Potential Dependence of Global Warming on the Residence Time (RT

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Potential Dependence of Global Warming on the Residence Time (RT) in the Atmosphere of Anthropogenically Sourced Carbon Dioxide Robert H. Essenhigh* Department of Mechanical Engineering, The Ohio State UniVersity, Columbus, Ohio 43210 ReceiVed July 23, 2008. ReVised Manuscript ReceiVed March 6, 2009

The driver for this study is the wide-ranging published values of the CO2 atmospheric residence time (RT), τ, with the values differing by more than an order of magnitude, where the significance of the difference relates to decisions on whether (1) to attempt control of combustion-sourced (anthropogenic) CO2 emissions, if τ > 100 years, or (2) not to attempt control, if τ ∼ 10 years. This given difference is particularly evident in the IPCC First 1990 Climate Change Report where, in the opening policymakers summary of the report, the RT is stated to be in the range of 50-200 years, and (largely) on the basis of that, it was also concluded in the report and from subsequent related studies that the current rising level of CO2 was due to combustion of fossil fuels, thus carrying the, now widely accepted, rider that CO2 emissions from combustion should therefore be curbed. However, the actual data in the text of the IPCC report separately states a value of 4 years. The differential of these two times is then clearly identified in the relevant supporting documents of the report as being, separately (1) a long-term (∼100 years) adjustment or response time to accommodate imbalance increases in CO2 emissions from all sources and (2) the actual RT in the atmosphere of ∼4 years. As a check on that differentiation and its alternative outcome, the definition and determination of RT thus defined the need for and focus of this study. In this study, using the combustion/chemical-engineering perfectly stirred reactor (PSR) mixing structure or 0D box for the model basis, as an alternative to the more commonly used global circulation models (GCMs), to define and determine the RT in the atmosphere and then using data from the IPCC and other sources for model validation and numerical determination, the data (1) support the validity of the PSR model application in this context and, (2) from the analysis, provide (quasi-equilibrium) RTs for CO2 of ∼5 years carrying C12 and ∼16 years carrying C14, with both values essentially in agreement with the IPCC short-term (4 year) value and, separately, in agreement with most other data sources, notably, a 1998 listing by Segalstad of 36 other published values, also in the range of 5-15 years. Additionally, the analytical results also then support the IPCC analysis and data on the longer “adjustment time” (∼100 years) governing the long-term rising “quasi-equilibrium” concentration of CO2 in the atmosphere. For principal verification of the adopted PSR model, the data source used was the outcome of the injection of excess 14CO2 into the atmosphere during the A-bomb tests in the 1950s/1960s, which generated an initial increase of approximately 1000% above the normal value and which then declined substantially exponentially with time, with τ ) 16 years, in accordance with the (unsteady-state) prediction from and jointly providing validation for the PSR analysis. With the short (5-15 year) RT results shown to be in quasi-equilibrium, this then supports the (independently based) conclusion that the long-term (∼100 year) rising atmospheric CO2 concentration is not from anthropogenic sources but, in accordance with conclusions from other studies, is most likely the outcome of the rising atmospheric temperature, which is due to other natural factors. This further supports the conclusion that global warming is not anthropogenically driven as an outcome of combustion. The economic and political significance of that conclusion will be self-evident.

1. Introduction In fossil-fuel combustion studies related to global warming and climate change, the debatable relevance of anthropogenic sources of CO2 as a driver for global warming and the presumed corresponding need for control of those combustion emissions1-4 principally derives from three widely accepted but questionable assumptions. * To whom correspondence should be addressed. Fax: 614-292-3163. E-mail: [email protected]. (1) Halman, M. M.; Steinberg, M. Greenhouse Gas Carbon Dioxide Mitigation; Lewis Publishers, CRC Press, LLC: Boca Raton, FL, 1999. (2) Intergovernmental Panel on Climate Change. Policymakers summary. In Climate Change; Houghton, J. T., Jenkins, G. J., Ephraums, J. J., Eds.; The IPCC Scientific Assessment, Cambridge University Press: Cambridge, U.K., 1990.

The first assumption is that the rising concentration of CO2, particularly in the last 250 years, has been and is currently the primary (radiation) driver for the concurrent global warming,4,5 cf.6 The alternative proposition is the reverse: that it is the rising temperature that is the driver for the rising CO2.7-10 The second (3) Watson, R. T.; Rodhe, H.; Oeschger, H.; Siegenthaler, U. Greenhouse gases and aerosols. In Climate Change; The IPCC Scientific Assessment, Cambridge University Press: Cambridge, U.K., 1990; Chapter 1. (4) Shine, K. P.; Derwent, R. G.; Wuebbles, D. J.; Morcrete, J.-J. Radiative forcing of climate. In Climate Change; The IPCC Scientific Assessment, Cambridge University Press: Cambridge, U.K., 1990; Chapter 2. (5) Houghton, J. T.; Meira Filho, L. G.; Bruce, J.; Lee, H.; Callander, B. A.; Haites, E.; Harris, N.; Maskell, K. Radiative forcing of climate change and an evaluation of the IPCC IS92 emission scenarios. In Climate Change: 1994; IPCC, Cambridge University Press: Cambridge, U.K., 1995.

10.1021/ef800581r CCC: $40.75  2009 American Chemical Society Published on Web 04/01/2009

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remains in the atmosphere “indefinitely”,11,12 thus introducing the factor of residence time (RT), determined jointly by the CO2 “mixing structure” in the atmosphere and, predominantly, its rate of exchange between the atmosphere and its surroundings (principally vegetation and the sea). With an infinite or even “long-term” RT, it could then be responsible for the continuing rise in CO2, specifically in the last 250 years, although this then again requires neglect of the natural-sourced CO2 (Figure 1). Of these three assumptions, the second and third, particularly, identify the focus source of this study, which is the determination of the RT of carbon dioxide in the atmosphere as the basis for differentiating between an anthropogenic global warming because of a long-term RT or a naturally based warming if the RT is short, with long and short numerically defined, following as >100 years or ∼10 years, respectively. The political and economic significance of this differentiation is then whether or not to attempt to control anthropogenic CO2 emissions (as defined in the Kyoto and Bali Protocol procedures13-15), e.g., by carbon capture and sequestration, at costs generally estimated to exceed $1 trillion per year.16 The primary focus of the study is, therefore, the selection and test of an appropriate atmospheric mixing and exchangerate model (section 3) to analytically define and determine the CO2 RT. This is set first in the context of background data for numerical comparison, particularly, with reported RTs. 2. Background Context

Figure 1. Sankey diagram for carbon flow in 1990 through atmosphere, vegetation, and (4) ocean control volumes (CVs). Carbon-flow flux rates are in units of gigatons of C per year. The band width is proportional to quantity flowing. Capacity values are in units of gigatons of C. Data source(s) are from refs 1 and 2.

assumption is that the primary source for the rising atmospheric CO2 in that time period and currently is anthropogenic, because of the combustion of fossil fuels, although this ignores the separate but substantial CO2 contribution from natural sources, notably vegetation and the sea, schematically illustrated in the carbon-flow (Sankey) diagram of Figure 1 (data from refs 1 and 2). The third assumption is that the anthropogenic CO2

(6) Essenhigh, R. H. Prediction of the standard atmosphere profiles of temperature, pressure, and density with height for the lower atmosphere by solution of the (S-S) integral equations of transfer and evaluation of the potential for profile perturbation by combustion emissions. Energy Fuels 2006, 20, 1057–1067. (7) Kuo, C.; Lindberg, C.; Thompson, D. J. Coherence established between atmospheric carbon dioxide and global temperature. Nature 1990, 343, 709–714. (8) Ahlbeck, J. Increase of the atmospheric carbon dioxide concentration due to ocean warming. Abo Akademi University, Finland, 1999. (9) Essenhigh, R. H. Does CO2 really drive global warming? Chem. InnoVations 2001, 31, 44–46, 62-64, and 66-68. Re-published in Essenhigh, R. H. Does CO2 really drive global warming? Energy EnViron. 2001, 12 (4), 351–355. (10) Rorsch, A.; Courtney, R. S.; Thoenes, D. Global warming and the accumulation of carbon dioxide in the atmosphere. Energy EnViron. 2005, 16, 101–125.

