A theory for the ocean-atmosphere partitioning of anthropogenic carbon dioxide on centennial timescales is presented. The partial pressure of atmospheric CO2 (PCO2) is related to the external CO2 input (ΔΣC) at air-sea equilibrium by: PCO2 = 280 ppm exp(ΔΣC/[IA + IO/R]), where IA, IO, and R are the pre-industrial values of the atmospheric CO2 inventory, the oceanic dissolved inorganic carbon inventory, and the Revelle buffer factor of seawater, respectively. This analytical expression is tested with two- and three-box ocean models, as well as for a version of the Massachusetts Institute of Technology general circulation model (MIT GCM) with a constant circulation field, and found to be valid by at least 10% accuracy for emissions lower than 4500 GtC. This relationship provides the stable level that PCO2 reaches for a given emission size, until atmospheric carbon is reduced on weathering timescales. On the basis of the MIT GCM, future carbon emissions must be restricted to a total of 700 GtC to achieve PCO2 stabilization at present-day transient levels.
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 Given current anthropogenic emissions of CO2, it is unclear how the additional carbon will be partitioned between the atmosphere and ocean. If changes in the terrestrial and ocean biospheres are ignored, there are three crucial time periods [Archer et al., 1997]: (1) an initial period where CO2 emissions are ongoing and air-sea equilibrium has not been reached, lasting several hundred years, 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 tens of thousands of years; and (3) a hypothetical final equilibrium where, after many tens of thousands of years, weathering and sediment interactions have occurred altering the total ocean-atmosphere carbon budget.
 In this study, a clear analytical theory is developed to describe atmospheric carbon partitioning when time period 2, air-sea equilibrium, is reached. The theory is tested and compared to two previously applied analytical approaches using a hierarchy of ocean models which include explicit methods of solution of the full system carbonate chemistry equations and a relatively realistic, three-dimensional ocean circulation and biogeochemistry model. The theory developed here is found to be more accurate than the previously used techniques and the reasons for this improvement are identified. The theory provides a clean and accurate approachfor representing the partitioning of fossil fuel carbon dioxide in climate and earth system modeling studies.
 The partitioning of carbon dioxide in the atmosphere and ocean is at equilibrium when there is no annual area-integrated flux of CO2 across the air-sea boundary. The flux of CO2 across the air-sea interface is controlled by the difference in partial pressures of carbon dioxide between both sides of the interface. The partial pressure of CO2 in the atmosphere is linearly related to the atmospheric inventory of carbon dioxide. However, on timescales longer than several years, the ocean carbon inventory also needs to be considered. In the ocean, CO2 exists as dissolved inorganic carbon (DIC, concentration CDIC) which comprises carbonate and bicarbonate ions, as well as charge neutral CO2 and carbonic acid. The partial pressure of CO2 in the ocean is dependent only on the contribution of the charge neutral fraction. As CO2 is added to the air-sea system, the ocean partitioning of DIC alters (Figure 1), making finding the equilibration state nontrivial. In this study an idealized theory is developed which relates changes in the atmospheric partial pressure, ΔPCO2, to changes in the carbon inventory of the atmosphere-ocean system, ΔΣC. This theory is shown to accurately and succinctly describe more complex models where the full carbonate system has been solved numerically [Follows et al., 2006].
 Our approach complements the study of Ito and Follows , which used an analogous approach to describe the relationship between atmospheric pCO2 and the efficiency of the ocean's soft tissue biological pump. Here, instead, the effect of carbon budget changes on carbon partitioning is investigated, assuming ocean circulation and biological pumps are constant. Three computational models are employed: firstly a two-box ocean model, in which globally averaged carbon parameters can be applied exactly; secondly a three-box ocean model with a simple meridional circulation is used; and thirdly the MIT general circulation model (GCM). All models represent a soft tissue biological sequestration of carbon in the ocean that is assumed to remain constant with time.
