Numerical studies have indicated that the steady-state ocean-atmosphere partitioning of carbon will change profoundly as emissions continue. In particular, the globally averaged Revelle buffer factor will first increase and then decrease at higher emissions. Furthermore, atmospheric carbon will initially grow exponentially with emission size, after which it will depend linearly on emissions at higher emission totals. In this article, we explain this behavior by means of an analytical theory based on simple carbonate chemistry. A cornerstone of the theory is a newly defined dimensionless factor, O. We show that the qualitative changes are connected with different regimes in ocean chemistry: if the air-sea partitioning of carbon is determined by the carbonate ion, then the Revelle factor increases with emissions, whereas the buffer factor decreases with emission size, when dissolved carbon dioxide determines the partitioning. Currently, the ocean carbonate chemistry is dominated by the carbonate ion response, but at high total emissions, the response of dissolved carbon dioxide takes on this role.
 Due to concern about the potentially significant impact of global warming in the foreseeable future, it is of crucial importance to understand how much of the anthropogenic carbon dioxide is likely to accumulate in the atmosphere and how much will be taken up by the oceans. This issue was initially considered by Revelle and Suess  who suggested that most of the carbon dioxide would be absorbed by the oceans within decades after emission. The ocean uptake of carbon dioxide is complicated by the fact that it reacts with dissolved carbonate ions to form bicarbonate. Such chemical effects were considered in a thorough analysis by Bolin and Eriksson  which led to the conclusion that only a small fraction of anthropogenic carbon can be absorbed on a decadal timescale. Furthermore, Bolin and Eriksson  explicitly defined what is now usually referred to as the ‘Revelle buffer factor’: the relative change in atmospheric carbon, given a relative change in oceanic carbon. An explicit expression for this buffer factor in terms of oceanic carbon was given by Sundquist et al. . Further analysis was provided by Volk and Hoffert  who introduced the concept of carbon pumps. However, after numerical models of the oceanic carbon cycle were first used [Knox and McElroy, 1984; Sarmiento and Toggweiler, 1984; Siegenthaler and Wenk, 1984], ocean uptake of carbon has mainly been investigated by means of simulations. Although this approach has provided valuable insights [Fasham, 2003], it seems to have diverted the attention from the development of conceptual models and basic, analytical theory.
 The analytical approach was taken up again by Ito and Follows  who derived a first-order differential equation describing the impact of the organic carbon pump on the atmospheric carbon content, but they assumed a constant Revelle buffer factor. This assumption was relieved by Goodwin et al.  who found an approximate solution to the differential equation by introducing a quantity called the ‘buffered carbon’ IB. Under the assumption that IB is constant (which is approximately the case over a large range of atmospheric pCO2), it was shown that the atmospheric carbon content depends exponentially on the total amount of carbon in the ocean-atmosphere system. Marinov et al. [2008a, 2008b] then accounted for small variations in IB, expressing atmospheric pCO2 as a sum of exponentials.
Ito and Follows  and Marinov et al. [2008a, 2008b] studied changes in the organic carbon pump that yield relatively modest variations in the atmospheric carbon content. Much larger changes are expected to result from current and future anthropogenic carbon emissions. With respect to the oceanic uptake of anthropogenic carbon, three crucial time periods can be distinguished [Archer et al., 1997]: (1) an initial period where CO2 emissions are ongoing and air-sea equilibrium has not been reached which is the present situation; (2) a quasi-steady state, where emissions have ceased and ocean and atmospheric carbon reservoirs are equilibrated though the fluid system has not yet come into equilibrium with the carbonate sediments. This period would extend for several thousands of years; and (3) a final equilibrium where, after many tens of thousands of years, weathering and sediment interactions have occurred altering the total ocean-atmosphere carbon budget. A recent study by Goodwin and Ridgwell  has focused on ocean-atmosphere partitioning of carbon during the last phase; centennial/millennial timescales (phase 2) were investigated by Goodwin et al. [2007, 2008, 2009] and by Egleston et al. . The results of the centennial/millennial simulations by Goodwin et al. indicate that large emissions of carbon lead to regime shifts in the air-sea carbon partitioning. To a good approximation, atmospheric pCO2 turns out to depend exponentially on total carbon emissions up to total emissions of 5000 Gt (consistent with the assumption of a constant IB); above 10000 Gt, the relationship becomes linear. Furthermore, the globally averaged Revelle buffer factor increases with emission size up to 8000 Gt, reaching a maximum of 18; for higher total emissions, it decreases again.
