Temperature effects on the atmospheric carbon dioxide level
by Gösta Pettersson, University of Lund, Sweden (February 24, 2014)
A simple two-box model has been designed to describe the main kinetic characteristics of carbon dioxide exchange between the atmosphere and the hydrosphere. The model is based on reported equilibrium data for the system, includes reported temperature changes and rates of anthropogenic emissions as perturbation factors, and uses the turnover time of airborne carbon dioxide and the activation energy for outgassing of waterborn carbon dioxide as adjustable parameters.
Analysis of the temperature dependent multiannual fluctuations of the Keeling curve provides evidence that they derive from thermal outgassing with an activation energy of 165 kJ/mol/K and a turnover time of 14 years. For these values of the adjustable parameters, the model gives an almost perfect description of the long-term trend of the Keeling curve and attributes the rising concentration of airborne carbon dioxide during the industrial era to approximately equal contributions from anthropogenic emissions and thermal outgassing of carbon dioxide from the hydrosphere. During the last two decades, thermal outgassing seems to have contributed more than anthropogenic emissions to the increased levels of airborne carbon dioxide.
The results indicate that the extensively discussed missing sink problem regarding anthropogenic carbon dioxide emissions derives from an underestimation of the sink capacity of the hydrosphere due to neglect of thermal outgassing. Projections of climate models that do not take thermal outgassing into account will gravely overestimate anthropogenic effects on future temperatures.
The Keeling curve describes the time-dependence of the atmospheric carbon dioxide concentration, as determined by C. D. Keeling and collaborators at the Mauna Loa observatory since 1958 . The curve exhibits both seasonal and multiannual fluctuations around its long-term trend towards higher values. In a thorough analysis of such fluctuations, Keeling found that the seasonal ones may reflect such a multitude of effects of uncertain strength and partly non-global character that their precise origin cannot yet be unequivocally established .
The multiannual fluctuations, however, were found to be strongly and positively correlated to global temperature changes associated with the El Niño–Southern Oscillation cycle. Rapidly rising air levels of carbon dioxide typically occur during El Niño events with their global scale temperature increases, as was first noted by Bacastow . Subsequent analyses have shown that fluctuations of the atmospheric carbon dioxide level lag several months behind the temperature changes . This means that the fluctuations are caused by the temperature changes and hence are of thermal origin.
Keeling pointed out that the multiannual fluctuations are unrelated to the rates of anthropogenic emissions of carbon dioxide , and volcanic emissions can be anticipated to remain unaffected by global temperatures. Consequently, the multiannual fluctuations are likely to derive from thermal effects on the rate of exchange of airborne carbon dioxide with the hydrosphere or the biosphere. Globally representative effects of increasing temperature on the biosphere are associated with increased growth and carbon dioxide consumption. Thermal effects on the biosphere, therefore, should be negatively correlated to changes of the carbon dioxide level and cannot account for the observed positive correlation.
Gas exchange between air and water, however, has been firmly established to be thermally affected in the direction required to explain the observed positive correlation. Warm water holds less dissolved carbon dioxide than cold water. This means that a temperature increase by necessity will trigger an outflow of carbon dioxide from the hydrosphere to the air and implies that such outgassing must contribute to the multiannual fluctuations of the Keeling curve. The analyses and model studies now reported were undertaken to estimate the magnitude of that thermal effect and hence to obtain information on the extent to which thermal outgassing has contributed to the long-term trend of the Keeling curve.
Scheme 1 shows a simple two-box reaction system describing the main processes contributing to the relaxation of airborne carbon dioxide. The rate constant k characterises the transfer of carbon dioxide from the atmosphere to the hydrosphere; its reciprocal value stands for the turnover time of airborne carbon dioxide, as defined by the IPCC . The rate constant d characterises the reverse process of degassing of the hydrosphere and refers to the total amount of aquatic carbon dioxide, i. e. to the dissolved gas and its hydrated derivatives carbonic acid, bicarbonate ions and carbonate ions. The latter four species are assumed to equilibrate rapidly over the time-scales considered and, therefore, need not be treated individually.
