MOTIVATION AND EXAMPLES
The amount of radiocarbon in atmospheric carbon dioxide (14CO2) is of fundamental importance to inference of the budget of radiocarbon and atmospheric CO2 and of natural and anthropogenic changes in both quantities; here the term “amount” denotes any measure of the quantity of the substance under consideration. Historically the use of radiocarbon to infer the extent of anthropogenic perturbation of the amount of atmospheric CO2 goes back to Suess (Reference Suess1955), who famously demonstrated the decrease in 14C/C of atmospheric 14CO2 with time over the nineteenth century as inferred from dendrochronologically dated wood samples; this decrease was attributed to emissions into the atmosphere of 14C-free CO2 from fossil fuel combustion. Although Suess recognized that the magnitude of the perturbation of 14C/C would be decreased by exchange of atmospheric CO2 with the oceans (what is now characterized as “disequilibrium isotope flux”), he underestimated by an order of magnitude the extent of increase of what he characterized as “contamination of Earth’s atmosphere by artificial CO2.” Nonetheless that work set the scene for use of 14CO2 to infer the extent of such exchange.
Historically and at present the 14C/C ratio of atmospheric CO2 has been reported most commonly as Δ14CO2, the depletion or enrichment of 14C in sample relative to an internationally accepted absolute reference standard, normalized for mass-dependent isotope fractionation and corrected for decay from date of sampling (atmospheric CO2) or of growth (tree rings), i.e., the quantity denoted Δ by Stuiver and Polach (Reference Stuiver and Polach1977), commonly presented in parts per thousand or “per mil.” The use of per mil notation reflects the generally small differences of fractionation- and decay-corrected relative 14C/C ratio in different ambient carbon reservoirs and the gradients within these reservoirs that result from relatively small perturbations together with rather rapid redistribution of 14C via air-sea gas exchange, photosynthesis and respiration, and physical mixing. Much larger differences and gradients occur when the system is substantially perturbed. Such perturbations have included the large excess of 14CO2 originating from above-ground testing of nuclear weapons in the 1950s and early 1960s (commonly referred to as “bomb radiocarbon”) and the anomalously high natural production of 14C and dramatically altered carbon cycle dynamics during the last ice age (Hughen et al. Reference Hughen, Lehman, Southon, Overpeck, Marchal, Herring and Turnbull2004).
The decrease in Δ14C of atmospheric CO2 that had been discovered by Suess (Reference Suess1955) from just a handful of tree-ring samples was examined much more systematically by Stuiver and colleagues (Stuiver and Quay Reference Stuiver and Quay1981; Stuiver et al. Reference Stuiver, Reimer and Braziunas1998), again using dendrochronologically determined ages and is presented as Δ14CO2, as is conventional, in Figure 1a. Figure 1b shows the time series of the amount of atmospheric 14CO2 in the global atmosphere as mole fraction relative to dry air, x 14CO2. Figure 1c shows the time series of the mole fraction of CO2 in Earth’s atmosphere x CO2. Here the designation of the several time series in Figure 1 as representative of the global atmosphere should be qualified as approximate, as the quantities Δ14CO2, x CO2, and x 14CO2, are obtained from measurements at different locations and thus do not take into account slight spatial variation.
Figure 1 Alternative presentations of the amount of 14CO2 in the global atmosphere over the first half of the twentieth century: a, as departure (in units of parts per thousand, “per mil,” ‰) of the ratio of 14CO2 to total CO2 in the atmosphere corrected for fractionation and year of growth from the ratio of 14C to C in the absolute standard, Δ14CO2 (samples from Douglas Fir and Noble Fir trees from the US Pacific Northwest (43°7'–47°46'N, 121°45'–124°06'W), and an Alaskan Sitka spruce tree (58°N, 153°W); data tabulated in Stuiver et al. Reference Stuiver, Reimer and Braziunas1998); and b, as mole fraction (mole 14CO2 per mole of dry air), x 14CO2; the unit amol/mol denotes part per 1018. Red points denote values calculated with observation-derived δ13CO2 (fit to data of Francey et al. (Reference Francey, Allison, Etheridge, Trudinger, Enting, Leuenberger, Langenfelds, Michel and Steele1999), Appendix B, Figure B2); green points denote values calculated for δ13CO2 taken as constant, –7‰. c, Mole fraction of atmospheric CO2 (mole CO2 per mole of dry air), x CO2, in units of parts per million, ppm (data from Law Dome, Antarctica; Etheridge et al. Reference Etheridge, Steele, Langenfelds, Francey, Barnola and Morgan1996). Because of limited spatial coverage of the measurements the quantities shown should be considered only approximate global averages.
