Sinking and subduction of organic material removes carbon from the surface ocean and stores it in inorganic form after remineralization. The wind-driven upwelling of deep waters, notably in the Southern Ocean, counteracts the biological carbon sequestration by returning excess carbon from the abyss, potentially releasing it back to the atmosphere. Numerical models have shown that significant fraction of the excess carbon in the Antarctic Surface Water is not degassed to the atmosphere but reenters into the deep ocean due to the incomplete air-sea equilibration, effectively increasing the efficiency of biological carbon storage in the deep ocean. We develop a simple theory to consider the controls on this effect. The theory predicts a strong coupling between biological carbon sequestration and air-sea disequilibrium expressed as a linear relationship between the biological carbon pump and the degree of supersaturation in the deep ocean. Sensitivity experiments with a three-dimensional ocean biogeochemistry model support this prediction and demonstrate that the disequilibrium pump almost doubles the efficiency of biological carbon sequestration, relative to the effect of nutrient utilization.
 The global ocean carbon inventory is more than 50 times that of the atmospheric reservoir; thus, the oceanic carbon storage and its evolution play crucial roles in regulating the atmospheric carbon dioxide (CO2) and the Earth's climate system [Siegenthaler and Sarmiento, 1993]. Photosynthesis and production of sinking, or subducted, organic material are major pathways by which carbon dioxide is sequestered in the deep ocean, controlling the vertical gradient of dissolved inorganic carbon and atmospheric CO2 [Volk and Hoffert, 1985]. Previous theory and simulations [Ito and Follows, 2005; Marinov et al., 2006; Goodwin et al., 2008] quantified the strength of the organic component of the biological carbon pump in terms of the efficiency of nutrient utilization: What fraction of nutrients entrained into the euphotic zone return to depth in organic form, as opposed to subduction of the unutilized inorganic form? The fraction of a macronutrient (often phosphorus) that is simply resubducted to the interior is called “preformed nutrient.” If the global nutrient inventory is conserved, a decrease in global mean preformed nutrient reflects an increase in biologically sequestered nutrient and the associated, biologically sequestered carbon α . We may write that . For simplicity, we neglect the carbonate component of Cbio. Then, the global-scale efficiency of biological carbon sequestration can be expressed in terms of a measure of nutrient utilization:
 In a similar manner, total inorganic carbon, C, in the interior ocean can be expressed as the sum of the equilibrium carbon, Csat, the degree of saturation, ΔC, and the biologically regenerated carbon, Cbio. Csat is set by the equilibrium carbonate chemistry as a function of temperature, salinity, alkalinity, and the pCO2 of the overlying atmosphere. Cbio primarily reflects the accumulated remineralization of organic material along circulation pathways. Dissolution of calcium carbonate particles can also increase the accumulation of Cbio; however, its effect on the surface alkalinity counteracts its additional carbon storage. Here we focus on the perturbations to the oceanic carbon reservoirs through the organic carbon pump.
 Assuming a constant ocean-atmosphere carbon inventory, mean ocean solubility and stoichiometric ratio between phosphorus and carbon, a simple scaling relationship can be derived between P* (the global efficiency of nutrient utilization) ∆C, and equilibrium atmospheric CO2 (pCO2atm) [Ito and Follows, 2005; Goodwin et al., 2008].
where Vocn is the volume of global oceans and IB is the sum of atmospheric carbon inventory and the oceanic carbon inventory scaled by the buffer factor [Goodwin et al., 2007]. Over a wide range of appropriate parameter space, IB is effectively constant. Here RCP is the stoichiometric ratio between carbon and phosphorus, which is assumed constant for biological processes. If the global mean ΔC is constant, these assumptions imply that the steady state atmospheric carbon dioxide could vary by approximately 300 ppm over the full range of variation in P* (0 to 1) [Ito and Follows, 2005]. However, we expect global mean ΔC to vary in association with changes to the other carbon reservoirs, both solubility and biological pumps. Here we seek to interpret and illustrate the relationship between the disequilibrium and biological pumps. Since ΔC is a preformed property of the subsurface ocean (i.e., it is a conservative tracer in the interior, and its concentration is set at the time of subduction), its distribution can be related to its regional variations in the surface ocean and especially sites of mode, deep, and bottom water formation.