Scheme 1. Kinetic two-box model for the exchange of carbon between the atmosphere
and the hydrosphere. Em(t) stands for the emission rate of anthropogenic carbon
If the atmospheric and hydrospheric carbon contents are denoted x and y, respectively, the law of mass action presccribes that their time-dependence will be governed by the simultaneous differential equations
dx/dt = Em(t) – k x + d (c – x) (1)
dy/dt = k x – d y (2)
where Em(t) represents the rate of emission of anthropogenic carbon dioxide to the atmosphere. Explicite sixth degree polynomial expressions for Em(t) were obtained by regression analysis of data reported by the Carbon Dioxide Information Analysis Center (CDIAC) for anthropogenic emissions of fossil carbon dioxide 1850–2009  and of emissions due to land-use changes 1850–2005. Emissions occurring prior to 1850 were assumed to be of negligible magnitude. Emissions due to land-use changes over the period 2006–2010 were put equal to 1.48 Giga ton carbon (GtC) per year, the mean value for the period 2001–2005 according to the CDIAC.
The rate constants k and d define the corresponding equilibrium constant Q as
Q = d/k (3)
According to the carbon cycle data presented by the Intergovernmental Panel on Climate Change (IPCC) , pre-industrial equilibrium values of x and y are given approximately by 600 GtC and 38 000 GtC respectively. The pre-industrial value of the equilibrium constant, therefore, was assumed to be given by Q’ = 600/38000.
The temperature dependence of the carbon exchange process was simulated with the assumption that the thermal effect derives from the degassing rate constant d and conforms to the Arrhenius equation
d = A Exp[–E*/RT(t)] (4)
where T(t) represents the absolute temperature, R the gas constant, and E* the activation energy of the degassing process. The temperature T(t) was described as a best-fit sixth degree polynomial in t, obtained by regression analysis of reported global temperature anomalies vs. time (normally the HadSST3-series ). These anomalies refer to a reference temperature of 14.0 ˚C , for which reason 287.15 K was added to convert them into absolute temperatures.
The time-course of x was evaluated by solving Eqns. (1–2) numerically for selected periods of time starting with the year 1850. The above preindustrial values of x, y and Q’ were taken to refer to that year and to the absolute temperature T(t)‘ = 286.76 K, which corresponds to the temperature anomaly of –0.39 ˚C that was found to apply for 1850 according to the above regression analysis. The proportionality constant A in Eqn. (4) was eliminated using year 1850 as a reference point, i. e. by using the relationship
A = k Q’/Exp[–E*/RT(t)‘] (5)
which can be readily derived from Eqns. (3–4). This means that Eqns. (1–5) constitute a kinetic model that defines the time-dependence of x using the rate constant k and the activation energy E* as the only adjustable parameters.
In climatology, the rate constant k usually is referred to in terms of its reciprocal value, the turnover time. The outcome of the model calculations, therefore, will be presented by stating what activation energy and turnover time they are based on.
The El Niño fluctuations of the Keeling curve
Fig. 1 describes the correlation between global temperatures and the atmospheric carbon dioxide level, as reported by Keeling for the period 1985–2000 . The El Niño event occurring around 1998 was exceptionally strong and caused a clearcut response in form of a corresponding single peak in the rate of change curve. The amplitude of that peak, therefore, should be exceptionally well suited for analysis of the strength of the temperature effect.
Figure 1. Data presented by Keeling  to illustrate the correlation between El Niño
increases of the temperature and rapid increases of the rate of change of the atmospheric
carbon dioxide level. The blue curve shows the longterm trend of the rate of change.
Kinetic analysis of the El Niño fluctuations
The kinetic model presented in the theory section is based on the assumption that the level of airborne carbon dioxide is controlled mainly by the rates of gas exchange between the atmosphere and the hydrosphere, which are determined by the atmospheric turnover time of the gas and the equilibrium constant for the exchange process. The model equations describe how the level of airborne carbon dioxide will be affected by anthropogenic emissions and temperature changes, when the thermal effects are attributed to the step of carbon dioxide outgassing from the hydrosphere and controlled by the activation energy E* for that step.
Fig. 2 illustrates some basic properties of the model predictions by example of results obtained for a fixed turnover time of 14 years and varied values of the activation energy E*. The featureless grey bottom curve in Fig. 2A was calculated for E* = 0. It provides evidence that the observed fluctuations of the rate of change of atmospheric carbon dioxide cannot be attributed to anthropogenic emissions and will not occur in the assumed absence of thermal outgassing.
Figure 2. Black curves iterate the empirical data in Fig. 1. Other curves in panel A
indicate the modelled rate of change, as calculated for a turn-over time of 14 years
and varied values of the activation energy E*. The blue curve i panel B represents the
temperature anomaly, as described by the polynoms used in the model calculations.