Conversion of measured Δ14CO2 to x 14CO2 is quite straightforward by Eq. (1),
$${x_{14{\rm{CO2}}}} = f{\left[ {{{1 + {\delta ^{13}}{\rm{C}}{{\rm{O}}_2}} \over {1 - 0.025}}} \right]^2}(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}){x_{{\rm{C}}{{\rm{O}}2}}},$$
provided that the associated CO2 mole fraction, x CO2, is also known, as developed in Appendix A; the factor f = 1.176 × 10–12. The conversion is also weakly dependent on δ13CO2 to account for mass-dependent fractionation. The mole fraction x 14CO2 is entirely analogous to the commonly used mole fraction of CO2 itself, x CO2, typically expressed as ppm ≡ µmol mol–1, and to mole fractions of other trace gases in the global atmosphere (ppm, ppb etc.), all relative to the amount of dry air. In view of the magnitude of x 14CO2 it would seem convenient to express x 14CO2 in the unit amol mol–1, where amol (attomole) is 10–18 mol. This measure of the amount of 14CO2 is denoted here as an absolute measure, as distinguished from the relative measure, i.e., Δ14CO2, in which the amount of 14CO2 is expressed relative to the amount of CO2. The mole fraction of 14CO2 (or any other gas) in Earth’s atmosphere is readily converted to moles by means of the essentially constant amount of air in the dry global atmosphere (1.765 × 1020 moles, uncertain to 0.5%; Prather et al. Reference Prather, Holmes and Hsu2012) or to molecules, by further use of the Avogadro constant.
The time series for Δ14CO2 and x 14CO2 in Figures 1a and b show an important qualitative difference between these two measures of the amount of 14CO2 in Earth’s atmosphere. In contrast to the familiar decrease of Δ14CO2 with time (Figure 1a), x 14CO2 yields a very different picture (Figure 1b), that of a systematic increase. The reason for the qualitatively different picture in the time series is the substantial increase of total atmospheric CO2 over the same time period (Figure 1c). The relative increase in CO2 mole fraction x CO2 over this time period, ∼5%, outweighs the relative decrease in 1 + Δ14CO2 [i.e., 1 + Δ14CO2(‰)/1000(‰)], ∼25‰ (Stuiver and Polach Reference Stuiver and Polach1977) or ∼2.5%, the measure of 14CO2 amount that enters into the calculation of absolute amount, resulting in the increase in x 14CO2 of ∼2.8%. The strong influence of the increase of x CO2, on x 14CO2, is apparent in the near congruence of the time profiles of these quantities. A strength of x 14CO2 as a measure of 14CO2 amount is that as an absolute measure of amount as opposed to a relative measure, that is, relative to the amount of atmospheric CO2, it is not influenced by the increase of CO2 amount over this time period.
Emission of fossil-fuel CO2 over the first half of the twentieth century has affected the evolution of both Δ14CO2 and x 14CO2. Δ14CO2 (or 14C/C ratio) has decreased mainly because of the increase in the atmospheric inventory of 14C-free CO2 from fossil fuel combustion. In contrast, x 14CO2 has increased mainly because the increase in 14C-free CO2 in the atmosphere has induced isotopic exchange with carbon in the 14C-containing ocean and terrestrial reservoirs that has resulted in net transfer of 14CO2 into the atmosphere, resulting in increase in x 14CO2 over this period. Here it might also be noted that x 14CO2 is only weakly dependent on δ13CO2, the dependence manifested as the difference between the red points in Figure 3b calculated for observation-derived values of δ13CO2, and the green points calculated for δ13CO2 taken as constant, –7‰.
A second example of the qualitative difference between time series of radiocarbon amount expressed as Δ14CO2 and x 14CO2 is manifested in Figure 2 for the period during and after the introduction of excess radiocarbon from above-ground nuclear weapons testing. This testing resulted in a large increase in atmospheric 14CO2 over the time period 1955–1964 (e.g., Levin et al. Reference Levin, Kromer, Schoch-Fischer, Bruns, Münnich, Berdau, Vogel and Münnich1985, Reference Levin, Naegler, Kromer, Diehl, Francey, Gomez-Pelaez, Steele, Wagenbach, Weller and Worthy2010; Turnbull et al. Reference Turnbull, Mikaloff Fletcher, Ansell, Brailsford, Moss, Norris and Steinkamp2017). Following nearly complete cessation of testing in 1964 the amount of atmospheric 14CO2 rather rapidly decreased, mainly because of isotopic exchange with other reservoirs in the carbon system. Δ14CO2 exhibits a monotonic decrease from its peak value in the early 1960s that continues up to the present time. Also shown in the figure is x 14CO2, as calculated by Eq. (1). In contrast to Δ14CO2, x 14CO2 reached a minimum around 2000, increasing thereafter. This contrast in the time series of the relative and absolute amounts of Δ14CO2 versus x 14CO2 over this time period, presented recently also by Andrews (Reference Andrews2020) and Andrews and Tans (Reference Andrews and Tans2022), highlights the distinction between these two measures of the amount of 14CO2 in Earth’s atmosphere.
