On the in ﬂ uence of marine biogeochemical processes over CO 2 exchange between the atmosphere and ocean

The ocean holds a large reservoir of carbon dioxide (CO 2 ), and mitigates climate change through uptake of anthropogenic CO 2 . Fluxes of CO 2 between the atmosphere and surface ocean are regulated by a number of physical and biogeochemical processes, resulting in a spatiotemporally heterogeneous CO 2 distribution. Determining the in ﬂ uence of each individual process is useful for interpreting marine carbonate system ob- servations, and is also necessary to investigate how changes in these drivers could a ﬀ ect air-sea CO 2 exchange. Biogeochemical processes exert an in ﬂ uence primarily through modifying seawater dissolved inorganic carbon ( C T ) and total alkalinity ( A T ), thus changing the seawater partial pressure of CO 2 ( p sw ). Here, we propose a novel conceptual framework through which the size of the CO 2 source or sink generated by any biogeochemical process, denoted Φ , can be evaluated. This is based on the ‘ isocapnic quotient ’ ( Q ), which de ﬁ nes the trajectory through ( A T , C T ) phase space for which there is no change in p sw . We discuss the limitations and uncertainties inherent in this technique, which are negligible for most practical purposes, and its links with existing, related approaches. We investigate the e ﬀ ect on Φ of spatiotemporal heterogeneity in Q in the present day surface ocean for several key biogeochemical processes. This leads the magnitude of the CO 2 source or sink generated by processes that modify A T to vary spatiotemporally. Finally, we consider how the strength of each process as a CO 2 source or sink may change in a warmer, higher-CO 2 future ocean.


Introduction
The global ocean hosts a substantial reservoir of carbon dioxide (CO 2 ) in the form of dissolved inorganic carbon (C T ), which can buffer changes in the atmospheric CO 2 concentration (Le Quéré et al., 2016) and its climatic consequences (IPCC, 2013). Although the global surface ocean is presently a net CO 2 sink, the spatiotemporal distribution of airsea CO 2 exchange is heterogeneous. To first order, CO 2 is supplied to the atmosphere in the tropics and upwelling zones, and taken up by the ocean in subpolar regions (Takahashi et al., 2009). The magnitude and phase of seasonal cycles in air-sea CO 2 exchange are also spatially variable. These patterns emerge from the interactions between a number of physical and biogeochemical processes. There are thus several reasons why we need to understand how each individual process affects air-sea CO 2 exchange, including interpreting in situ observations of seawater chemistry, and projecting how changes in these drivers could affect the future oceanic CO 2 sink.
Marine processes can affect air-sea CO 2 exchange by altering the seawater partial pressure of CO 2 (p sw ), with increasing/decreasing p sw tending to create a CO 2 source/sink, respectively, for atmospheric CO 2 . One way that this can happen is through changes in seawater temperature and/or salinity (Weiss et al., 1982). Additionally, biogeochemical processes can modify p sw , primarily through their effects on C T and total alkalinity (A T ). The impact of changes in C T is intuitive: if C T is removed from solution (e.g. by conversion into organic matter), this creates the potential for additional CO 2 to be taken up from the atmosphere, thus reducing p sw . A T can qualitatively be thought of as the capacity of seawater to store C T for a given value of p sw . Processes that decrease A T therefore increase p sw and act as CO 2 sources to the atmosphere, while increasing A T constitutes a CO 2 sink (Fig. 1).
Consequently, there exists some ratio of changes in A T and C T that cancel out each other's effect on p sw , resulting in zero net p sw change. We refer to this p sw -neutral ratio as the 'isocapnic quotient' (Q). Its value varies through (A T ,C T ) phase space, and is equal to the slope of the p sw isoline (or 'isocap') at any given point. We have appropriated the word 'isocapnic' from medical science, where it refers to a constant dissolved CO 2 concentration in blood, as there was no oceanographic term for this concept in seawater.
Here, we derive mathematical expressions for Q, and show how its value depends upon the state of the marine carbonate system. The changes in A T and C T driven by any biogeochemical process define its 'biogeochemical process vector' (q). We introduce the parameter Φ, which combines Q and q to quantify the potential size of the CO 2 source or sink associated with the biogeochemical process. Φ thereby indicates the amount of C T that must be lost or gained by the seawater (e.g. through air-sea CO 2 exchange), following the action of a biogeochemical process, in order to return to the p sw value from before the process occurred. The 'released CO 2 to precipitated carbonate ratio' (Ψ) defined by Frankignoulle et al. (1994) is a special case of Φ, applicable only to calcification. We discuss the limitations and uncertainties associated with our approach, such as the assumption of isocap linearity, and timescale considerations. Our novel technique indicates that the CO 2 source or sink strength of any biogeochemical process that modifies A T is not constant, but rather depends upon the local seawater conditions, in particular temperature and p sw . We investigate the effect of spatiotemporal heterogeneity in Q in the surface ocean on Φ for different biogeochemical processes, using a climatological dataset that represents near-present day conditions (Takahashi et al., 2014b). Finally, we consider how these effects may change in a warmer, higher-CO 2 future ocean.

