{a}) is the time scale characterizing the decay of an instantaneous pulse input into the reservoir.

The concept of a single, characteristic atmospheric lifetime is not applicable to CO_{2}.[31] (p. 473)

No single lifetime can be given [for CO_{2}]. The impulse response function for CO_{2}from Joos et al. (2013) [42] has been used.[31] (p. 737)

_{2}” without specifying which ones [32] (p. 302, Table 2.2; p. 1017, Table 7.15).

_{2}by IPCC may be inferred from what follows.

#### 3.2. Separate Treatment of CO_{2} Depending on Its Origin

_{2}in the atmosphere depends on its origin and that CO

_{2}emitted by anthropogenic fossil fuel combustion has higher residence time than when naturally emitted. This is clear in the IPCC AR5:

Simulations with climate–carbon cycle models show multi-millennial lifetime of the anthropogenic CO_{2}in the atmosphere.[31] (p. 435)

This delay between a peak in emissions and a decrease in concentration is a manifestation of the very long lifetime of CO_{2}in the atmosphere; part of the CO_{2}emitted by humans remains in the atmosphere for centuries to millennia.[32] (p. 642 FAQ 4.2)

_{2}is different from that of the natural CO

_{2}. For example, Joos et al. [43] stated the following:

When considering the fate of anthropogenic CO_{2}, the emission into the atmosphere can be considered as a series of consecutive pulse inputs.

_{2}”, Archer and Brovkin [44] stated,

The largest fraction of the CO_{2}recovery will take place on time scales of centuries, as CO_{2}invades the ocean, but a significant fraction of the fossil fuel CO_{2}, ranging in published models in the literature from 20–60%, remains airborne for a thousand years or longer.

The models agree that 20–35% of the CO_{2}remains in the atmosphere after equilibration with the ocean (2–20 centuries).

Estimates for how long carbon dioxide (CO_{2}) lasts in the atmosphere […] are often intentionally vague, ranging anywhere from hundreds to thousands of years. […] As it stands, says [Ed] Boyle, human-generated carbon dioxide is expected to continue warming the planet for tens of thousands of years [46].Once [carbon dioxide is] added to the atmosphere, it hangs around, for a long time: between 300 to 1000 years. Thus, as humans change the atmosphere by emitting carbon dioxide, those changes will endure on the timescale of many human lives [47].

_{2}molecules in the atmosphere.

#### 3.3. Modeling Approach

_{2}exchange is based on the so-called Bern modeling approach (Joos et al. [42,43]; Myhre et al. [49]; Strassmann and Joos [50]; Luo et al. [51]). It is reflected in the following expression of the IRF as the sum of three exponential functions and a constant term:

_{2}is no more than 4 years, as admitted even by IPCC [32] (p. 2237):

Carbon dioxide (CO_{2}) is an extreme example. Its turnover time is only about 4 years because of the rapid exchange between the atmosphere and the ocean and terrestrial biota. However, a large part of that CO_{2}is returned to the atmosphere within a few years. The adjustment time of CO_{2}in the atmosphere is determined from the rates of removal of carbon by a range of processes with time scales from months to hundreds of thousands of years. As a result, 15 to 40% of an emitted CO_{2}pulse will remain in the atmosphere longer than 1000 years, 10 to 25% will remain about ten thousand years, and the rest will be removed over several hundred thousand years.

_{2}molecule remains in the atmosphere for such a long time.

## 4. RRR Application to the Atmospheric Component of the Carbon Cycle

#### 4.1. Data

_{2}have been made since 1958 [52] by the Scripps CO

_{2}Program of the Scripps Institution of Oceanography, University of California, and are available online [53,54,55]. The data include observations of CO

_{2}concentration (in micro-moles CO

_{2}per mole, or parts per million—ppm), and are processed to extract monthly values, filled in in case of missing data. Here, the monthly time series have been retrieved and processed for two stations, namely, Mauna Loa Observatory, Hawaii (19.5° N, 155.6° W, 3397 m a.s.l., 1958–present), and Barrow (recently renamed to Utqiagvik), Alaska (71.3° N, 156.6° W, 11 m a.s.l., 1961–present).

- From mass of C to mass of CO
_{2}, we multiply by 44/12 = 3.67 kg CO_{2}/kg C (where 44 and 12 are the molecular masses of CO_{2}and C). - From atmospheric CO
_{2}concentration in ppm to total atmospheric mass in Gt CO_{2}, we multiply by 7.8 Gt CO_{2}/ppm CO_{2}.

#### 4.2. Premises of the Application

_{2}.