 Observed surface carbon dioxide concentrations significantly deviate from the local equilibrium (Figure 1). Surface waters in the upwelling regions such as the equatorial Pacific are supersaturated (Figure 1) [Takahashi et al., 2009], and carbon dioxide is degassed to the atmosphere. In contrast, the midlatitude and northern North Atlantic oceans are undersaturated, where carbon dioxide is absorbed into the surface oceans. The subantarctic region about 40°S is a region of undersaturation and carbon uptake, but the polar Southern Ocean appears to be a region of supersaturation. What makes the northern and southern polar outcrop behave so differently? The strong undersaturation in the newly formed North Atlantic Deep Water (NADW) is caused by the intense heat loss and the small size of the polar outcrop [Toggweiler et al., 2003a]. In the Antarctic, the subtle balance between the heat loss, biological carbon uptake, and the upwelling of regenerated carbon regulates the ΔC of newly formed water masses [Murnane et al., 1999; Gruber and Sarmiento, 2002; Toggweiler et al., 2003b]. The effect of upwelling is stronger to the south of the Polar Front, leading to the supersaturation there. Furthermore, Stephens and Keeling  pointed out that sea ice coverage could significantly influence the supersaturation of newly formed Antarctic Bottom Water, which may have contributed to increased carbon storage in the glacial deep ocean. To better understand the control of ΔC in the Southern Ocean, we perform theoretical and numerical analyses to understand the physical and biogeochemical balances and interactions responsible for determining the global distribution of ΔC and its relationship to biological carbon pump, extending the theory for steady state atmospheric CO2.
 The goal of this paper is to understand what controls the air-sea disequilibrium of carbon (ΔC) in the deep ocean and to generalize the theory predicting the steady state atmospheric CO2 [Ito and Follows, 2005; Goodwin et al., 2008] to include the effect of incomplete air-sea equilibration in relation to the biological carbon pump. To achieve this goal, we must first understand and model the coupling of these carbon reservoirs. To that end, in the next section, we derive the governing equation for ΔC as a function of physical and biogeochemical influences on the saturation state of carbon dioxide (section 2). Using that framework, we develop a conceptual model for ΔC, which predicts that it is linearly proportional to the soft tissue pump, Cbio (section 3). Finally, we perform numerical simulations of preindustrial carbon cycle using an ocean circulation and biogeochemistry model and quantify the distribution and the sensitivity of ΔC (section 4). The numerical simulations support the idealized model's prediction that the disequilibrium pump is proportional to, and amplifies, the soft tissue pump.
 We start by focusing on surface ocean carbon dynamics. Following a surface water parcel, provided that vertical and lateral mixing can be neglected, the evolution of dissolved inorganic carbon (DC/Dt), depends on transport, air-sea gas transfer, and biological carbon uptake,
where h is the depth of the mixed layer, is the air-sea gas flux (positive into the ocean), and Φorg is the sinking flux of organic material. The Φcarb is the export of mineral calcium carbonate evaluated at the base of mixed layer. (E − P) is the net evaporation minus precipitation. In the mixed layer, Cbio is defined to be zero, and the dissolved inorganic carbon can be decomposed into the saturation component, Csat, and the disequilibrium component, ΔC:
 Substituting (5) into (4), linearizing changes in Csat with respect to changes in temperature, salinity, and alkalinity and describing the relationship between pCO2 and Csat using Revelle buffer factor we can describe changes in ΔC as follows (a detailed derivation appears in the Appendix A):
 The first term on the right encapsulates the effect of air-sea gas exchange, which effectively relaxes surface ΔC toward zero. The second term describes the forcing of anomalies due to the formation and export of organic and inorganic particulate carbon. The third term describes changes in disequilibrium driven by surface heat fluxes, H, and their effect on solubility. The fourth term indicates the effect of dilution due to the net effect of evaporation and precipitation (E − P). The last term on the right of (6) parameterizes the entrainment of regenerated carbon and regenerated alkalinity from below the mixed layer, which generally leads to supersaturation. A detailed derivation of (6), including definitions of the parameters and thermodynamic coefficients, ai, is given in the Appendix A.
3 A Conceptual Model of the ΔC Dynamics in the Southern Ocean
 The transformation of ΔC occurs almost entirely in the surface mixed layer through air-sea interaction and biological carbon uptake. It is conservative (i.e., considered a “preformed” quantity) following subduction into the thermocline. Hence, we use the description of ΔC dynamics in equation (6) to interpret its regulation in the surface branches of the meridional overturning circulations. In particular, we focus on the balance of processes that set ΔC in the large volume of bottom waters formed in the Southern Hemisphere. Figure 2a shows a conceptual view of an ocean overturning where a parcel of water makes a loop starting from the polar sinking, abyssal flow, upwelling through the middepth thermocline, and the surface return flow. We consider two cases. In the first case (solid line in Figure 2a), the region of upwelling is widely separated from the region of downwelling. In this case, it takes a relatively long time for the water parcel to travel from the region of upwelling to that of sinking, which is analogous to the NADW cell. The second case (dashed line in Figure 2a) is analogous to the Antarctic Bottom Water (AABW) cell where the region of upwelling and sinking are close to one another. The difference between the two overturning cells can effectively be measured by the difference in the surface residence time (τres). Here we focus on the latter regime (depicted in Figure 2b) in particular, its implications for the disequilibrium pump in the bottom waters of the ocean.