Figure 2 Alternative presentations of the amount of 14CO2 in the global atmosphere over the second half of the twentieth century to the present, as departure of the ratio of 14CO2 to total CO2 in the atmosphere from that ratio in the absolute standard, Δ14CO2, blue, left axis (Hua et al. Reference Hua, Turnbull, Santos, Rakowski, Ancapichún, De Pol-Holz, Hammer, Lehman, Levin, Miller, Palmer and Turney2022); and as mole fraction relative to dry air x 14CO2, red, right axis. Values of x CO2 needed to calculate x 14CO2 are from measurements in ice cores at Law Dome, Antarctica and air at Cape Grim, Tasmania (Etheridge et al. Reference Etheridge, Steele, Langenfelds, Francey, Barnola and Morgan1996) and from measurements in air (Keeling et al. Reference Keeling, Bacastow, Bainbridge, Ekdahl, Guenther, Waterman and Chin1976, Reference Keeling, Piper, Bacastow, Wahlen, Whorf, Heimann and Meijer2001 as updated, and Ballantyne et al. Reference Ballantyne, Smith, Anderegg, Kauppi, Sarmiento, Tans, Shevliakova, Pan, Poulter, Anav and Friedlingstein2017 as updated by Dlugokencky and Tans Reference Dlugokencky and Tans2018) as tabulated by Le Quéré et al. (Reference Le Quéré2018). Values of δ13CO2 used to calculate x 14CO2 are from a linear fit to data of Francey et al. (Reference Francey, Allison, Etheridge, Trudinger, Enting, Leuenberger, Langenfelds, Michel and Steele1999), shown in Appendix B, Figure B1; also shown, larger green markers, right axis, are values of x 14CO2 calculated with δ13CO2 taken as constant, –7‰. Because of limited spatial coverage of the measurements the quantities shown should be considered approximate rather than true global averages.
Figure 3 Comparison of alternative presentations of atmospheric 14CO2 amount and controlling quantities in Northern and Southern Hemisphere summers. a, Δ14CO2, at Niwot Ridge (NWR, Colorado, USA) for summer months (May-August) in the NH and at Baring Head (BHD, New Zealand) in the SH (November-February), similar to Hua et al. (Reference Hua, Turnbull, Santos, Rakowski, Ancapichún, De Pol-Holz, Hammer, Lehman, Levin, Miller, Palmer and Turney2022). b, δ13CO2. c, x CO2. d, x 14CO2; e, difference in Δ14CO2 and x 14CO2 between the BHD and NWR sites.
As shown in Figure 2, Δ14CO2 is now decreasing below zero, the approximate value in the preindustrial atmosphere. Δ14CO2 can be expected to continue to decrease because of continued emissions of 14C–free CO2 from fossil fuel combustion. In contrast, x 14CO2, already increasing subsequent to about year 1995, would be expected to continue to increase, mainly because isotopic equilibration results in net transfer of 14C from the ocean and the terrestrial biosphere into the atmosphere in response to the increasing amount of 14C–free atmospheric CO2, together with slight emissions of 14CO2 from nuclear power production.
