Starting from PI steady-state conditions, the model is first run for 20 kyr under constant LGM forcing and then for an additional 20 kyr under transient forcing from the LGM to PI. Forcings (Fig. ) include radiative forcing imposed by CO2, N2O, and CH4 and variations in orbital parameters . Radiative forcing of CO2 is prescribed and biogeochemistry is determined with interactive CO2. Ice sheet extent and related changes in albedo are prescribed based on benthic δ18O data and ice sheet reconstructions . Additionally, freshwater (FW) pulses are prescribed in the North Atlantic as detailed in , also accounting for the LGM to PI change in salinity of about 1 PSU. Coral reef regrowth is implemented from 14 ka onward by uniformly removing the respective amount of alkalinity, carbon, and 13C from the uppermost ocean grid cells. These forcings (Fig. a–d) are termed standard forcings.

Standard transient deglacial forcings in the Bern3D model are complemented by changes in seven additional mechanisms, grouped into generic classes and introduced below (see top of Fig. ). A set of 50 factorial experiments (Table ), covering the same time interval as the model run with standard transient forcings, is conducted to quantify the sensitivities of the carbon cycle to changes in these seven mechanisms. The forcing history for changes in Southern Ocean wind stress, Southern Ocean gas transfer rate, the rain ratio, the remineralization profile, and the organic weathering rate are prescribed following an idealized glacial–interglacial evolution. PI values are set to LGM values at 40 ka, kept constant from 40 to 18 ka, scaled back to the PI value over the termination (18 to 11 ka), and kept constant thereafter (Fig. f and Table ). The sensitivities of carbon cycle properties are obtained by subtracting the standard forcing run from the runs with complementary changes in the seven mechanisms. This also yields characteristic response relationships between different carbon cycle properties, e.g., changes in atmospheric CO2 versus changes in whole-ocean δ13C, for each process. The objective is to schematically represent the space of plausible property–property relationships for key proxies. In other words, the selected mechanisms are assumed to be representative on the global scale for the myriad of processes influencing the carbon cycle and will be outlined below.

Physical mechanisms. Changes in ocean circulation, for example in response to altered wind stress, stratification, eddy mixing, or buoyancy forcing, affect the marine carbon inventory e.g., by altering the cycling of organic and inorganic carbon. The Southern Ocean has been identified as an important region modulating CO2 changes, and a reduced Antarctic overturning circulation has been invoked to, at least partly, explain low glacial atmospheric CO2 . Under standard forcing, Southern Ocean circulation changes are weak in the Bern3D model. In this study, the circulation is modified by uniformly scaling the wind stress field over the Southern Ocean (>48∘ S, south of South America) see also to reduce deep ocean ventilation. We do not consider such large changes in wind stress to be realistic, but rather use prescribed Southern Ocean wind stress as a way to vary deep ocean ventilation in the Bern3D model. Consequently, we do not scale air–sea gas transfer rates with the changes in wind stress but vary transfer rates independently.

Proxy reconstructions suggest a larger sea-ice extent in the Southern Ocean during glacial times . Sea ice hinders air–sea gas exchange and thus exerts a large impact on atmospheric δ13CO2 . Additionally, changes in Southern Ocean winds may have affected glacial Southern Ocean air–sea gas transfer rates. Here, the standard air–sea gas transfer rate is scaled (Fig. f; Table ) in the Southern Ocean to estimate related carbon cycle sensitivities. We note that changes in sea ice and a corresponding change in air–sea gas transfer rates are reasonably well simulated by the Bern3D model. The effect of changes in ocean temperature and salinity on the solubility of CO2 is discussed elsewhere and is part of the standard forcing.

Oceanic carbonate. The partial pressure of CO2 in the ocean depends on alkalinity and processes that alter surface ocean alkalinity, thus influencing atmospheric CO2. The sedimentary burial of CaCO3 removes twice as much alkalinity as carbon from the ocean, leading to oceanic CO2 outgassing. Increased CaCO3 burial over the deglacial has been put forward as the coral reef hypothesis to explain increases in atmospheric CO2. reconstructed coral reef growth history and provided a lower-limit estimate of about 380 GtC for the amount of shallow water carbonate deposition over the deglacial period. Other estimates of CaCO3 deposition amount to 1200 GtC or even more . This yields a large range of possible scenarios with substantial impacts on atmospheric CO2. Here, we scale the reconstructed deposition history based on (Fig. d) by applying a constant scaling to vary CaCO3 deposition within the published range.