 In section 2.1 a simple atmosphere-ocean system at equilibrium is considered where small perturbations are added to the total system carbon. An integral relation between ΔPCO2 and ΣC is found. In section 2.2 three analytical methods, two previously used and one developed here, of solving this integral relation are analyzed and tested against two-box and three-box ocean models. The most accurate analytical method, a constant buffered carbon framework, is found to predict ΔPCO2 to within 6% while emissions remain below 5000 GtC. In section 3, the three methods of solving the integral relation between ΔPCO2 and ΣC are tested against a general circulation model. Again the constant buffered carbon framework is found to be a more accurate representation of model output than the other methods. In section 4 the wider implications of the study are discussed.
2. CO2 Relations
2.1. Air-Sea Carbon Dioxide Partitioning
 Changes in atmospheric CO2 are investigated from a pre-industrial starting point, with anthropogenic emissions changing the total carbon inventory. Firstly some basic carbonate chemistry is reviewed and, secondly, a perturbation to the total air-sea carbon inventory is introduced.
2.1.1. Basic Carbonate Chemistry
 The partial pressure of CO2 in the atmosphere is linearly related to the atmospheric inventory of carbon dioxide. However, carbon is exchanged between the atmosphere and ocean on a timescale of years so that the ocean inventory needs to be considered. In the ocean, carbon exists as dissolved inorganic carbon (DIC, concentration CDIC), which comprises carbonate and bicarbonate ions as well as charge neutral CO2 and carbonic acid,
where the bulk of DIC in the present day ocean comprises bicarbonate ions (Figure 1). The partial pressure of carbon dioxide in the ocean is dependent only on the contribution of charge neutral species,
where [CO′2] is the combined concentration of CO2(aq) and H2CO3(aq), and K0 is a function of temperature and salinity. The air-sea partitioning of CO2 for a uniform ocean model is controlled by the relative concentrations of the dissolved inorganic carbon constituents. As CO2 is added into the ocean, it reacts with water to form carbonic acid and then dissociates into carbonate and bicarbonate ions,
 Reactions (3b) and (3c) reach a steady state according to: K′1 = [H+] and K′2 = [H+], where K′1 and K′2 are functions of temperature and salinity. As charge neutral CO2 is added to the ocean and dissociates the concentration of H+ increases (Figure 2b). Thus the steady state ratios of [HCO3−]/[CO′2] and [CO32−]/[HCO3−] must decrease, resulting in a greater proportion of DIC being in the forms CO2(aq) and H2CO3(aq). Thus adding charge neutral carbon dioxide to seawater causes a larger fractional change in CO′2 and PCO2 than the fractional change in dissolved inorganic carbon (Figure 2a). The sensitivity of changes in PCO2 to changes in CDIC is measured by the Revelle buffer factor, R,
which varies with temperature, salinity and alkalinity, and is on the order of 10 for present day ocean waters. As carbon dioxide is added to a pre-industrial air-sea system with a well-mixed ocean (temperature 15°C, salinity 34.7 psu, alkalinity 2.4 molar equivalence m−3), R increases from 10 for the pre-industrial case to a maximum value of around 18 when emissions total around 8000 GtC (Figure 2c). As emissions rise above 8000 GtC the value of R then falls until it reaches a value of about 8 when emissions reach 25,000 GtC.
 In reality the value of R is not constant across the ocean's surface, since R is temperature, salinity and alkalinity dependent. In addition DIC concentrations are not entirely set by carbonate chemistry, but are also determined by the biological pump and the distance from saturation with the atmosphere [Ito and Follows, 2005]. More generally the Revelle buffer factor can be reexpressed as a global average,
where is the global average DIC concentration. The use of Rglobal allows for the consideration of spatial variations in water temperature, salinity and alkalinity, biological pumps and water masses not being at saturation.