2.1. RT Definition and Graphical Formulation. Defining the atmosphere as a control volume (CV), the RT as defined in standard formulation is the ratio of the total (carbon) content in the (atmospheric) CV divided by the input/output (carbon) flux rate2,3,17-19 (cf. Figure 11-3). As discussed more fully below, this RT definition, with certain qualifications, is also alternatively used in two of the papers2,3 in the 1990 Intergovernmental Panel on Climate Change (IPCC) First Climate Change Report, equivalently to define both a turnover time and a lifetime. At steady state, the input rate equals the output rate and, as seen in the Sankey diagram of Figure 1, at steady state, this definition gives a RT in the atmosphere of 5 years, as set out in more detail in section 2.3. When a time dependence is incorporated, the definition, as will be shown, also applies to (potentially significant) unsteady-state or transient conditions. Numerically, for the anthropogenic addition to the atmosphere to be significant, this requires a long-term RT, of at least 100 years. This is illustrated in Figure 2, which compares plots of rising CO2 concentration from 1750 for the historical recorded rise (line A3) (11) Revelle, R.; Munk, W. The carbon dioxide cycle and the biosphere. Energy and Climate: Studies in Geophysics; National Academy of Sciences: Washinton, D.C., 1977; pp 140-158. (12) Rotmans, J. IMAGE (Integrated Model to Assess the Greenhouse Effect); Kluwer Academic Publishers: Dordrecht, The Netherlands, 1990. (13) Morissey, W. A.; Justus, J. R. Issue Brief for Congress: IB89005: Global Climate Change. Congressional Research Service (CRS): Resources, Science, and Industry Division, Washington, D.C., March 2000; http:// www.cnie.org/nle/clim-2.html. (14) Bolin, B. Science 1998, 279, 330–331. (15) Parker, L.; Blodgett, J. Global Climate Change: Reducing Greenhouse GasessHow Much from What Baseline? Congressional Research Service, Environment and Natural Resources Policy Division, Report for Congress, Washington, D.C., March 1998; http://www.cnie.org/nle/clim13.html. (16) Mannne, A.; Richels, R. Buying Greenhouse Insurance: The Economic Costs of Carbon Dioxide Emission Limits; MIT Press: Cambridge, MA, 1992; cited in EPRI Journal, Dec 1992. (17) Thring, M. W. Science of Flames and Furnaces; John Wiley: New York, 1962. (18) Levenspiel, O. Chemical Reaction Engineering: An Introduction to Design of Chemical Reactors; Wiley: New York, 1962. (19) Nauman, E. B. Residence time theory. Ind. Eng. Chem. Res. 2008, 47, 3752–3766.

Anthropogenically Sourced Carbon Dioxide

Figure 2. Variation of CO2 concentration in the atmosphere with time (1750-2005). The left-hand scale is in units of parts per million of CO2 in air (ppmv), and the right-hand scale is in units of the weight of carbon in atmosphere (gigatons). For conversion factors, see Table 1. Line A, actual concentration (data source, ref 3; reference values are given at 1 year intervals but plotted at 5 year intervals for clarity); line B, calculated concentration (increase) from anthropogenic (combustion) CO2 supply, assuming infinite RT (cf. Figure 3; data source, refs 20 and 21); line B*, calculated concentration (increase) from anthropogenic (combustion) CO2 supply, assuming 100 year RT (cf. Figure 3; data source, refs 20 and 21); line C, calculated concentration (increase) from anthropogenic (combustion) CO2 supply, assuming 10 year RT (cf. Figure 3; data source, refs 20 and 21).

Figure 3. Variation of the anthopogenic carbon emission rate with time (1750-2005). Line X, actual rates (gigatons per year; data source, refs 20 and 21); line Y, net rate calculated from line A of Figure 1 (allowing for the return to sea and vegetation).

compared to the summation calculations from line X in Figure 3 based on reported emission rates20,21 for the same time period, assuming (for the lines in Figure 2) an infinite RT for calculating line B, a 100 year RT for line B*, and a 10 year RT for line C. As seen in the figure, the infinite and 100 year lines (B and B*) are (20) Marland, G.; Boden, T. A.; Andres, R. J. Global, regional, and national CO2 emissions. In Trends: A Compendium of Data on Global Change; Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, United States Department of Energy: Oak Ridge, TN, 2003. (21) Etheridge, D. M.; Steele, L. P.; Langenfelds, R. L.; Francey, R. J.; Barnola, J. M.; Morgan, V. I. Division of Atmospheric Research, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia, June 1998; http://cdiac.ornl.gov/ftp/trends/co2/lawdome.smoothed.yr20.

Energy & Fuels, Vol. 23, 2009 2775 substantially superimposed; the major change(s) are from 100 to ∼10 year RTs. 2.2. Graphical RT Data Comparisons. For the long-term RTs in Figure 2 (lines B and B*), the rise in CO2 is evidently somewhat higher than the actual rise but, being of the same order, that rise can, consequently, be broadly considered to be, at least primarily, anthropogenically sourced. However, as a (back) calculation from the actual concentration (line A in Figure 2) to obtain the equivalent emission rate, this is shown as line Y in Figure 3, and although also still of the same order as line X, this shows, nevertheless, a substantially lower effective emission rate. This line Y in Figure 3, although separately calculated here, also corresponds almost exactly to a “total flux” curve in a greenhouse gas paper on a current Wikipedia site.22 As the line shows, the “current” rate in 2005 is at about 4.5 gigatons/year, which, in comparison to the sum of the total (line X) reported rates of about 7.5 gigatons/year, is ∼40% less, where this lower value is broadly interpreted as a result of the continuing removal of CO2 from the atmosphere by returns to vegetation and the sea. In comparison and contrast, for the short term, 10 year, line C (in Figure 2), the anthropogenic supply is clearly inadequate to supply the given rise, in support of the alternative positions illustrated in Figure 1 that the RT is “short” (∼ (10 years); hence, the sources of the rising CO2 cannot be primarily from combustion and therefore must be, primarily, natural. Thus, comparing these long and short time extremes (as discussed more fully, following) the anthropogenic CO2-driver conclusion is reasonably supported at a RT of ∼100+ years but is essentially denied at the shorter time of ∼10 years. Numerical values from other sources are compared next. 2.3. Numerical RT Values. Figure 1 RT, of 5 years, uses the IPCC 1990 data as given in the figure,1,2 showing a carbon content in the atmosphere of 750 gigatons and an input/output rate of 150 gigatons/year. From these data, the RT, on the basis of the section 2.1 RT definition, is 750/150 ) 5 years, which self-evidently is short-term. Alternatively, using the currently (2009) higher carbon storage value of 900 gigatons and assuming the same input/output rate of 150 gigatons/year, this would increase the RT to 6 years. This increase from 5 to 6 years is still, nevertheless, of minor account. These values of 5 and 6 years are then in agreement, particularly, with the lower values of a data set of 36 studies from 1957 to 1992, summarized by Segalstad23 in 1998 and which reports RTs mostly in the range of 5-15 years. These values are also in close agreement with a RT for CO2 of 4 years in a table set (of different gases) reported by Mueller in 1971,24 and a corresponding figure of 4 years is also given in the text of the Watson IPCC paper (in section 1.2.1),3 although defined as the turnover time as already noted but nevertheless based on essentially the same RT definition. This short time is further emphasized by the associated commentary in the paper:3 “This means that on average it takes only a few years before a CO2 molecule in the atmosphere is taken up by plants or dissolved in the ocean”. This is followed up by the further comment “This short (4-year) time scale must not be confused with the time it takes for the atmospheric CO2 level to adjust to a new equilibrium if sources or sinks change”, with this “adjustment time” then estimated3 at 50-200 years using a “GCM” procedure (ref 3 and section 3.2.1, below). 2.4. Lifetime/Adjustment Time. Other literature citations of RT, however, are substantially more variable, with values ranging up (22) http://www.globalwarmingart.com/wiki/Image:Carbon_History_and_ Flux_Rev_png. (23) Segalstad, T. V. Carbon cycle modeling and the residence time of natural and anthropogenic atmospheric CO2: On the construction of the “Greenhouse Global Warming” dogma. In Global Warming: The Continuing Debate; Bate, R., Ed.; Cambridge University Press: Cambridge, U.K., 1998; pp 184-219. (24) Mueller, P. K. Detection and analysis of atmospheric pollutants. In Combustion Generated Air Pollution; Starkman, E. S., Ed.; Plenum Press: New York, 1971; pp 83-144.