Methods
Throughout this article, square brackets indicate the dissolved concentration of the enclosed chemical species in mol kg -1 , where kg -1 is of seawater (not H 2 O). In some instances, we use s and h to denote [CO 2(aq)  ion concentrations (Zeebe and Wolf-Gladrow, 2001): The reactions for the dynamic equilibria between the carbonate species in Eq. (1), and definitions of the relevant dissociation constants, are given in Appendix B. Roughly speaking, 'total alkalinity' (A T ) quantifies the deficit of protons in solution relative to a 'zero-level' at pH around 4.5. More precisely, it is the excess of proton acceptors (like bicarbonate and carbonate ions) over proton donors with respect to a 'zero level of protons' (Dickson, 1981). The following simplified equation captures > 99.8% of its components in typical surface ocean seawater, which is sufficient for our purposes: (2) A T is therefore influenced by chemical species additional to those directly related to CO 2 . The relevant equations and reactions are given in Appendix B. All of the marine carbonate system variables (e.g. C T , A T , p sw or s, h) can be calculated if the values of any pair are known (Zeebe and Wolf-Gladrow, 2001).

Isocapnic quotient
The isocapnic quotient (Q) is defined as the rate of change of A T relative to C T at constant s (or equivalently p sw ), under constant salinity, temperature and pressure (STP) conditions: 2.2.1. Simple approximation Before deriving the 'exact' equation for Q, we developed an informative approximation. Starting from the following simplified expressions for C T and A T : and practical salinity = 35 using CO 2 SYS (van Heuven et al., 2011). The black contours, called isocaps, are drawn at the p sw values labelled on the colour bar. (b) Schematic trajectories of some example unit biogeochemical process vectors (q ). Vectors in the CO 2 'sources' region (red shading) increase p sw , while CO 2 'sinks' (blue shading) decrease p sw . The boundary between these regions is an isocap, and its slope is equal to the isocapnic quotient Q. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) we derived the following approximations for Q (i.e. Q x ), which are identical to each other, but stated in terms of p sw and s respectively: The full derivation of Eqs. (6) and (7) is provided in Appendix C.

Full derivation
We derived the exact equation for Q from the definitions of C T and A T in Eqs. (1) and (2), as shown in Appendix D: We can thus use Eq. (8) to calculate Q, given any pair of marine carbonate system variables (Zeebe and Wolf-Gladrow, 2001).

Biogeochemical process vectors
The change in A T relative to C T during any given biogeochemical process can be quantified as a vector q = (A q , C q ) through (A T , C T ) phase space (Fig. 2). We use the notationq to refer to the unit vector case (i.e. where q has been normalised such that A q 2 + C q 2 = 1). In applying our method, it would normally be appropriate to use q; the unitq simply provides a convenient way to normalise the different processes for our analysis. Previous studies have investigated these changes in detail (e.g. Wolf-Gladrow et al., 2007), so here we only provide A q and C q values associated with the selection of common processes described below, as examples (Table 1). Air-sea CO 2 exchange increases or decreases C T by the amount of CO 2 transferred (i.e. C q ), with no change in A T (hence A q = 0).
Autotrophic production converts C T into particulate organic carbon (POC), so C q for this process is stoichiometrically equal to the amount of POC formed. The associated increase in seawater A T (i.e. A q ), of +0.21 mol per mol-C converted into POC, results from simultaneous nutrient uptake (Wolf-Gladrow et al., 2007). Calcification converts C T into CaCO 3 , which decreases A T by 2 mol per mol CaCO 3 formed (Eq. (2)). Remineralisation and CaCO 3 dissolution have opposite effects to autotrophic production and calcification respectively. Denitrification/ anammox and N 2 fixation do not affect C T (hence C q = 0), but they alter A T through nitrate uptake or production, in the same way as autotrophic production/remineralisation. N 2 fixation itself only converts N 2 into organic matter; this must be followed by remineralisation if A T is to be decreased by nitrate production (Wolf-Gladrow et al., 2007). Therefore 'N 2 fixation', here and throughout this article, refers to the complete process of N 2 fixation, followed by remineralisation and hence nitrate production.