Egleston et al.  defined a number of different buffer factors (some of which are equivalent to factors derived by Frankignoulle ), quantifying the change in atmospheric CO2, oceanic pH, and carbonate saturation state (Ω) with increasing oceanic DIC concentrations. A regime shift is also suggested by the behavior of these new buffer factors: they all reach an extremum approximately at the point where the DIC concentration becomes equal to the ocean alkalinity.
 In this article, we focus on explaining the different regimes identified by Goodwin et al. and Egleston et al. by formulating a complementary theory based on elementary carbonate chemistry. The regimes turn out to be related to the compound dominating the response of the ocean carbon chemistry: at relatively low emissions, this compound is CO32−, whereas at higher emissions, dissolved CO2 becomes more important.
 We make the following general assumptions:
 (1) Both the ocean and the atmosphere are assumed to be well-mixed boxes. Carbon is fully equilibrated between these compartments, but there has not (yet) been exchange of carbonate with the sediments. This implies that we are concerned with centennial to millennial timescales, or time period (2) as defined above.
 (2) The oceanic temperature and preformed alkalinity, as well as the strength of the carbon pumps and the land biosphere, are assumed constant. However, any change in either the carbon pumps or carbon in the terrestrial biosphere can simply be formulated as a change in the total carbon inventory and thus, its impact can be predicted.
 In the following section, we will introduce a new quantity, analogous to the well-known Revelle factor, which reveals useful insights.
2. Ocean-Atmosphere Carbon Partitioning
 The oceanic carbonate chemistry can be characterized by means of two quantities [Chester, 2000]: the DIC concentration (C) and the total alkalinity (A). Throughout this article, we will consider situations with pH ≈ 6–9 where the main bases in the ocean are [HCO3−] and [CO32−] and we will therefore identify A with the carbonate alkalinity. This inevitably leads to imprecision (since bases such as OH− and B(OH)4− constitute a few percent of total alkalinity), but it will turn out that we can explain well the general features of the ocean-atmosphere carbon partitioning regimes using this approximation. The impact of this and other approximations that we make in the calculations are discussed in section 3. Thus, we define C and A as:
 The relationship between the atmospheric pCO2 and C at the ocean surface is commonly described by means of the Revelle buffer factor (R) defined as:
If R is constant, then the relationship between pCO2 and C is given by a simple power law:
Unfortunately, R is not constant: it varies between approximately 8 and 15 at the ocean surface [Watson and Liss, 1998]. Furthermore, the globally averaged Revelle buffer factor depends strongly on the total amount of carbon in the ocean-atmosphere system [e.g., Goodwin et al., 2007]. We now derive an alternative index that is more constant than R.
2.1. A New Quantity to Describe Ocean Carbon Chemistry
 In the ocean, the main chemical reactions involving carbon are dissolution of carbon dioxide from the atmosphere, the subsequent formation of carbonic acid H2CO3 by reaction with water, dissociation of H2CO3 into bicarbonate HCO3− and H+, and dissociation of HCO3− into carbonate CO32− and H+ [Bolin and Eriksson, 1959; Broecker and Peng, 1982]. If the air-sea partitioning of carbon dioxide as well as the carbonate reaction system are at equilibrium, we have:
where KH, K0, K1, K2 are dissociation constants. Eliminating [H+] from the above equations, one obtains:
Here, K is a composite chemical equilibrium constant equivalent to .