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Table 1. Atmospheric Mass and Conversion Factors (cf. refs 6 and 8 Thickness of the atmosphere at constant density: Experimental data show the air density in the atmosphere falling approximately exponentially with height. At constant density with height, the equivalent height (Ho) containing the air would be 8.05 km (5 miles). Air mass ) area × density × Ho Surface area of earth ) 201 × 106 miles2 standard temperature and pressure air properties density temperature pressure 1.225 kg/m3 25 °C (288 K) 101.325 kPa 0.076 lbm/ft3 ∼60 °F (520 °R) ∼14.7 psi Mass of air in atmosphere ) 5.135 × 1018 kg; 5.66 × 1015 megatons Mass conversion factors for calculating CO2 and C in the atmosphere defining CO2 concentration as ppm(v) For CO2, gigatons of CO2 ) 8.6 × ppm(v); gigatons of C ) 2.35 × ppm(v) At 380 ppm(v), CO2 mass ) 3268 gigatons; C mass ) 891 gigatons (∼900 gigatons)

to 400 years (e.g., ref 25) and with possible extension to 1000 years.26 This long-term range in values also includes the (given) 50-200 years adjustment time data set;3 this also essentially identifies a significant source of controversy regarding anthropogenic warming, as the result, essentially, of confusion between definitions of RT, lifetime, and adjustment time. In the text of the 1990 IPCC Watson paper,3 the turnover time, equivalent to (section 2.1) definition to RT, is given as 4 years, as already cited, but in Table 1.1 of that paper and, correspondingly, in Table 1 by the Houghton et al.2 policymakers summary paper of the same report, the atmospheric lifetime, also generally identified as equivalent to RT and based on the same (section 2.1) definition, is stated to be 50-200 years, although defined as already noted in the text3 as the “adjustment time” (cf. section 3.5) and governing, as also noted, time to adjust to a new (higher or lower) CO2 (eq) value. This could also account for the (longer timed) “delayed return” to the original equilibrium CO2 concentration illustrated in Figure 1.2 in ref 3 derived from CO2-exchange analyses reported by MaierReimer and Hasselmann27 and Siegenthaler and Oeschger.28 This “long-term” 50-200 years lifetime, in association with the Hansen value of 1000 years,26 is then commonly cited as a basis for acceptance of anthropogenic global warming. This confusion in definition(s) between lifetime and adjustment time (and RT) thus, also, introduces a questionable interpretation of the relevant data by the IPCC, as examined further in section 3.5. For clarification, this again puts emphasis on the target of determining numerical values of RTs, requiring appropriate selection of a relevant atmospheric mixing model, as follows in the next section.

mental data showing different concentrations in the two hemispheres, with corresponding and time-dependent exchange between them, as amplified as follows in the next section. 3.1. Schematic Basis. In the atmospheric CV (Figure 4), the mixing structure determines the way in which the injected gases, e.g., CO2, are distributed through the CV with the mixing distribution structure, in addition determines the RT and/or residence time distribution (RTD), and then also possibly influences or controls both the carbon dioxide exchange rate (from and to the atmosphere) and, consequently, the radiative balance (cf. refs 4-6). The exchange process is the movement of CO2 from and to the atmospheric CV across the control surface (CS) interface between the atmosphere and its (land and sea) surroundings (cf. Figure 4A); this is the basis for defining the input/ output flux rates required for calculating the RT as defined in section 2.1 and illustrated on Figure 1. Figure 4B extends this scope, as noted, to allow for different CO2 concentrations in the northern and southern hemispheres. As further defined in the analytical development (section 4), the exchange rates across the CS are then determined by the differences in the CO2 concentration between the CV and the land/sea surroundings. These, again, are potentially influenced by the degree of complete or incomplete mixing, but in the analysis developed

3. Atmospheric Mixing Models The defined CV selected for formulation of an appropriate atmospheric mixing and exchange-rate model, as a basis for calculating the CO2 RT, is schematically illustrated in Figure 4, where Figure 4A defines the atmosphere as a single integral control volume (ICV)29-31 and Figure 4B redefines it as split into northern and southern hemispheres to allow for and accommodate experi(25) Hileman, B. Chem. Eng. News 1992, 70 (17), 7–19. (26) Hansen, J.; Sato, M.; Ruedy, R.; Kharecha, P.; Lacis, A.; Miller, R.; Nazarenko, K.; Lo, K.; Schmidt, G. A.; Russell, G.; Aleinov, I.; Bauer, S.; Baum, E.; Cairns, B.; Canuto, V.; Chandler, M.; Cheng, Y.; Cohen, A.; Del Genio, A.; Faluvegi, G.; Fleming, E.; Friend, A.; Hall, T.; Jackman, C.; Jonas, J.; Kelley, M.; Kiang, N. Y.; Koch, D.; Labow, G.; Lerner, J.; Menon, S.; Novakov, T.; Oinas, V.; Perlwitz, Ja.; Perlwitz, Ju.; Rind, D.; Romanou, A.; Schmunk, R.; Shindell, D.; Stone, P.; Sun, S.; Streets, D.; Tausnev, N.; Thresher, D.; Unge, N.; Yao, M.; Zhang, S. Dangerous humanmade interference with climate: A GISS modelE study. Atmos. Chem. Phys. 2007, 7, 2287–2312. (27) Maier-Reimer, E.; Hasselmann, H. Transport and storage of carbon dioxide in the ocean, and in organic ocean-circulation carbon cycle model. J. Atmos. Sci. 1978, 35, 1340–1374. (28) Siegenthaler, U.; Oeschger, H. Biospheric CO2 emissions during the past 200 years reconstructed by deconvolution of ice core data. Tellus, Ser. B 1987, 39, 140–154. (29) Fox, R. W.; McDonald, A. T. Introduction to Fluid Mechanics; John Wiley: New York, 1992. (30) Moran, M. J.; Shapiro, H. N. Fundamentals of Engineering Thermodynamics; John Wiley: New York, 1995. (31) Essenhigh, R. H. An introduction to stirred reactor theory applied to design of combustion chambers. In Combustion Technology: Some Modern DeVelopments; Academic Press: New York, 1974; Chapter 14.

Figure 4. (A) Specification of input and output carbon flux rates for the total atmosphere. (B) Specification of input and output flux rates of carbon for northern and southern hemispheres with north/south exchange atmosphere CV.