CO 2 source/sink magnitude
We can use Q and any process vector q to determine the excess of C T generated by any biogeochemical process. In this context, the 'excess of C T ' (denoted Φ) is defined as the amount of C T that would need to be lost from the seawater following the action of a biogeochemical process, through an A T -neutral process like air-sea gas exchange, in order for p sw to return to its initial value (Fig. 2). This can be considered to represent the magnitude of the CO 2 source (Φ > 0) or sink (Φ < 0) generated by the process.
The geometry of Fig. 2 shows that Q satisfies: which can trivially be rearranged to define Φ: The units of Φ are the same as the units used for the inputs A q and C q to Eq. (10), which should be the same as each other (Appendix A). The notation Φ  is used where the CO 2 source/sink magnitude has been evaluated for the unit vector case (i.e. usingq ).

Investigating the isocapnic quotient distribution
We used the Takahashi et al. (2014b) surface ocean carbonate chemistry climatology to investigate the distribution of Q in the global surface ocean at the 'present day' (this climatology is normalised to the reference year 2005). Although this dataset is complete for sea surface temperature (SST), salinity and p sw , it excludes A T and C T in the equatorial Pacific Ocean, because of high temporal variability driven by El Niño Southern Oscillation (Takahashi et al., 2014a). We filled this gap using CO 2 SYS v1.1 (van Heuven et al., 2011) as follows. We used Fig. 2. Schematic of the geometric relationship between Q and Φ for three example biogeochemical processes. The dashed lines show the vector components A q and C q , dotted lines show the overall trajectories of q (not shown for process 3), and solid lines show the corresponding Φ. Process 1 (purple, analogous to calcification) has negative A q1 and C q1 . This leads to an increase in C T relative to the isocap value, and therefore positive Φ 1a CO 2 source. Process 2 (green, similar to primary production) also has negative C q2 but positive A q2 , leading to negative Φ 2 and a CO 2 sink. Process 3 (orange, like air-to-sea CO 2 transfer) has zero A q3 , and therefore C q3 = Φ 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 A q and C q coefficients for the selection of biogeochemical processes schematically illustrated in Fig. 1b, following Wolf-Gladrow et al. (2007). Calculated both as q (in this case, normalised to C q or A q = ± 1) andq (normalised by definition such that A q 2 + C q 2 = 1). qq a Denitrification and/or anammox. b N 2 fixation, followed by remineralisation and nitrate generation.
the climatological SST and salinity fields to predict A T following Lee et al. (2006). C T was calculated from these predicted A T values, and p sw , SST and salinity from the climatology. Silicate and phosphate concentrations, and pressure, were all set to zero. We thence evaluated s from p sw , SST and salinity using Henry's law (B.11). Next, h was calculated from s and C T using Eq. (D.1). Finally, we determined both Q x and Q from h and C T using Eqs. (6) and (8) respectively, using the approximation C x = C T in the former case Eq. (4). The coefficients necessary for these calculations were quantified following Weiss (1974) for K 0 , Lueker et al. (2000) for K 1 and K 2 , Dickson (1990b) for K B , Lee et al. (2010) for B T , and Dickson et al. (2007) for K w . All dissociation constants reported on the Total pH scale were converted to the Free pH scale (Zeebe and Wolf-Gladrow, 2001) using the sulfate-to-chlorinity ratio of Morris and Riley (1966) and the bisulfate dissociation constant of Dickson (1990a).
We determined the relative importance of SST seasonality on Q at each grid point in the Takahashi et al. (2014b) dataset by evaluating Q using the monthly values for the SST, but with all other values held at their annual mean values. A similar procedure was followed to compute the seasonal influence of the marine carbonate system (i.e. p sw and C T ) component, with SST instead held at its annual mean value, while p sw and C T were allowed to vary.
A caveat of our analysis based on this climatological dataset is that the spatiotemporal variability in Q in any specific individual year could exceed or fall short of that described here, because the dataset represents climatological mean conditions, normalised to the year 2005 (Takahashi et al., 2014a).

Results and discussion
3.1. The isocapnic quotient

Initial analysis of the approximation
We begin by briefly considering the Q x approximation (Eq. (6)), which helps to later interpret the global distribution of Q. In the surface open ocean, C T varies by a maximum of about 10% of its global mean concentration, whereas p sw can vary by over 100% of its mean value (Takahashi et al., 2014a). Therefore Eq. (6) leads us to expect p sw to be the dominant marine carbonate system control on Q, and we expect the spatiotemporal distributions of p sw and Q to be inversely proportional to each other. However, Q x is also dependent on the coefficients K 0 , K 1 and K 2 . Through their influence, we also expect temperature and salinity to modulate the Q distribution . The factor K 2 /K 0 K 1 in Eq. (6) is positively correlated with both seawater temperature and salinity. Temperature has a stronger effect than salinity: across the range of temperatures commonly encountered in the global surface ocean, K 2 /K 0 K 1 increases by a factor of 6-7, while it grows by a factor of less than 2 across the equivalent salinity range. This suggests that SST will influence the Q distribution 6-7 times as strongly as either salinity or p sw do.
We note that Q could also vary with depth through the influence of pressure on the equilibrium constants (Culberson et al., 1967). We do not investigate this pressure effect in detail here, because our focus is on air-sea CO 2 exchange in the near-surface ocean. In any case, this effect is small: moving from the sea surface to the deepest point in the ocean (Mariana Trench, pressure ≈ 1.1 × 10 4 dbar) reduces Q by < 3%, if all other conditions (i.e. temperature, salinity, C T and p sw ) are held constant. Accounting for this pressure influence, for both Q x and Q, does not require any alterations to our equations. Rather, the effect of pressure should be included when evaluating the equilibrium/dissociation constants.