For non-vanishing values of E*, the model curves in Fig. 2A do exhibit temperature dependent fluctuations. These fluctuations invariably mimic the temperature curve (Fig. 2B), showing that the El Niño fluctuations of the Keeling curve indeed may derive from thermal outgassing. The modelled amplitude of the fluctuations is strongly dependent on (almost proportional to) the magnitude of the activation energy. E* = 83 kJ/mol/K (green curve) underestimates the amplitude of the observed fluctuations. E* = 330 kJ/mol/K (violet curve) overestimates the amplitudes. The best description of the amplitudes in general, and that of the 1998 El Niño peak in particular, was obtained for E* = 165 kJ/mol/K (red curve).
The model system was found to behave similarly when the turnover time was set to other values. For any postulated turnover time, a corresponding value of E* can be found that gives a satisfactory reproduction of the observed peak-to-throught amplitude of the 1998 El Niño peak of the rate of carbon dioxide change. This means that there is an infinite number of sets of joint turnover time and E* values that may account for the observed strength of the temperature effect by this criterion. Three representative sets of that kind are listed in Table 1.
Model predictions regarding the longterm trend of the Keeling curve
The different sets of joint turnover time and E* values that account for the amplitude of the 1998 El Niño peak result in widely different model predictions as to the carbon dioxide levels that will be reached due to temperature changes and anthropogenic carbon dioxide emissions since 1850. This is illustrated by Fig. 3, which shows that the modelled carbon dioxide level in 1985 (the midpoint of the time range presently covered by the Keeling curve) steadily increases with increasing values of the turnover time moiety of such parameter sets. The experimentally observed level in 1985 is reproduced only by the parameter set in which the turnover time equals 14 years (the middle set in Table 1). As shown in Fig. 4, this parameter set actually provides an almost perfect description of the entire Keeling curve. It also accounts satisfactorily for the carbon dioxide levels indicated by proxy values for the time period 1850–1958 .
Figure 3. Variation of the modelled carbon dioxide level in 1985 with the magnitude of
the turnover time moiety of parameter sets accounting for the amplitude of the 1998
El Niño peak. The experimentally observed level is reproduced only by the set where
the turnover time equals 14 years. .
Figure 4. Variation of the atmospheric carbon dioxide level since 1850, as indicated by
empirical data (green) and the model now described. Model curves were calculated
for the three parameter sets listed in Table 1.
Interrelationship between anthropogenic emissions and thermal outgassing
Perturbations caused by temperature changes on one hand and anthropogenic emissions on the other give rise to relaxation fluxes that, according to Scheme 1, must show a kinetic interdependence.
To obtain information on the significance of this interdependence, the model calculation resulting in the red curve in Fig. 4 was repeated, first with the hypothetical assumption that the temperature has remained unchanged since 1850 and then with the assumption that there has been no anthropogenic carbon dioxide emissions since 1850. In other words, the effect of each perturbation factor was examined separately in the assumed absence of the other.
The results are given in Fig. 5, where the blue curve shows how the level of atmospheric carbon dioxide level would be expected to vary due to thermal outgassing alone when the turnover time is 14 years and E* = 165 kJ/mol/K. The black curve shows the analogous response of the modelled carbon dioxide level to anthropogenic emissions alone. The separately calculated carbon dioxide increases indicated by the black and blue curves add up to the red curve in Fig. 4. The latter curve agrees within 0.4 ppm with the red curve in Fig. 3, i. e. with the curve describing the combined effect of the two perturbation factors.
Figure 5. Separately estimated effects of anthropogenic emissions and thermal
outgassing add up to a curve that is practically indistinguishabe from the model
curve obtained for the combined effects of the two perturbation factors
These observations provide evidence that the kinetic interdependence of anthropogenic emissions and thermal outgassing is practically negligible. The effects of these two perturbation factors seem to be largely additive, such that they can be estimated separately as indicated in Fig. 5.
Table 2 expresses the results in Fig. 5 numerically with regard to the relative importance of the two perturbation factors over some representative periods of time. The numerical data provide the main inference that anthropogenic emissions and thermal outgassing have been of approximately equal importance as contributors to the rising atmospheric carbon dioxide levels since 1850. During the last two modelled decades, thermal outgassing has contributed more than anthropogenic emissions to the long-term trend of the Keeling curve (the slope of the blue curve is larger than that of the black curve after 1990).
Carbon cycle budget considerations
Summing up the above results, 124 GtC (23% ) of the cumulative total amount of anthropogenic carbon dioxide emissions (531 GtC) since 1850 remained airborne in 2010. The rest (407 GtC) had been removed from the atmosphere. Concomitantly, 103 GtC of carbon dioxide had been released to the atmosphere through thermal outgassing driven by global warming.