 Here circumpolar deep waters, enriched in regenerated carbon (i.e., high Cbio) are brought to the surface by the residual mean upwelling associated with the Antarctic Circumpolar Current. In the surface, diabatic processes facilitate water parcels to move equatorward (associated with warming and/or freshening) or poleward (associated with cooling and/or enhanced salinity). In the surface, Cbio is zero by definition, so the entrainment of subsurface regenerated carbon is effectively a positive source of ΔC. We will consider the idealized case where the forcing of ΔC in the surface waters involves air-sea equilibration, entrainment of subsurface Cbio, and biological uptake acting between the region of upwelling and that of subduction (depicted schematically in Figure 2b). In order to consider the change from the point in time shortly following entrainment up until the time of subsequent resubduction, (6) can be simplified to
where the effect of entrainment is included in the initial condition for ΔC and includes a contribution from the preformed disequilibrium at the last time of subduction and a contribution from the accumulation of regenerated carbon over the period in the subsurface ocean: i.e., ΔC(0) = ΔCpre + Cbio. When the subsurface water is entrained into the mixed layer, there is an abrupt increase of ΔC due to the conversion from the subsurface Cbio. In fact, the preformed disequilibrium of the upwelling deep water, ΔCpre, is set by the mixing of undersaturated NADW and supersaturated AABW end members. In this highly idealized conceptual view, ΔCpre at the time of entrainment will be assumed negligible relative to Cbio. Thus, we set ΔC(0) = Cbio = RCPN0P*, where P* describes the efficiency of the soft tissue pump in terms of nutrient utilization. The climatological distribution of Apparent Oxygen Utilization (AOU) indicates that on the global average, approximately 36% of macronutrients are in the regenerated pool (P* = 0.36) [Ito and Follows, 2005]. We parameterize the air-sea CO2 flux with a characteristic time scale τgas such that gas exchange erodes the disequilibrium on a time scale of τgas, which depends upon the mixed layer thickness, gas transfer velocity, and carbonate system parameters (Appendix A) [Ito et al., 2004]:
 We parameterize biological carbon export, Φorg, as a function of the change in the abundance of macronutrients in the water parcel:
 We assume that the uptake and export of carbon is proportional to that of a macronutrient, N, here phosphate. The biological consumption and export of the macronutrient are parameterized as an exponential decay with characteristic time scale, τbio, which represents the net effect of a complex set of food web processes. Entrainment supplies the water parcel with subsurface nutrient concentration, N(0), the concentration in upwelling circumpolar deep water, which is close to the global mean nutrient concentration.
 Substituting parameterizations of gas exchange and biological export, from (8) to (9) into (7), we can solve the resulting ordinary differential equation for ΔC(t) as an initial value problem where ΔC(0) and N(0) are the disequilibrium carbon and macronutrient concentration immediately following entrainment into the surface mixed layer. We solve for the disequilibrium at the point of resubduction, ΔC(τres), after spending a period τres (the surface residence time) in contact with the atmosphere.
 Over that period, the strength of biological uptake is measured by the nondimensional number, , indicating the relative time scale between surface flow and biological uptake. In this simple model, the contrast between NADW and AABW cells is determined by the magnitude of ηbio. The long distance between the upwelling and sinking (large τres) with vigorous biological uptake (small τbio) tend to deplete the preformed nutrient of NADW. The short distance (small τres) and relatively weak biological uptake (large τbio) potentially due to the iron and/or light limitation maintain a high-preformed nutrient of AABW. Similarly, describes the efficiency with which air-sea gas exchange erodes the disequilibrium while the water parcel is in contact with the atmosphere. The solution for ΔC when t = τres is
which depends on the upwelling Cbio, here encapsulated in P*. The solution has a singularity when the two nondimensional numbers are equal to one another, in which case the solution must be calculated separately.