Annual mean state
Moving on to the 'exact' calculation (Eq. (8)), our analysis of 'present day' Q, calculated using the Takahashi et al. (2014b) climatological dataset, showed that the dominant controls on annual mean Q throughout the global surface ocean are indeed SST and p sw . Q has the expected positive correlation with SST, and inverse relationship with p sw . Annual mean Q varies from a minimum around 1.08 at high latitudes to a maximum of nearly 1.28 in the tropics (Fig. 3a). This first order spatial distribution is dominantly driven by SST, but certain regions are also visibly influenced by p sw . For example, high p sw resulting from local upwelling (Wang et al., 2006) is responsible for the relatively low Q values observed in the eastern equatorial Pacific Ocean.
The approximation Q x has a similar pattern to annual mean Q (Fig. 3b). They have roughly equal values at high latitudes, but Q x tends to overestimate Q at higher SST values towards the equator, by up to 0.09 (c. 7%).

Seasonal variability
Some locations exhibit strong seasonal variability in Q, as quantified by the Q 'annual range' (Fig. 3c). This was defined as the difference between the maximum and minimum monthly Q at each grid point in the climatological dataset. The maximum Q annual range is about 0.13, which is almost half of the spatial range in the annual mean.
One striking feature is that the Q annual range is close to zero across large regions, for example the oligotrophic subtropical gyres (Fig. 3c). In these regions, the seasonal cycle of p sw is dominantly controlled by SST-driven changes in K 0 , so SST and p sw are positively correlated (Takahashi et al., 2009). We know from Section 3.1.1 that SST is positively correlated with Q, while p sw has an inverse relationship with both. We would therefore expect this coupling of the SST and p sw seasonal cycles to lead each to have opposing effects on Q, which could cancel each other out to some extent. Returning again the approximation Q x , we found that the majority of the temperature sensitivity of the K 2 /K 0 K 1 factor in Eq. (6) is driven by K 0 . This therefore exactly cancels out the SST-driven K 0 effect on p sw , as was further indicated by Eq. (7). The factor K 2 /K 1 in Eq. (7) has much lower temperature sensitivity than K 2 /K 0 K 1 , and consequently the amplitude of Q seasonality is relatively low where SST is the dominant control on the seasonal p sw cycle (Fig. 4a).
The isocapnic quotient therefore has a small seasonal range where p sw seasonality is mainly controlled by SST, while large seasonal ranges are found where the p sw and SST seasonal cycles are decoupled. Regions featuring seasonal upwelling of relatively cold, high-p sw subsurface waters, such as the eastern equatorial Pacific (Wang et al., 2006), exhibit such decoupling (Fig. 4b). It is also observed in regions with high seasonal biological uptake of CO 2 , because primary production is more intense there during the warmer, lighter spring and summer months (Takahashi et al., 2009). Where the SST and p sw cycles are not exactly in antiphase, this can also lead to multiple Q maxima within a single year (Fig. 4c).

CO 2 source/sink magnitude
The value of Φ quantifies the magnitude of the CO 2 source or sink. This is the amount of C T that would need to be taken out of seawater (for positive Φ) in order to return to its original p sw value, following the changes in A T and C T specified by q. As changes to p sw create the chemical gradients necessary to drive air-sea CO 2 fluxes, we have stated that Φ can be considered to represent the size of the CO 2 source or sink driven by q. However, this interpretation assumes that the seawater is then given time to re-equilibrate back to its original p sw value through air-sea CO 2 exchange, which may not always be the case. In Southern Ocean waters, for example, rapid subduction of water masses from the surface layer could terminate air-sea CO 2 exchange before the full CO 2 source or sink indicated by Φ has been realised (Ito and Follows, 2013). We might therefore more accurately describe Φ as the potential CO 2 source or sink. However, such fast subduction is not typical for the majority of the ocean, and CO 2 equilibration timescales for the ocean surface layer are typically on the order of a few months to a year (Jones et al., 2014), fast enough for re-equilibration to occur. Also, on much longer timescales, when these subducted waters eventually are returned to the ocean surface, the biogeochemically driven change in their CO 2 source or sink capacity should still be present. The rest of our discussion is therefore framed by the assumption that Φ does accurately represent the eventual magnitude of the CO 2 source or sink.

