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Keywords:

  • carbon dioxide;
  • carbon cycle;
  • ocean processes

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] Carbon perturbations leading to an increase in atmospheric CO2 are partly offset by the carbon uptake by the oceans and the rest of the climate system. Atmospheric CO2 approaches a new equilibrium state, reached after ocean invasion ceases after typically 1000 years, given by PCO2 = P0exp(δIχ/IB), where P0 and PCO2 are the initial and final partial pressures of atmospheric CO2, δIχ is a CO2 perturbation, and IB is the buffered carbon inventory of the air-sea system. The perturbation, δIχ, includes carbon emissions and changes in the terrestrial reservoir, as well as ocean changes in the surface carbon disequilibrium and fallout of organic soft tissue material. Changes in marine calcium carbonate, δICaCO3, lead to a more complex relationship with atmospheric CO2, where PCO2 is changed by the ratio PCO2 = P0{IO(AC)/(IO(AC)δICaCO3)} and then modified by a similar exponential relationship, where IO(AC) is the difference between the inventories of titration alkalinity and dissolved inorganic carbon. The overall atmospheric PCO2 response to a range of perturbations is sensitive to their nonlinear interactions, depending on the product of the separate amplification factors for each perturbation.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] Atmospheric carbon dioxide concentrations are currently rising from anthropogenic emissions, which are partly offset by the exchange of carbon with the terrestrial biosphere, the ocean and, eventually, through the weathering of rocks. The ocean uptake is particularly important in reducing the impact of emissions on timescales of decades to thousands of years [Archer et al., 1997; Sabine et al., 2004]. Our aim is to elucidate the role of ocean processes in modifying the atmospheric response to carbon emissions by developing a new analytical framework. The analytical relationships provide insight into how the carbon system operates by explicitly revealing the atmospheric CO2 dependence of different variables and are ideal to investigate parameter space. The analytical relations provide quantitative predictions for long-term atmospheric CO2 and, thus, provide a simpler reference point to more detailed numerical investigations, such as those by Lenton et al. [2006], Plattner et al. [2001] and Matear and Hirst [1999].

[3] The coupling of the atmosphere and ocean carbon systems is achieved in a rather complex and disjointed manner (Figure 1). While the surface mixed layer is in direct contact with the atmosphere, the timescale for air-sea exchange of carbon dioxide is generally too slow to keep pace with seasonal-forced physical and biological changes [Broecker and Peng, 1982]. Thus, a local equilibrium between the atmospheric and oceanic partial pressure for carbon dioxide is rarely achieved and usually a disequilibrium exists. In turn, the carbon concentrations in the ocean interior are determined by the physical and biological transfer of carbon from the surface mixed layer, which occur in an intermittent manner. The physical transfer is achieved via convection within the mixed layer and then subduction into the stratified thermocline during late winter [Follows et al., 1996]. The biological transfer involves the gravitational fallout of organic matter from the surface sunlit ocean, usually peaking during a spring bloom, with the fallout containing soft tissue, organic carbon and hard tissue, calcium carbonate material. While increased export of organic carbon leads to an increased ocean drawdown of CO2, increased export of calcium carbonate instead alters the charge balance of dissolved inorganic carbon species in the surface ocean and leads to an ocean outflux of CO2.

image

Figure 1. A schematic section depicting the processes transferring carbon within the ocean. Air-sea exchange of carbon dioxide occurs between the atmosphere and the ocean mixed layer (black, open arrows), which varies seasonally and spatially over the globe. Carbon is subsequently transferred into the underlying stratified ocean through a combination of physical and biological processes. The physical transfer is achieved through subduction into the thermocline and overturning in the deep ocean (black, solid arrows). The biological transfer is achieved through the production of organic material in the surface sunlit ocean, which gravitationally fall out and remineralize in the underlying ocean interior. This biological transfer is separated in terms of the fallout of soft tissue and calcium carbonate material (gray curly solid and open arrows, respectively). The atmospheric inventory of carbon is altered through the exchange of carbon with the ocean, as well as through the combination of emissions and the terrestrial exchange (gray, solid arrow).

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[4] Given the complexity of the ocean processes transferring carbon, this study extends a new analytical framework to address two related questions: 1. What is the effect of separate ocean processes on the long-term atmospheric concentration of carbon dioxide? 2. How do these different ocean processes interact with increasing carbon emissions and combine together to effect the long-term atmospheric concentration of carbon dioxide?