A third example deals with differences in amounts of atmospheric 14CO2 between the Northern and Southern Hemispheres as quantified by x 14CO2 versus Δ14CO2. The differences in Δ14CO2 were examined by Levin et al. (Reference Levin, Naegler, Kromer, Diehl, Francey, Gomez-Pelaez, Steele, Wagenbach, Weller and Worthy2010) and Graven et al. (Reference Graven, Guilderson and Keeling2012), with the finding that subsequent to about year 2000, Δ14CO2 in the Southern Hemisphere (SH) systematically exceeded that in the Northern Hemisphere (NH). Examining hemispheric and sub-hemispheric measurements in the summer months in each hemisphere, Hua et al. (Reference Hua, Turnbull, Santos, Rakowski, Ancapichún, De Pol-Holz, Hammer, Lehman, Levin, Miller, Palmer and Turney2022) similarly found that over this time period Δ14CO2 in the SH was systematically greater than in the NH. The data examined here in Figure 3 for the summer months (in the two hemispheres) of years 1990–2020 at specific locations, Niwot Ridge (NWR, Colorado, USA; Lehman et al. Reference Lehman, Miller, Wolak, Southon, Tans, Montzka, Sweeney, Andrews, LaFranchi, Guilderson and Turnbull2013, updated) in the NH and Baring Head (BHD, New Zealand; Turnbull et al. Reference Turnbull, Mikaloff Fletcher, Ansell, Brailsford, Moss, Norris and Steinkamp2017, updated) in the SH, likewise show systematically greater Δ14CO2 in the SH, Figure 3a. However, a different picture emerges for x 14CO2, Figure 3d (for some years for which x 14CO2 could not be derived from BHD measurements, measurements at Cape Grim Observatory, Tasmania, Australia (Levin et al. Reference Levin, Kromer and Francey1996, Reference Levin, Kromer and Francey1999, Reference Levin, Kromer, Steele and Porter2011; Turnbull et al. Reference Turnbull, Mikaloff Fletcher, Ansell, Brailsford, Moss, Norris and Steinkamp2017) were substituted). In contrast to Δ14CO2, x 14CO2 was comparable to or even slightly greater at the NH site than at the SH site. Figure 3e presents differences between Δ14CO2 and x 14CO2 at the two sites; as the summertime measurements are staggered by half a year, the differences were taken for values obtained by interpolation. These differences explicitly show that whereas summertime Δ14CO2 at BHD systematically exceeds that at NWR, values of x 14CO2 at BHD are essentially the same as, or less than those at NWR over this time period; that is, a difference, even in sign, between the interhemispheric differences of Δ14CO2 versus x 14CO2.
PRIOR PRESENTATION OF ABSOLUTE AMOUNT OF ATMOSPHERIC 14CO2
Although the amount of atmospheric 14CO2 has most commonly been presented as Δ14CO2, there are more than a few precedents in which this amount has been presented as an absolute quantity (number of molecules or moles in the global atmosphere). Much of this usage has been to examine the disposition of bomb radiocarbon, where use of absolute amount permits comparison of the total bomb yield of 14C with the amounts in the global atmosphere, the world ocean, and the terrestrial biosphere, (e.g., Levin and colleagues (Hesshaimer et al. Reference Hesshaimer, Heimann and Levin1994; Naegler and Levin Reference Naegler and Levin2006: their Figure 4a reproduced here as Figure 4); Broecker et al. Reference Broecker, Sutherland, Smethie, Peng and Ostlund1995; Lassey et al. Reference Lassey, Enting and Trudinger1996; Joos and Bruno Reference Joos and Bruno1998; Caldeira et al. Reference Caldeira, Rau and Duffy1998; Key et al. Reference Key, Kozyr, Sabine, Lee, Wanninkhof, Bullister, Feely, Millero, Mordy and Peng2004; Peacock Reference Peacock2004; Mouchet 2013) or comparison just of amounts in the several reservoirs (Graven Reference Graven2015: figure S1).
Figure 4 Evolution of the inventories of anthropogenic radiocarbon in the stratosphere, the troposphere, the world ocean, and the terrestrial biosphere, as given by Naegler and Levin (Reference Naegler and Levin2006); units are 1026 atoms (left ordinate) and kmol (right ordinate). Solid line denotes estimated total production amount based on the Yang et al. (Reference Yang, North and Romney2000) compilation of atmospheric nuclear detonation. Symbols denote measurements; for identification see the original paper. Curves denote modeled amounts in the several reservoirs. Reproduced with permission of the American Geophysical Union.
In another application Roth and Joos (Reference Roth and Joos2013, their Figure 2, modified and presented here as Figure 5), examined the rate of natural production of 14C and the disposition of this 14C among the several geophysical reservoirs and its distribution under the influence of a changing carbon cycle. Those investigators presented the amount of 14CO2 in the global atmosphere over the past 21 kyr both as Δ14CO2 and as absolute amount (moles of 14CO2 in the global atmosphere). Comparison of the two time series shows that the absolute amount over this time period was more or less constant, in contrast to Δ14CO2, which exhibited a substantial systematic decrease. The decrease in Δ14CO2 must therefore be ascribed largely to increase in the amount of atmospheric CO2; this is confirmed by the rather close conformance of Δ14CO2 and CO2 mole fraction x CO2, the latter shown on an inverted scale. The relative constancy of 14CO2 inventory that results from the cancellation of these trends has also been noted by Köhler et al. (Reference Köhler, Adolphi, Butzin and Muscheler2022; Figure 1c), who stress the central role of the trend of the amount of atmospheric CO2 as the source of the trend in Δ14CO2 over this time period.