 Generally, there are much larger relative variations in [CO32−] than in [HCO3−]. Hence, to a fair approximation, [HCO3−] is constant which implies KpCO2 ∝ [CO32−]−1: a power-law dependence of pCO2 on [CO32−], rather than on C. However, because [HCO3−] is not exactly constant, it is suggestive that a more accurate way of formulating the power-law relationship between pCO2 and [CO32−] would be:
In the situations considered in this paper, K is constant, which means that we can also write:
To find an explicit expression for O, we express relationship (3) in terms of the carbonate alkalinity A and [CO32−]:
which can be rewritten as:
Alkalinity is conserved when carbon is exchanged between the ocean and the atmosphere. We can thus assume constant A when describing the ocean-atmosphere partitioning of carbon as carbon dioxide is added to the atmosphere. After differentiating equation (6) with respect to ln([CO32−]) and resubstituting [HCO3−] +2[CO32−] for A, we arrive at:
Because is only about 0.1, it can easily be seen that O must be almost constant. O varies between about 1.2 and 1.6 at the surface of the World Ocean. This is illustrated in Figure 1 where we show O and R as a function of latitude along a transect at 170°W in the Pacific. Most notably, R increases, whereas O decreases towards higher latitudes. Moreover, the relative variation as a function of latitude is more than twice as large for R than it is for O: the ratio of the standard deviation of O to its median value equals 0.11, whereas this ratio is 0.25 for R.
 Another important advantage of O is that it is a simple function of [CO32−] and [HCO3−]. Therefore, the dependence of O on oceanic carbon and alkalinity is transparent. O decreases monotonically with increasing DIC concentration, because decreases monotonically; O increases monotonically with increasing alkalinity, because increases monotonically. O has a simpler and more transparent behavior than both R and IB, particularly when a large range of carbon emissions is considered. As mentioned in the Introduction, R changes strongly and non-monotonically if more carbon is added to the ocean-atmosphere system. The buffered carbon, IB, has the advantage of staying rather constant up to a total emission of 5000 Gt. However, O remains at approximately the same value, even if the total emission far exceeds 5000 Gt: it is about 1.4 in the pre-industrial situation, and it approaches 1, when total carbon emissions become very high.
with KCa a solubility constant. Now, [Ca2+] stays constant when carbon is exchanged between the atmosphere and the ocean, and therefore:
Thus, can be interpreted as the relative decrease in the ocean carbonate saturation state with increase of pCO2.
2.2. Dependence of the Revelle Factor on Oceanic Carbon
 Simulations presented by Goodwin et al.  have shown that the Revelle buffer factor R increases as carbon is added to the ocean-atmosphere system up to total emissions of about 8000 Gt C; at higher emissions, it decreases again. We can use O to understand why this is the case. We reformulate R in terms of [CO2], [CO32−], O, and the DIC concentration C:
Now, we use C = A − [CO32−] + [CO2], so with A constant:
 Changes in the minor species [CO32−] and [CO2] are relatively much larger than in the total C and therefore, the behavior of R is dominated by the denominator term + [CO2]. If ≫ [CO2], then R ≈ . This means that as long as O can be considered constant (i.e., [CO32−] ≪ [HCO3−]), R must increase with increasing pCO2, since [CO32−] decreases with increasing pCO2. If, on the other hand, [CO2] ≫ , then R ≈ which means that R decreases with increasing pCO2, because [CO2] increases with increasing pCO2. In the intermediate regime, R must attain a maximum. In this range, most DIC is in the form of [HCO3−]; thus, C ≈ A which suggests that we can write:
If the denominator reaches a minimum, then R reaches its maximum value. To find the location of the maximum of R, we can therefore write:
which leads to
Using that [HCO3−] ≈ A and that pCO2 = KH [CO2], we can rewrite equation (3) into:
with Rmax the maximum value that R attains. Before we consider the above relationship in a quantitative sense, we need to realize that O is not entirely constant. In fact, at the relatively high carbon concentration for which R reaches its maximum, O must be lower than in the pre-industrial ocean. To estimate O, we use relationships (7) and (13); we again assume that [HCO3−] ≈ A:
Filling in this value for O in equations (13) and (14), we find that when R is maximized, Rmax = 19 and [CO32−] = 75 μM, [CO2] = 59 μM, pCO2 = 1540 ppm.