Potential changes in the rain ratio (CaCO3 : Corg) affect both alkalinity and CO2 e.g.,. An increase in the export of CaCO3 relative to Corg from surface waters has similar consequences for CO2 as shallow water carbonate deposition . There is a lack of estimates on how the rain ratio has varied in the past. Here, we vary the global rain ratio for LGM conditions (see also Appendix ; Fig. f; Table ) between 0.045 and 0.098, while the rain ratio is 0.083 in the standard setup and for PI conditions.

Oceanic organic matter. There is a broad range of mechanisms, in addition to circulation changes, that affect the cycling of organic matter within the ocean, the whole-ocean nutrient inventories, and thereby surface ocean nutrient and DIC concentrations as well as atmospheric CO2. Several studies have suggested changes in the remineralization length scale of organic carbon due to colder ocean temperatures see, e.g.,. A deeper remineralization of organic matter leads to the removal of nutrients and DIC from the shallow subsurface ocean and, via ocean–sediment interactions, to globally reduced nutrient and increased alkalinity inventories , resulting in a decrease in atmospheric CO2 . Weathering–burial feedbacks lead to an amplification of changes in atmospheric and oceanic δ13C as shown by and . Here, the remineralization profile in the upper 2 km is varied between the standard Martin curve (Eq.  in Appendix) and a linear profile (75 to 2000 m). The modification in the remineralization profile of particulate organic matter is different than those applied by and in simulations with the Bern3D model. Here, only the remineralization profile in the upper 2000 m is changed, and, in contrast to these earlier studies, it is assumed that the fraction of exported particles that reach the sediments below 2000 m is not altered. The implicit assumption is that cooler temperatures at the LGM only significantly affect the dissolution of relatively small or labile particles in the upper ocean, while large or refractory particles sink fast enough to reach ocean sediments both under LGM and PI conditions.

Oceanic organic carbon and nutrient inventories are further affected by changes in the weathering flux compensating for the burial of organic matter . A negative balance between the riverine input of PO4, DIC, DI13C, and Alk, originally buried as organic material, and sedimentary burial of particulate organic matter leads to a transient decrease in atmospheric CO2 and also affects atmospheric and marine δ13C. The weathering flux to the ocean (Table ) is kept constant in the standard setup of the Bern3D (see also Appendix ) but might have varied with changes in climate and CO2. To account for this possibility, we varied the input of PO4, DIC, DI13C, and Alk that compensates for the burial of particulate organic material in the sediment in factorial experiments (Fig. f; Table ). This variation in the input will be referred to as changes in the organic weathering flux.

Another widely discussed mechanism relates to iron fertilization in the Southern Ocean . Iron fertilization leads to changes in the cycling of marine organic material and to related changes in the whole-ocean inventory of δ13CDIC and other variables. These changes are broadly similar to the changes caused by altering the organic matter remineralization depth or the weathering input flux and implicitly included in the Monte Carlo simulations. Similarly, other processes, such as changes in the ratio of nutrient to carbon uptake by marine organisms or changes in the association of POC with ballast minerals , may have affected the efficiency of the marine biological cycle in reducing carbon in the surface ocean and the atmosphere.

Land biosphere carbon inventory. Changes in land biosphere carbon stock are prescribed following the evolution illustrated in Fig. e. We assume that the change in land carbon stock over the Holocene is well constrained and amounts to an uptake of 290 GtC at 11–5 ka and a release of 36 GtC thereafter . Scenarios with different cumulative land carbon change imply a corresponding uptake or release of terrestrial carbon during the last glacial termination (18–11 ka). The δ13C signature of terrestrial carbon is set to -24 ‰.