 For a given value of P*, the resubducted saturation, ΔC(τres), is determined by the efficiency of air-sea CO2 exchange (ηgas) and biological nutrient uptake (ηbio). Figure 3 illustrates the relationship between preformed ΔC, ηgas and ηbio for several values of P*. When ηgas and ηbio are varied over two orders of magnitude the dynamic range of ΔC(τres) is between −125 μM and 50 μM for the control case, P* = 0.36, roughly in accord with today's ocean.
 The first term on the right-hand side (RHS) of equations (10) and (11) is always positive, representing the upwelling of regenerated carbon. Stronger upwelling of regenerated carbon, either by increasing P* and Cbio (moving from Figure 3 (top) to Figure 3 (bottom)) or by increasing the overturning rate (reducing τres moving to the lower left corner of each panel in Figure 3) leads ultimately to supersaturation of subducted waters. In contrast, a relatively long surface residence time leads to well-equilibrated surface ΔC (large ηgas) regardless of the magnitude of P* (for all panels in Figure 3). When the gas exchange is relatively weak (ηgas < 1), a strong biological carbon uptake (large ηbio) leads to undersaturation regardless of the magnitude of P*. This is due to the biological carbon uptake as represented in the second term on the RHS of equations (10) and (11), which is always negative. Theoretically, it is possible to drive strong undersaturation by this mechanism (ηbio > > 1), but in three-dimensional simulations of the ocean, it is difficult to achieve this limit due to the small size of polar outcrop and the seasonality in the biological uptake and ocean ventilation [Ito and Follows, 2005].
 This simple model assumes a constant stoichiometric ratio between carbon and macronutrient for both the biological uptake and the previously regenerated organic material which drives the relative rates of upwelling of carbon and macronutrients. We note that elemental ratios of marine phytoplankton are variable in time and space depending on taxanomic differences and the physiological state of the cells. Notably, diatoms, which are prevalent in the Southern Ocean, have been observed with distinct stoichiometric ratios [Sweeney et al., 2000] where the C:P ratio of diatom is about half than that of the Redfield ratio. This effect primarily influences the productivity-driven undersaturation (the second term of the equations (10) and (11)), as the composition of the upwelling water (the first term) is likely less variable due to the long memory of deep waters averaging the remineralization along the circulation pathway. We speculate that the simplified theory underestimates the supersaturation in the diatom-dominated biome due to the overestimation of carbon export when assuming Redfieldian carbon uptake per unit phosphorus consumption.
 In the global integral sense, a key parameter describing the sensitivity of ΔC to the biological pump is γ, the partial derivative of ΔC with respect to Cbio, which, from equation (10), is defined
 If the gas exchange is slow (small ηgas) relative to the circulation, the partial derivative can be significantly greater than zero, indicating that an increase in upwelling of regenerated carbon in the Southern Ocean will enhance supersaturation in the bottom waters formed there. If the gas exchange were efficient (large ηgas), there would be little coupling between the air-sea disequilibrium and biological carbon pump. This limit case scenario was demonstrated in the simulations of Marinov et al. . Since the residence time of water in the surface of the Southern Ocean and the time scale for air-sea equilibration of carbon are all on the order of 1 year, we estimate that ηgas is approximately of order 1. This suggests that the air-sea disequilibrium may enhance the effect of biological pump by tens of percent based on equation (12).
 Combining equations (3) and (12), we can derive a new scaling for the steady state atmospheric pCO2, which includes an accounting for the resubduction of nonequilibrated waters and the explicit connection between the soft tissue and disequilibrium pumps:
 The simple model makes two key predictions: (i) The global disequilibrium pump should be linearly related to the global soft tissue pump with a coefficient determined by the relative rates of overturning, biological uptake, and air-sea gas exchange. (ii) The significant role of this amplification of the soft tissue pump associated with the formation of bottom waters in the vicinity of Antarctica causes a significant net enhancement of the sequestration of carbon in the global ocean compared to the “efficient gas exchange” case. In the following section, we examine these hypotheses in the context of a more realistic, three-dimensional global ocean circulation, and biogeochemistry model.