With the reasonable simplifying assumption that the removed amount of carbon dioxide emissions has been taken up mainly in the hydrosphere, these data indicate that the net flux of carbon dioxide from the atmosphere to the hydrosphere has been 407–103 = 304 GtC during the examined time period (1850–2010). This corresponds to an average net flux of 1.9 GtC/year, consistent with the IPCC assessment that the net flux was 1.8 ± 0.8 GtC/year in the 1980s and 2.2 ± 0.4 GtC/year in the 1990s .
Effect of temperature on the Keeling curve
The reaction system defined in Scheme 1 includes the carbon dioxide pools and reaction steps that minimally have to be considered for a quantitative evaluation of thermal outgassing under realistic conditions. The model derived by strict kinetic analysis of this system shows that combined effects of temperature changes and anthropogenic emissions on the level of atmospheric carbon dioxide will be a function of two critical parameters (the carbon dioxide turnover time and the activation energy E* that is assumed to charaterise the strength of the temperature effect in accordance with the Arrhenius equation). Figs. 2–3 illustrate that the model can account for the observed amplitude of the 1998 El Niño peak of the rate of carbon dioxide change and the observed carbon dioxide level in 1985 when, and only when, the turnover time equals 14 years and E* = 165 kJ/mol/K.
It is not surprising that a fit of the model to two particular characteristics of the Keeling curve will provide best fit estimates of the model’s two adjustable parameters. But it is highly remarkable that the model for the parameter estimates thus obtained gives an almost perfect description of the entire Keeling curve (Fig. 4) and accounts satisfactorily for the El Niño fluctuations of the curve over the entire time period they have been recorded . This provides clear evidence that the model equations adequately describe how the examined two disparate and rather irregular perturbation events (temperature changes and anthropogenic emissions) affect the atmospheric carbon dioxide level.
Results now reported establish that the simple two-box system in Scheme 1 is sufficient to explain and quantitatively account for the El Niño fluctuations of the Keeling curve in terms of thermal outgassing. The observed fluctuations are by no means minor, but document the existence of a very strong temperature effect that drastically affects the observed rates of carbon dioxide change (see Fig. 1). Evidence has been previously presented which suggests that this temperature effect, due to global warming, has contributed most significantly also to the long-term trend of the Keeling curve [13,14]. The model data in Figs. 4–5 confirm that such is the case and provide quantitative estimates of the magnitude of the contributions.
The turnover time estimate of 14 years obtained by application of the model falls within the range of turnover time values determined experimentally by various methods . This lends credence to the model and supports the conclusion of Starr  that amplitudes of the seasonal fluctuations of the Keeling curve are too large to be consistent with turnover times of the order of 100 years. Table 1 lists representative sets of joint parameter values that account for the amplitude of the 1998 El Niño fluctuation. As illustrated by Fig. 4, such sets fail to reproduce the long-term trend of the Keeling curve when the turnover time moiety differs significantly from 14 years.
The model now described is the only one so far presented that is capable of reproducing both the long-term trend of the Keeling curve and the El Niño fluctuations of it. The excellent agreement between the observed and the modelled Keeling curve corroborates that carbon dioxide exchange between the atmosphere and the hydrosphere represents the largely predominant mechanistic factor controlling the air level of the gas. The combined effects of anthropogenic emissions and thermal outgassing, as now modelled, are sufficient to account for the observed increases of airborne carbon dioxide. This provides the additional inference that factors such as volcanism  and feedbacks involving the biosphere  are unlikely to have had any appreciable influence on the long-term atmospheric carbon dioxide level during the industrial era.
According to Fig. 5 and Table 2, thermal outgassing has been almost as important as, and during the last two decades more important than, anthropogenic emissions as a source of the increasing levels of atmospheric carbon dioxide. The large thermal contribution calculated for 2000–2009 might seem difficult to reconcile with the fact that global temperatures have not changed over that time period, but is fully consistent with the expected response of the system to previous temperature changes.
To understand that, one has to recognize that temperature changes perturb equilibrium conditions and trigger relaxation processes tending to adjust concentration variables to the new conditions. The effect of temperature on individual rate constants is instantaneous. This explains why the observed and modelled rate of change of carbon oxide (dx/dt) mimics the temperature curves without any time delay (Figs. 1–2). The consequent change of the carbon dioxide level (x), however, will be time-dependent and reflect the turnover time of the gas . A turnover time of 14 years implies that it will take several decades for the system to adjust to a temperature change. The large thermal contribution to the increased carbon dioxide levels over the period 2000–2009 derives mainly from the global warming that occurred during the preceeding two decennia.