_{2}exchanges that follow:

- Human activities are responsible for only 4% of carbon emissions.
- The vast majority of changes in the atmosphere since 1750 (red bars in the graph) are due to natural processes, respiration and photosynthesis.
- The increases in both CO
_{2}emissions and sinks are due to the temperature increase, which expands the biosphere and makes it more productive. - The terrestrial biosphere processes are much stronger than the maritime ones in terms of both production and absorption of CO
_{2}. - The CO
_{2}emissions by merely the ocean biosphere are much larger than human emissions. - The modern (post 1750) CO
_{2}additions to pre-industrial quantities (red bars in the right half of the graph, corresponding to positive values) exceed the human emissions by a factor of ~4.5. In the most recent 65 years, covered by measurements, the rate of natural emissions is ~3.5 times greater than the CO_{2}emissions from fossil fuels.

_{2}concentration that is opposite to the popularly assumed one, which is also the one assumed and embedded in climate models. Indeed, according to conventional wisdom, it is the increased atmospheric carbon dioxide concentration ([CO

_{2}]) that caused the increase in temperature (T). However, this was questioned by Koutsoyiannis and Kundzewicz [59], while later Koutsoyiannis et al. [11,33,34] provided evidence, based on analyses of instrumental measurements of the last seven decades, for a unidirectional, potentially causal link between T as the cause and [CO

_{2}] as the effect. The same causality direction was confirmed for the entire Phanerozoic by using several proxy data series [35].

_{2}] on the CO

_{2}inflow to the atmosphere is depicted in Figure 10. The two endpoints, corresponding to 1750 and 2017, were determined from the quantities given in Figure 9, with the additional information that [CO

_{2}] was 280.0 and 404.6 in these two years, respectively.

_{2}] observed values at a decadal time step. The figure also shows a power law between [CO

_{2}] and input (emitted) CO

_{2}, with an exponent of 0.7, determined from the two endpoints. The intermediate points do not perfectly agree with this power law, although they show a rising trend. If we increase the ${Q}_{10}$ values by 30% (e.g., to account indirectly for processes that are not explicitly considered, such as ocean outgassing driven by Henry’s Law, which states that as water temperature increases, the solubility of CO

_{2}in water decreases), the agreement is improved, as also shown in the figure. In addition, the figure shows two points of CO

_{2}outflow from the atmosphere corresponding to 1750 and 2017, which were again determined from the quantities given in Figure 9. These points show a similarity with those of the input, with a greater rise, yielding an exponent of a power law equal to 0.77.

#### 4.3. Model and Its Fitting Methodology

_{2}, we represent the storage $S\left(t\right)$ as the atmospheric [CO

_{2}] (in ppm) and the inflow $I\left(t\right)$ and outflow $Q\left(t\right)$ as the emissions and sinks, respectively (ppm/year). An eminent characteristic of the atmospheric CO

_{2}exchange is its seasonality, implying seasonal variation of the characteristic residence time. To take seasonality into account in a parsimonious manner, we modify Equation (2) by substituting ${S}_{0}/{W}_{0}$ for ${Q}_{0}$ and then replacing ${W}_{0}$ with a periodic function of time, $W\left(t\right)$:

#### 4.4. Results of Final Modeling

#### 4.5. Results for Imaginary Cases

- Human emissions are disregarded, and only natural processes are considered.
- The natural processes are neglected, and only the anthropogenic emissions are considered.
- In addition to anthropogenic emissions, natural outputs (but no inputs) are also considered.
- All processes are considered, but the biosphere expansion is neglected.

#### 4.6. RRR Validation

_{2}balance is heavily studied and also officially reported in IPCC Assessment Reports. However, possible agreement of the RRR framework results with those of IPCC would not validate the former because of the severity of the problems in the latter, which are discussed in Section 3 and in Appendix B and Appendix C. In particular, Appendix C offers an indirect (not formal) validation of the RRR results by enrolling additional data, namely isotopic data of atmospheric

^{14}C. These data reflect an accidental real-world experiment, not designed as such but related to nuclear weapons testing, in the 1950s and 1960s, which stopped afterwards. The injection of a series of

^{14}C impulses in the atmosphere made a real-world situation close to an ideal to estimate an IRF of the

^{14}CO

_{2}dynamics. The analysis in Appendix C shows that the observed

^{14}CO

_{2}dynamics are compatible with the RRR results and blatantly incompatible with the IPCC results.

_{2}] in the last few years. Figure 21 does not have any discernible visual difference in net inflow, $I-Q$, from Figure 15, in which the calibration was for the entire observation period. Table 2 shows that the parameter values changed only slightly with the change in the calibration period. Finally, Table 3 shows slight decreases of the performance indices in the period 2003–2023 when the fitting is made in the period 1958–2002. The decrease is about 3.5% in [CO

_{2}] and 1–1.5% in $I-Q$ when compared to the values of the fitting on the entire observation period. Overall, the validation results are deemed satisfactory.