4 Numerical Experiments
 We performed a suite of three-dimensional numerical simulations in order to quantitatively evaluate the interaction between biological carbon uptake, air-sea disequilibrium of CO2, and biological carbon pump, and how the surface ΔC enters into the interior oceans. We employ the MITgcm ocean model [Marshall et al., 1997a, 1997b] including representations of the cycling of carbon, phosphate, and oxygen with a simple, coupled atmosphere-ocean biogeochemistry scheme. The ocean circulation and biogeochemistry model was configured with lateral resolution of 2.8° and 24 vertical layers and was coupled to a well-mixed atmospheric reservoir of CO2. The circulation of the model was spun-up for several thousand years forced by monthly climatological surface winds stress, air-sea heat, and freshwater fluxes. The effect of unresolved ocean eddies was parameterized using the isopycnal thickness diffusion scheme [Gent and Mcwilliams, 1990]. The biogeochemical parameterizations includes the transport of dissolved inorganic carbon, alkalinity, phosphate, dissolved organic phosphate, oxygen, and explicit tracer representing preformed phosphate, Ppre which is set equal to phosphate in the surface layer, and is transported as a passive tracer below the surface [Ito and Follows, 2005]. This allowed us to precisely determine the simulated three-dimensional distributions of the carbon components, Csat, ΔC, and Cbio.
 We examined a suite of numerical simulations: First, we qualitatively compared a “control case” in which biological consumption and export was represented by Newtonian relaxation of surface (top 50 m) PO4 toward the monthly climatological distribution with a relaxation time scale of 1 month, and an “efficient export” case in which surface phosphate is relaxed toward a concentration of zero on a time scale of 1 month. Then we examined the sensitivity of the ocean's carbon reservoirs and atmospheric CO2 to a range of damping time scales, where the target surface phosphate concentration is zero.
 Figure 4a shows the change in the distribution of ΔC between the “efficient export” and “control” simulations, revealing the models sensitivity to an increase in the efficiency of biological carbon uptake in the surface ocean when ocean circulation and gas exchange rate coefficient are unchanged.
 In the efficient export case, stronger biological carbon uptake increases Cbio globally, which can be measured by the increase of global mean P*. When volumetrically averaged, the change in the global mean Cbio is about 43 μM, indicating that rapid uptake and export of surface nutrient significantly increased biological carbon sequestration.
 The effect of air-sea disequilibrium can either enhance or diminish biological carbon sequestration. Indeed, the response of ΔC was of opposite sign in the model's NADW and AABW (Figure 4a). The stronger biology decreases surface ΔC in the northern North Atlantic by about 20 μM, and increases it in the surface Southern Ocean by about 60 μM, and the net effect on the global mean ΔC is an increase of 32 μM dominated by the response of AABW. Figure 4a shows the different surface ΔC change within the Southern Ocean where the Weddell Sea is significantly more supersaturated than the Ross Sea. Clearly the global sensitivity of ΔC to changes in nutrient utilization and the soft tissue pump, γ = ∂ΔC/∂Cbio, are complex. Even within the modeled AABW, the two major source regions can have different sensitivities and imprints on the newly formed deep water.
 What controls the magnitude of ΔC in the region of water mass formation? Consider the results of the simulations in Figure 4a in the context of the conceptual model in Figure 3: In the conceptual view, the effect of upwelling regenerated carbon (Cbio) and the nutrient uptake rate (ηbio) are treated independently. In the numerical simulations (Figure 4a), these changes are coupled as the rate of nutrient uptake ultimately controls the regenerated carbon of the upwelling waters, and their relationship can be complex. Upwelling of subsurface waters immediately increases ΔC due to the entrainment of excess carbon in the form of Cbio. The upwelling of carbon is also associated with the supply of nutrients, which fuels the biological carbon uptake, leading to a decrease in ΔC over the biological time scale, τbio. The former effect dominates in the AABW, and the latter dominates in the NADW. In the regime where the gas exchange is strong (ηgas > > 1), ΔC is weakly negative, and there is little coupling between the biological pump and air-sea disequilibrium (γ ~ 0). When the gas exchange is weak (ηgas < < 1), γ can significantly deviate from zero. If the biological carbon uptake is relatively efficient (ηbio > > 1), ΔC can be undersaturated because the biological uptake driven by the upwelling of excess nutrient overwhelms the effect of the upwelling of Cbio. This regime is realized in the North Atlantic where the biological productivity is more active. In contrast, if the biological carbon uptake is weak (ηbio < < 1), ΔC is supersaturated due to the upwelling of excess carbon. This is a more relevant regime for the high-nutrient, low-chlorophyll condition of the Southern Ocean where the efficiency of biological nutrient utilization is relatively weak due, in part, to the limited iron supply [Boyd et al., 2000; Martin et al., 1990; Parekh et al., 2006b].