Sensitivity to Q
We can determine the sensitivity of Φ to changes in Q by considering Eq. (10). First, we can see that Φ→ C q as A q → 0. In other words, if C T is added to seawater with no change in A T , then the amount of C T that must be removed to return to the original p sw (i.e. Φ) is simply equal to the amount of C T that was added. Second, differentiation of Eq. (10) gives dΦ/dQ = A q /Q 2 . Φ is therefore sensitive to Q for biogeochemical processes that modify A T , and this sensitivity increases with the absolute value of A q . Biogeochemical processes with greater absolute A q /C q thus exhibit greater spatiotemporal variability in Φ as a fraction of its mean value (Fig. 5).
Two key properties of Φ  can thus be intuitively predicted from A q / C q . First, the magnitude of Φ  is dictated by the geometric relationship betweenq and Q (Fig. 2). Φ  is zero when the biogeochemical process vector is parallel to the isocap field, and therefore the slope of q is equal to Q. Φ  reaches a maximum whenq is perpendicular to the isocap, when A q /C q = −1/Q. Second, the magnitude of the spatiotemporal variability in Φ  (driven by variability in Q) increases with the absolute value of A q /C q , and Φ  is homogeneous for processes with A q = 0.
These properties can be illustrated by considering the latitudinal distribution of Φ  calculated from the climatological dataset for the CO 2 source processes shown in Fig. 1b and Table 1 (Fig. 5). In other words, we describe how the spatial distribution of Q in the climatology affects Φ  for each biogeochemical process in turn. Air-sea CO 2 exchange does not affect A T , so its Φ  is uniformly equal to C q (i.e. 1 for the unit vector caseq ) throughout the global ocean. The change in A T associated with remineralisation is small relative to its C q , leading to a modest range of Φ  (i. e. 1.13-1.17). The A q value in this case movesq more perpendicular to Q, resulting in greater mean Φ  than for air-sea CO 2 exchange. Fig. 4. Drivers of seasonal change in Q at selected sites from Fig. 3. (a) In the oligotrophic North Atlantic, the p sw seasonal cycle is dominantly driven by SST, so their influences on Q compensate each other, leading to a small seasonal amplitude in Q. (b) Seasonal upwelling in the eastern equatorial Pacific drives high C T and p sw to accompany low SST, and these factors combine to drive relatively strong Q seasonality. (c) In the subpolar North Pacific, strong seasonal cycles in both SST and the marine carbonate system are slightly out of phase, leading to strong Q seasonality with dual maxima.  Table 1. Remineralisation is the strongest CO 2 source because itsq vector is closest to being perpendicular to Q, while calcification is the weakest as it is nearest to being parallel to Q. N 2 fixation has the greatest range, as it has the greatest A q /C q (i.e. −∞), while air to sea CO 2 transfer is always exactly 1, because A q = 0 and therefore Φ = C q (Fig. 2). The plot for the equivalent CO 2 sink processes has the same appearance, but with all Φ  values multiplied by −1.
Calcification and N 2 fixation both have greater A q coefficients, and consequently greater ranges of Φ  (0.24-0.38 and 0.77-0.92 respectively). However, the mean Φ  for calcification is significantly smaller than for N 2 fixation, becauseq for the former process is closer to being parallel to Q.

Timescale considerations
A key component of our conceptual framework is the biogeochemical process vector q. Some example values are suggested in Table 1, but these may not always be appropriate to use, depending on the timescale and context.
For example, consider the effect of combined organic matter and CaCO 3 production in the sea surface layer. On timescales of up to the lifetime of the particulate matter that is produced (e.g. days to weeks), the appropriate q would simply be the combination of the calcification and autotrophic production vectors from Table 1. However, on longer timescales (e.g. weeks to months), some of the particulate material may undergo remineralisation within the surface layer, while other components may sink into the abyss. On these timescales, material that is remineralised within the surface layer should be excluded from the calculation of q, as from the perspective of the seawater chemistry it has not been used up. For the hypothetical case where both primary production and calcification were occurring, but all organic matter was remineralised within the surface layer while all CaCO 3 was exported to depth, the appropriate q to use would simply be that for calcification.
Consequently, the correct combination of A q and C q values depends on (in these examples) export efficiencies and timescales of interest. Equivalent considerations also need to be made for other processes; for example, for N 2 fixation, the accuracy of our simplifying assumption that all fixed N 2 is remineralised into nitrate should be taken into account. The correct choice of A q and C q thus varies depending upon the nature of the scientific question being investigated, and is therefore left to the reader's discretion.