Figure 5 a. Reconstructed Δ14C of atmospheric CO2 (blue, left axis) and absolute inventory of atmospheric radiocarbon (red, right axis) over the past 21 kyr, modified from Figure 2 of Roth and Joos (Reference Roth and Joos2013). Added to the figure, far left axis (inverted scale), is mole fraction of atmospheric CO2 inferred from the EPICA Dome (Antarctica) ice core (Bereiter et al. Reference Bereiter, Eggleston, Schmitt, Nehrbass-Ahles, Stocker, Fischer, Kipfstuhl and Chappellaz2015), green, and from multiple ice cores (MacFarling Meure Reference MacFarling Meure2004; MacFarling Meure et al. Reference MacFarling Meure, Etheridge, Trudinger, Steele, Langenfelds, van Ommen, Smith and Elkins2006; Etheridge et al. Reference Etheridge, Steele, Langenfelds, Francey, Barnola and Morgan1996 as tabulated by Etheridge et al. Reference Etheridge2010), brown, again on an inverted scale. b. Last 600 years of the several time series, denoted by cyan rectangle in a, with 5-fold expansion of horizontal scale.
Although not explicitly noted by Roth and Joos, the increase in the absolute amount of 14CO2 inventory over the first half of the twentieth century (shown above in Figure 1b) is evident also in the most recent part of the data presented by those investigators (50 to 0 year cal. BP), shown on an expanded time scale in Figure 5b, along with the increase in mole fraction of atmospheric CO2 and near constancy of Δ14CO2 over this time period.
Reporting amounts of radiocarbon in terms of absolute amount rather than as departure from a standard is not uncommon in other contexts, for example the amount of 14CO in the global atmosphere (Jöckel et al. Reference Jöckel, Brenninkmeijer, Lawrence, Jeuken and van Velthoven2002; Manning et al. Reference Manning, Lowe, Moss, Bodeker and Allan2005; Hmiel et al. Reference Hmiel, Petrenko, Dyonisius, Buizert, Smith, Place, Harth, Beaudette, Hua, Yang and Vimont2020).
DISCUSSION
Based on the examples presented it is suggested here that presentation of 14CO2 amount as x 14CO2 readily allows assessment of time trends or spatial gradients of 14CO2 amount and that such presentation, alongside Δ14CO2, the directly measured and more widely presented measure of 14CO2 amount in the atmosphere, may lend additional insight. Importantly, the roughly constant value of x 14CO2 between 1995 and 2010 shown in Figure 2 shows that in these years loss of 14CO2 from the atmosphere was closely balanced by addition, whereas earlier, the amount of 14CO2 in the atmosphere had been decreasing, and at present that amount is increasing. This situation is not at all evident in the time trace of Δ14CO2, which shows monotonic decrease over this period.
On the other hand, presentation of atmospheric 14CO2 data as Δ14CO2 readily permits examination of time series of the departure from isotopic equilibrium between atmospheric 14CO2 and 14C in other reservoirs. For example, Andrews et al. (Reference Andrews, Siciliano, Potts, DeMartini and Covarrubias2016) and Wu et al. (Reference Wu, Fallon, Cantin and Lough2021) compared time series of Δ14CO2 with time series of Δ14C in the oceanic mixed layer (ML), the latter as determined from measurements of Δ14C in dated coral and fish otolith samples and directly in ML water samples in the North and South Pacific Gyres.
A key motivation of study of the amount of 14CO2 in the global atmosphere is qualitative and quantitative understanding of the processes that govern the changes in this amount and the amounts of 14C in other reservoirs of the biogeosphere. In some contexts process understanding is enhanced by presenting and examining the amount of atmospheric 14CO2 as absolute amount, as quantified by atoms, moles, or mole fraction in air (x 14CO2), in addition to relative amount, as quantified by departure of isotopic ratio from the absolute standard, in Δ14C units. As noted above (Figure 4) examination of absolute amount permits immediate comparison of the amounts of 14C in different reservoirs of the biogeosphere. This comparison has been of great value in understanding the disposition of the bomb perturbation. Likewise, examination of time series of x 14CO2 yields a picture of the temporal evolution of the amount of 14CO2 in the global atmosphere that is qualitatively different from that exhibited by time series of Δ14CO2, which is influenced by changes in amounts of both 14CO2 and total CO2.