2.1.2. Introducing a Perturbation to the Atmosphere-Ocean System
 Consider an atmosphere, with atmospheric carbon dioxide partial pressure PCO2, in steady state with an ocean with average dissolved inorganic carbon concentration . The total amount of carbon in the system, ΣC, is given by
where V is the volume of the ocean, and M is the molar volume of the atmosphere and IA and IO are the atmospheric and oceanic carbon inventories respectively. Adding an infinitesimal change to total carbon, δΣC, leads to a change in PCO2 given by the identity
and the change in δΣC is given by
 Expanding the change in total system carbon in the identity (7) gives
which may be rewritten by multiplying through by PCO2/δPCO2, and using the definition of Rglobal to give
 Thus the infinitesimal change in PCO2 is related to the infinitesimal addition of carbon with the response controlled by the background PCO2 and the buffered amount of carbon in the system, IA + IO/Rglobal,
 The buffered amount of carbon represents the available CO2 for redistribution between the atmosphere and ocean. An addition of carbon to the system reaches a steady state CO2 partitioning between the atmosphere and ocean as if there were IA moles of carbon in the atmosphere and IO/Rglobal moles of nondissociating carbon in the ocean. For a hypothetical air-sea system, in which all oceanic DIC exists in the form CO2(aq), a steady state is reached when an addition of carbon is partitioned between the atmosphere and ocean in the same ratio as the existing carbon inventories (IA and IO). Conversely in the real air-sea system, in which oceanic DIC dissociates between different forms, a steady state is reached when an infinitesimal carbon emission is partitioned between the atmosphere and ocean in the same ratio as the existing atmospheric inventory (IA) and the effective nondissociating oceanic inventory (IO/Rglobal). Rewriting (10b) in integral form gives
 The variables in (11) cannot be separated to integrate without making assumptions or approximations. This leaves two methods for solving the steady state air-sea partitioning of carbon emissions, (11): (1) Use a numerical model in which the carbonate chemistry is explicitly solved, or (2) make assumptions in order to simplify, and integrate, equation (11). In the next section box model representations of the atmosphere-ocean system, with explicit solutions of the carbonate chemistry system [Follows et al., 2006], are compared to three methods for approximating an analytical solution to the relationship between steady state PCO2 and ΔΣC (equation (11)).
2.2. Comparing Approximations for Relating PCO2 to ΣC With Numerical Models
 Three analytical methods for approximating a relationship between ΔPCO2 and ΔΣC are now discussed, and compared to the numerical solutions for two-box and three-box ocean models (Figure 3) with explicit numerical solution of the carbonate system [Follows et al., 2006]. Of the three approximate methods outlined, the first two frameworks have been used previously [e.g., Archer, 2005; Lackner, 2002], then a new framework is presented which remains accurate for larger additions of carbon.
2.2.1. Methodology: Testing the Relationship Between PCO2 and ΣC
 In order to test approximate solutions to the steady state air-sea partitioning equation (11) against model output, the following methodology is adopted: (1) The size of the inventories (IA and IO) and the global Revelle factor (Rglobal) are identified for each model for a single reference air-sea equilibrium case, such as the pre-industrial (Table 1). (2) The numerical models are separately integrated for different external inputs of C to air-sea equilibrium. (3) Analytical relations approximating (11) are derived and evaluated using the identified pre-industrial values for the inventories and Rglobal for each model. The analytical relations are then compared to numerical model results.
Table 1. Pre-Industrial Values for IA, IO, Rglobal, and IB Evaluated For the Atmosphere-Ocean Modelsa
Pre-Industrial Carbon Pools and Global Revelle Factors for Air-Sea Models
Atmosphere IA, GtC
Ocean IO, GtC
Revelle Factor Rglobal
Total Buffered IB, GtC
Where IB = IO/R + IA.
Evaluated by fitting the extrapolation framework to a 500 GtC emission.
Evaluated by fitting the constant Revelle framework to a 500 GtC emission.
Evaluated by fitting the constant buffered carbon framework to a 500 GtC emission.