The missing sink problem
The IPCC has presented the carbon budget data in Table 3 to account for the fate of anthropogenic carbon dioxide emissions in the 1990s . According to these data, the biosphere acts as a source of atmospheric carbon dioxide due land use changes. This leaves the hydrosphere as the only truly identified carbon dioxide sink. The estimated emissions, however, greatly exceed the estimated sum of carbon dioxide taken up by the hydrosphere and carbon dioxide remaining airborne. About one third of the emissions appears to have been taken up by an unidentified ‘missing sink’.
The missing sink problem was discussed by the IPCC in the first assessment report , where it was presented as a budget imbalance issue. In the fourth IPCC report, the name of the missing sink was changed to ‘residual terrestrial sink’. The missing sink still remained unidentified, however, and its contribution was still calculated from the imbalance between identified carbon dioxide sources and sinks. As stated by the IPCC in its first report , “the question therefore arises whether an important mechanism has been overlooked”.
The present investigation identifies thermal outgassing of carbon dioxide as the important mechanistic factor that has been overlooked. The IPCC budget in Table 3 is based on the unjustified presumption that long-term changes of the atmospheric carbon dioxide levels derive almost exclusively from anthropogenic emissions. The results in Fig. 5 indicate that thermal outgassing on the average has been an equally important contributor to the increasing carbon dioxide levels.
According to the model data in Table 2, anthropogenic emissions and thermal outgassing have contributed, respectively, 1.5 and 2.3 GtC/year to the increase of airborn carbon dioxide during the 1990s. This adds up to 3.8 GtC/year, agreeing with the increase indicated by the Keeling curve . The IPCC budget in Table 2 presumes that this atmospheric increase represents the amount of anthropogenic emissions remaining airborne. Such attribution of the increase to the less important one of the two contributing perturbation factors leads to an underestimation of the rate at which the emissions actually have been removed from the atmosphere.
The model results now reported show that the part of the emissions that does not remain airborne (6.5 GtC/year in the 1990s) has been taken up by the hydrosphere, the only sink considered in the model because of its predominance. This indicates that the IPCC budget in Table 2 underestimates the rate of uptake of emissions by the hydrosphere. The budget estimate is based on the unjustified presumption that empirically determined net fluxes of carbon dioxide from the atmosphere to the hydrosphere provide direct measures of the rate of uptake of anthropogenic emissions. In reality, the observed fluxes must be assumed to represent the net outcome of emission uptake and thermal outgassing, which affect the atmospheric carbon dioxide level in opposite directions. A large part of the uptake of anthropogenic emissions will be hidden by thermal outgassing during periods characterised by long-term global warming.
This would appear to solve the missing sink problem. The major part of the ‘residual terrestrial sink’ can be identified as the hydrosphere, the sink capacity of which has been underestimated due to neglect of thermal outgassing. The results in Fig. 5 and Table 2 establish that no carbon budget imbalances are encountered when the combined effects of emissions and temperature are evaluated using a model that takes thermal outgassing into account and that is based on empirically estimated kinetic parameter values for the relaxation processes.
Thermal outgassing and climate modelling
Carbon dioxide is a greenhouse gas. Climate model calculations of future effecs of anthropogenic carbon dioxide emissions require estimates of future concentrations of anthropogenic carbon dioxide as input data. Such estimates invariably have been calculated for postulated emission scenarios using carbon cycle models which presume that the observed increases of airborne carbon dioxide derive exclusively from anthropogenic emissions. Results now presented indicate that such models gravely underestimate the rate of removal of anthropogenic carbon dioxide from the air and render the climate model predictions unacceptably biased already at the level of data input.
According to Table 2, anthropogenic emissions account only for about half of the observed increase of atmospheric carbon dioxide during the twentieth century. The other half derives from thermal outgassing of the hydrosphere. This supports the view that the increased carbon dioxide levels to a large extent have been a consequence, rather than the source, of global waming.
The strong temperature effect on the atmospheric carbon dioxide level implies that calculations of future carbon dioxide levels not only require assumptions about future emission rates, but also about future temperatures. Climate modellers are facing the delicate problem that they have to know what the future temperatures will be before they can predict them by calculation of the greenhouse effect caused by future carbon dioxide levels. This complication can possibly be circumvented by extensive modification of the calculation procedures, but it certainly cannot be ignored.
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