[5] In order to illustrate these questions prior to developing our analytical framework, consider a series of separate, carbon perturbations applied to a simple numerical ocean box model (Appendix A; Sarmiento and Toggwieler [1984]): an anthropogenic emission of carbon into the atmosphere (Figure 2a, gray solid line), an increase in biological fallout of soft tissue and calcium carbonate material (Figure 2a, gray dashed and dotted lines, respectively). For the carbon emissions, there is an initial peak in atmospheric concentrations and then a decline to a background state when ocean invasion ceases after typically 1000 years. For the increased fallout of organic carbon, there is a reduction in the atmospheric CO2 from the soft tissue fallout, but a slight increase from the calcium carbonate drawdown. The integrated effect of these different processes differ according to whether each process is treated separately in the model and then linearly summed or, more realistically, allowed to vary together at the same time in the model (Figure 2b, dashed and solid lines, respectively).

image

Figure 2. Atmospheric PCO2 (ppm) versus time (years) for a numerical model of the air-sea system with three ocean boxes for different carbon perturbations: (a) model PCO2 over time when three carbon perturbations are applied separately, a 2000 GtC emission, lasting 400 years (gray solid line), an increase in organic marine fallout (gray dashed line), and CaCO3 fallout (gray dotted line) with both increased at the start by a factor of 1.5; (b) model PCO2 over time when the three carbon perturbations are applied simultaneously (black solid line), and PCO2 over time for a linear sum of the PCO2 changes resulting from the perturbations in Figure 2a being applied separately (black dashed line).

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[6] This study, extending a new analytical framework to understand long-term carbon cycling, is structured in the following manner. Analytical relations revealing the effects of charge neutral carbon cycle changes upon atmospheric CO2 are derived in section 2, which combine a buffered carbon inventory approach [Goodwin et al., 2007] with a process-driven, carbon storage view [Ito and Follows, 2005]. This framework is extended to incorporate changes in the marine CaCO3 cycle including perturbations in the overall surface charge balance in section 3. The analytical framework is used to demonstrate how the amplifying feedbacks combine between different perturbation mechanisms in section 4 and, finally, the implications of the study are discussed in section 5.

2. Developing an Analytical Framework for CO2 Perturbations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[7] Consider an atmosphere, with atmospheric partial pressure PCO2, for carbon dioxide in a global equilibrium with an ocean with an average dissolved inorganic carbon (DIC) concentration equation image. The total amount of carbon in the system, ΣC, is given by

  • equation image

where IA and IO are the atmospheric and oceanic carbon inventories respectively, M is the molar volume of the atmosphere, V is the volume of the ocean, and CDIC = [CO2] + [H2CO3] + [HCO3] + [CO32−]. The effective PCO2 of the ocean is only dependent upon the uncharged constituents of CDIC (Appendix B), but the speciation of DIC in seawater makes calculating this new steady state nontrivial.

[8] Ocean DIC concentrations, CDIC, can be separated into component concentrations due to different processes [Brewer, 1978],

  • equation image

In order to understand this separation, consider a parcel of water, initially at the surface of the ocean in contact with the atmosphere (Figure 1). If the water is in equilibrium with the atmosphere, the DIC concentration is equal to the saturation concentration, Csat. More typically, if there is an air-sea exchange of CO2, then the concentration of DIC is equal to the saturation concentration plus the disequilibrium concentration, Cdis. If the water parcel is now subducted, no further exchange with the atmosphere is possible and so the disequilibrium concentration of the parcel is fixed until the water parcel resurfaces. While the water parcel is in the deep ocean, remineralization of biological soft tissue increases the DIC concentration by Cbio. Finally, dissolution of falling CaCO3 from the hard tissue of an organism increases CDIC of the water parcel further by CCaCO3, as well as increasing the titration alkalinity of the water parcel.

[9] This mechanistic view can now be expressed in terms of a global inventory equation, combining (1) and (2) [Ito and Follows, 2005]:

  • equation image

where an overbar represents a global average. If small perturbations to the inventory equation (3) are now considered:

  • equation image

where, for example, δequation image represents a change in the ocean storage of carbon due to a change in biological nutrient utilization. Changes in the total air-sea carbon inventory (δΣC) on the right-hand side of (4) may be due to anthropogenic carbon emissions (Iem) and exchanges with the terrestrial carbon reservoir (Iter):

  • equation image

There is a minus sign for the δIter term, since an expansion of the terrestrial carbon reservoir decreases the total amount of carbon in the air-sea system.

[10] In order to solve for the change in atmospheric PCO2 on a millennial timescale, (5) can be rearranged (ignoring changes to CCaCO3, which are addressed in section 3) to yield:

  • equation image

where the right-hand side of (6) represents perturbations imparted upon the system, and the left-hand side represents the response of the system on a millennial timescale. Using δIχ to represent the combined effects of δIem, −δIter, −equation image and −equation image, (6) is rewritten as

  • equation image

During the response to a imposed perturbation, δIχ, the only term in DIC (2) that changes is Csat with δequation image = δCDIC. Thus, the term (PCO2δequation image)/equation imageδPCO2) can be reexpressed as (PCO2δequation image)/equation imageδPCO2) = 1/Bglobal, where Bglobal is the globally averaged Revelle buffer factor of seawater. Hence, (7) can be rewritten in terms of the buffered carbon inventory of the air-sea system, IB [Goodwin et al., 2007], where IB = MPCO2 + (Vequation image/Bglobal) = IA + (IO/Bglobal), giving

  • equation image

which relates an infinitesimal perturbation in δIχ to the air-sea system response in PCO2 after ocean invasion. Integrating (8), assuming that IB is unchanged as the system is perturbed, δIBIB, [Goodwin et al., 2007], gives