In other contexts expressing the amount of 14C in Δ14C units also promotes process-level understanding. As the relative amount of 14C in a sample is based on the activity of the sample relative to that of a standard, or its equivalent determined by accelerator mass spectrometry (AMS), it is appropriate that for geochemical samples this amount be reported as the fractionation- and decay-corrected Δ14C (the quantity Δ of Stuiver and Polach Reference Stuiver and Polach1977; Stuiver Reference Stuiver1980) or as fractionation-corrected fraction modern (the quantity F14C of Reimer et al. Reference Reimer, Brown and Reimer2004; Millard Reference Millard2014). Under the condition of isotopic equilibrium, the several reservoirs of the biogeosphere (e.g., atmosphere, ocean, terrestrial biosphere) will all, in the absence of appreciable decay, exhibit the same Δ14C values or will exhibit Δ14C values that differ by a simple expression that accounts for decay and turnover time. A major driver of changes in atmospheric radiocarbon is thus isotopic disequilibrium with the oceans and the land biosphere. For example, how might the increase in the mole fraction of atmospheric 14CO2 that arises from adding radiocarbon-free CO2 to the atmosphere, as in Figure 1b, be understood? Using Δ14C units provides a straightforward qualitative answer: the increase results from upsetting the preindustrial isotopic equilibrium causing, e.g., ecosystem respiration to be “hotter” than photosynthesis, i.e., characterized by a greater Δ14C value. The disequilibrium flux and concomitant impact on Δ14CO2 stop when the Δ14C values converge between the reservoirs. Similarly, how might the controls on the interhemispheric gradient or the long-term trend in radiocarbon be understood? The answer is again relatively simple from a Δ14C perspective: the dominant controls are fossil-fuel burning and the disequilibrium fluxes. Additionally, much of the variability imposed by the land biosphere net fluxes (i.e., net ecosystem exchange) conveniently drops out because of the 13C normalization. Hence, because of the important contribution from disequilibrium fluxes, an understanding of the controls on the mole fraction of atmospheric radiocarbon sensibly starts with understanding of the controls on Δ14C.
As x 14CO2 obtained from measured Δ14CO2 by Eq. (1) is proportional to x CO2, it is essential that the value of x CO2 used in the conversion be specified. Although, as shown in Figure 2, x 14CO2 calculated with δ13C taken as constant –7‰ is quite accurate, it may be necessary in precise work to use the concurrently measured value of δ13C, Figure 3. Hence it is recommended that when x 14CO2 is presented, the value of δ13C used in the conversion be presented. The uncertainty associated with x 14CO2 is readily determined from propagation of uncertainties in Eq. (1).
At the end of the day, is one means of expressing amounts of 14CO2, “better” or more suitable than the other? It is not clear that there is an unambiguous answer. It would seem that for studies examining the consequences of addition of fossil fuel CO2 to the atmosphere, use of Δ14C values might be more appropriate, because it is not necessary to know the amount of CO2 precisely (which amount may not be known, particularly when 14C content is derived from plant material or NaOH absorption samples). On the other hand, for modeling studies, absolute amounts (x CO2, x 13CO2, and x 14CO2) might be more suitable because these quantities are measures of what actually gets transported. Also, especially in time series of the amount of atmospheric 14CO2, absolute amount (mole fraction) presents the actual change in the amount of 14CO2 in the atmosphere as opposed to relative amount (isotope ratio), which can be influenced more by change in the amount of total CO2 than by change in the amount of 14CO2.
AUTHOR CONTRIBUTIONS
This discussion paper was stimulated by discussion among several authors (DA, RFK, HAJM, SES, JCT) arising out of comments (Andrews and Tans Reference Andrews and Tans2022; Schwartz et al. Reference Schwartz, Keeling, Meijer and Turnbull2022) on a publication that had misinterpreted the rate of decrease of atmospheric bomb 14CO2 as a measure of the rate of removal of anthropogenic CO2 from the biogeosphere. SES prepared the initial draft and figures and led revisions. QH provided data for Figure 3. All authors contributed to preparation and revision of the manuscript.
ACKNOWLEDGMENTS
We thank the Editors and three Reviewers for valuable comments and suggestions. Work by SES was conducted mainly while at Brookhaven National Laboratory and supported in part by the US Department of Energy under Contract No. DE-SC0012704; views expressed here do not necessarily represent the views of BNL or DOE.