 The pre-industrial value of Rglobal is diagnosed using equation (5). For each model a 5 ppm perturbation is applied to PCO2 at pre-industrial spin-up and the change in ocean carbon inventory when air-sea equilibrium is reached is observed. No knowledge of how Rglobal changes from its pre-industrial level is required in order to calculate each approximate solution to (11).
2.2.2. Framework 1: Extrapolate From an Infinitesimal CO2 Emission
 The infinitesimal change equation (10a) can be used to estimate how PCO2 will change under the addition of CO2 at steady state. The approximation can be achieved by treating a CO2 emission of any size as if it were infinitesimal to find a linear relationship [e.g., Archer, 2005]. The infinitesimal relationship (10b) can be rearranged and expanded to finite changes to give a linear approximation,
which is valid as long as
during the interval ΔΣC. As carbon is added to the air-sea system, IA increases faster than IO because the Revelle buffer factor is greater than unity. Thus this approximated extrapolation only holds if either: (1) ΔΣC is small in (12) or (2) as carbon is added to the air-sea system, Rglobal decreases in (13). From a pre-industrial starting point the value of Rglobal increases as carbon is added to the air-sea system (Figure 2c). Thus (12) is only valid for small carbon emissions (Figure 4).
2.2.3. Framework 2: Assume a Constant Revelle Buffer Factor
 If the Revelle buffer factor is assumed to remain constant [e.g., Lackner, 2002] then the definition of Rglobal, (5), can be integrated to relate PCO2 and ΣC,
which leads to
where k is a constant of integration. Under this assumption of a constant Rglobal, the total system carbon (ΣC) can be written in terms of the partial pressure of atmospheric CO2 by replacing CDIC,
 Thus the value of PCO2 can be calculated for any imposed ΣC assuming a constant R, which is valid as long as
is satisfied. The constant Rglobal approximation (16) remains though a better approximation for larger additions of CO2 than the extrapolation method (12), for the box models (Figure 4). However, from a pre-industrial starting point the Revelle buffer factor, Rglobal, increases as carbon is added to the system (Figure 2c).
2.2.4. Framework 3: Assume a Constant Air-Sea Buffered Carbon Inventory
 Both previous frameworks are useful for a small range of emissions but become inaccurate before the plausible limit of conventional fossil fuel emissions [Rogner, 1997] is reached since the Revelle buffer factor, Rglobal, increases as charge neutral CO2 is added to the air-sea system (Figure 2c). To improve on frameworks 1 and 2, an approximation is required that allows Rglobal to increase as carbon is added to the air-sea system. Again consider the integral relation (11), this time rewritten as
 We define the total air-sea “buffered” carbon inventory, IB, as the 'total carbon inventory of the atmosphere plus the total “buffered” carbon inventory of the ocean,'
where the steady state atmospheric PCO2 responds to an infinitesimal perturbation as if the atmosphere-ocean system contains a nondissociating carbon inventory of size IB. If it is then assumed that IB remains constant, (18) can be integrated to relate PCO2 and ΣC simply as
where Pi is the initial partial pressure of carbon dioxide. This constant buffered carbon approximation (20) remains valid as long as
for the interval ΔΣC. This condition can be met under two circumstances: (1) ΔΣC is small in (20) or (2) the value of R increases as carbon is added to the system in such as way as to restore IB in (19). The larger the carbon addition (ΔΣC), the more closely condition 2 has to be met in order for the constant buffered carbon method to remain valid. The constant buffered carbon method (20) remains a good approximation for larger emissions of CO2 than the constant R(16) or extrapolation (12) methods for the box models (Figure 4).
 As a starting point PCO2 can be related to ΣC either by extrapolation [e.g., Archer, 2005], or by assuming a constant Revelle factor [e.g., Lackner, 2002]. However a more rigorous and accurate analytical relationship between PCO2 and ΣC is obtained if the total buffered carbon is assumed constant. This constant buffered carbon approximation holds up to emissions of around 5000 GtC, while Rglobal increases fast enough with further emission to satisfy condition (21).