Anthropogenically Sourced Carbon Dioxide here, with certain (to be given) qualifications, the mixing is assumed ab initio to be “perfect” or, equivalently, instantaneous. 3.2. Alternative Analytical Models. 3.2.1. Global Circulation Models. In the majority of past studies, the generally preferred approach to analysis of the mixing structure in the atmosphere has been use of (alternative versions of) the global circulation models (GCMs).27,28,32-35 These are complex, predominantly mechanistic, flow-equation sets (that implicitly define the mixing structure), and as previously set out in 1934 by Brunt,36,37 the theoretical and analytical basis for the GCMs has a long history (cf. refs 37 and 38). However, they generally require the choice of mechanistically based but often open-ended assumptions, notably the common need for empirical selection of both equations to be included or omitted and, more particularly, values of parameter constants required for solution, with numerical predictions then commonly dependent, as emphasized particularly by Rorsch,10 upon the selected equations and parameter values. Consequently, without providing a independent basis for essentially identifying or defining what could be the more likely accurate predictions(s), for example, by test against independently controlled experiments, the results also remain openended (e.g., compare Figure 1.2 in ref 3 with Figure 9 in ref 23). 3.2.2. Perfectly Stirred Reactor (PSR)/Continuous Stirred Tank Reactor (CSTR): 0D Box Model. To avoid this empirical selection problem, the alternative, as applied here, is based on the phenomenological PSR17 or CSTR18 CV model, principally developed in combustion and chemical engineering mixing and reaction studies, and on the basis of the initial assumption of instantaneous and consequently perfect mixing in the CV, with the mixing speed so fast relative to any reaction that, as a first approximation and even at nonsteady state, all concentrations are both essentially constant at any moment in time and uniform throughout the CV. As such, it also essentially corresponds in conceptual principle to the perfect-mixing Eulerian 0D box model,39 separately used in atmospheric physics as an alternative to the (more open-ended) GCMs, where the perfect mixing is the outcome of sufficiently intense turbulence in the CV. At steady state, standard atmospheric measurements of CO2 directly support, at a first approximation, an overall concentration uniformity, with the average currently given as about 380 ppm. In more detailed measurements in both space and time, local variations do exist, notably the annual fluctuations in certain locations,40 of which the 45 year, 5 ppm variation at Mauna Loa is probably the best known sequence (Figure 1.4 in ref 3), although with others ranging up to a 10 ppm variation. Mechanistically, the predominant mixing mechanisms in the atmosphere are the thermally driven vertical and horizontal atmospheric convection flows, noting that, although time-dependent, these convection flows have generally sufficiently shorter time constants compared to the CO2 RT, thus allowing for acceptance of the limiting assumption of “instantaneous” mixing; this averaging includes the Mauna Loa and similar annual oscillations. 3.3. Mixing Uniformity: Determination by Transient Injection and Time Dependence. As a test for validation of any mixing model, it is standard procedure, in either reacting or nonreacting systems, to inject tracers. In nonreacting studies, because the tracers are designed to determine only the mixing (32) Rasool, S. I.; Schneider, S. H. Atmospheric carbon dioxide and aerosols: Effects of large increases on global climate. Science 1971, 173, 138–141. (33) Mahlman, J. D.; Moxim, W. J. Tracer simulation using a general circulation model. J. Atmos. Sci. 1978, 35, 1340–1374. (34) Garrett, C. W. On global climate change, carbon dioxide, and fossil fuel combustion. Prog. Energy Combust. Sci. 1992, 18, 369–407. (35) Joos, F.; Bruno, M.; Fink, R.; Stocker, T. F.; Siegenthaler, U.; Le Quere, C.; Sarmiento, J. L. An efficient and accurate representation of complex oceanic and biospheric models of anthropogenic carbon uptake. Tellus, Ser. B 1996, 58, 603–613. (36) Brunt, D. Physical and Dynamical Meteorology; Cambridge University Press: Cambridge, U.K., 1934. (37) Brunt, D. Weather Study; Thomas Nelson: London, U.K., 1942. (38) Sutton, O. G. Atmospheric Turbulence; Methuen Monographs: London, U.K., 1949.

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Figure 5. Pulse input and decay of atmospheric CO2 (C14) concentration with time from 1950s/1960s A-bomb tests (data source, ref 44). Upper line, northern hemisphere; lower line, southern hemisphere; solid lines, back-calculated from the analytical model and data reduction (sections 4-6).

structure, they are commonly carried out in cold flow.41,42 In the case of the atmosphere, CO2 then acts as a natural and appropriate tracer in both steady and unsteady states. In standard experiments, the tracers are then used in either continuous steady-state injection to determine the extent of mixing uniformity in the CV or unsteady-state, pulse, or “cut-off” injection, with the concentration decay then followed in time, commonly at different locations in the chamber (e.g., ref 43), directly to determine the RT. In the case of the atmosphere, however, with a single (unplanned) exception, such independently controlled experiments cannot generally be carried out. The unplanned experiment, with the data source used here as the validation target for the analytical predictions, was the unsteady state, massive increase of atmospheric CO2 (carrying C14), as the result of A-bomb tests in the 1950s/ 1960s.44 As illustrated in Figure 5, this specifically introduces the factors of both unsteady-state response and a north/south hemisphere exchange, with this exchange accommodated by the CV definition extension illustrated in Figure 4B. 3.4. Flow, Space, and Time Dependence. In this application to the atmosphere, this PSR/CSTR perfect-mixing model does require extension qualifications on two counts. First, there is the absence of a continuously flowing “carrier” fluid, which, although not included in the box model, is central to the combustor or tank reactor17,18 systems, with continuous injection, flow through, and removal of the reacting and nonreacting fluids. Second, on the factor of perfect mixing, with qualifications on time-dependent CO2 fluctuations, already identified, absolute concentrations in the (39) Seinfeld, J. H.; Pandis, S. N. Atmospheric chemistry and physics. From Air Pollution to Climate Change; Wiley: New York, 1998. (40) Atmospheric Carbon Dioxide Trends. Oak Ridge National Laboratories, Oak Ridge, TN; http://cdiac.ornl.gov/trends/co2/sio-keel.htm. (41) Johnstone, R. E.; Thring, M. W. Pilot Plants, Models, and Scaleup Methods in Chemical Engineering; McGraw-Hill: Columbus, OH, 1957. (42) Winter, E. F. Flow Visualization Techniques. In Progress in Combustion Science and Technology; Ducarme, J., Gerstein, M., Lefebvre, A. H., Eds.; Pergamon Press: Elmsford, NY, 1960; Vol. 1, Chapter 1. (43) Zeinalov, M. A. O.; Kuwata, M.; Essenhigh, R. H. Stirring factors in combustion chambers: A finite-element model of mixing along an “information flow path”. Proceedings of the 14th International Symposium on Combustion, The Combustion Institute, Pittsburgh, PA, 1973; pp 575583. (44) Broecker, W. S.; Peng, T.-H. Greenhouse Puzzles: Part I; LamontDoherty Earth Observatory/New York Online Access to Health (NOAH): Palisades, NY, 1993.

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atmosphere, defined as moles per unit volume, do vary with altitude. However, with the exception of water, the relative concentrations (mol/mol) are, on average, substantially constant in space (and time), as required for the first application of the perfect-mixing model.31 As previously discussed elsewhere6 in some detail, the (relative) water concentration does fall with altitude (in an analytically known manner) but the variation is not sufficient to invalidate the overall perfect-mixing assumption at a first approximation. In principle, these altitude variations essentially then correspond, alternatively, to the 1D box (column) model,39 although further detailed extension to this 1D model is not an immediate necessary requirement for this study. As will be shown, the requirement here is to know the relevant gas concentrations at the land/sea exchange surface(s) (cf. Figure 4) and the related factor, as in all corresponding models, is the (concentration-governed) exchange rates between the CV and the surroundings, as set out in the analysis to follow (section 4). As shown in Figure 5, the A-bomb experiments generated an initial increase in 14CO2 of approximately 1000% above the normal value, which then decayed substantially exponentially with time, introducing the factor of time dependence, in accordance with the model predictions developed in section 4 and included as back plots as shown on Figure 5, with the back-fit curves calculated (section 6) from the analytical model after an appropriate equation test and data reduction. 3.5. Adjustment Time. As a primary time dependence factor, the major pulse-driven unsteady-state element is self-evident in the figure but the further time factor, already identified as the adjustment time as outlined in section 2.4 (cf. section 5.2) and developed from the analysis, was the time dependence for the rise of the original “equilibrium” CO2 concentration value to a slightly higher value, because of the long-term rise in the background atmospheric CO2 equilibrium concentration. In the two summary tables (Tables 1 and 1.1) of the IPCC First Climate Change Report, this adjustment time, of 50-200 years, was then, confusingly, (re)defined as the lifetime (cf. sections 4.3.2 and 6.4, below).

exchanging back across the CS to return to the vegetation and sea. Defining the (total) atmospheric mass contained in the CV as MA (Figure 4A and Table 1), defining the relative atmospheric concentration of the carbon (either C12 and/or C14), carried as CO2, as a (mass) ratio, C, (mass of carbon per mass of air), and then, in unsteady state, ab initio assuming Fi > Fe, the atmospheric (mass) concentration will increase by the amount δC in the time δt, so that the unsteady-state, time-based C-balance equation, in standard formulation17,18,31 is

4. PSR/CSTR Analytical Structure

Fi ≈ constant ) F oi

(2a)