  • equation image

where δIχ again represents either δIem, −δIter, −equation image and −equation image. This relationship is now employed to predict how atmospheric PCO2 will rise exponentially on a millennial timescale if carbon is added into the atmosphere through (1) an emission of fossil fuels (δIem > 0) or a contraction of the terrestrial carbon reservoir (δIter < 0), (Figure 3a, dashed line); and (2) a reduction in air-sea disequilibrium (δequation image < 0) or a reduction in the carbon stored in the deep ocean due to weakening in biological drawdown (δequation image < 0) (Figure 3b, dashed line).

image

Figure 3. Atmospheric PCO2 (ppm) on a millennial timescale for a range of different carbon perturbations, as predicted by the analytical relations (black dashed line) and compared with a 3 box ocean model (gray solid line): (a) PCO2 perturbed by carbon emissions (δIem) and changes in the terrestrial carbon reservoir size (δIter) (testing (9) with 280 < PCO2 < 1080ppm); (b) PCO2 perturbed by changes in biological soft tissue drawdown (δCbio) and the mean state of ocean disequilibrium (δCdis) (testing (9) with nutrient utilization efficiency ranging from 0 to 97%); (c) PCO2 perturbed by changes to the drawdown of carbon by marine calcification (δCCaCO3) (testing (21) with the rain ratio of CaCO3 to organic carbon in falling matter ranging from 0 to 0.5). (d) PCO2 perturbed by CaCO3 dissolution and precipitation imbalances to alter the ocean titration alkalinity and total air-sea carbon inventories (where δIopen is equal to the carbon inventory change, and half the titration alkalinity inventory change) (testing (23) from −1000 < δIopen < +3000 GtC). Note that in Figures 3b and 3c, a change in carbon concentration of 0.1 gCm−3 is equivalent to a globally averaged carbon inventory change of ∼1500 GtC.

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[11] Why can the buffered carbon inventory, IB, be considered constant in the integration of (8)? In preindustrial ocean conditions the majority of DIC exists in the bicarbonate form, with carbonate dominating over the uncharged forms (collectively labeled CO2*) among the minor constituents (Figure 4a). If a charge neutral carbon perturbation is imparted on the system, δIχ, such that PCO2 (and therefore IA) increases, the proportion of DIC in the form CO2* increases and the carbonate proportion decreases, while the majority of DIC remains in the bicarbonate form. These DIC constituent changes cause the globally averaged buffer factor of ocean waters, Bglobal, to increase from a preindustrial value of around 12 to a maximum of around 20 when [CO2*] ≈ [CO32-] (Figure 4b). An increase in PCO2 acts to increase the atmospheric inventory, IA, but at the same time acts to increase Bglobal, thus leading to relatively small changes in the buffered carbon inventory IB (Figure 4c, up to 4000 GtC). However, further emissions in excess of 4000 GtC leads to both a further increase in IA, as well as a decrease in Bglobal, thus leading to an eventual increase in IB (Figure 4c).

image

Figure 4. Sensitivity of the carbon system to emissions (GtC) evaluated at a steady state from a 3 box ocean model (Appendix A): (a) Fractional concentration of DIC species (CO2*, solid line; HCO3, dashed line; CO32−, dotted line) versus emissions; (b) global average Revelle buffer factor, Bglobal, versus emissions; (c) buffered carbon inventory, IB, versus emissions. In the present-day regime up to accumulated emissions of 4000 GtC, where CO32− ≫ CO2*, Bglobal increases as emissions increase, causing the buffered carbon inventory, IB = IA + (IO/Bglobal), to remain close to its preindustrial value. When emissions exceed ∼5000 GtC, CO2* ≫ CO32− and Bglobal decreases as emissions increase, causing IB to increase significantly above its preindustrial value.

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[12] These analytical relations for PCO2(9) not only provide insight as to the effect of CO2 perturbations when ocean invasion ceases, they also provide quantitative skill. In order to illustrate this predictive ability, the analytical relations are compared against a numerical 3 ocean box model (Appendix A) based on Sarmiento and Toggweiler [1984]. The numerical box model includes crude representations of meridional overturning, ocean exchanges and ocean biological fallout and remineralization, but does not contain representations of sediment or weathering interactions. The box model is perturbed by additions of carbon and changes to biological nutrient utilization. The final PCO2 predicted by the analytical relations (9) are in close agreement with the numerical calculations of the box model (Figures 3a and 3b), confirming the skill of the analytical relations for this range of parameter space (as well as the condition δIBIB being satisfied). For accumulated carbon emissions up to 4000 GtC, the analytical relationship (9) has already been shown to have skill in comparing with a series of 3000 year integrations of the MIT global circulation and carbon model [Goodwin et al., 2007].

[13] Given the skill of the analytical relationship for CO2 perturbations, the more complex problem of the calcification cycle is considered.

3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[14] Changes in the marine calcification cycle alter the charge balance of ocean waters, where the formation or dissolution of CaCO3 in seawater is given by,

  • equation image

On the left-hand side, there is 1 unit of DIC which carry 0 units of charge, while on the right-hand side, there is 2 units of DIC which collectively carry 2 units of charge. Therefore, as dissolution of CaCO3 occurs and (10) proceeds to the right, the DIC concentration increases by one unit and the charge concentration, the carbonate alkalinity, AC, increases by two units [Bolin and Eriksson, 1959].