 Additional perturbation experiments are performed to quantify this relationship by varying the inverse time scale of nutrient restoring over a wide range from 0 to 1 month−1. This series of numerical simulations reveal a quasi-linear relationship between the global mean ΔC and Cbio supporting the prediction of the highly idealized view of surface interactions described by equations (10) and (11). This set of global nutrient depletion experiments also suggested that the associated amplification of the soft tissue pump is very significant in these simulations; the contribution of the soft tissue pump to ocean carbon storage was enhanced by about 70% (γ = 0.71, inferred from the least squares fit in Figure 4b). This result has a direct implication to the sensitivity of atmospheric CO2 to changes in the biological carbon pump. Figure 4b shows the steady state response of globally (volume weighted) averaged ΔC to the changes in globally averaged Cbio. P* and Cbio are linearly related to one another as . Previous theory [Ito and Follows, 2005] assumed that ΔC is independent of biological carbon pump and Ppre, leading to a much weaker sensitivity of atmospheric CO2. The sensitivity experiments are highly idealized where the rate of nutrient uptake was uniformly perturbed. These simulations also emphasize the role of the Southern Ocean on the global scale: The globally integrated result reflects the volume weighted effect of the opposing effects in different water masses (e.g., NADW versus AABW) but was clearly dominated in this model by the response of the Southern Ocean and its impact on AABW and intermediate waters.
5 Sensitivity of Atmospheric CO2 to the Biological Pump
 Theoretical prediction for the sensitivity of atmospheric CO2 crucially depends on the coupling between the air-sea disequilibrium and biological carbon sequestration. Including the effect of air-sea disequilibrium, the sensitivity of atmospheric CO2 to global mean Cbio increases by a factor of γ as shown in equation (13). Figure 5 shows relationship between atmospheric CO2 and Ppre based on the theoretical prediction with a specific magnitude of γ. The three lines (red, blue, and black) are the theoretical values with γ = 0, 1, and 2. Triangular dots are the idealized experiments shown in Figure 4, and the circular dots are the results from sensitivity experiments using a more complex biogeochemistry model where aerosol iron deposition has been perturbed over a wide range [Parekh et al., 2006a]. The numerical experiments are close to the theoretical values for γ = 1 for a large perturbation in Ppre, but the iron perturbation experiments are rather close to the γ = 2 case when the magnitude of perturbations is moderate.
 The direct effect of biological carbon sequestration is measured by the distance between the horizontal line and the theoretical line (red) with γ = 0. Air-sea disequilibrium responds to the change in the biological carbon uptake, and the change in global mean ΔC correlates with the global inventory of Cbio as measured by P* (and Ppre). The circulation and air-sea gas exchange rates are held constant across all sensitivity experiments; therefore, any departure from this “Cbio effect” is due to the coupling between air-sea disequilibrium and the biological carbon sequestration. All numerical experiments show significant enhancement of the sensitivity due to the “ΔC effect,” primarily driven by the covariation of ΔC and Cbio in the Southern Ocean. A stronger sensitivity in the iron perturbation experiments [Parekh et al., 2006a] confirms the dominant role of the Southern Ocean since the perturbation in the iron deposition most strongly stimulates the biological productivity there.
 We found that the numerical experiment supports the theoretical prediction based on the simple Lagrangian model that there is a linear relationship between ΔC and Cbio. This coupling leads to an enhancement of current biological pump by about 70%, driven by short residence time and inefficient air-sea CO2 equilibration in the Southern Ocean with respect to the Antarctic Bottom Water formation.
6 Concluding Remarks
 Deep waters are enriched in dissolved inorganic carbon due to the integrated effect of remineralization of organic material and calcium carbonate shells. When the deep water upwells into the surface mixed layer, the excess carbon is exposed to the atmosphere leading to a supersaturation and degassing to the atmosphere. There are two effects of inefficient air-sea equilibration in the Southern Ocean. First, the upwelling deep waters do not completely release the excess CO2 to the atmosphere, retaining the excess carbon in the surface water. Second, the newly formed Antarctic Bottom Water (AABW; see Figure 4a) is enriched in ΔC due to the proximity between the region of upwelling and sinking, and the excess carbon can resubduct into the deep ocean, which enhances the biological carbon sequestration.
 In this study, we quantified the impact of incomplete air-sea equilibration on the deep ocean carbon storage in the form of resubducted supersaturation (ΔC) in the AABW (equations (10) and (11)). The magnitude of preformed ΔC depends on the upwelling regenerated carbon (Cbio or equivalently, P*) in the circumpolar deep water and the relative strength of air-sea gas exchange. The latter effect can be quantified by the nondimensional number (ηgas), defined as the ratio between the surface residence time and air-sea CO2 exchange time scale. The surface residence time of water parcels at the water mass formation region is crucial for setting both preformed nutrient and ΔC [Toggweiler et al., 2003a, 2003b]. Considering the surface branch of the AABW circulation, the regions of upwelling and sinking are both in the high-latitude Southern Hemisphere oceans. Due to the proximity of the upwelling and sinking regions and the winter-time sea ice coverage, the upwelled deep water has a very narrow window to interact with the atmosphere before it sinks back down to the deep ocean.