The PSR/CSTR mixing analysis as now developed focuses initially on the unsteady-state behavior, with the steady-state relations then obtained by reduction from the unsteady-state analytical results, and then, as already noted, using the experimental data of Figure 5 as a target for verification of the analytical (PSR) model structure and predictions. For the unsteady (transient or pulse injection) state, e.g., defining response to the (massive) injection of 14CO2 (Figure 5), this is developed in two stages: first, for the single CV (Figure 4A), and then, extended to the twin CVs (Figure 4B), to incorporate the north/south hemisphere exchange. In Figure 4A, the exchange surface between the atmosphere contained in the CV and the (land/sea) surroundings is the base of the CV, with carbon-exchange flux rates, across the exchange surface, defined as shown. In the corresponding schematic for the two north/south hemispheres (Figure 4B), they have a common interface between the two (north/south) CVs but alternatively defining the left faces of the (joint) boxes as the atmosphere-surroundings exchange surface. These two figures provide the schematic basis for the following analytical development. 4.1. Basis Analysis: Single CV. In Figure 4A, with CO2 as the carbon carrier, the factors Fi and Fe are defined as the carbon input and output flux rates (mass of C/unit time), with the CO2 carrying the carbon initially exchanging into the atmosphere, principally from vegetation and the sea, across the (land/sea) earth atmosphere CS interface, being dispersed (essentially) uniformly through the CV that is the atmosphere, and then

Fe ) keMAC

(2b)

MAδC ) (Fi - Fe)δt

(1)

Phenomenologically, we assume as previously defined that, in this formulation, the relative concentration (C) inside the defining (atmospheric) CV is (overall) uniform because of rapid mixing, so that details of the mixing mechanism are not relevant for this analysis; however, crossing the earth/atmosphere CS interface, some consideration of mechanisms is appropriate. For the input flux rate, Fi, from land/sea to the atmosphere, this can, initially, be set as a constant, i.e., Fi ≈ Fio, even when including (short-time) periodic fluctuations in concentration, for example, the annual 5 ppm fluctuation at Mauna Loa,40 which nevertheless, can be averaged out over an appropriately defined time period (e.g., 1 year). For longer time periods, notably allowing for a time increase in the “equilibrium” CO2 concentration governed by the adjustment time, a time-based rate can be rewritten as Fi(t), as further developed below (section 4.3.2); it does also then depend upon a definition of a “local” (absolute) carbon concentration, and this dependence develops from the analysis. For the return flux, Fe, then, using the standard analytical ab initio assumption31 of being a first-order process, i.e., proportional to the atmospheric concentration, C, we can initially write for the joint pair

where ke is the velocity constant for the return-flux process, also separately defined10 as the “recycle ratio” factor. Again, as a phenomenological analysis, it is not necessary to define the physical mechanistic nature of the process governing the return flux, but it is generally interpreted, largely, as “washout” by rain because of the absorption of CO2 in water vapor condensing as rain. The initial focus here, however, is on the base or “platform” analysis, assuming that ke is a constant that can then be modified and extended in later analysis and studies if so needed. It also provides a useful possible numerical target for independent (parallel) prediction from (the mechanistically based) GCM analyses. This formulation also further identifies the base difference as indicated earlier between this procedure and the standard (combustion and chemical) PSR analyses, in which there is a “carrier” fluid, which is the sum of all fluids flowing through the (combustion or reactor) chamber, with the outcome that, with a carrier fluid, all constituents, if nonreacting, then have the same RT. In this atmosphere formulation, however, there is no carrier fluid, so that the individual components, e.g., 12CO2 and 14CO2, can have different RTs, as will be seen in the developed results. Despite the absence of a carrier fluid, the solutions to the equation formulations, nevertheless, have the same PSR analytical structure, as shown, to follow (section 5). 4.2. DE Format. In formal DE format and introducing as a boundary condition at the dynamic (quasi-)equilibrium limit that



C ) Ceq when dC/dt ) 0, then, using eqs 2a and 2b, eq 1 can be rewritten in the alternative forms MA(dC/dt) ) F oi - MAkeC

(3a)

or (dC/dt) + (C/τ) )

(F oi /MA)

) (Ceq/τ)

(3b)

(MA/2)(dCN/dt) ) (1/2)F oi - ke(MA/2)CN - kc(MA/2) × (CN - CS) (5a)

(3c)

(MA/2)(dCN/dt) ) (1/2)F oi - ke(MA/2)CS + kc(MA/2) × (CN - CS) (5b)

thus defining ke ) 1/τ ) (F oi /MACeq)

difference; thus, kc(MA/2)(CN - CS), where kc is defined as a (phenomenological) convective-exchange velocity constant, which is (assumed to be) common for both exchange directions. The DEs as modified forms of eq 4a then take the joint alternative forms

where the recycle-ratio parameter ke, is formalized by eq 3c as the reciprocal of τ. As shown to follow (section 5.2), the characteristic time parameter, τ, also corresponds, as a standard PSR/CSTR result, to the RT, also separately identified in other studies as the “fill” time38 or the “turnover time”.3 Rewriting eq 3c, we obtain defining expressions for (1) Ceq, the terminal (quasi-)steady-state equilibrium concentration value (at dC/dt ) 0) and for (2) Foi , the (constant) input rate; thus Ceq ) (F oi τ/MA) ) (F oi /keMA)

(4a)

F oi ) keMACeq

(4b)

and

This terminal value, Ceq, defines a presumed, long-term, steady-state “equilibrium” concentration (reached in the analytical formalism at t ) ∞), and for conditions such that Ceq does change only marginally with time, eq 4b still then provides the basis for, initially, setting Fi ≈ F io as a first approximation. However, because Ceq is also proportional to the time constant, τ, this, separately, provides an analytical basis for evaluation of long-term changes in the otherwise initially assumed “constant” input flux rate, Foi , when redefined as Fi(t), which, as already noted in section 4.1 (cf. section 4.3.2), is the most likely alternative mechanism driver for the time-based (and temperature-determined) increasing (quasi-equilibrium) CO2 concentration. Thus, if Ceq and, hence, Fi (slowly) increase (as the result of, for example, increasing atmospheric temperature), this reasonably accounts, independently, for the rising CO2 concentration (cf. Introduction; refs 6-9), governed by the adjustment time introduced in section 2.4 and developed further below (section 4.3.2). Equation 4a then also shows, as an important distinction, that, because Ceq is dependent upon (proportional to) the influx rate parameter, Fi, it is a dynamic and not a static equilibrium concentration, even when Fi is a constant. Using eq 4b, showing F oi proportional to Ceq, this (proportional) dependence, likewise, generates the necessary correspondence between the carbon influx rate across the CS and the “effective” carbon concentration at the CS, thus amplifying/extending eq 2a; it also provides a parallel to eq 2b (which defines Fe as proportional to the potentially variable atmospheric concentration, C). 4.3. Analytical Extensions. 4.3.1. Twin CVs. For the two CVs in Figure 4B, the equation format of eq 3b still applies with only modification by addition of a north-south exchange term. To incorporate that exchange, with the atmosphere split into two halves as shown in the figure, the mass in each half is defined as MA/2. Redefining the carbon concentrations for the two (north and south) hemispheres as, respectively, CN and CS, the (convective) exchange between the hemispheres can be defined by an exchange term proportional to the concentration

As shown, these have essentially identical structure, with the difference only in the sign of the north-south convectiveexchange contribution. Additional assumptions are that Foi and ke (the input flux rate and the earth/atmosphere interfaceexchange velocity constant) have the same values in both hemispheres, although these assumptions can be modified if there is independent experimental supporting evidence of the need for any changes. 4.3.2. Variable Influx Rate: Adjustment Time. If there are significant changes in Fi over “long” time periods (e.g., >50 years), the analytical outcome of such changes, as essentially defined by eq 4a, will result in a (proportional) increase (or decrease) in Ceq. Analytically, adopting the separately given postulate3 that Ceq would effectively rise with time on a (1 exp) format, from an initial value, Coeq, at a (defined) t ) 0, to a final value, C ∞eq, then a “quasi-equilibrium” but time-dependent concentration Ceq can be defined as o ∞ o [C(t) eq - Ceq] ) (Ceq - Ceq)[1 - exp(-t/τ**]