Deviation from linearity
Our equation for Φ (Eq. (10)) is such that the magnitude of Φ depends on the absolute A q and C q values, not just their ratio. For example, Φ calculated for (A q , C q ) = (4, 2) would be double that calculated for (2, 1), despite both cases having the same A q : C q . Whatever the values of A q and C q , our calculation assumes that isocaps are linear in (A T , C T ) phase space. This assumption is also implicit in the related concept of Ψ (Frankignoulle et al., 1994). However, Eq. (10) technically returns the instantaneous value of Φ, for infinitesimal A q and C q , and isocaps are in fact slightly curved. When calculated from large values of A q and C q , Φ therefore deviates from the correct result, which we call the 'deviation from linearity' (DfL, Fig. 6). Because of the direction of isocap curvature, the DfL is always negative, so Φ slightly underestimates the true CO 2 source/sink magnitude (Fig. 6b).
However, the magnitude of the DfL is likely to be negligible for many practical applications in open-ocean seawater. The DfL magnitude increased with the absolute value of A q (Fig. 6b), and was insensitive to C q . These properties were anticipated from our earlier discussion of the sensitivity of Q to A q (Section 3.2.1). The DfL typically remains less than 1 μmol kg -1 for A q values exceeding ± 200 μmol kg -1 (Fig. 6b). Given that an analytical precision of ± 2 μmol kg -1 is considered to represent a high quality oceanographic A T and/or C T measurement , and that uncertainties in the rates of biological processes typically far exceed this value, we consider the uncertainty in Φ to be negligible in this context. The DfL is slightly greater (but still unimportant) at lower p sw values (Fig. 6b), because the isocap curvature is greater under these conditions. The DfL is also slightly greater at low A T values (by a factor of ≤ 2), but still negligible for most practical applications.

Lower limit for Q
The approximation Q x helps to understand the reduced deviation from linearity at high p sw (Section 3.2.3), and also indicates a lower limit for Q as follows. All of the variables in Eqs. (6) and (7) are always positive. Furthermore, [CO 2(aq) ] cannot exceed C T , by definition (1). Given these two facts, our approximation shows that the minimum possible value for Q occurs when the entirety of C T is in aqueous CO 2 form, with no bicarbonate or carbonate ions present. Towards this point, Q x → 1 + 2K 2 /K 1 , which is virtually equal to 1 (more precisely, 1.0013 at 15°C and practical salinity = 35). This represents a 'minimum buffering' condition, occurring where C T ≫ A T , and p sw would be correspondingly very high. The Q lower limit of c. 1 makes sense intuitively as follows. Under these poorly buffered conditions, A T would be dominated by the − HCO 3 term, and changes in s and C T would be directly proportional to each other. An increase in A T by 1 unit would thereby permit 1 unit of CO 2(aq) to be converted into HCO − 3 . Returning to the original s, and therefore to the original p sw , would therefore require 1 unit of CO 2 to be added to the seawater through airsea gas exchange. For comparison, under more typical seawater conditions where A T ≈ C T , the −

[HCO ]
3 component represents only a fraction of any overall A T change. A 1 unit increase in A T would therefore involve conversion of less than 1 unit of CO 2(aq) , so less than 1 unit of CO 2 uptake will return p sw to its original value, and therefore Q > 1. In typical open-ocean seawater, − [HCO ] 3 represents about 80% of A T , so its Q is about 20% greater than 1.