 When comparing analytical solutions to the two-box model output, the error in ΔPCO2 rises above 10% when emissions exceed 900 GtC and 2500 GtC for the extrapolation and constant Revelle methods, respectively (Figure 4a). In contrast the error in ΔPCO2 given by the constant buffered carbon method remains less than 5% for emissions up to 5000 GtC. When compared to the three-box model output, the error in ΔPCO2 given by the extrapolation and constant R methods rise above 10% when emissions reach 600 GtC and 2000 GtC, respectively (Figure 4b). In contrast the error in ΔPCO2 given by the constant buffered carbon method remains less than 6% for emissions up to 5000 GtC.
 As an aside, when emissions exceed 5000 GtC there appears to be a linear link between PCO2 and ΣC (Figure 5) as noted by Lenton . For a linear approximation to be valid the extrapolation condition (13) has to be met. When the anthropogenic carbon emission exceeds around 8000 GtC the steady state value of Rglobal decreases as more carbon is added to the air-sea system (Figure 2c), making the extrapolation condition (13) valid for a large range of ΔΣC. For emissions larger than 8000 GtC, the extrapolation equation (12), where IA, IO, and Rglobal are reevaluated for a new steady state where total air-sea carbon is at least 8000 GtC more than pre-industrial levels, reasonably describes the partitioning of further carbon emissions at air-sea equilibrium (Figure 5). Section 3 tests the constant buffered carbon framework for the MIT general circulation model, which includes ocean overturning, biology and seasonality.
3. Assessing Air-Sea Partitioning Within a More Complex Model Environment
 This section tests the constant buffered carbon framework in a more complex environment, as represented by a general circulation model, in order to assess whether the framework can be applied to the real atmosphere-ocean system.
3.1. GCM Model Description
 The MIT ocean circulation model [Marshall et al., 1997a, 1997b], is configured globally at coarse resolution (2.8 × 2.8 degrees, 15 vertical levels) and is forced with a climatological annual cycle of surface wind stresses, surface heat and freshwater fluxes with additional relaxation toward climatological sea surface temperature and salinity. The large-scale effects of mesoscale eddy transfers are parameterized following Gent and McWilliams  and a globally uniform, background vertical mixing rate is imposed for tracers (salt, temperature and biogeochemical) of 5 × 10−5 m2 s−1. The biogeochemical cycles of carbon, phosphorus, oxygen and alkalinity are coupled to the circulation model (fully described by McKinley et al.  and Dutkiewicz et al. ; see details and sources therein).
 Two thirds of net production is assumed to enter the dissolved organic pool which has a remineralization timescale of 6 months and the remaining fraction of organic production is exported to depth as sinking particles where it is remineralized according to the empirical power law of Martin et al. . Transformation of carbon and oxygen to and from organic form are linked to those of phosphorus assuming fixed Redfield stoichiometry. The calcium carbonate cycle is also parameterized using a fixed rain ratio. The partitioning of dissolved inorganic carbon is solved explicitly [Follows et al., 2006] and the air-sea exchange of CO2 is parameterized with a wind-speed-dependent gas transfer coefficient. The model is spun up to steady state (where annual mean air-sea exchange of CO2 is zero) with an imposed pre-industrial CO2 concentration of 278 ppmv. Once in steady state, the ocean is coupled to a well-mixed atmospheric box of CO2, and ocean and atmosphere carbon cycles interact. The model is run further with this interactive atmosphere and remains in steady state, with PCO2 stabilizing at 280 ppm. A small drift in PCO2 of ∼0.1 ppm/1000 years occurs from this state. All experiments discussed in this paper are initialized from this spin-up.
3.2. Testing the Analytical Relations for ΣC and PCO2 for the GCM
 The methodology for testing the relationship between ΣC and PCO2 in the GCM is as described in section 2.2.1, with the exception that pre-industrial value of Rglobal cannot be diagnosed by perturbing the system and using (5). If a small perturbation is used the drift in PCO2 during air-sea equilibration becomes a significant part of the overall PCO2 change, whereas if a large perturbation is used the value of Rglobal increases from the pre-industrial value that is being measured. For each analytical framework, the pre-industrial value of Rglobal is set to predict the steady state PCO2 reached after a 500 GtC emission without error (Table 1), the frameworks can then be compared by observing how they diverge from model results for emissions greater than 500 GtC.