We consider four globally relevant observational constraints (Table ) to estimate the PI–LGM change in the land biosphere carbon inventory, Δland. These are the PI–LGM difference (Δ) in (i) atmospheric CO2 (ΔCO2), (ii) atmospheric δ13C(CO2) (Δδ13Catm), (iii) mean ocean δ13C(DIC) (Δδ13CDIC), and (iv) deep (equatorial) Pacific carbonate ion concentration (ΔCO32-) (Table ). The carbon isotope constraints are directly relevant for the isotopic mass balance used to infer Δland. The carbonate ion concentration difference constrains changes in the ocean's alkalinity budget, and the change in atmospheric CO2 constrains the integrated effect of deglacial carbon cycle changes. We allow for a relatively wide uncertainty range of 20 ppm for ΔCO2, which is admittedly wider than the proxy uncertainty, to avoid an overfitting of model outputs. The Δδ13Catm range for the constraint is 0.1±0.05 ‰ and based on ice core measurements . compiled 480 benthic foraminiferal records and report a whole-ocean PI minus LGM change in δ13CDIC of 0.34±0.19 ‰, and we use their range (0.15 ‰ to 0.53 ‰) as a constraint. Proxy reconstructions of ΔCO32- based on measured B/Ca ratios in foraminifera show little change from LGM to PI in the equatorial deep Pacific . Based on a different approach, also conclude on negligible change in ΔCO32- in the western tropical Pacific across the last glacial termination. In a recent study, report, based on CaCO3 reconstructions in the South China Sea, that carbonate ion concentrations in the mid-depth Pacific hardly changed from LGM to PI. All these studies point to a small change in ΔCO32-, providing a further constraint on deglacial simulations. However, the uncertainty in these reconstructions is of the same order as the change itself. The target range for ΔCO32- is chosen here from -6 to 5 µmolkg-1.

Sensitivities of the four target proxies (ΔCO2, Δδ13Catm, Δδ13CDIC, and ΔCO32-) to changes in the selected seven forcings are computed based on 50 factorial experiments. For each forcing, i, and related parameter change, Δpi (Table ), the corresponding sensitivity, i.e., the change in target per unit change in parameter, is computed. These discrete sensitivities are fitted linearly or quadratically for each target, T, and forcing to get a continuous function for the sensitivity, SiT(Δpi). These sensitivity functions are used to build a simple representation to compute changes in target variables for a combination of changes in parameters: 1ΔTΔp1,…,Δp7=∑i=17Δpi×SiTΔpi. In order to evaluate the assumed linear additivity of the seven mechanisms, a 60-member ensemble of parameter combinations is calculated both with the Bern3D model and Eq. () (Fig. ). The seven parameters are sampled uniformly using Latin hypercube sampling LHS;. The results from the Bern3D model and Eq. () are regressed against each other for each target variable (Fig. ). The corresponding linear fit is used to account for nonadditive behavior when combining forcings. This then yields the following emulator: 2ΔTΔp1,…,Δp7=aT+bT×∑i=17Δpi×SiTΔpi,

where aT is the offset and bT the slope of the respective linear fit as given in Appendix Fig. . The root mean square deviation between the emulator (Eq. ) and the Bern3D model is 8.8 ppm for ΔCO2, 0.06 ‰ for Δδ13Catm, 0.02 ‰ for Δδ13CDIC, and 4 µmolkg-1 for ΔCO32-. The seven parameters of the LGM to PI sensitivity experiments are sampled uniformly by drawing random samples over their employed ranges (Table ) to generate a 500 000-member Monte Carlo ensemble. The above emulator is used for each of these members to estimate the corresponding changes in the four targets. Members that yield changes in the targets that lie outside the defined ranges (Table ) are excluded from the analysis; all remaining members are used to calculate median and confidence ranges in Δland and other variables of interest.

The mean δ13C signature of the ocean–atmosphere system changes in response to potential deglacial processes (Fig. a–b). For simplicity, we focus on the change in mean ocean δ13CDIC (Fig. a) as the ocean holds about 20 times more carbon than the atmosphere and interactive land biosphere together. The factorial simulations reveal a Δδ13CDIC (PI–LGM) between -0.3 ‰ and more than 0.6 ‰ when halving or doubling the weathering flux of organic material, respectively. Prescribed changes in Southern Ocean (SO) wind stress inducing changes in ocean circulation, the organic matter remineralization profile, and in the rain ratio lead to a change in the ocean signature of about -0.2 ‰, while changes in the SO air–sea gas transfer rate and in coral reef growth have a small influence on Δδ13CDIC. The influence of these mechanisms on δ13CDIC cannot be ignored when estimating Δland based on reconstructed changes in δ13C.