Future impacts
Global mean surface ocean SST and p sw are presently increasing (Wu et al., 2011;Tjiputra et al., 2014). Although these changes have opposing effects on Q, the p sw trend dominates the Q response for an equilibrium climate sensitivity (i.e. the increase in global mean surface temperature induced by a doubling of p atm ) in the range from 1.5 to Fig. 6. Examples of the deviation from linearity (DfL) in Φ resulting from isocap curvature, at practical salinity = 35, temperature = 15°C, and A T = 2300 μmol kg -1 . (a) Φ calculated from Eq. (10) across a range of A q with C q = 0 μmol kg -1 , at p sw values increasing in intervals of 200 μatm from 200 (dark blue) to 1000 μatm (light blue). (b) Difference between Φ calculated from Eq. (10) and its exact value (i.e. the deviation from linearity, DfL), for the same data shown in (a). The deviation from linearity is zero when A q = 0. At higher p sw , isocaps are more linear, so the percentage error in Φ is smaller. Panel (b) would not be modified by changing C q . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 4.5°C (IPCC, 2013). Global mean Q is therefore likely to decrease as the ocean progresses towards a warmer, higher-CO 2 state (Fig. 7), at least until anthropogenic CO 2 emissions cease. After this point, Q may increase further as SST initially continues to rise, and then falls more slowly than p atm (Solomon et al., 2009). Biogeochemical processes that decrease seawater A T may thus become stronger sources of CO 2 . This positive feedback on p sw has previously been identified specifically for calcification (Frankignoulle et al., 1994). Because of the opposing effects of SST and p sw on Q, the smaller the climate sensitivity, the greater this positive feedback could be (Humphreys, 2017). Nevertheless, the size of this feedback moving into the future is likely to be modest, with global mean Q expected to decline from a mean value of about 1.18 at the Takahashi et al. (2014b) global mean p sw of c. 360 μatm to 1.10 for a p sw of 850 μatm (taking into account the associated warming), which is representative of the year 2100 under the 'medium stabilisation' anthropogenic CO 2 emissions scenario RCP6 (van Vuuren et al., 2011). For processes that only modify A T (i.e. those that are most sensitive to Q, e.g. N 2 fixation and denitrification/anammox), this represents a 7.3% increase in the absolute value of Φ  , while processes with smaller absolute A q /C q would be relatively less affected: 4.6% for calcification/ CaCO 3 dissolution, and 1.6% for primary production/remineralisation (see also Section 3.2.1).
However, this change applies to the magnitude of Φ, and in both directions: CO 2 sources with positive Φ will become stronger, but CO 2 sinks will also become stronger, with more negative Φ values. On timescales up to a few hundred years (i.e. shorter than whole-ocean mixing), the overall effect on air-sea CO 2 exchange may therefore depend on the depth distribution of each process relative to its counterpart. Marine primary production is focused within the surface mixed layer of the ocean, while a non-zero fraction of organic matter remineralisation typically occurs beneath this layer (Henson et al., 2012), from where it cannot directly influence air-sea gas exchange. The response to changing Q may therefore be a stronger CO 2 sink, driven by surface net community production, providing a negative feedback on p atm . Similarly, net calcification mostly occurs within the euphotic surface layer (Poulton et al., 2006;Balch et al., 2011), while in the open ocean CaCO 3 dissolution is theoretically confined to relatively deep waters where CaCO 3 minerals are undersaturated (Morse et al., 2007). The CaCO 3 cycle may therefore act as a positive feedback on air-sea CO 2 exchange, dominated by the calcification (not dissolution) response to changes in Q. N 2 fixation is particularly concentrated in the thermocline of the (sub)tropical oceans (Gruber and Sarmiento, 1997;Capone et al., 1997;Moisander et al., 2010). However, A T is not generated by this process until the organic matter thus formed has been remineralised into NO − 3 (Wolf- Gladrow et al., 2007), which may occur deeper still in the water column. The opposite processes of denitrification and anammox (Arrigo, 2005) are typically associated with poorly ventilated, subsurface oxygen minimum zones (Gruber and Sarmiento, 1997). It therefore seems unlikely that either process will strongly influence the ocean's surface mixed layer, and it is unclear whether a positive or negative feedback would occur for this pair of processes.

Conclusions
We constructed a novel framework, built upon the 'isocapnic quotient' concept (i.e. Q), that allows us to systematically analyse the influence of any biogeochemical process that modifies the marine carbonate system over air-sea CO 2 exchange. The size of the CO 2 source or sink (to/from the atmosphere) generated by any such process (i.e. Φ) can be calculated from Q and the biogeochemical process vector q = (A q ,C q ). The vector q quantifies the changes in seawater A T and C T driven by the biogeochemical process.
Analysis of 'present day' Q, calculated from a climatological dataset (Takahashi et al., 2014b), revealed spatial and seasonal variability in Q throughout the global surface ocean, mostly as a function of SST and p sw . This means that the magnitude of the CO 2 source or sink generated by any specific biogeochemical process is similarly heterogeneous. Where the p sw seasonal cycle is dominantly driven by SST, their effects on Q approximately cancel each other out, leading to a small seasonal range in Q. Stronger seasonal variability in Q can be observed where the p sw and SST cycles are decoupled, for example by seasonal primary production or upwelling.
The sensitivity of Φ to Q increases with the absolute value of A q /C q , and Φ is simply equal to C q when A q is zero. Processes with greater A q / C q therefore have greater spatiotemporal variability in Φ  . Processes withq perpendicular to Q are the strongest CO 2 sources or sinks, while processes withq parallel to Q are p sw -neutral, and thus do not affect airsea CO 2 exchange.
The buffer factor Ψ (Frankignoulle et al., 1994) is a special case of Φ, normalised to C q = −1, and only applicable to calcification. Our Φ method, and its Ψ precursor, assume that isocaps are linear in (A T , C T ) phase space, but they are actually slightly curved. However, the error in Φ resulting from this curvature is negligible for most practical applications, typically remaining under 0.5% for changes in A T exceeding 100 μmol kg -1 .
We note that a similar approach to that taken here could be also applied to calculate the influence of biogeochemical processes over other marine carbonate system variables such as pH, − [CO ] 3 2 , and CaCO 3 mineral saturation states. However, it would be less obvious how to interpret the physical significance of the equivalent of Φ in these cases.
Based on an equilibrium climate sensitivity of up to 4.5°C, global mean Q is likely to decrease as anthropogenic CO 2 emissions continue. This will lead to biogeochemical processes that decrease seawater A T becoming stronger sources of CO 2 to the atmosphere, while processes that increase A T will become stronger CO 2 sinks. Appendix A. Summary of symbols Table A.2 Summary of symbols and abbreviations used in this manuscript.