## 5. Discussion and Further Results

#### 5.1. Residence Times

#### 5.2. Anthropogenic Emissions Remaining in the Atmosphere: Total Mass

_{2}or 20.9 ppm, while ${M}_{\mathrm{A}}=2612$ Gt CO

_{2}or 334.9 ppm, so that ${M}_{\mathrm{R}}/{M}_{\mathrm{A}}=6\%$, comparable to (somewhat smaller than) the estimate ~10% by Stallinga [41] and also slightly smaller than the cumulative emissions of the last 4 years (as is reasonable). This contradicts the IPCC assertion [32] (p. 676, also repeated many times in AR6), which follows:

Over the past six decades, the average fraction of anthropogenic CO_{2}emissions that has accumulated in the atmosphere (referred to as the airborne fraction) has remained nearly constant at approximately 44%.

#### 5.3. Anthropogenic Emissions Remaining in the Atmosphere: Probabilistic Assessment of Characteristic Times

_{2}pulse will remain in the atmosphere longer than 1000 years, 10 to 25% will remain about ten thousand years, and the rest will be removed over several hundred thousand years”. We examine it also in connection with the IPCC statement that the “turnover time is only about 4 years”, which we deem correct, as it agrees with the results of this study. We make the following calculations:

- The probability that a molecule remains after 1000 years is $p=1-{F}_{\underset{\_}{W}}\left(1000\right)=1-{F}_{\underset{\_}{w}}\left(1000/4\right)={e}^{-250}={10}^{-108.6}$, where we have used Equation (29) to evaluate the ${F}_{\underset{\_}{w}}\left(w\right)$.
- The probability that out of $N$ molecules none remain after 1000 years is ${(1-p)}^{N}$, and the probability that at least one molecule remains is ${{p}_{1}=1-(1-p)}^{N}$. Given that as $p\to 0$, $\left({1-\left(1-p\right)}^{N}\right)/pN\to 1$, for small $p$ (as in our case), we have ${p}_{1}=pN$.
- According to IPCC [32] (Figure 5.12), the atmospheric CO
_{2}amounts to 870 Pg C = $8.7\times {10}^{17}$ g C. Thus, the mass of CO_{2}is $8.7\times {10}^{17}\times \left(44/12\right)=3.2\times {10}^{18}$ g (where 44 and 12 are the molecular masses of CO_{2}and C, respectively). The number of moles is $3.2\times {10}^{18}\mathrm{g}/\left(44\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}\right)=7.1\times {10}^{16}\mathrm{m}\mathrm{o}\mathrm{l}$. - The Avogadro constant is $6.022\times {10}^{23}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, and thus the number of CO
_{2}molecules in the atmosphere is $N=7.3\times {10}^{16}\mathrm{m}\mathrm{o}\mathrm{l}\times 6.022\times {10}^{23}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}=4.4\times {10}^{40}={10}^{40.6}$. - Hence, the probability that after 1000 years, at least one out of the $N={10}^{40.6}$ molecules remains in the atmosphere is ${p}_{1}=pN={10}^{-108.6}\times {10}^{40.6}={10}^{-68}$.
- A probability ${10}^{-68}$ is virtually no different from an impossibility. Hence, we can be certain that none of the molecules existing in the atmosphere now, whether due to an “emitted CO
_{2}pulse” or existing before it, will remain after 1000 years—let alone after “ten thousand years” or after “several hundred thousand years”. - To make this probability a reasonable rarity of 1% (${10}^{-2}$) that a single molecule out of the $N={10}^{40.6}$ remains in the atmosphere, we need to make $p={p}_{1}/N={10}^{-2}/{10}^{40.6}={10}^{-42.6}$. This would occur at time $t$ such that $1-{F}_{\underset{\_}{W}}\left(t\right)=1-{F}_{\underset{\_}{w}}\left(t/4\right)={e}^{-t/4}={10}^{-42.6}$, which yields $t=392$ years.

_{2}pulse will remain in the atmosphere longer than 1000 years, 10 to 25% will remain about ten thousand years, and the rest will be removed over several hundred thousand years” needs to be corrected to “not even one molecule from an emitted CO

_{2}pulse will remain in the atmosphere longer than 400 years, even if that emitted pulse amounts to the entire current atmospheric CO

_{2}content”.

## 6. Conclusions

- It defines and clarifies the relevant quantities, including the characteristic time lags, such as residence and response times, which are often confused in the literature. (The Glossary presented below summarizes the related concepts and their definitions.)
- It refines the case of a reservoir with linear dynamics, which admits analytical solutions for all related variables, and rederives and streamlines these analytical solutions.
- It classifies the cases of a reservoir with nonlinear dynamics, studies some special cases that admit analytical solutions, and provides working approximations of the outflow and the residence time, including its probability distribution and statistical characteristics.
- It provides an exact solution for the instantaneous response function and the response time, whether for the linear or nonlinear case.
- It proposes a framework for model fitting, based on observed data, for several cases, whether with linear or nonlinear dynamics.