Reconstructions of the change in δ13C in the atmosphere and ocean differ in magnitude. The change in the atmosphere (0.1±0.05 ‰) is about one-third of the oceanic change (0.34±0.19 ‰). Changes in the organic weathering flux and terrestrial carbon storage affect Δδ13CDIC and Δδ13Catm in an identical way. Changes in SO wind stress, rain ratio, coral reef growth, and the remineralization of organic material have a similar influence on the two isotope variables. In contrast, changes in the SO air–sea gas transfer rate alter Δδ13Catm, while hardly changing Δδ13CDIC. The strong sensitivity of δ13Catm to changes in SO gas transfer rate is due to the large disequilibrium between the atmosphere and surface ocean. This makes this process, in addition to temperature-driven changes in fractionation, potentially important in setting atmospheric δ13C independently from mean ocean δ13CDIC change (see also Fig. b).

Considering ΔCO2 and ΔCO32-, the changes in investigated mechanisms significantly influence atmospheric CO2 and deep Pacific carbonate ion concentration (with the exception of air–sea gas transfer rate). A general relationship becomes apparent. A positive change in ΔCO2 is always accompanied by a negative change in ΔCO32- and vice versa (Fig. c–d). Thus, the explanation for the observed deglacial increase in CO2 of ∼90 ppm requires a deglacial decrease in deep Pacific carbonate ion concentration (Fig. 6d), at least in our model and considering the mechanisms implemented in the standard and factorial simulations. Changes in alkalinity and DIC (PI–LGM) for the standard forcing amount to about 500 Gt and 365 GtC, respectively. In the case of rain ratio, coral reef growth, and changes in the remineralization profile, changes in alkalinity and DIC are largest and occur close to a 2:1 ratio. Changes in land carbon uptake remove carbon from the system, leading to changes in alkalinity via CaCO3 compensation, whereas in the case of the organic weathering flux, not only carbon is added and/or removed from the system but also nutrients and alkalinity, following Redfield ratios (see Appendix ). Changes in alkalinity and DIC resulting from changes in Southern Ocean wind stress and gas transfer rate are small.

The results for the target variables are plotted against each other to gain further insight into the relative importance of the different mechanisms to meet the observational constraints (Fig. ). This yields a characteristic slope for each target pair and mechanism. For example, this slope is negative and equates to about -90 ppm ‰-1 for the pair ΔCO2 and Δδ13CDIC and the mechanism of land carbon storage (grey line in Fig. a). Roughly similar slopes are found for variations of the organic weathering flux, SO wind stress, and SO air–sea gas transfer rate. Figure a reveals that variations in these four processes cannot prompt the model results to agree with these two observational constraints. The slopes are too small in magnitude and either ΔCO2 or Δδ13CDIC remains outside the observational range. The four processes are effective in varying Δδ13CDIC but relatively ineffective in varying ΔCO2. Processes that have a much larger impact on ΔCO2 than on Δδ13CDIC are related to the CaCO3 cycle (coral reef growth and rain ratio) and changes in the upper ocean remineralization of particulate organic matter. This implies that variations in Δland, organic weathering, and ocean circulation (SO wind) are required to meet the Δδ13CDIC target, while variations in coral reef regrowth, rain ratio, and upper ocean remineralization depth are necessary to meet the ΔCO2 target. Indeed, the median changes for the parameters in the observationally constrained ensemble (Fig. ) imply a deeper remineralization of organic matter and a reduced rain ratio at LGM than PI and a 3 to 4 times higher amount of CaCO3 deposition during coral reef regrowth than applied in the standard run. These, together with a reduced SO wind stress at the LGM, all contribute to a deglacial increase in CO2. The associated decrease in Δδ13CDIC from these changes is in most Monte Carlo realizations more than compensated for by a deglacial increase in land carbon inventory and in many realizations by an increased glacial organic weathering flux (Figs. , f–g). In general, the smaller the change in Δland, the larger the increase required in the glacial flux of organic weathering.