 To add carbon emissions into the model and integrate to air-sea equilibrium the following procedure is applied: Emissions are added into the atmosphere at 10 GtC per year. For each choice of emission size, the model is then integrated for at least 3000 years, for air-sea equilibration to be effectively reached. For example, in the last century of integration, PCO2 changed by just ∼0.01 ppm for the 500 GtC emission, and ∼1 ppm for the 5000 GtC emission, showing the extent to which air-sea equilibration had been attained.
 The constant buffered carbon framework again remains a valid approximation for larger emission sizes compared with the extrapolation and constant Revelle frameworks (Figure 6). The error in ΔPCO2 given by the extrapolation and constant Revelle methods rise above 10% when emissions reach 1500 GtC and 2000 GtC, respectively. In contrast the error in ΔPCO2 given by the constant buffered carbon framework remains below 10% for emissions up to and including 4500 GtC. In summary the GCM results support the application of the constant buffered carbon framework to predict the equilibrium partitioning of anthropogenic carbon.
3.3. Synthesis of Model Assessments
 The use of the constant buffered carbon framework has been supported by both the box models and GCM. The crucial role the buffered carbon inventory (IB) plays in the steady state air-sea partitioning of anthropogenic carbon has been identified. However, each model tested has a different pre-industrial steady state value of IB (Table 1). Here the factors that affect the value of IB are investigated, and the relationship between the buffered carbon inventories of the models and that of the real atmosphere-ocean system are discussed.
 Rewriting the definition of the buffered carbon inventory (19),
it can be observed that the value of M and V and all factors that affect the global average DIC and Rglobal at a pre-industrial steady state will affect a models buffered carbon inventory, IB. The values of Rglobal and the global average DIC, at a steady state spin-up, are affected by: the properties of surface ocean waters, ocean circulation, and the biological pump.
 When spun up to a steady state with the atmosphere, the strength of the biological pump affects the value of both ocean average DIC and Rglobal. For an infinitesimal perturbation acting on the two-box model, in which all surface waters are perfectly saturated with the atmosphere, the change in steady state ocean average DIC is independent of biological pump strength. Thus the term = , and therefore IB, are also independent of the strength of the fixed biological pump. However, in the more general case, the strength of the biological pump affects the distance from saturation of the ocean waters at steady state. Consequently, the steady state average DIC for an infinitesimal perturbation is affected by the strength of the biological pump, leading to different model values of IB for different biological pump strengths. However, the effect of the biological pump strength on the distance from saturation of ocean waters is small in comparison to the overall value of DIC [Ito and Follows, 2005]; thus the effect of using a different fixed biological pump strength on the value of IB is expected to be of secondary importance (Figure 7a).
 At steady state with an atmosphere, water properties affect both ocean average DIC and Rglobal. For example, colder waters have higher Revelle factors, thus a colder ocean will have a smaller value of IB. The ocean ventilation process is important through its determination of deep water mass properties. A model with ventilation preferentially occurring through warm, low-latitude, surface waters will have a larger IB than a model with ventilation preferentially occurring through cold, high-latitude waters (Figure 7b).
3.4. How Can IB Be Evaluated for the Real Atmosphere-Ocean System?
 To evaluate the value of the buffered carbon inventory, IB, for the real pre-industrial atmosphere-ocean system, a model with accurate representations of M, V, surface ocean properties, ocean circulation and the biological pump must be used. The ocean circulation and carbon cycle of the MIT GCM is relatively realistic, at least compared with box models, making the GCM potentially suitable for estimating the real world pre-industrial IB. The MIT GCM model circulation reveals recirculating gyres within the ocean basins with a transport reaching ±40 Sv, as well as a zonal circumpolar current with a transport reaching 140 Sv (Figure 8a). For the pre-industrial spin-up, the annual air-sea flux of CO2 is directed into the ocean over the northern high latitudes reaching −3 molC m−2 year−1 and is directed out of the ocean over the tropics and Southern Ocean reaching 2 molC m−2 year−1 (Figure 8b).