(6)

where the time constant, τ**, is the formal definition of the adjustment time as given in the cited4,3 earlier studies for the CO2 re-equilibration. [The actuality of the currently recognized rising Ceq is mechanistically interpreted in the Watson et al.3 study as a result of an incomplete or partial resorption or recovery of CO2 by the surface ocean, resulting in a decrease in the CO2 concentration absorbed in the surface (but not the deep) ocean, and a corresponding increase in atmospheric CO2, defined as Ceq. Alternatively, however, the fall in the surface ocean concentration is more likely due to a falling equilibrium constant for CO2 absorption in water (cf. ref 8) as an outcome of the rising temperature. With the reduced CO2 ocean capacity, the balance remains in the atmosphere, thus accounting for the rise in CO2 concentration with temperature. The exact mechanistic interpretation is not significant at this point, however; the significant factor, phenomenologically, is recognition of the probable or actual rise, in the long term, of Ceq with time.] Introducing that time-dependent assumption in the DE of eq 3b and defining ∆C ) (C - Coeq), ∆Co ) (Co - Coeq), and ∆C∞ ) (Co - C∞eq), the DE can be rewritten in the modified form d(∆C)/dt + (∆C/τ) ) [∆Co - ∆C∞eq/τ][1 - exp(-t/τ**)] (7) If the re-equilibration/adjustment time (τ**) is infinite, effectively corresponding to a time-independent value of Ceq, eq 7 reduces to eq 3a, showing the formal equation correspondence at that limit. 5. Analytical Solutions and RT 5.1. Solutions. Solutions for eqs 3b, 5a, and 5b provide the general basis required. Solutions for eqs 5a and 5b are obtained

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after reformatting, using the standard procedure for such a joint DE pair, determined separately after addition and subtraction. Thus, redefining the concentrations as sum and difference terms: Csum ) (CN + CS)/2, and Cdif ) (CN - CS), the DE relations take the forms

Finally, for the variable influx-rate formulation (eq 7), the solution can be written ∆C/∆Co ) Zoexp(-t/τ) + [(1 - (∆C∞/∆Co)] × [1 - exp(-t/τ**)/(1 - τ/τ**)] (14a)

(d/dt)(CN + CS) ) d(Csum)/dt ) Foi (MA/2) - ke(CN + CS) ) 2Foi /(MA) - (Csum)/τ

∆CS ) (1/2)(∆CNo + ∆CSo)exp(-t/τ) - (∆CNo - ∆CSo) × exp(-t/τ*) (13)

(8a)

where Zo ) 1 + [(1 - (∆C∞/Co)][(τ/τ**)/(1 - τ/τ**)] ≈ 1 (14b)

and (d/dt)(CN - CS) ) ) ) )

d(Cdif)/dt -2ke(CN - CS) - 2kc(CN - CS) -2(ke + kc)(Cdif) -(Cdif)/τ* (8b)

with ke ) 1/τ, as before, and 2(ke + kc) ) (1/τ*). Equations 3b and 8a, self-evidently, have exactly the same form and, correspondingly, the same (standard-form) solution structure. For eq 3b, the solution is C ) Ceq + (Co - Ceq)exp(-t/τ)

(9a)

where Co is the initial (pulse-injection) excess concentration at t ) 0 or redefining in difference terms above the equilibrium value, Ceq, as used in Figure 5 (C - Ceq) ) ∆C ) (∆Co)exp(-t/τ)

(9b)

thus recovering the standard PSR/CSTR prediction of the exponential decay of the injected excess. For eqs 8a, the solutions have similar form. For eq 8a, we obtain Csum ) (CN + CS)/2

Reduction of the parameter group Zo to the order of unity is the outcome that, numerically, the ratio pairs ∆C∞/∆Co and τ/τ** are of the order, respectively, of 0.9 and 0.05. The solution (eq 14a) then has the same primary form as eq 9b but with the add-on adjustment time or “corrector” function (second RHS term in eq 14a) to account for an increase in the baseline Ceq from Coeq to C∞eq and, as shown, to incorporate the adjustment time. This equation reverts to the eq 9a structure as the adjustment time, τ** f ∞. Separately, it also recovers the basis for the plot-format structure of Figure 1.2 in ref 10. 5.2. RT Basis: Analytical Definition. These equations and solutions provide the formal, analytical basis for defining the (average or mean) RT, τav, relating it to τ, defined in eq 3c. For the exponential decay in the concentration given by eq 9a and then using the standard PSR formulation31 that the total mass loss because of the concentration decay in infinite time equals the mass loss at the initial loss rate held constant for the RT, t ) τav, we have mass )

∆Csum ) (∆CN + ∆CS)/2 ) (1/2)(∆CNo + ∆CSo)exp(-t/τ) (10b) again showing a zero limit of the excess of C above Ceq at t ) ∞ and, importantly, with the same characteristic decay time, τ. The solution to eq 8b is similar and again with a zero limit at infinite time; however, with the different (substantially smaller) time constant (τ*) and, in difference terms, we obtain ∆Cdif ) (∆CN - ∆CS) ) (∆CNo - ∆CSo)exp(-t/τ*) (11) Equations 9a, 10, and 11 are the required format for model validation, developed to follow (section 6), using the Figure 5 data. For the plotted back calculations as already shown in Figure 5, the separate expressions for CN and CS are then required and these are obtained from eqs 10b and 11 by appropriate sums and differences, giving the solutions ∆CN ) (1/2)(∆CNo + ∆CSo)exp(-t/τ) + (∆CNo - ∆CSo) × exp(-t/τ*) (12)

e

- Foi )dt ) (Foe - Foi )τav

(15a)

For the integral term in eq 15a, using eqs 2b and 9b to obtain Fe ) keMAC ) keMA(Ceq - ∆C) mass )

) Ceq + [(1/2)(CNo + CSo) - 2Ceq]exp(-t/τ) (10a) or in difference terms above Ceq

∫ (F

)

∫ (F

e

- Foi )dt

∫ [k M C e

A eq

- Foi ]dt -

∫ k M ∆Cdt e

A

(15b)

In this equation, because we have, from eq 4a, [keMACeq - Foi ] ) 0, then, substituting for ∆C in the second RHS term of eq 15b, using eq 9b, and integrating wrt t from t ) 0 to t, we obtain mass(t) ) -

∫ k M ∆Cdt ) k M (∆C )(τ)[1 - exp(-t/τ)] o

e

A

e

A

Using the relation of eq 3c giving keτ ) 1, then, at t ) ∞, we obtain, as expected mass(∞) ) (keτ)MA∆Co ) MA∆Co

(15c)

Next, expanding the RHS term of eq 15a, using eq 2b to give Foe ) keMACo, at t ) 0, and using eq 4b to substitute for F oi and eq 3c to substitute for ke, we obtain mass ) keMA(Co - Ceq)τav ) keMA∆Coτav ) MA∆Co(τav/τ) (15d) Equating eqs 15c and 15d, we then obtain the relation between the decay time (τ) and RT (τav) τ ) τav

(16)



Figure 6. Log-linear plot of Figure 1 data for variation of log [C14] against time (year) for northern and southern hemispheres: Validation of eq 9a.

This is the standard result for a PSR/CSTR analysis, equating τ with τav, but it also re-establishes this relationship for the special conditions set here for the PSR mixing conditions without a carrier fluid; this extension of the standard PSR/CSTR analysis to a non-carrier system and, particularly, its application to the atmosphere is believed to be original here, although the result would appear to be quite widely used. Importantly, this also identifies the RT as independent of whether the inflow/ outflow process is in (dynamic) equilibrium or non-equilibrium status. With this correspondence between the characteristic decay time, τ, and the defined RT, τav, eq 4a can be rewritten in the form τav )

MACeq Foi

)

atmospheric carbon capacity input/output rate of carbon exchange (17)

where, at steady state, the input rate (Foi ) ) the output rate (Fe) ) the throughput flux rate (F). This formulation then corresponds to and is identical with the formulation set out in section 2.1 and, also, to the IPCC definitions,3 given, as noted (section 2.1), as “lifetime” and “turnover time”. 6. RT of CO2 in the Atmosphere: Equation Validation and Numerical Values 6.1. Equation Validation. Test and validation of the derived equations, supporting the proposition of (approximation to) perfect mixing, with corresponding determination of the (C14) RT, is set out in Figure 6 for eq 9a and in Figure 7 for eqs 10b and 11, using the C14 data of Figure 5. Notably, the validation is direct, not requiring prior (empirical) selection of any numerical constants. 6.1.1. Single CV. As a test of eq 9b, initially using the solution for a single CV, the graph lines in Figure 6 are (loglinear) plots of the log ∆C data against time (t) for, separately, the northern and southern hemisphere data values shown in Figure 5, i.e., not yet incorporating the analytical format of the