_{2}gives useful insights in terms of residence and response times, which have been an issue of controversy. The theoretical framework results in excellent agreement with real-world data on carbon dioxide concentration. The atmosphere appears to behave as a linear reservoir in terms of the atmospheric CO

_{2}, whose exchange is clearly dominated by the biosphere processes, with human emissions playing a minor role. The quantification of the atmospheric CO

_{2}exchange with the RRR framework yields reliable and intuitive results, complying with observations, in contrast to the results of complex climate models, which are shown to be inconsistent with reality. The mean residence time of atmospheric CO

_{2}is about four years, and the mean response time is smaller than that, thus contradicting the mainstream estimates, which suggest times of hundreds or thousands of years, or even longer.

_{2}molecules at their pertinent scales, without waiting for the slow or the very slow processes to act.

_{2}observational data are not consistent with the climate narrative. They rather contradict it. In this, the present study complements earlier studies in that (a) causality direction between temperature and atmospheric CO

_{2}is opposite to that commonly assumed [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34], (b) climate models misrepresent the causality direction that is identified by the data [11], (c) there are no discernible signs of anthropogenic CO

_{2}emissions on the greenhouse effect, which is dominated by water vapor and clouds [64], and (d) there are no discernible signs of change in the isotopic synthesis of atmospheric CO

_{2}sources and sinks, which is determined by the biosphere processes [65].

## Funding

## Data Availability Statement

## Acknowledgments

^{14}C, which helped me to think about it and prepare Appendix C. Three anonymous reviewers provided comments, most of which were constructive and helped me to improve and expand the paper. Finally, I thank the Editorial Office staff for the processing of the paper, the editors involved in the final assessment, and particularly the Water’s Editor-in-Chief for the favorable decision.

## Conflicts of Interest

## Glossary

## Appendix A. Alternative Approximations of a Sublinear or Superlinear Reservoir

## Appendix B. Notes on the Sum of Exponential Functions as a Response Function

Term | $\mathit{i}\mathbf{=}\mathbf{0}$ | $\mathit{i}\mathbf{=}\mathbf{1}$ | $\mathit{i}\mathbf{=}\mathbf{2}$ | $\mathit{i}\mathbf{=}\mathbf{3}$ |
---|---|---|---|---|

$\mathrm{Coefficient}{a}_{i}$ | 0.2173 | 0.224 | 0.2824 | 0.2763 |

$\mathrm{Coefficient}{W}_{i}$ (years) | ∞ | 394.4 | 36.54 | 4.304 |

## Appendix C. Indirect Validation of the RRR Results Using ^{14}C Isotopic Data

_{2}and thereby assess whether the claimed time lags by IPPC, reaching “several hundred thousand years”, can have any relevance to reality or, alternatively, whether the reality is that these time lags are of the order of a few years, as found in this paper.

^{12}C and

^{13}C at percentages of 99% and 1%, respectively [66]. It also appears in the unstable isotopic form

^{14}C, known as radiocarbon but in trace amounts (of the order of 1 × 10

^{−12}). As detailed by Hua et al. [67], radiocarbon is naturally produced in the upper atmosphere by the interaction of the secondary neutron flux from cosmic rays with atmospheric nitrogen isotope

^{14}N. Following its production and oxidation to CO

_{2},

^{14}C enters the biosphere and oceans via photosynthesis and air-sea gas exchange, respectively, providing a supply that approximately compensates for the decay of the existing

^{14}C in terrestrial and marine reservoirs.

^{14}C was dramatically increased due to nuclear weapons testing. This produced large fluxes of thermal neutrons, which reacted with atmospheric

^{14}N to form

^{14}C. These were mostly injected into the stratosphere and subsequently transported to the troposphere. Since about 1965, the

^{14}C concentration in the atmosphere has been dropping rapidly. Given that the half life of

^{14}C is about 5700 years [68], this drop was not due to radioactive decay but due to CO

_{2}absorption by other reservoirs. Hence, the radioactive decay during these few decades can be neglected.

^{14}C data to estimate the atmospheric CO

_{2}residence time is not new, as it appears that it has been pioneered by Starr (1993) [69], who noted the following:

This study explores the plausibility of this concept, which results in much shorter atmospheric residence times, 4-5 years, than the magnitude larger outcomes of the usual global carbon cycle models which are adjusted to fit the assumption that anthropogenic emissions are primarily the cause of the observed rise in atmospheric CO_{2}. The continuum concept is consistent with the record of the seasonal photosynthesis swing of atmospheric CO_{2}which supports a residence time of about 5 years, as also does the bomb C^{14}decay history. The short residence time suggests that anthropogenic emissions contribute only a fraction of the observed atmospheric rise, and that other sources need be sought.