[15] Consequently, if the rate of CaCO3 formation, fallout and dissolution at depth increases, there is a 2:1 increase in AC and CDIC in the deep waters, as well as an opposing 2:1 decrease in AC and CDIC in the surface waters (since globally alkalinity and DIC are conserved). This enhanced reduction in surface ocean alkalinity causes the PCO2 of surface waters in the ocean to rise, leading to an ocean outgassing of CO2 until the surface ocean PCO2 and the atmospheric PCO2 reequilibrate.

[16] The analytical framework for PCO2 is now extended to include perturbations in the marine calcification cycle with changes in alkalinity and DIC incorporated in a 2:1 ratio and allowing air-sea CO2 exchange.

3.1. Effect of a CaCO3 Perturbation on Seawater PCO2

[17] Approximating titration alkalinity with carbonate alkalinity, equations for the carbonate alkalinity system (Appendix B) can be combined to form a quadratic in [CO2*]:

  • equation image

where a, b, and c are terms containing CDIC, the carbonate alkalinity, AC, and the first and second dissociation constants of CO2 in seawater, K1 and K2 (Appendix B). Implicitly differentiating this quadratic (11) assuming constant temperature, T, and salinity, S, leaves an expression relating infinitesimal changes to AC and CDIC to the resulting change in [CO2*] of the form:

  • equation image

where δT,S indicates a small change with T and S held constant. Performing this implicit differentiation loses the information required to analytically approximate an initial PCO2 value, but keeps the information required to calculate a change in PCO2. Assuming δAC = 2δCDIC, this relation (12) can be rearranged for CaCO3 perturbations to give

  • equation image

This relationship can be simplified in the following manner. First, K2/K1 is small, implying that

  • equation image

and, second, [CO2*] is very small in relation to CDIC implying

  • equation image

which approximates to ∣2δ[CO2*]∣ ≪ ∣δCDIC∣, since for a CaCO3 perturbation δAC = 2δCDIC. Thus, whenever conditions (14) and (15) are met, (13) can be simplified, as well as combined with seawater PCO2 being proportional to [CO2*], to relate an infinitesimal CaCO3 perturbation to the resulting infinitesimal change in seawater PCO2:

  • equation image

which can be integrated to give

  • equation image

relating the change in the PCO2 of a water parcel due to a 2:1 change in AC and CDIC from CaCO3 changes. As CaCO3 is dissolved, CDIC increases, but PCO2 decreases because of the greater addition of alkalinity (Figure 5; dots); the log change in PCO2 being given by the negative of the log change in AC − CDIC.

image

Figure 5. Water parcel PCO2 (ppm) for dissolution or precipitation of CaCO3 versus change in DIC (moles m−3) for a range of temperatures and initial PCO2 values, as predicted by analytical expression (17) (dots) and an explicit numerical carbonate system model (solid line) after Follows et al. [2006]: water temperatures of (a) 20°C, (b) 12°C, and (c) 5°C, as well as initial PCO2 of 360 ppm (light gray line), 280 ppm (dark gray line), and 180 ppm (black line). In each case, the unperturbed titration alkalinity = 2.4 mol m−3, and salinity = 34.7 psu. The precipitation or dissolution of CaCO3 leads to alkalinity and DIC changes in a 2:1 ratio.

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[18] This relationship (17) is valid for surface ocean conditions with PCO2 typical of the present-day, Holocene or last glacial maximum levels (as illustrated in the model test in Figure 5), while conditions (14) and (15) are met and titration alkalinity is approximated by carbonate alkalinity, AC; eventually, (17) breaks down when there is very high PCO2, CDIC approaches AC, and (15) becomes invalid.

[19] In comparison, Broecker and Peng [1982] derived an analytical prediction for the value of PCO2 of ocean waters at given values of CDIC and AC, but their prediction requires knowledge of T and S and ignores the CO2* component of DIC in the carbonate chemistry equations. In this alternative relation (17), the CO2* component of DIC is retained in its derivation, and knowledge of T and S is not required (as the terms in (13) containing the dissociation constants of CO2 are insignificant), but (17) does require a known initial value of PCO2 in order to approximate analytically the change due to a CaCO3 perturbation.

3.2. Closed System Perturbations: Reorganization of the Vertical Alkalinity Gradient

[20] The analytical framework is now applied to examine the effects of two types of perturbation to the ocean calcium carbonate cycle: Closed system changes, where the oceanic calcium ion budget is fixed, but its distribution rearranged (this section), and open system changes where external sources and sinks of calcium ions are included (section 3.3).

[21] Closed system changes are relevant for internal ocean changes, such as a global change in phytoplankton community structure affecting the amount of calcium carbonate production and export. For example, it has been hypothesized that glacial periods may have been characterized by a leakage of silica from the Southern Oceans enhancing diatom production globally and reducing coccolithophore production [Brzezinski et al., 2002; Matsumoto et al., 2002]. The effects of such a perturbation (on a submillenial timescale) can be viewed as a closed system response, with a reduction in the rain ratio of CaCO3 to organic tissue in falling matter, a decrease in the mean vertical gradient of alkalinity, and a global increase in surface alkalinities.