Figure 7. Log-linear plots of function of [C14] against time (year): Validation of eqs 10b and 11. (A) Variation of the concentration sum with time for northern and southern hemispheres. The correlation line is identical to that on Figure 6 for hemisphere concentrations plotted independently. Replot merges data for 1965 to 1970. (B) Variation of concentration difference with time for northern and southern hemispheres. The correlation line defines the exchange time between northern and southern hemispheres.

north-south exchange. In this plot (Figure 6), the difference between the two hemispheres for the earliest time period, 1965-1970, is more clearly evident than in Figure 5 but also with a clear convergence in both cases beyond that date, so that, from 1970 to 1990 and with a small departure from 1990 to 1995, both sets are good log-linear approximations, in initial support of the prediction from eq 9b and, likewise, providing additional support for the validity of the PSR/CSTR mixing model. From this plot, the time constant, τ, as determined from the linear slope, is ∼16 years (the higher end of the Segalstad23 data set of 36 values). This result is more fully validated by Figure 7. 6.1.2. North/South CV. Figure 7 is the corresponding test validation for the more complete eqs 10b and 11, again in loglinear format, representing, respectively, the sum (Figure 7A) and difference (Figure 7B) formulations, which take into account the north-south hemisphere exchanges. In both cases, validation of the equations is again shown by the linearity of the relevant plots, with the linearity range for Figure 7A, compared to Figure 6, extended back to 1965 (although again with the small departure from 1990 to 1995) and giving time values of 16.3 years for τ and 2.2 years for τ*. From the two plots, the values for the initial (pulse) increases (∆Co) defined for 1965 and corresponding to t ) 0 were obtained as ∆CNo ) 902 and ∆CSo ) 640 (with ∆Ceq ) 0, by definition). A minor difference between the two figures, other than the values of the slopes, is the degree of scatter in the

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plots which, however, is to be expected because Figure 7A is based on summed values and Figure 7B is for difference values, so that, for a given absolute error, the proportional error is increased as expected. However, the adopted PSR model is, nevertheless, clearly validated by the graphical linearity of both plots. 6.2. Alternative (Graphical) Procedures. The solution equations derived in section 5 also provide alternative analytical bases for defining or calculating the RT that have been used in past studies, generally given with alternative (some previously identified) definition names, e.g., fill time, decay time, e-fold time, turnover time, lifetime, and reciprocal recycle ratio. The two most commonly used procedures44,45 both depend upon (the assumption of the validity of) the exponential-decay solution of eq 9b, thus implying, when used, adoption of and/or support for the PSR/CSTR perfect-mixing model (although this correspondence is not commonly identified). Likewise, because the procedures only assume validity but do not directly validate the equation structures, the results, consequently, can be regarded as somewhat open to question. Of the two procedures, the more commonly used is directly based on eq 9b for defining an “e-fold” time factor (e.g., ref 45). Setting t ) τ in eq 10b, we obtain ∆C ) ∆Co/e, i.e., corresponding to a reduction in ∆C by the factor 1/e in the time period τ, and thus identifying e-fold t ≡ τ ) τav as the “e-fold” time. Numerical determination of the value of τ ) t at ∆C ) ∆Co/e is then obtained, either from data tabulations or from a graph of ∆C versus t (cf. Figure 5) with direct read off from the graph at the defined concentration. Applying this procedure to Figure 5, taking 1975 as the t ) 0 condition, with ∆C ) 400, then ∆Co/e ) 147, which, corresponding to 1991, gives τ ) 1991 - 1975 ) 16 years, in essential agreement with the values above. The alternative is also a graphical procedure, again using the (nonlinear) graph of C versus t (Figure 5); this then requires drawing in the tangency to the curve at t ) 0 to obtain the initial gradient because, by differentiation, (dC/dt)o ) -(∆C)/ τ, so that, for a linear gradient drawn on the graph, the time intercept, at ∆C ) 0, is τ. Applying this procedure to Figure 5, the intercept to a tangent to the curve at 1975 (not included here for space), was obtained as 17 years, again closely corresponding to the previous values. As this shows, these procedures depend upon a very limited number of the experimental data values and, to that extent, can be considered somewhat open-ended. Nevertheless, the background concepts and specifically the implied PSR mixing structure are broadly in agreement and in accordance with the developments given here and which, correspondingly, provides support for their actual wide use in these forms. 6.3. C12 and C14 Numerical Values Comparison. With this validation of the PSR model, the CO2 RTs carrying C12 are now compared to CO2 carrying C14. For the 12CO2 RT, this is obtained as outlined in section 2.3 using the steady-state IPCC data of 1990.2,1 As listed in Table 1, using a (then current) carbon (predominantly C12) storage in the atmosphere of MACeq ) 750 gigatons and a (natural) in/ out exchange rate of Foi ) Fe ) 150 gigatons/year, the ratio of the two, using eq 17, gives the (already cited) value for the RT of 5 years, as also already given in Figure 1. A comparison to other published sources is already reviewed in section 2.3, except for a similar value obtained by both Dietze45 by direct read off

from the C14 graphs using the (above-defined) e-fold procedure and Rorsch10 from the reciprocal “recycle ratio” (eq 4c) parameter, ke ) 1/τ, where, for Rorsch’s value of ke ) 0.2 per year, then τ ) 5 years. For the 14CO2, the RTs obtained from the slopes (section 6.1) and gradients (section 6.2) of parts A and B of Figure 7 have values, as given, of 16.3, 16, and 17 years for τ and 2.2 years for the north-south exchange time, τ*. [In the 1959 novel On the Beach, the author (aeronautical engineer), Nevil Shute (Norway), introduced, as a central factor of the story, the time constant for north-south exchange of atomic radiating products injected by destructive atom-bomb explosions into the northern atmosphere, initially posing, in the novel, two alternatives: a long time constant for the north-south exchange that would allow for such sufficient radiation decay that the southern hemisphere would survive or such a short exchange time that life would be terminated. His choice in the novel was the second. The relevant parameter for comparison here is the τ* factor of 2.2 years. This is also consistent with the hemisphere exchange time estimated from the outcome of the 1815 Tambora volcano in Indonesia resulting, a year later 1816, in “the year without a summer” in the northern hemisphere.] These values then provided the numerical basis for the Figure 5 back-calculated curves of ∆CN-t and ∆CS-t, calculated from eqs 12 and 13. The extent to which this difference in the C12 and C14 RT values, which is quite small, is statistical because of mechanistic differences in exchange behavior is undeterminable at this time, particularly when compared to the wider range of values discussed earlier. Nevertheless, considering, as a mechanism, the dependence of ke on the (average) atmospheric humidity as a result of the possible absorption (wash out) of CO2 in water (rain), then, as a material-dependent variable, this could probably account, mechanistically, for the real difference in the τ and ke values for the two molecules, although further development of that factor is outside the immediate scope of this study. 6.4. Long-Term Values. Considering, next, for comparison, the longer term values, generally in the range of 100-1000 years, these are open to question on several grounds. The most significant and highly quoted value, as already reviewed, is the range of 50-200 years reported in the First 1990 IPCC Climate Change Report2,3 and given in the two tables as the lifetime based on a definition that equates to RT but with the numerical values defined and determined in the text as the adjustment time. This requires no further comment other than noting the (unstated) change in definition as a source of continuing confusion. Of other long-term results, the basis for the differences is generally not given, but where they are given, it would appear to be alternative use of the relevant data. The most common and widely used procedure (e.g., ref 46) would seem to be exclusion of the natural (sea and vegetation) flux rates, with selection only of the anthropogenic carbon sources for the input flux rate. Thus, using the 1990 data value of 5 gigatons/year for the carbon-influx rate and 750 gigatons for the capacity, the RT would then be 150 years. At the higher carbon-storage value of 900 gigatons and 6-7 gigatons/year influx rate, the RT is marginally but not significantly reduced. In those studies, however, where these values are used, the basis for ignoring the natural flux inputs is not stated. Because independent treatment of the atmospheric carbon components generated separately from natural sources on the one hand and from

(45) Dietze, P. IPCC’s Most Essential Model Errors, http://www.johndaly.com/forcing/moderr.htm; Carbon Model Calculations, http://www.johndaly.com/dietze/cmodcalc.htm.