^{14}C and Δ

^{14}C denote the so-called “fraction modern” and “the per mil difference of the normalized sample/modern-carbon ratio from unity”, respectively, and are defined in [76,77,78]. As we consistently refer to atmospheric CO

_{2}, and since both quantities express ratios, the symbols Δ

^{14}C and Δ

^{14}CO

_{2}are used here interchangeably (and likewise for F

^{14}C).

^{14}CO

_{2}] per se, because the latter also depends on the total [CO

_{2}] in the atmosphere, which has been increasing for more than a century. Here, the question we deal with is how fast the excess

^{14}C was removed by the biosphere, and, therefore, we should isolate the study of that question from the modern increase of the total $\left[{\mathrm{C}\mathrm{O}}_{2}\right]$. To see that this is the reasonable approach, let us consider the imaginary case that throughout the examined period, the concentration of ${[}^{14}{\mathrm{C}\mathrm{O}}_{2}]$ was constant, while the fraction ${\mathrm{F}}^{14}\mathrm{C}$ was decreasing, e.g., at the observed rate. This would happen if the

^{14}CO

_{2}absorbed by the biosphere, ${{[}^{14}{\mathrm{C}\mathrm{O}}_{2}]}_{\mathrm{A}\mathrm{B}\mathrm{S}}$, was replaced by that added through the total CO

_{2}inflow, ${\left[{\mathrm{C}\mathrm{O}}_{2}\right]}_{\mathrm{I}\mathrm{N}}$, that is, if ${{[}^{14}{\mathrm{C}\mathrm{O}}_{2}]}_{\mathrm{A}\mathrm{B}\mathrm{S}}={{[\mathrm{F}}^{14}{\mathrm{C}\mathrm{O}}_{2}]}_{\mathsf{{\rm I}}\mathsf{{\rm N}}}\times {\left[{\mathrm{C}\mathrm{O}}_{2}\right]}_{\mathrm{I}\mathrm{N}}$, where the ${{[\mathrm{F}}^{14}{\mathrm{C}\mathrm{O}}_{2}]}_{\mathsf{{\rm I}}\mathsf{{\rm N}}}$ is the isotope-14 fraction in the input CO

_{2}. Clearly, if in this imaginary case we considered the concentration ${[}^{14}{\mathrm{C}\mathrm{O}}_{2}]$ in our calculations, we would conclude that the residence time of

^{14}CO

_{2}would be infinite, because ${[}^{14}{\mathrm{C}\mathrm{O}}_{2}]$ would be constant. This is absurd, because the biosphere in fact removes

^{14}CO

_{2}, as shown by the decrease in ${\mathrm{F}}^{14}\mathrm{C}$.

^{14}CO

_{2}dynamics. It is remembered that by definition, an IRF assumes zero input after the impulse, and this is precisely consistent with the above explanation as to why we should not consider the ${\left[{\mathrm{C}\mathrm{O}}_{2}\right]}_{\mathrm{I}\mathrm{N}}$ and hence the ${[}^{14}{\mathrm{C}\mathrm{O}}_{2}]$ in our estimation.

^{14}CO

_{2}dynamics by considering either of the quantities (relative differences):

**Figure A1.**Comparison of $\mathrm{D}\left[{\mathrm{F}}^{14}\mathrm{C}\right]$ and $\mathrm{D}\left[{\mathsf{\Delta}}^{14}\mathrm{C}\right]$ derived from the Hua et al. [67] data for NH zone 1.

Decreases in atmospheric Δ^{14}C from the mid-1960s to mid-1980s are mainly due to rapid exchange between the atmosphere and the biosphere and oceans […], while combustion of fossil fuels free of^{14}C is the main causal factor for the Δ^{14}C decline since the late 1980s and early 1990s […]. Since the early and late 2000s, the atmospheric Δ^{14}C values have been lower than those of the surface waters in the North and South Pacific Gyres, respectively, indicating the oceans might become a net^{14}C source (instead of a net^{14}C sink) of the atmosphere […]The last data points in our compiled monthly data at 2019.375 have respective F^{14}C values of 1.0084 and 1.0195 for the NH and SH (see Supplementary Tables 2a–e), which are very close to the pre-bomb F^{14}C value of slightly lower than 1. This indicates that clean-air F^{14}C is likely to reach the pre-bomb value in the early 2020s […].

^{14}C by bomb testing disappeared in about 55 years.