 The modeled pre-industrial DIC distributions in both the Atlantic (Figure 9a) and Pacific (Figure 9b) basins show low concentrations in the upper thermocline, reflecting the low solubility of CO2 in the tropical surface waters and the wind driven ventilation of the upper thermocline [Marshall et al., 1993]. The high concentration of DIC in high-latitude surface waters and at depth reflects the effect of both the solubility and biological pumps.
 The present air-sea system is in a transient state, with total emissions so far estimated at around 300 GtC [Intergovernmental Panel on Climate Change, 2001]. To stabilize PCO2 in the MIT GCM to 388 ppm, a little over current levels, total carbon emissions over time must be restricted to 1000 GtC. After PCO2 equilibration at present-day transient levels, in the MIT GCM, the stabilized concentration of anthropogenic DIC in the surface of the Atlantic (Figure 9c) and Pacific (Figure 9d) oceans is similar to that observed in the present ocean by Sabine et al. , indicating that the present surface ocean is close to being in equilibrium with the atmosphere. The latitudinal variation in surface ocean carbon uptake observed in the present transient state [Sabine et al., 2004] is similar to the air-sea equilibrated MIT GCM results (Figure 9), and reflects how the latitudinal variation in surface Revelle factors is controlling the air-sea exchange: There is enhanced carbon uptake in the tropics where the Revelle factor R is low.
 The MIT GCM includes relatively realistic representations of the ocean circulation, biological pump and surface ocean carbon solubility properties. Thus the steady state pre-industrial total buffered carbon inventory of the MIT GCM is presented as a first estimate of the corresponding IB of the real atmosphere-ocean system, and is evaluated as: IB = 3100 GtC. Future comparison studies with other GCMs will be required in order to refine this estimate.
 Anthropogenic carbon changes are posing a significant perturbation to the Earth's climate. Our concern here is how the extra carbon is partitioned between the atmosphere and ocean at air-sea equilibrium. The partial pressure of atmospheric CO2 (PCO2) over the next century is heavily dependent upon the emission scenario. After air-sea equilibrium PCO2 levels are determined by the total emission, ΔΣC, by
where Pi is the initial pre-industrial value of partial pressure of carbon dioxide and IB is the steady state, pre-industrial, total buffered carbon inventory; evaluated as IB = 3100 GtC for the MIT GCM. This relationship remains valid for emissions reaching 5000 GtC. Rearranging this relation allows us to address the question of how much of the available fossil fuel resources can be utilized to allow atmospheric carbon levels to stabilize at a given PCO2 after air-sea equilibration,
 For atmospheric carbon levels to stabilize to the given PCO2, this value of ΔΣC cannot be exceeded until weathering effects alter ocean chemistry on timescales of thousands of years [Archer et al., 1997]. To achieve PCO2 stabilization at present day levels for the MIT GCM requires limiting future CO2 emissions to 700 GtC.
 In summary, an analytical relationship predicting the partial pressure of CO2 is provided at air-sea equilibrium which only requires knowledge of the external carbon inputs and the global carbon inventories and buffer factor for a pre-industrial reference point. This relationship should prove useful for developing simplified, but accurate, climate models to be used in policy decisions about how the future climate state will evolve given known anthropogenic perturbations.
 This research was funded by the NERC RAPID grant NER/T/S/2002/00439 and the associated tied studentship NER/S/S/2004/13006. We are grateful for discussions with J. Marshall, and for constructive reviews from T. Lenton and D. Archer which improved the quality of the manuscript. M. J. F. and S. D. are grateful for funding from NSF OCE-336839.