[22] The impact of such closed system changes are now considered in terms of the analytical framework. In reality, when the calcium carbonate cycle is perturbed, changes in deep ocean storage of alkalinity and DIC occur simultaneously with an air-sea exchange in CO2. However, for simplicity, consider this adjustment process to occur in two separate hypothetical stages.

[23] 1. In the first stage, the system is assumed to be at a steady state with a partial pressure of P0, and ocean saturation concentration of Csat, with the carbon inventory given by

  • equation image

When CCaCO3 is perturbed, assume that the deep ocean storage of alkalinity and DIC and the effective PCO2 of the ocean adjusts, but no air-sea exchange of CO2 is yet permitted. At this transient stage, atmospheric PCO2 remains at P0 while the saturation carbon concentration of the ocean is altered to Csat(Pocean). For this state, P0 and Pocean can be related using (17) with the globally averaged concentrations of preformed DIC, equation image, and preformed alkalinity, equation image,

  • equation image

where an increase in equation image reduces equation image and equation image in a 2:1 ratio by redistributing alkalinity and DIC away from the surface ocean and into the ocean interior. This hypothetical stage creates a charge neutral carbon anomaly of δIχ = M(P0Pocean) with the ocean now saturated at Csat = Csat(Pocean), but the atmosphere still having a CO2 partial pressure of P0.

[24] 2. In the second stage, air-sea exchange of CO2 proceeds until the charge neutral carbon anomaly is removed and there is no further net annual air-sea carbon exchange. Once stage 2 has completed the system reaches a final steady state with the atmospheric CO2 partial pressure equal to Pfinal and Csat reaching Csat(Pfinal). The resulting charge neutral carbon anomaly, Pfinal can be related to Pocean by (9),

  • equation image

[25] Combining (19) and (20) then allows the change in atmospheric PCO2 to be related to a small perturbation in global average CCaCO3,

  • equation image

where the initial value is P0 and the final value after air-sea exchange is Pfinal. This equation can be written more succinctly by defining IO(AC) as the initial difference between global inventories of preformed alkalinity and DIC, and by writing PCO2 = Pfinal:

  • equation image

An increase in CCaCO3 acts to raise atmospheric PCO2 (Figure 3c, dashed line) due to the 2:1 reduction in preformed alkalinity and DIC, from the ratio term, ∣IO(A−C)/(IO(A−C)equation image)∣ in (21b); there is larger increase in PCO2 when the inventory IO(A−C) is small. This increase in PCO2 is partly opposed by the resulting air-sea exchange with a damping though the exponential term. Thus, in contrast to perturbations in Iem, Iter, Cbio and Cdis, a given perturbation in CCaCO3 causes a larger change in PCO2 if IB is large, because the damping due to air-sea exchange is decreased.

[26] To exploit this analytical relationship (21b) to understand how the sensitivity of PCO2 to CaCO3 perturbations varies, consider two air-sea systems in steady state, both with CaCO3 weathering and sediment cycles in equilibrium and with the same deep sea carbonate ion concentration: (1) The system that has a higher vertical gradient of DIC, e.g., because of having a stronger biological drawdown of carbon, will have a larger value of IO(AC) and its atmospheric carbon levels will be correspondingly less sensitive to changes in CCaCO3. (2) The system that has a lower vertical gradient of DIC, because of having a weaker biological drawdown of carbon, will have a smaller value of IO(AC) and its atmospheric carbon levels will be correspondingly more sensitive to changes in CCaCO3.

[27] Hence, the analytical framework suggests that a more efficient drawdown of carbon by biological soft tissue can make atmospheric CO2 levels less sensitive to changes in the marine calcification cycle.

[28] This analytical relationship (21b) provides reasonable quantitative skill with predictions in close agreement (with a typical accuracy of 4%) with the final PCO2 reached by the 3 box ocean model (Figure 3c and Appendix A).

3.3. Open System Perturbations: Sources and Sinks of Alkalinity to the Global Ocean

[29] On geological timescales of many millennia and longer, the combined ocean-atmosphere cannot be considered a closed system with respect to carbon and alkalinity due to weathering and interactions with the carbonate sediments [Ridgwell and Zeebe, 2005]. Dissolved CaCO3 is constantly being added to the ocean from the weathering of rocks on land, which adds titration alkalinity and DIC. This source is offset by a deposition of precipitated CaCO3 on the ocean floor, which removes titration alkalinity and DIC. An imbalance between weathering and deposition of CaCO3 can alter the ocean inventories of titration alkalinity and DIC over many thousands of years. For example, the long-term response to anthropogenic emissions implies an acidification of the ocean and a reduced rate of calcium carbonate being buried in sediments. This reduction in burial rate in turn leads to an imbalance between supply and removal of dissolved CaCO3, resulting in a 2:1 increase in the whole ocean titration alkalinity and DIC inventories until carbonate ion concentrations are restored [Archer, 2005; Archer et al., 1997]. The analytical framework is now extended to account for such “open system” perturbations.