**Figure A2.**Global ${\mathsf{\Delta}}^{14}\mathrm{C}$ time series, as provided by Hua et al. [67], and fitted linear reservoir model (Equation (A21)).

_{2}]. These are the IPCC model (Equation (50) with the coefficients shown in Table A1) and the linear reservoir model of the present study with ${\mu}_{h}=4$ years. The absolute incompatibility of the IPCC model with reality, as demonstrated through the Δ

^{14}C data (and the model fitted to them, which is in perfect agreement with the data) is obvious (Figure A3).

^{14}C data and model are compatible with the linear reservoir model of the present study. The answer is affirmative, and the longer mean response time in the Δ

^{14}C case (${\mu}_{h}=$ 17.2 years) compared to the total [CO

_{2}] case (${\mu}_{h}=4$ years) is expected. There are three very strong reasons for this increase in the response time of Δ

^{14}C:

- The absorption of the heavier isotope
^{14}C is subject to a function known as fractionation, that is, isotope discrimination. In particular, photosynthesis, during the exchange of O_{2}and CO_{2}, discriminates against the heavier isotopes, and, as a result,^{14}C remains in the atmosphere for longer periods. - As already noted above, most of the
^{14}C produced by nuclear weapons testing was injected into the stratosphere, and the transport from the stratosphere to the troposphere is a slow process, substantially increasing the time lags. - While, by its definition, the IRF presupposes zero inflows after the impulse, in reality, there were additional
^{14}C inflows due to anomalous neutron flux (corresponding to a systematic increase of 5–10% over the last 30 years, according to Harde and Salby [74]). The fact that these^{14}C inflows were not considered in the model led to an artificial increase in the actual response time.

^{14}C analysis offers an indirect validation of the RRR results by determining an upper bound of the response time, which the RRR model respects, while the IPCC model blatantly violates it.

**Figure A3.**Comparison of two total [CO

_{2}] IRFs, i.e., (a) the IPCC model (Equation (50) with the coefficients shown in Table A1), (b) the results from the present study (a linear reservoir model with ${\mu}_{h}=4$ years), and two $\mathrm{D}\left[{\mathsf{\Delta}}^{14}{\mathrm{C}\mathrm{O}}_{2}\right]$ IRFs, i.e., (c) empirical (from data) and (d) modeled (from Equation (A21) with ${\mu}_{h}=$ 17.2 years).

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**Figure 1.**Comparison of the exact (lines) and approximate (dots) solutions for the indicated cases and for constant inflow. The exact solutions are derived from (

**left**) Equation (19) (superlinear reservoir, $b=2$) and (

**right**) Equation (20) (sublinear reservoir, $b=1/2$). The approximate solutions are derived from Equation (11).

**Figure 2.**Comparison of the exact (lines) and approximate (points) solutions for $b=2$ and (

**left**) ${q}_{0}=0.8$ and (

**right**) ${q}_{0}=1.25$ and for the indicated cases. In the case of the constant inflow ($a=0$), the exact solution is derived from Equation (19) and the approximate solution from Equation (11). The cases of increasing and decreasing flow correspond to $a=0.1$ and $a=-0.1$, respectively, and the exact curves were determined by numerical integration, while the approximate solutions were determined by Equation (15).

**Figure 3.**Probability plots of the distribution function of the dimensionless residence time for the indicated cases and constant inflow. The exact distributions (lines) are determined by numerical integration of Equations (29)–(31) and are compared to approximate analytical solutions (dots) determined by Equation (34).

**Figure 4.**Comparison of the mean dimensionless residence time from exact benchmark solutions (Equations (30) and (31) with numerical integration to find ${\mu}_{w}$) and approximations 1 and 2 (Equations (39) and (41), respectively).

**Figure 5.**Comparison of the mean dimensionless residence time for the indicated cases. Continuous lines for constant inflow ($a=0$) were obtained from Equations (30) and (31) (with numerical integration to find ${\mu}_{w}$), and those for increasing ($a=0.1$) and decreasing ($a=-0.1$) inflow were found by numerical integration of the exact differential equations (considering a finite upper limit of integration, determined such as to avoid numerical instabilities). Beside each exact curve, that of approximation 2 (Equation (41)) is also plotted as a dashed line.

**Figure 6.**Comparison of characteristic dimensionless time lags for the indicated cases. Mean and median residence times are determined from approximation 2 (Equation (41)). Response times are determined from the exact solution (Equation (44)). The mean response time in the bottom left graph ($b=2$) is infinite.

**Figure 7.**Dimensionless outflow for zero inflow in the indicated cases, representing an impulse response function, with comparison of exact solutions (Equation (17)) and approximate ones (Equation (14)). In the right panel (sublinear reservoir, $b=1/2$), according to the exact solution, the reservoir empties at time $\tau =1/\left(1-b\right)=2$, while according to the linear reservoir approximation, the emptying time is $\tau =-\mathrm{ln}\left(1-b\right)/b=1.39$; beyond emptying time, the outflow is zero.