 Our results quantitatively relate the inefficient air-sea equilibration (measured by ηgas) and the sensitivity of atmospheric CO2 to the biological carbon pump by extending the P* theory of Ito and Follows  and Goodwin et al. . Steady state atmospheric CO2 decreases with increased biological nutrient utilization (measured by P*), but the constant of proportionality depends on the relative strength of air-sea equilibration (ηgas), as quantified by equation (13). The nondimensional γ factor measures the strength of the coupling between the P* and the resubducted supersaturation (ΔC).
 Southern Ocean biological cycling has more potential to alter atmospheric CO2 than any other ocean basin, not only due to its large pool to unutilized surface nutrients but also because of its ability to change the saturation state of the abyssal waters. Our global nutrient depletion experiments suggest that the modern ocean operates close to γ ~ 0.7 where the air-sea disequilibrium increases the effect of biological carbon sequestration by 70%. The actual magnitude of the γ factor is dynamic, and any change depends on the nature of the perturbation. For example, the iron addition/subtraction experiment by Parekh et al. [2006a] shows much stronger sensitivity γ ~ 2 due to its focused perturbation in the Southern Ocean as shown in Figure 5.
 In the modern climate, both Ppre and ΔC are low in the NADW, and they are both high in the AABW. Holding the surface nutrients and ΔC constant, increasing the relative volume AABW leads to an increase in both the global mean Ppre and ΔC, and they have compensating effects on atmospheric CO2. Therefore, model errors in the representation of deep circulation tend to produce compensating anomalies in the deep ocean Cbio and ΔC, hiding potential problems in the climatological mean carbon cycle. The problem could manifest itself in the model sensitivity and climate change projections. For example, while holding the ocean circulation the same, we may perturb the Southern Ocean biology, causing some nutrient drawdown there. There will be a strong response in atmospheric CO2 due to the reinforcing responses of air-sea disequilibrium and biological carbon sequestration. However, the magnitude of the CO2 response will depend on the relative volume of NADW and AABW. Erroneous representation of deep circulation, especially in the Southern Ocean, likely impacts on the sensitivity of atmospheric CO2, more than the mean state.
 Previous marine carbon cycle model intercomparison projects have found major model-model differences in the Southern Ocean due to the varying representation of the Southern Ocean circulation and eddy-mean flow interactions [Doney et al., 2004; Najjar et al., 2007; Orr et al., 2001]. Even though the end-members of preformed nutrient are constrained in the nutrient-restoring schemes, the circulation differences lead to a wide range of Cbio as seen in different representation of subsurface oxygen. Model-model comparisons, so far, have been limited to the mean state and the anthropogenic perturbation. Differences may be even greater for the sensitivity with varying biological pump depending on the coupled, amplifying responses in Cbio and ΔC. Analysis of model output in terms of different carbon pump components would be beneficial for our understanding in each carbon pump processes, and the coupling among different components could play crucial role in the behavior of the state-of-the-art, coupled climate and carbon cycle models.
Appendix A: The ΔC Equation
 Here we provide a detailed derivation of the ΔC equation to supplement the main text. We use the following notation for the lateral advection and vertical entrainment of arbitrary tracer concentration, χ, in the mixed layer.
 We use linear operators for the horizontal advection, , and the entrainment, . The depth of the mixed layer is h. Horizontal velocity of the flow is (u, v), and the vertical velocity (w) is evaluated at the base of the mixed layer. Λ is a Heaviside step function. Λ is 1, if w is positive and is 0 otherwise. χth is the tracer concentration in the upper thermocline, and Sχ is the source of the tracer.
 This notation is applied to the dissolved inorganic carbon. The concentration of surface C can be changed subject to the transport, air-sea gas transfer, and biological carbon uptake. Sources of C include air-sea gas flux (GCO2, positive into the ocean), sinking flux of organic material (EPorg) and carbonate shell (EPcarb) evaluated at the base of mixed layer, and the net evaporation (E − P), leading to the Eulerian representation of equation (4).