[30] Now extend the analytical framework to include marine calcification changes for an open system with an increase in the DIC inventory, δIopen, from reduced precipitation or enhanced weathering input of CaCO3. Following the same method as for the closed system, the hypothetical change in ocean partial pressure of CO2 with no air-sea exchange is given by

  • equation image

and the final change to atmospheric CO2 with air-sea exchange included is given by

  • equation image

These relations (22) and (23) are modified from (19) and (21) by the inclusion of the additional inventory source of DIC, δIopen, from the global dissolution of CaCO3 exceeding global precipitation (also equal to half the titration alkalinity inventory change). In addition, the sign of the term −CCaCO3 in (21) is changed to +δIopen in (23), since adding 2 units of titration alkalinity and 1 unit of DIC increases the difference between preformed concentrations.

[31] The analytical theory (23) predicts that increasing the titration alkalinity and DIC inventories, due to total global CaCO3 dissolution exceeding precipitation, (δIopen > 0) will decrease steady state PCO2 (Figure 3d, dashed line). Within the range −1000 < δIopen < +3000 GtC, the analytical theory predicts steady state PCO2 to within 10% of a numerical box model comparison (Figure 3d).

4. Generalizing the Analytical Framework for the Carbon System

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[32] The simultaneous effect of a set of perturbations on atmospheric PCO2 is now considered, rather than treat each perturbation in isolation. Combining together the generic exponential relation (9), the closed (21) and open system (23) relations for calcium carbonate changes gives:

  • equation image

This combined relationship for atmospheric PCO2 provides a concise summary of how carbon system responds to a range of perturbations. Atmospheric PCO2 increases if carbon is added to the atmosphere by emissions or a contraction of the terrestrial carbon reservoir, while there is a decrease in PCO2 for an increase in organic biological drawdown (Figure 6a, dashed lines). Atmospheric PCO2 is also increased through an increase in the marine drawdown of CaCO3 (Figure 6b, dashed lines) or a removal of titration alkalinity and DIC in a 2:1 ratio by a net precipitation of CaCO3 within the ocean (Figure 6c, dashed lines).

image

Figure 6. Contours of atmospheric PCO2 (ppm) for ocean biological changes, δCbio + δCdis or δCCaCO3 (mol m−3), and external CaCO3 sources, δIopen (GtC), versus external carbon additions, δIemδIter (GtC), as predicted from the analytical relationship (24) (black dashed lines) and compared with a numerical calculation at steady state from a 3 box model (gray solid lines). Adding extra carbon into the air-sea system, because of changes in Iem and Iter, increases steady state PCO2. Increasing the efficiency of soft tissue carbon drawdown increases Cbio + Cdis and lowers steady state PCO2 (a), while increasing the strength of the marine calcification cycle increases CCaCO3 and increases steady state PCO2 (b). Adding titration alkalinity and DIC in a 2:1 ratio from CaCO3 dissolution/precipitation imbalances effects increases Iopen and decreases steady state PCO2 (c).

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[33] In terms of quantitative skill, the analytical predictions for PCO2 are in reasonable accord with the numerical calculations from the simplified box model (Figure 6). While PCO2 remains below 1080ppm, or ∼4 times preindustrial levels, the analytically predicted PCO2 agrees with model output to within 4% within the ranges of δIem, δIter, δCdis, δCbio and δCCaCO3 tested. In the extreme limit when atmospheric PCO2 begins to exceed 4 times preindustrial levels, then the buffered carbon inventory can no longer be considered constant, and (24) is no longer valid. When considering “open system” effects, errors become significant when both δIemδIter exceeds 3000 GtC and δIopen exceeds 2000 GtC (Figure 6c) for a combination of reasons: IB cannot be assumed constant with such large combined changes to titration alkalinity and carbon inventories (and also then the extra assumptions made when altering titration alkalinity, (14) and (15), become invalid).

[34] In reality, PCO2 is affected by the combination of the multiple carbon processes considered, rather than by a single carbon perturbation acting alone. The overall change in PCO2 is not accurately given by a linear summation of the individual perturbations acting in isolation, as revealed at the outset in the box model illustration in Figure 2b. The analytical relationship for the combined perturbations (24) can be expanded to reveal how the separate perturbations δIem, δequation image, δequation image, δequation image, δIter and δIopen combine to alter PCO2

  • equation image

and after dividing by P0 can be equivalently written as

  • equation image

where the terms labeled by PCO2/P0 for the changes in Iem, Iter, Cbio and Cdis are given by (9), the change in CCaCO3 is given by (21) and the change in Iopen is given by (23). Therefore, the overall amplification of PCO2 above the initial value P0 is given by multiplying together each of the separate amplifications of PCO2 when each carbon perturbation acts separately.

[35] These analytical predictions for the final equilibrium state are supported by the numerical box model calculations (Table 1 and Figure 7, gray and black solid lines, respectively), explaining the nonlinear interaction. In addition, this framework allows the changes in PCO2 from a single model integration to be attributed to each of the separate perturbations considered.

image

Figure 7. Modeled atmospheric PCO2 (ppm) versus time (years) for three carbon perturbations, including a 2000 GtC emission and an increase in organic marine and CaCO3 fallout (as in Figure 2a): when the perturbations are numerically modeled separately and the PCO2 changes linearly combined (black dashed line), and when the perturbations are numerically modeled separately and the results combined according to the analytical prediction (26) (gray solid line), compared with numerical model output when the perturbations are applied simultaneously (black solid line).