**Figure 8.**Probability plots of the distribution function of dimensionless residence time for the indicated cases and zero inflow, with comparison of the exact solution (Equation (38)) with the linear reservoir approximation (Equation (14), which, combined with Equation (27), yields ${F}_{\underset{\_}{w}}\left(w\right)=\mathrm{min}(1,(1-{e}^{-bw})/b)$).

**Figure 9.**Annual carbon balance in the Earth’s atmosphere, in ppm CO

_{2}/year, based on the IPCC estimates [32] (Figure 5.12). The balance of 2.4 ppm CO

_{2}/year is the annual CO

_{2}accumulation in the atmosphere. The total of the modern natural additions (64.2 + 36.5 − (52.2 + 25.6)) = 22.9 ppm is 4.4 times larger than the human emissions (4.4 + 0.8 = 5.2 ppm). (Adapted from [11]).

**Figure 10.**Atmospheric carbon dioxide inflows and outflows as a function of carbon dioxide concentration. Large circles correspond to IPCC estimates for 1750 and the recent decade (assigned to 2017), and the line joining them is a power law with slope $b=0.7$. The other circles have been determined using the ${Q}_{10}$ method, as in Koutsoyiannis et al. [11], and the triangles by the same method but using 30% higher values of ${Q}_{10}$. A power law for output (not drawn in the graph) has a slope of 0.77.

**Figure 11.**Optimized values of ${b}_{I}$ (the exponent of the power law of inflow) as a function of the values of $b$ (the exponent of the power law of outflow). The optimization was made by fixing $b$ to a specified value and letting all other parameters free.

**Figure 12.**Achieved values of the indicated explained variances, after maximizing their sum by fixing $b$ to a specified value and letting all other parameters free.

**Figure 13.**Comparison of observed and simulated storage, $S\left(t\right)\equiv \left[\mathrm{C}{\mathrm{O}}_{2}\right]$, for (

**upper**) Mauna Loa and (

**lower**) Barrow. The insets show the seasonal variation of the characteristic times $W,{W}_{I}$.

**Figure 15.**Comparison of observed and simulated net inflows for (

**upper**) Mauna Loa and (

**lower**) Barrow.

**Figure 16.**Comparison of observed and simulated (

**upper**) storages and (

**lower**) net inflows for Barrow, in the case that the cosine function of inflow is replaced by an arbitrary function, defined through 24 coordinates.

**Figure 17.**Comparison of observed and simulated storage, $S\left(t\right)\equiv \left[\mathrm{C}{\mathrm{O}}_{2}\right]$, for (

**upper**) Mauna Loa and (

**lower**) Barrow, as in Figure 13, but omitting anthropogenic emissions (imaginary case 1). The insets show the seasonal variation of the characteristic times $W,{W}_{I}$.

**Figure 18.**Comparison of observed and simulated storage, $S\left(t\right)\equiv \left[\mathrm{C}{\mathrm{O}}_{2}\right]$, for (

**upper**) Mauna Loa and (

**lower**) Barrow, as in Figure 13 but for the indicated imaginary cases. The insets show the seasonal variation of the characteristic times $W,{W}_{I}$ for the case of all emissions without biosphere expansion (imaginary case 4). The explained variance noted is for the same case, while it is smaller in the other indicated cases, and decreased to less than –100% for the worst case of human emissions only.

**Figure 19.**Comparison of observed and simulated net inflows for (

**upper**) Mauna Loa and (

**lower**) Barrow as in Figure 15 but without biosphere expansion (imaginary case 4).

**Figure 20.**Comparison of observed and simulated storage, $S\left(t\right)\equiv \left[\mathrm{C}{\mathrm{O}}_{2}\right]$, as in Figure 13 but for the calibration period 1958–2002: (

**upper**) Mauna Loa and (

**lower**) Barrow. The insets show the seasonal variation of the characteristic times $W,{W}_{I}$.

**Figure 21.**Comparison of observed and simulated net inflows as in Figure 15 but for the calibration period 1958–2002: (

**upper**) Mauna Loa and (

**lower**) Barrow.

Site | $\mathit{b}$ | $\mathit{\phi}$ | $\mathit{A}$ (Years) | $\mathit{\psi}$ | ${\mathit{b}}_{\mathit{I}}$ | ${\mathit{\phi}}_{\mathit{I}}$ | ${\mathit{A}}_{\mathit{I}}$ (Years) | ${\mathit{\psi}}_{\mathit{I}}$ |
---|---|---|---|---|---|---|---|---|