SUPPLEMENTARY MATERIAL
To view supplementary material for this article, please visit https://doi.org/10.1017/RDC.2024.27.
APPENDIX A Relation of measured activity of atmospheric CO2, reported as atmospheric Δ14CO2, and absolute amount of atmospheric 14CO2
The expression for conversion of measured Δ14CO2 to x 14CO2 given as Eq. (1) of the main text and presented earlier by Karlén et al. Reference Karlén, Olsson, Kållburg and Kilicci1964 and Stuiver Reference Stuiver1980,
$${x_{14{\rm{CO2}}}} = f{\left[ {{{1 + {\delta ^{13}}{\rm{C}}{{\rm{O}}_2}} \over {1 - 0.025}}} \right]^2}(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}){x_{{\rm{C}}{{\rm{O}}_2}}},$$
is developed here. The quantities Δ and δ, representing departure of isotopic ratio from that of a standard are given as decimal fractions; e.g., for Δ14CO2 = 1‰, the value in the conversion expression is 0.001. The factor
$$f \equiv {{{M_{\rm{C}}}{A^{{\rm{ABS}}}}} \over {\lambda {N_A}}} = 1.176 \times {10^{ - 12}}$$
is a constant; M C is the molecular weight of carbon, AABS is the activity of the absolute standard, λ is the decay constant of 14C, and N A is the Avogadro constant.
The relation between the amount of 14CO2 in a sample and the relative activity of 14C in the sample reported as Δ14C follows the conventions of Stuiver and Polach (Reference Stuiver and Polach1977) and Stuiver (Reference Stuiver1980). These conventions, which were developed initially for reporting Δ14C from activity measurements, have since been adapted for application to the now more widely used accelerator mass spectrometry (AMS) approach (Donahue et al. Reference Donahue, Linick and Jull1990; Mook and van der Plicht Reference Mook and Van der Plicht1999). From Stuiver (Reference Stuiver1980, Eq. 1) the depletion or enrichment of 14C relative to an absolute standard corrected for mass-dependent fractionation is defined as
$${\rm\Delta ^{14}}{\rm{C}} = (1 + {\delta ^{14}}{\rm{C}})\left[ {{{1 - 0.025} \over {1 + {\delta ^{13}}{\rm{C}}}}} \right] - 1,$$
where δ13C and δ14C denote the fractional depletion or enrichment with respect to the respective standards. Combining Eq. (A2) with the expression relating δ 14C to the specific activities of the absolute standard and the air sample, A ABS and A S, respectively (Stuiver and Polach Reference Stuiver and Polach1977)
$${\delta ^{14}}{\rm{C}} = {{{A^S}{e^{\lambda (y - x)}}} \over {{A^{{\rm{ABS}}}}}} - 1,$$
where A ABS= 0.226 Bq g(C) –1 (Mook and van der Plicht Reference Mook and Van der Plicht1999), λ is the decay constant of 14C (3.8332 × 10–12 s–1) corresponding to a geophysical half-life of 5730 ± 40 years (Godwin Reference Godwin1962) and x and y denote year of growth (for tree-ring samples) and year of measurement, respectively, yields specific activity of an air sample A S in terms of Δ14CO2 and δ13CO2 as
$${A^{\rm S}} = {A^{{\rm{ABS}}}}(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}){\left[ {{{1 + {\delta ^{13}}{\rm{C}}{{\rm{O}}_2}} \over {1 - 0.025}}} \right]^2} = g{A^{{\rm{ABS}}}}(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}).$$
The factor g in Eq. (A4), defined as
$$g = {\left[ {{{1 + {\delta ^{13}}{\rm{C}}{{\rm{O}}_2}} \over {1 - 0.025}}} \right]^2} \approx 1.0373,$$
accounts for fractionation, where the numerical value (1.0373) is for δ13CO2 taken as –7‰. As that value of δ13CO2 is representative for the past several centuries (e.g., Francey et al. Reference Francey, Allison, Etheridge, Trudinger, Enting, Leuenberger, Langenfelds, Michel and Steele1999; Levin et al. Reference Levin, Naegler, Kromer, Diehl, Francey, Gomez-Pelaez, Steele, Wagenbach, Weller and Worthy2010; Graven et al. Reference Graven, Allison, Etheridge, Hammer, Keeling, Levin, Meijer, Rubino, Tans, Trudinger and Vaughn2017), the factor g can, to good approximation, be treated as constant, as seen by the near equality of the red and green data points in Figures 1b and 2 of the main text. For the range of δ13CO2 from preindustrial time to the present, –6.4‰ to –8.4‰, the corresponding range of g is 1.0385 to 1.0343, Appendix B), supporting the use of a constant value of g if measurements of δ13CO2 are not available. For precise work, such as comparison of
$x_{14{\rm{CO}}2}^{{\rm{air}}}$
at different locations (e.g., Figure 3), variation of g would need to be taken into account.