(46) Kolbert, E. The climate of man: Parts I, II, and III. The New Yorker; EBSCO Publishing Company: Ipswich, MA, 2005; Vol. 81, issues 10, 11, and 12.


anthropogenic sources on the other hand is analytically invalid, the outcomes in terms of the alternative RTs is then arguable. In the absence of any supportable basis for a long RT (>100 years), all reviewed data clearly support a value of ∼10 years. 6.5. Adjustment Time. Finally, it will be noted that use of eq 10b for Figure 7A excludes the adjustment-time “corrector” term in eq 12, and this omission can possibly account for the slight departure from linearity in the last 5 years (1990-1995) of the Figure 7A plot, although the departure is substantially inside the random variation calculated from the SD of the plot analysis. However, inclusion of the corrector term in the back calculations shown in Figure 5, using a value of 100 years for the (τ**) adjustment time, did close the departure sufficiently, as seen on Figure 5, to provide support for both the reality of this adjustment time and agreement with the range of 50-200 years (obtained by GCM analytical extensions) reported in the text of Chapter 1 of the Watson et al.3 paper (although then redefined in their Table 1.1, as noted, as the “lifetime”). 7. Discussion These results overall relate to both substantiation of the validity of the adopted PSR/CSTR/0D (box) mixing model and, correspondingly, support for the determined short-term (∼10 years) RT values. With regard to the first point, the validity of the perfectmixing model at a first approximation and in the absence of contrary results is well-supported, even with the particular modifications for this application of the absence of a carrier fluid, the extension to include the split into the northern and southern hemispheres, and the averaging out of such regular short-term variations as the Mauna Loa oscillations. The further, potentially significant, outcome of this phenomenological mixing model structure is, then, the absence of the need, initially, for mechanistically defining either the mixing process or the input/ output processes. The most specific outcome of this lack of mechanistic requirements is that different carbon carriers, separately, have different RTs, as shown, but these differences do not affect the overall outcome results and/or their interpretation and also support the validity of this phenomenological rather than mechanistic approach for this initial analysis. Likewise, considering the assumed long-term time dependence of the input flux rate, Fi, this input rate is more completely the sum of the different sources (the sea and vegetation being the two primary ones) that, separately, can have very different time constants for their carbon processing (cf. Figure 1). However, from the point of view of the base analysis, these processing (“buffering”47) sources are in CVs that, as seen in Figure 1, are outside the primary atmospheric CV, so that their processing time constants will have no influence on the (shorter term) atmosphere time constant, although this differentiation is evidently not considered in many other studies. Related to this is the determination of the 50-200 years adjustment time based, as reported,3 on the results of the two detailed GCM models27,28 and which also involved analytical extensions to include the kinetics of the ocean absorption. However, inclusion of the ocean absorption likewise expands the analysis outside the scope of the atmospheric CV, so that the “RT” result no longer applies uniquely to the atmosphere, although the basis for such analysis extension is seen in eq 14a. (47) Bolin, B.; Eriksson, E. Changes in the carbon dioxide content of the atmosphere and sea due to fossil fuel combustion. In The Atmosphere and the Sea in Motion (Scientific Contributions to the Rossby Memorial Volume); Bolin, B., Ed.; The Rockefeller Institute Press: New York, 1959; pp 130-142.


On the longer time scale, the carbon processing in these external CVs can, nevertheless, affect the (sum of the) input flux rates [Fi(t)], as outlined in section 4.1. Specifically, the data of Figure 5 and its associated analysis do support the wellestablished conclusion3 that the C14 equilibrium concentration returns to a slightly higher Ceq value than the value preceding the atom-bomb tests, thus accounting for the minor but clearly evident data departure in Figure 6 after 1990, as discussed in section 6.5; at a value of ∼100 years for τ**, this is in numerical agreement, as shown, with the Watson3 adjustment time. Correspondingly, this more certainly supports and separates the IPCC long-term adjustment time from their short-term (∼4 years) turnover, i.e., RT, with the RT values from all sources broadly in the range of 10 ( 5 years, as discussed. Returning then to the primary driver for this study, which is the relevance of RT to anthropogenic carbon sources, given a RT of the order of 5-15 years, these results support the conclusion from Figure 2 that it shows the relatively insignificant net contribution of carbon supply from combustion sources to the CO2 rise in the last 250 years. The further outcome of that conclusion is then the further support for vegetation and the sea as the primary sources of CO2 as illustrated in Figure 1, thus voiding the need for or value of CO2 sequestration and control. The conclusions to be drawn from this, particularly from the numerical results, are largely self-evident. Most significantly, the short-term RT notably provides no basis for control of CO2 emissions as proposed in the Kyoto and Bali Protocol procedures.13-15 Nevertheless, proposed control procedures are still being politically implemented and amplified. These include conversion or replacement of current energy systems by instituting, for example, transition from fossil fuels, as currently used for most energy generation, to converted fuels and/or to alternative energy sources: wind, solar, and other. These additionally carry significant transition and associated conversion or replacement costs, which, on account of their financialresources diversion, are also commonly identified, as already noted, as a highly probable and significant retarding factor in future economic growth.16 In the past context of the (unsupported) long-term RT assumption, the further concern, although now eliminated by these results, was that, because of the transition time (potentially of decades) required for the move from current to future energy procedures, the further expected outcome would be associated continuing and initially uncontrollable build-up of (possibly damaging) atmospheric CO2 during the transition, if CO2 was the warming driver and the RT was long enough (>100 years). With the identified short (∼10-year) RT, these build-up concerns are substantially vitiated or eliminated, but the concern over continuing build-up is still currently being given major political focus. With combustion emissions thus eliminated as the primary reason for the rising carbon concentration in the atmosphere, then, lacking any adverse analysis, this additionally supports the alternative conclusion7-10 that it is the rising temperature that is driving up the carbon emissions and not the rising emissions driving up the temperature. As further extension of that factor, it is also reasonable to assume that, regarding current ice-sheet melt off, the overall contracting ice sheets are the result of the annual snow-ice supply being, currently, overdriven by the competitive melt off, even with the temperature oscillating but with the average temperature rising accordingly as less energy from the sun is reflected back to space and is, instead, absorbed by the earth. This, nevertheless, could reverse if the snow supply increases sufficiently because of, for example, sufficient or total unfreezing of the Arctic Ocean, which would

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then provide the necessary moisture source to override the melt off, as assumed to be the case in past millennia48 and which quite possibly or probably was the driver source for the Little Ice Age. 8. Conclusions The results considered to be of principal significance are as follows: (1) As a basis for the interpretation of the data considered, the analytical results applied to the C14 (Figure 5) response data broadly support the non-carrier PSR/CSTR “perfect” mixing model as applicable to the atmosphere, at least to a first order. This provides the necessary physical and analytical basis for defining, calculating, and interpreting the RTs obtained from the available data sources, as summarized to follow. (2) For the (current) quasi-steady-state conditions, interpreted as primarily defining the RT of C12 carbon (carried as CO2), the RT of 5-6 years obtained here based on the IPCC data source is in very good agreement with the other cited sources,23,24 notably including the value of 4 years explicitly stated in Chapter 1 in the Watson et al.3 paper in the 1990 IPCC First Climate Change Report,2 and the further data sources as overviewed in section 2.3. (3) The longer RT for the C14, of 16 years, is of the same order of magnitude compared to the C12 results, but it is still short compared to the alternatively published (but questionable) values (for C12) of, broadly, >100 years. The (small) difference between the C12 and C14 data is separately interpreted (mechanistically) as possibly a result of differences in the absorption or solution rate of the two different gases (governed by the velocity constant for the rate flux process, ke), although this is not a factor in any way affecting the conclusions given here. (4) Such a short RT (of 5-15 years) is emphasized particularly in the First IPCC Climate Change Report,3 with the text comment: “This means that on average it takes only a few years before a CO2 molecule in the atmosphere is taken up by plants or dissolved in the ocean”. (5) This short RT is also then consistent with fairly rapid natural accommodation to the increased carbon input from fossil fuel combustion, so that attribution of the recognized rising concentration of CO2 in the last century is not consistent with supply from anthropogenic carbon sources, which is supported particularly by line C of Figure 2. (6) As an extension of point 5, the further significant and related factor is the proportionately small (

Potential Dependence of Global Warming on the Residence Time (RT

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