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Table 1. Model Output After 2000 Years for Three Carbon Perturbations Applied Separately and Togethera
ForcingCarbon Cycle ResponsePerturbation (GtC)Model ΔPCO2 (ppm)equation image
  • a

    Scenarios as in Figures 2 and 7. Linearly adding the effects of multiple carbon cycle changes does not accurately predict the overall PCO2 change. Instead, applying the analytical framework using the amplification factors for each carbon perturbation leads to a much smaller error in PCO2.

  • b

    The numerical model ΔPCO2 values for the separate forcings are added linearly.

  • c

    The numerical model (PCO2/P0) values are multiplied according to (26).

Soft tissue fallout changeequation image+767−530.81
CaCO3 fallout changeequation image+154+91.03
Carbon emissionδIem+2000+2191.78
 Prediction for combined forcing+175b1.49c
Emission + soft tissue + CaCO3Model output for combined forcing+1321.47
 ERROR+44ppm+6ppm

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[36] This study provides an analytical framework to understand how the ocean modifies the atmospheric response to carbon emissions on a millennial timescale. The final and initial atmospheric partial pressure of carbon dioxide are connected by a generic exponential relationship, PCO2 = P0 exp equation image, depending on the size of the carbon perturbation, δIχ, divided by the atmosphere and ocean buffered carbon inventory, IB.

[37] This relationship reveals how atmospheric PCO2 will rise exponentially on a millennial timescale if carbon is added into the atmosphere by an emission of fossil fuels, a contraction of the terrestrial carbon reservoir, a reduction in the carbon stored in the deep ocean due to weakening in biological drawdown or a reduction in local air-sea disequilibrium. This relationship becomes more complicated when calcium carbonate cycling is incorporated, altering the surface charge balance, with PCO2 increasing following either an increased export of calcium carbonate into the deep ocean or a global net precipitation of CaCO3.

[38] These analytical relationships provide insight as to how different processes operate in the carbon system when ocean invasion ceases, as well as quantitative predictions for atmospheric PCO2. The relationships are ideal to explore a large range of parameter space and, thus, an ideal complement to more sophisticated numerical studies of the carbon system, which are often computationally restricted to a smaller range. Thus, the analytical relationships are a useful tool to explore the long-term response of carbon perturbations for a wide range of carbon problems.

[39] The analytical relations support the following speculations as to how the carbon system is likely to change. Anthropogenic carbon emissions will cause PCO2 to be higher on a millennial timescale according to (9) (δIem > 0). However, the long-term response of the terrestrial carbon inventory to climate change remains uncertain [Lenton et al., 2006]. Ocean acidification due to emissions will make the precipitation of CaCO3 in surface waters less favorable, and could reduce marine calcification (δCCaCO3 < 0) [Ridgwell and Zeebe, 2005], which will tend to dampen the PCO2 increase, (21). Changes in wind-forcing due to climate change are also highly uncertain; however if wind-forcing of the surface ocean increases, ocean DIC is likely to become further from saturation and the Cdis pool will become more negative (δCdis < 0) [Ito and Follows, 2003, 2005], amplifying the increase in PCO2, (9). Changes in global ocean nutrient utilization in response to anthropogenic forcing are also highly uncertain; however if the Southern Ocean ventilation intensifies, increasing the supply of nutrients which remain unutilized, then the biological soft tissue carbon pool will decrease (Cbio < 0), further amplifying the increase in PCO2, (9). In addition on timescales of tens of thousands of years, ocean acidification due to emissions is predicted to cause a net addition of dissolved CaCO3 to the ocean (δIopen > 0), which will dampen the anthropogenic increase in PCO2, (23), until carbonate ion concentrations are restored.

[40] While each carbon perturbation can be understood and quantified in isolation, using (9), (21), and (23), in reality a range of carbon perturbations are occurring at the same time. Our framework predicts that the PCO2 after ocean invasion is accurately estimated by multiplying the amplification factors for each process using (26), rather than linearly combining the sum of each separate process. While a reasonable estimate of the carbon response from our box model studies is initially obtained by linearly combining the effects of different processes, an accurate estimate of the final equilibrium state is only obtained by accounting for the nonlinear amplification between the feedback processes implied by our analytical framework (Figure 7). Thus, our analytical framework can be used to provide insight into the nonlinear interactions between carbon perturbations.

[41] There are more complicated feedback processes that are not included in the analytical relationships, which ultimately need to be considered for a complete prediction. For example, temperature changes alter the solubility of CO2 and feedback on atmospheric concentrations, which is estimated to provide a ∼10% amplification for PCO2 [Archer, 2005]. The sensitivity of the ocean circulation to anthropogenic forcing remains poorly understood and any changes in ventilation or overturning can feedback and alter PCO2.

[42] While numerical models provide more detailed information about spatial and temporal variability, our analytical relations provide a transparent view of how the final equilibrium is controlled. Increasingly model hierarchies are needed to provide insight into how the climate system operates [Held, 2005], by revealing how the climate system changes as sources of complexity are added or removed. In the same way, the analytical relations presented here should form part of that model hierarchy for the carbon system.