Noting that the number of 14C atoms in a given sample
$n_{14{\rm{C}}}^{\rm{S} }= {A^{\rm{S}}m_C^{\rm{S}}/\lambda}$
, where
$m_C^{\rm{S}}$
is the mass of carbon in the sample and that the number of carbon atoms in the sample
$n_C^{\rm{S}} = m_C^{\rm{S}}{N_A}/{M_C},$
where N
A is the Avogadro constant (6.022 × 1023 mol–1) and M
C is the molecular weight of carbon (12.011 g mol–1) yields
$$n_{{\rm{14C}}}^{\rm{S}} = {{{M_C}{A^{{\rm{ABS}}}}} \over {\lambda {N_A}}}g(1 + {\rm\Delta ^{14}}{{\rm{C}}^{\rm{S}}})n_{\rm{C}}^{\rm{S}} = fg(1 + {\rm\Delta ^{14}}{{\rm{C}}^{\rm{S}}})n_{{\rm{C}},}^{\rm{S}}$$
where the factor f, Eq. (A1), is a constant. Specializing to the amount of 14CO2 in the global atmosphere, as mole fraction in dry air is proportional to number of molecules,
$${x_{1{\rm{4CO2}}}} = fg(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}){x_{{\rm{C}}{{\rm{O}}_2}}} = f{\left[ {{{1 + {\delta ^{13}}{\rm{C}}{{\rm{O}}_2}} \over {1 - 0.025}}} \right]^2}(1 + {\rm\Delta ^{14}}{\rm{C}}{{\rm{O}}_2}){x_{{\rm{C}}{{\rm{O}}_2}}},$$
as has been given previously, e.g., Karlén et al. (Reference Karlén, Olsson, Kållburg and Kilicci1964) and Stuiver (Reference Stuiver1980).
For δ13CO2 = –7‰ the product fg ≈ 1.220 × 10–12, the inverse of which was presented by Levin et al. (Reference Levin, Naegler, Kromer, Diehl, Francey, Gomez-Pelaez, Steele, Wagenbach, Weller and Worthy2010); this quantity is accurate to about 1% (Stuiver Reference Stuiver1980). For present (2022) dry-air mole fraction of atmospheric CO2, x CO2, ∼ 420 × 10–6 (420 parts per million, ppm) and Δ14CO2 ≈ 0, the mole fraction of atmospheric 14CO2, x 14CO2, is approximately 512 × 10–18 (512 amol mol–1), where amol mol–1 denotes attomoles (10–18 mol) per mole.
In addition to the dominant dependence of x
14CO2 on
${\rm\Delta ^{14}}{\rm{C}}$
and x
CO2, the fractionation factor g (Appendix A, Eq. A5), which is inferred from the amount of 13CO2 in the sample δ13CO2, exhibits a slight dependence on this amount, Figure B1. Although this dependence is slight, the systematic decrease in g of about 0.4% over the industrial era, Figure B2, should be accounted for in evaluation of x
14CO2 in precise work, rather than simply using a constant value of δ13CO2.
Figure B2 a. Time dependence of δ13C of atmospheric CO2 as compiled by Francey et al. (Reference Francey, Allison, Etheridge, Trudinger, Enting, Leuenberger, Langenfelds, Michel and Steele1999), data points and associated uncertainties are from the Cape Grim Air Archive, firn at DE08-2, and cores DE08, DE08-2 and DSS, Law Dome, Antarctica; thin black curve denotes spline fit. Red points denote measurements by Scripps Institution of Oceanography (Keeling et al. Reference Keeling, Piper, Bacastow, Wahlen, Whorf, Heimann and Meijer2001; https://scrippsco2.ucsd.edu/assets/data/atmospheric/stations/flask_isotopic/monthly/monthly_flask_c13_spo.csv. Downloaded 2022-0811). Green and blue lines in panel a denote values of δ13C employed in evaluation of isotopic fractionation factor g (panel b) used in evaluation of x 14CO2 presented in Figures 1 and 2.
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