Symbol Variable (and relevant equation) Units
A T Total alkalinity (Eq. (2)) mol kg -1 A q Change in A T associated with a process mol kg -1 A x Approximation of A T (Eq. (5)) mol kg -1 B T Dissolved boric acid (Eq. (B.5)) mol kg -1 C T Dissolved inorganic carbon (Eq. (1)) mol kg -1 C q Change in C T associated with a process mol kg -1 C x Approximation of C T (Eq. (4) Boric acid stoichiometric dissociation constant (Eq. (B.7)) mol kg -1 K w Stoichiometric ion product of water (Eq. (B.9)) mol 2 kg -2 p sw Seawater CO 2 partial pressure (Eq. (B.11)) atm p atm Atmospheric CO 2 partial pressure (Eq. (B.11)) atm q Biogeochemical process vector (A q ,C q ) mol kg -1 q Unit q (normalised such that A q 2 + C q 2 = 1) mol kg -1 s Aqueous CO 2 concentration, [CO 2(aq) ] mol kg -1 Q Isocapnic quotient (Eq. (D.5)) -Q x Approximation of Q based on A x and C x (Eq. (C.8)) -Tr Gas transfer velocity (Eq. (B.12)) mol atm -1 m -2 β Shorthand for p sw K 0 K 1 /K 2 (Eqs. (C.5)-(C.8)) mol kg -1 Φ Size of CO 2 source/sink driven by a process (Eq. (10) where the stoichiometric dissociation coefficients K 1 and K 2 , which can be estimated using empirical functions of temperature, salinity and pressure , are given by: Dissolution of CO 2 into seawater can be represented by the following reaction: The direction of net air-sea CO 2 transfer is determined by the partial pressure of CO 2 in seawater (p sw ) relative to its atmospheric value (p atm ).
Seawater p sw is directly proportional to [CO 2(aq) ], following Henry's law for CO 2 : where the solubility coefficient K 0 is a function of seawater temperature and salinity (Weiss, 1974). The net sea-to-air CO 2 flux (F) is then given by: where Tr is the gas transfer velocity (also known as the 'piston velocity'), which can be parameterised as a function of wind speed, temperature and salinity (e.g. Wanninkhof, 2014). Processes that increase p sw are therefore considered to be sources for atmospheric CO 2 , for they will shift the value of (p sw − p atm ) in favour of ocean to atmosphere CO 2 transfer. Conversely, processes that decrease p sw are CO 2 sinks, enhancing seawater CO 2 uptake.

Appendix C. Isocapnic quotient approximation
We derived the approximation to the isocapnic quotient Q (i.e. Q x ) as follows. First, we combined Eqs. Then, using the approximations (4) and (5) for C T and A T respectively, we derived equations for The approximations (4) and (5)  terms together typically represent over 99% of C T and over 97% of A T .
Next, we substituted Eqs. (C.2) and (C.3) into Eq. (C.1): This is a quadratic equation for A x , which we rearranged as follows: where β is shorthand for p sw K 0 K 1 /K 2 . We used the quadratic formula to solve Eq. (C.5): The solution of Eq. (C.6) with the negative square root term gives realistic values for A x . We simplified this further using a Taylor series expansion: Finally, we differentiated Eq. (C.7) with respect to C x to find an approximation for Q (i.e. Q x ): This can be converted into a function of s, rather than p sw , by using Eq. (B.11): Eqs. (C.8) and (C.9) are identical to Eqs. (6) and (7) respectively in the main text.

Appendix D. Exact isocapnic quotient
To derive an exact equation for Q, we first converted Eqs. (1) and (2)   We can thence use Eq. (D.5), which is identical to Eq. (8) in the main text, to calculate Q given any pair of marine carbonate system variables (Zeebe and Wolf-Gladrow, 2001).