Mauna Loa | 1 | 5.445 | 1.973 | 2.115 | 0.953 | 5.247 | 1.462 | 2.855 |

Barrow | 1 | 5.757 | 4.181 | 1.370 | 0.953 | 5.149 | 3.104 | 1.633 |

Site | $\mathit{b}$ | $\mathit{\phi}$ | $\mathit{A}$ (Years) | $\mathit{\psi}$ | ${\mathit{b}}_{\mathit{I}}$ | ${\mathit{\phi}}_{\mathit{I}}$ | ${\mathit{A}}_{\mathit{I}}$ (Years) | ${\mathit{\psi}}_{\mathit{I}}$ |
---|---|---|---|---|---|---|---|---|

Mauna Loa | 1 | 5.399 (5.445) | 2.126 (1.973) | 2.092 (2.115) | 0.935 (0.953) | 5.164 (5.247) | 1.578 (1.462) | 2.858 (2.855) |

Barrow | 1 | 5.710 (5.757) | 4.174 (4.181) | 1.368 (1.370) | 0.935 (0.953) | 5.134 (5.149) | 2.207 (3.104) | 1.594 (1.633) |

**Table 3.**Explained variances (%) as performance indices of the RRR method for the indicated applications.

↓Site | $\mathbf{Storage}\mathit{S}\equiv \left[\mathbf{C}{\mathbf{O}}_{2}\right]$ (ppm) | $\mathbf{Net}\mathbf{Inflow},\mathit{I}-\mathit{Q}$ (ppm/year) | ||||
---|---|---|---|---|---|---|

Period→ | All | 1958–2002 | 2003–2023 | All | 1958–2002 | 2003–2023 |

Calibration over the entire period | ||||||

Mauna Loa | 99.94 | 99.82 | 99.64 | 85.81 | 87.25 | 83.30 |

Barrow | 99.77 | 99.25 | 99.16 | 85.30 | 85.82 | 84.64 |

Calibration over the period 1958–2002 | ||||||

Mauna Loa | 99.73 | 99.90 | 96.24 | 85.57 | 87.46 | 82.25 |

Barrow | 99.55 | 99.44 | 95.88 | 84.85 | 86.18 | 83.13 |

**Table 4.**Mean residence times, seasonal (${W}_{\mathrm{m}\mathrm{i}\mathrm{n}},{W}_{\mathrm{m}\mathrm{a}\mathrm{x}}$) and annual ${W}_{\mathrm{m}}$ (in years).

Site | $\mathbf{Minimum},$ ${\mathit{W}}_{\mathbf{m}\mathbf{i}\mathbf{n}}=\mathit{A}\left(\mathit{\psi}-1\right)$ | $\mathbf{Maximum},$ ${\mathit{W}}_{\mathbf{m}\mathbf{a}\mathbf{x}}=\mathit{A}\left(\mathit{\psi}+1\right)$ | $\mathbf{Arithmetic}\mathbf{Average},\mathit{A}\mathit{\psi}$ | $\mathbf{Theoretical}\mathbf{Mean},{\mathit{W}}_{\mathbf{m}}=\sqrt{{\mathit{W}}_{\mathbf{m}\mathbf{i}\mathbf{n}}{\mathit{W}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ | $\mathbf{Empirical}\mathbf{Mean}{\mathit{W}}_{\mathbf{m}}$, Beginning Year | $\mathbf{Empirical}\mathbf{Mean}{\mathit{W}}_{\mathbf{m}}$, Ending Year |
---|---|---|---|---|---|---|

Calibration over the entire period | ||||||

Mauna Loa | 2.20 | 6.15 | 4.17 | 3.68 | 3.68 | 3.70 |

Barrow | 1.55 | 9.91 | 5.73 | 3.91 | 3.94 | 3.95 |

Calibration over period 1958–2002 | ||||||

Mauna Loa | 2.32 | 6.57 | 4.45 | 3.91 | 3.93 | 3.98 |

Barrow | 1.53 | 9.88 | 5.70 | 3.89 | 3.92 | 3.98 |

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Koutsoyiannis, D.
Refined Reservoir Routing (RRR) and Its Application to Atmospheric Carbon Dioxide Balance. *Water* **2024**, *16*, 2402.
https://doi.org/10.3390/w16172402

**AMA Style**

Koutsoyiannis D.
Refined Reservoir Routing (RRR) and Its Application to Atmospheric Carbon Dioxide Balance. *Water*. 2024; 16(17):2402.
https://doi.org/10.3390/w16172402

**Chicago/Turabian Style**

Koutsoyiannis, Demetris.
2024. "Refined Reservoir Routing (RRR) and Its Application to Atmospheric Carbon Dioxide Balance" *Water* 16, no. 17: 2402.
https://doi.org/10.3390/w16172402