 The maximum in R can be understood qualitatively from a change in the relative abundances of carbonate and carbon dioxide which is illustrated in Figure 2. At low total carbon, there is much more carbonate than carbon dioxide in the water; carbon entering the ocean from the atmosphere is essentially used to neutralize carbonate into bicarbonate. The resulting decrease of the carbonate concentration leads to an increase of the Revelle buffer factor. As more carbon is added to the ocean, there are fewer carbonate ions left to be neutralized; at some point, the extra carbon entering the ocean mostly stays in the form of carbon dioxide. In this regime, the Revelle factor decreases because of the increasing [CO2]. The different regimes of R are illustrated in Figure 3a.
 Another way of looking at this is by considering the bicarbonate fraction of DIC. According to Figure 2, this fraction attains a maximum at a DIC concentration of about 2.4 mM. Now, O ≈ 1, therefore:
If the bicarbonate fraction of DIC, equal to , reaches a maximum, then 1 − reaches a minimum which in turn implies that R attains a maximum.
 Recently, Egleston et al.  defined new buffer factors that behave in a way analogous to R as a function of the oceanic carbon concentration. We discuss some of these factors in Appendix B.
2.3. Carbon Partitioning in the Ocean-Atmosphere System
 Now we will apply the theory to derive the dependence of atmospheric pCO2 on total carbon in the ocean-atmosphere system. We will show that depending on the relative sizes of two different terms, three regimes can be distinguished, of which two were previously identified by Goodwin et al. . We start by writing down a balance equation for carbon in the ocean-atmosphere system (as given by Ito and Follows ):
with M the total gas content of the atmosphere (mol), V the volume of the ocean (l), C the average oceanic concentration of preformed dissolved inorganic carbon, Creg the regenerated carbon concentration (M), and Ct the total carbon content of the ocean-atmosphere system (m). Using that C = A − [CO32−] + [CO2] (where A is the preformed alkalinity averaged over the ocean) and that [CO2] and pCO2 are simply related through Henry's law, i.e. pCO2 = KH [CO2], we can write:
Now, we differentiate equation (18) with respect to Ct, while assuming A and Creg constant and using that = :
Using the definition of O, we can rearrange to obtain:
The term (M + )pCO2+ V is equivalent to the ‘buffered carbon’ IB introduced by Goodwin et al. , so we can write:
 Pleasingly, IB can be reinterpreted in this framework. It consists of two terms; one is proportional to the atmospheric and oceanic carbon dioxide concentrations, the other one is proportional to the oceanic carbonate ion concentration. Based on the relative importance of the terms, three regimes can be distinguished, the ranges of which are given in Tables 1 and 2:
Table 1. Regime Changes From Theory With Modern Titration Alkalinity
Table 2. Parameter Values
mol l−1 atm−1
 1) The low-carbon ocean-atmosphere system; we do not know of any geological period during which atmospheric and oceanic carbon were so low that this regime was relevant, but we have added it for completeness. In this regime, the atmospheric and oceanic carbon dioxide concentrations are low, the oceanic carbonate concentration is high and the partitioning is determined by the behavior of the carbonate ion. Specifically, (M + )pCO2 ≪ V, so equation (18) can be written as:
which can be rearranged into:
In this regime, we cannot suppose that O is constant, because 4 is rather large. However, an alternative quantity H ≡ (see Appendix A) remains quite close to 1 and therefore, we will assume that this factor is constant. We now combine the relationships [CO32−] = and pCO2 ∝ [H+]H with the above equation to find that
Hence, pCO2 appears to have a (complicated) power-law type of dependence on total carbon Ct in this regime.