Appendix A:: Numerical Box Model Description

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[43] To illustrate the predictive power of the analytical relations derived in this paper, the analytical relations are compared with a numerical model of the air-sea system based on Sarmiento and Toggweiler [1984]. The numerical model contains a well mixed atmosphere attached to a 3-box representation of the ocean, containing cold high-latitude surface ocean, warm low-latitude surface ocean and deep ocean boxes. A meridional overturning circulation is applied, where water sinks from the high-latitude surface ocean to the deep ocean, upwells in the low-latitude surface ocean and is transported back into the high-latitude surface ocean. An additional exchange is applied between the high-latitude surface ocean and the deep ocean boxes. The ocean carbonate system is solved after Follows et al. [2006]. Biological production occurs in the surface ocean boxes as a function of supplied phosphate, marine production of CaCO3 also occurs, removing dissolved inorganic carbon and titration alkalinity in a 1:2 ratio. All organic and marine CaCO3 fallout is dissolved in the deep ocean box, there is no sedimentation or weathering interactions. The preindustrial spin up is achieved by forcing the atmospheric PCO2 to 280ppm, air-sea exchange of CO2 is then allowed. The perturbations applied to the model consist of adding carbon into the atmosphere, changing the biological phosphate utilization and/or changing the marine CaCO3 fallout. The model reaches a steady state ∼1000 years after perturbation. The relevant parameter values used for the 3 box model are given in Table A1.

Table A1. Values of the Parameters Used in the 3 Box Ocean Model
Model Parameter DescriptionParameter Value
Volume of the ocean1.3 × 1018 m3
Surface area of the ocean3.49 × 1014 m2
Molar volume of the atmosphere1.77 × 1022 moles
Salinity of ocean34.7 psu
Alkalinity inventory3.08 × 1018 mole equivalents
Phosphate inventory2.86 × 1015 moles
Meridional overturning circulation25.6 Sv
Additional high-latitude-deep ocean exchange38.1 Sv
Ratio of phosphate to carbon in soft tissue fallout1:116
Area high-latitude surface ocean20% of total surface area
Area low-latitude surface ocean80% of total surface area
Volume high-latitude surface ocean1.3% of total volume
Volume low-latitude surface ocean2.1% of total volume
Temperature of high-latitude surface ocean2°C
Temperature of low-latitude surface ocean20°C
Preindustrial total carbon inventory (IA + IO)35600 GtC
Preindustrial pCO2280 ppm
Preindustrial biological phosphate utilization of surface ocean boxes38% of supplied phosphate
Preindustrial CaCO3 to organic C fallout rain ratio1:5
Buffered atmosphere and ocean carbon inventory, IB3530 GtC
IO(AC)2.66 × 1017 mole equivalents

Appendix B:: Carbonate Alkalinity System

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

[44] In the ocean, carbon dioxide exists as dissolved inorganic carbon, DIC, defined with concentration CDIC:

  • equation image

where [CO2*] is the combined concentration of aqueous CO2 and carbonic acid. The total charge of dissolved inorganic carbon species, the carbonate alkalinity, is defined by the concentration AC:

  • equation image

[45] The constituents of CDIC and AC at steady state are partitioned according to the dissociation equations:

  • equation image

and

  • equation image

where K1 and K2 are functions of temperature (T) and salinity (S).

[46] The effective partial pressure of CO2, PCO2, within a water parcel is determined by the concentration of uncharged constituents of CDIC by:

  • equation image

where k0 is a function of temperature and salinity. Thus, the charged species of DIC do not directly interact with the atmosphere. Locally, there will be no air-sea transfer of CO2 when the PCO2 of the air is equal to the PCO2 of the seawater.

[47] If carbonate alkalinity is used to approximate titration alkalinity, equations (B1) to (B5) describe a closed carbonate system. The equations can be combined, along with additional constraints, such as borate and silicate concentrations and the dissociation constant of water, to explicitly solve the system numerically [e.g., see Follows et al., 2006].

[48] Equations (B1) to (B4) can be combined to form a quadratic in [CO2*] by eliminating terms in [CO32−], [HCO3] and [H+] [Goodwin, 2007]

  • equation image

Implicitly differentiating this quadratic (B6) assuming constant temperature and salinity, and thus a constant K1 and K2, leaves an expression relating infinitesimal changes to AC and CDIC to the resulting change in [CO2*]:

  • equation image

This relation is equivalent to (12), with the coefficients a, b, and c explicitly stated in terms of K1, K2, AC, and CDIC.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Developing an Analytical Framework for CO2 Perturbations
  5. 3. Effect of Calcium Carbonate Cycling and Changes in Alkalinity
  6. 4. Generalizing the Analytical Framework for the Carbon System
  7. 5. Discussion
  8. Appendix A:: Numerical Box Model Description
  9. Appendix B:: Carbonate Alkalinity System
  10. Acknowledgments
  11. References
  12. Supporting Information
FilenameFormatSizeDescription
gbc1520-sup-0001-t01.txtplain text document1KTab-delimited Table 1.

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