 Following equation (5), C can be separated into two components in the surface mixed layer, C = Csat + ΔC, which can be substituted into equation (A2). While Cbio does not exist in the surface mixed layer, the vertical entrainment (third term of the left-hand side (LHS)) can transport subsurface water which contains nonzero Cbio into the mixed layer. The saturation carbon concentration, Csat, can be reexpressed as a linearization relative to (T), salinity (S), preformed alkalinity (Alk), and pCO2 of the overlying atmosphere (pCO2air).
 The governing equation for ΔC can be derived by substituting equation (5) into (A2) and further combining with the thermodynamic equation and tracer continuity equations for salinity and alkalinity. In order to eliminate Csat, we must use the thermodynamic equation and the two tracer continuity equations for salinity (S) and preformed alkalinity (Alk) as follows.
H is the net heat flux into the surface ocean, and ρ0 and cP are the reference density and specific heat of seawater. RNC is the stoichiometric ratio between nitrate and carbon, and Alkbio is the regenerated alkalinity due to the dissolution of calcium carbonate.
 While the linearization of carbonate chemistry (A3) is a powerful conceptual tool, its validity must be evaluated using the full nonlinear carbonate chemistry equations. We evaluated the derivatives of Csat as a function of T, S, Alk, and pCO2air, as shown in Figure A1. The variation of Csat can be treated as a linear function with respect to S and Alk to an excellent approximation, while its overall magnitude still depends on the background temperature and alkalinity. These relations are also evaluated at the surface pressure for obvious reason. Temperature sensitivity is moderately nonlinear [Goodwin and Lenton, 2009], where the temperature dependence of Csat changes by approximately 20% over the whole temperature range of the ocean (from 0°C to 30°C). The sensitivity to pCO2air is even more nonlinear, where the derivative of Csat changes by approximately 30% over the range of anthropogenic perturbation O (100 ppmv). However, we keep the value of pCO2air at a uniform constant; thus, this specific factor causes no error in our experiments. Due to the variations in the sensitivity of Csat with respect to the background conditions in T, S, and Alk, we prepared the Table A1 showing the representative values of the partial derivatives used in equation (A3).
Table A1. Coefficients Calculated From the Equilibrium Carbonate Chemistrya
aThe mean conditions have been adjusted for low latitudes (T = 25°C, S = 35 psu, Alk = 2.35 eq m−3), midlatitudes (T = 10°C, S = 34.5 psu, Alk = 2.25 eq m−3), and polar latitudes (T = 0°C, S = 34.5 psu, Alk = 2.25 eq m−3). All values are based on the preindustrial atmospheric pCO2 of 280 µatm.
molC m−3 K−1
molC m−3 psu−1
−1.01 · 10−2
−6.51 · 10−3
−8.38 · 10−3
−5.06 · 10−3
−7.51 · 10−3
−4.07 · 10−3
 Combining equations (A2) through (A6), we find the governing equation for ΔC in the preindustrial condition where atmospheric pCO2 is approximately uniform and constant in time.
 This equation integrates diverse surface processes into the evolution of the surface ΔC distribution. The LHS describes the effect of physical transport. The RHS describes the effects of air-sea gas transfer, entrainment of regenerated carbon and alkalinity, biological carbon uptake, and the air-sea heat and freshwater fluxes, where these terms are the sources and sinks of ΔC. The export production of organic and carbonate materials are combined using the carbonate rain ratio (r). The coefficients a0, a1, and a2 are defined as follows and their respective values are listed in Table A2.
aThe value of a0 is based on the stoichiometric ratio between nitrate and carbon of 16:106 and the carbonate to organic rain ratio of 0.07 [Yamanaka and Tajika, 1997].
−2.40 · 10−9
−2.00 · 10−9
−1.79 · 10−9
 The upwelling and entrainment of the subsurface waters (second term on the RHS of equation (A7)) is generally positive and increases the saturation state due to the upwelling of regenerated carbon. This effect is partially compensated by the upwelling of regenerated alkalinity. The biological carbon uptake (third term on the RHS) always lowers the saturation state. If averaged over wide spatial scale and over long time, the entrainment of regenerated carbon and biological uptake should cancel out with one another because the same upwelling ultimately controls the nutrient supply and the integrated biological productivity of the upper ocean. However, they may not cancel out locally as the physical upwelling and biological uptake can be decoupled due to the effect of lateral transport and temporal fluctuations in upper ocean nutrient inventory.
 We are thankful for the comments from David Munday on an earlier version of this manuscript. Anonymous reviewers also provided helpful comments on the manuscript. T. Ito is supported by the NSF grant OPP-1142009. M.J. Follows is grateful for support from NSF and NOAA.