 2) The intermediate regime (the current situation) where both carbon dioxide and carbonate play a role. Specifically, ≈ 0, so IB is approximately constant. As derived by [Goodwin et al., 2007], this implies an exponential dependence of pCO2 on total system carbon:
 3) The high-carbon ocean-atmosphere system where the oceanic carbonate ion concentration is negligible. This regime may have been relevant during geological periods with very high carbon levels such as the Cretaceous and the Eocene [Huber et al., 2000]. Specifically, (M + )pCO2 ≫ V; hence, IB ≈ (M + )pCO2. From equation (19), it can be seen that this leads to
which implies a linear relationship between pCO2 and Ct. According to equation (22), = 4.35*10−15 ppm/mol which corresponds to 0.362 ppm/Gt C. The simulations by Goodwin et al. suggest a slope in the linear regime of about 0.356 ppm/Gt C [Goodwin et al., 2007, Figure 5]. The exponential and linear regimes are illustrated in Figure 3b.
 In this article, we have developed a simple analytical theory to explain how the Revelle buffer factor (and new buffer factors defined by Egleston et al. ) as well as the ocean-atmosphere partitioning of carbon depend on total carbon. It complements earlier developments [Goodwin et al., 2007; Marinov et al., 2008b; Egleston et al., 2010] and it is consistent with the general features found from numerical simulations [Goodwin et al., 2007, 2008, 2009]. Both suggest that the Revelle factor increases with total carbon until it reaches a maximum, after which it starts to decrease. Furthermore, both the simulations and this framework predict that the atmospheric carbon dioxide concentration will first increase exponentially and then linearly with emissions. The maximum in the Revelle factor can be explained from the neutralization of dissolved carbonate into bicarbonate, combined with an increase in dissolved carbon dioxide, as more carbon enters the ocean. Once the carbonate concentration has become negligible, practically all carbon entering the ocean remains in the form of carbon dioxide; at this point, the atmospheric CO2 concentration starts to increase linearly with emissions. Thus, the anthropogenic carbon emissions can be thought of as driving a regime change in the oceanic carbon chemistry.
 There are some minor notable differences between the predictions from this theory and the Goodwin et al.  simulation results, especially concerning the behavior of the Revelle buffer factor. In fact, the theory suggests that this factor has its maximum at pCO2 = 1540 ppm, whereas the simulations suggest that the maximum of R rather occurs at pCO2 = 1800 ppm. Furthermore, we predict that Rmax = 19, whereas it is only 18 according to the simulations. Both these differences may be explained from our assumption that the carbonate alkalinity is equal to the total alkalinity since we did not take bases other than (bi-)carbonate into account (the most important one being B(OH)4− throughout the regime that we have considered). If we assume that the concentration of such bases scales linearly with the carbonate concentration, then the -term in the denominator of R is effectively multiplied by a value larger than 1. This means that R becomes slightly lower at every value of pCO2 (but especially in the low-carbon regime), and that [CO2], and thus pCO2, must be slightly higher at the maximum of R. In the same vein, one could argue that the V-term in IB should effectively be multiplied by a factor larger than 1. The effects on the different regimes defined in section 2.3 would be rather modest: the intermediate-carbon regime (where pCO2 depends on total carbon exponentially) would begin and end at slightly higher values of pCO2. At very high or very low pH (very low and very high pCO2, respectively), our assumption that the carbonate alkalinity is equal to the total alkalinity is certainly not valid: at pH < 5, [H+] accounts for a significant portion of total alkalinity, whereas at pH > 9, [OH−] becomes important.
 In addition, we made two further assumptions impacting our calculation of Rmax: we implicitly supposed that C is constant by assuming C ≈ A (equation 10) and we took O constant when calculating derivatives with respect to lnpCO2 in (11). Both these assumptions primarily affect the value of pCO2 for which R reaches its maximum. In section 2.2, we basically analyzed the denominator of R instead of R itself and we assumed that when the denominator reaches its minimum, R has its maximum. The fact that C increases with increasing pCO2 implies that when the denominator of R reaches its minimum, R will still be increasing. Therefore, R must actually have its maximum at a slightly higher pCO2 than we estimated. The fact that O decreases with increasing pCO2 implies that is less negative than if O were constant which in turn means that the denominator of R reaches its minimum for a lower value of pCO2. Hence, the effects of these two assumptions work in different directions: the constant C tends to give a too low estimate of pCO2 at Rmax, whereas the constant O tends to yield a too high value of pCO2 at Rmax.
 Overall, the ocean-atmosphere partitioning of carbon appears to be described well across a very wide range of regimes by a theory based on simple carbonate chemistry. The quantity O has turned out to be particularly useful for a number of reasons. First of all, it is much more constant than the Revelle buffer factor that has been extensively applied in theories of the ocean-atmosphere carbon partitioning. Second, O behaves transparently as a function of the total carbon in the ocean-atmosphere system: if total carbon increases, then O decreases, because the ratio goes down. Even at very high total emissions, the order of magnitude of this quantity remains the same: in the pre-industrial situation, O is about 1.4, whereas it approaches 1, if total carbon becomes exceedingly high. Hence, the framework presented appears to have a number of advantages: relative simplicity, transparency and applicability to different regimes. Therefore, we expect that it will turn out useful for obtaining a deeper understanding of various topics in ocean biogeochemistry, first of all the dependence of pCO2 on temperature and alkalinity. The framework presented here may also be useful for determining which carbon regime (exponential or linear) the system displayed during past geological periods, by using atmospheric pCO2 [Royer et al., 2001] and carbonate chemistry [Ridgwell, 2005] considerations.
Appendix A:: Factor H
 We define the quantity H as follows:
The above definition implies that if H can be considered constant, then:
To evaluate (A1), we rewrite the expression for pCO2(3) in terms of A and [H+], using relationship (2) between [H+], [HCO3−], and [CO32−]:
resubstitution then leads to:
As long as [CO32−] ≪ [HCO3−], H behaves qualitatively very similar to O: it is slightly above 1, approaching 1 as total carbon increases. However, in the low-carbon regime, O becomes large, whereas H remains close to 1, approaching 2, if [CO32−] ≫ [HCO3−]. Thus, H is more constant than O which can be considered as an advantage for using H over using O. On the other hand, O is more directly related to the most important compounds in the oceanic carbon system which makes analysis based on O more straightforward than analysis based on H.
Appendix B:: Behavior of Other Buffer Factors
Egleston et al.  defined six buffer factors: (γDIC), (βDIC), (ωDIC) at constant alkalinity, and (γAlk), (βAlk), (ωAlk) at constant DIC. Because our paper is concerned with what happens to the ocean carbon system as carbon is added to the ocean-atmosphere system, but alkalinity stays constant, we will only consider the constant-alkalinity buffer factors γDIC, βDIC, and ωDIC. In fact, γDIC is equal to the denominator term of R which we already analyzed. Furthermore, because of the definition of Ω:
As can be seen in Figure 2 of Egleston et al. , −ωDIC ≈ γDIC at high oceanic DIC concentrations which can be understood from the fact that O ≈ 1 in that regime. With decreasing C, the ratio of −ωDIC to γDIC becomes larger, because O increases with decreasing C. To relate βDIC to γDIC, we will use H, defined in (A1):
The relationship between βDIC and γDIC is qualitatively very similar to the relationship between γDIC and ωDIC: at high DIC, βDIC ≈ γDIC, since H ≈ 1; as DIC decreases, H increases and therefore, the ratio of βDIC to γDIC increases. However, βDIC is always smaller than −ωDIC, because H is always smaller than O.
 We would like to thank Stephanie Dutkiewicz, David Ferreira, George van Voorn, Ric Williams, and two anonymous reviewers for helpful discussions and comments. A.W.O. was financially supported through a Rubicon fellowship provided by the Netherlands Organisation for Scientific Research (NWO), P.G. was supported by the U.K. Natural Environment Research Council through the QUEST feedbacks project NE/F001657/1, and M.J.F. is grateful for support from NOAA.