Slopes for the target pair Δδ13Catm-ΔCO2 resemble the Δδ13CDIC-ΔCO2 pair (Fig. a–b). Variations in SO air–sea gas transfer rate, SO wind stress, Δland, and organic weathering flux show relatively high sensitivity for Δδ13Catm and low sensitivity for ΔCO2, whereas coral reef regrowth, rain ratio, and upper ocean remineralization changes exert a large impact on ΔCO2 and affect Δδ13Catm little (Fig. b). In contrast to the Δδ13CDIC-ΔCO2 target pair (Fig. a), the slopes for the Δδ13Catm-ΔCO2 pair do not fall on roughly two characteristic slopes but vary across processes (Fig. b).

The clear relationship between oceanic alkalinity and oceanic carbon uptake (Fig. c–d) is also visible in Fig. c. Higher alkalinity during the glacial (negative ΔCO32-) generally results in a larger drawdown of atmospheric CO2 (positive ΔCO2). An interesting exception is that a modest deepening of the remineralization of organic matter in the upper ocean leads to large changes in atmospheric CO2, while at the same time hardly changing deep Pacific CO32-, making it a potentially important process to fulfill the ΔCO2 target without shifting ΔCO32- away from the observationally constrained range (Fig. c). Accordingly, the remineralization depth of organic matter is deeper at the LGM than PI in all Monte Carlo realizations (Fig. e).

Next, we consider the Δδ13CDIC-Δδ13Catm target pair (Fig. d). The difficulty in simultaneously fulfilling the atmospheric and oceanic δ13C constraints with a negative Δland is becoming obvious. With negative Δland, Δδ13Catm and Δδ13CDIC are both moved towards negative values relative to standard forcings. Therefore, the isotopic constraints could only be fulfilled with an increased glacial organic weathering flux such that Δδ13Catm and Δδ13CDIC are both moved towards positive values relative to the standard forcings (Fig. d). Considering the ΔCO2 and ΔCO32- constraints (Fig. c) implies that negative Δland can only be offset by increased glacial organic weathering or rain ratio. However, reaching a ΔCO2 of 80–100 ppm remains difficult for negative Δland when combining information from all panels of Fig. . This explains why only very few solutions with negative Δland and no solutions with Δland smaller than -220 GtC are found in the Monte Carlo simulations (Sect. ).

In this section, we describe the spatial distribution of changes in Δδ13CDIC. The reference run with standard LGM forcings (Fig. a) shows substantial spatial changes in Δδ13CDIC from internal reorganization linked to ocean circulation changes. Reduced glacial ocean ventilation as evidenced in changes in ideal age leads to the accumulation of light carbon from marine export productivity in the ocean interior. Circulation changes hence explain the variability in the spatial pattern of Δδ13CDIC in Fig. a. In particular, the deepening and strengthening of the AMOC during the deglaciation lead to positive Δδ13CDIC changes in the deep Atlantic and to negative Δδ13CDIC changes in the upper Atlantic, consistent with proxy reconstructions.

The sensitivity simulations may be grouped by processes with small and large changes in the spatial pattern of Δδ13CDIC. Δland, coral reef regrowth, and organic weathering flux affect the input of carbon and carbon isotopes into the atmosphere–ocean system, leading to relatively uniform patterns of change (Fig. b, f, and h). Changes in SO wind stress, SO gas transfer rate, rain ratio, and upper ocean remineralization profile, on the other hand, mainly lead to a redistribution of carbon and carbon isotopes in the atmosphere–ocean system amplified by sediment interactions (Fig. c, d, e, g).

The uptake of carbon by the land biosphere leaves the atmosphere and ocean relatively enriched in δ13C (Fig. b). As a result of the carbon removal from the coupled system, changes in bottom water concentrations of CO32- and O2 may influence sedimentation fluxes of POC and CaCO3 locally and lead to small local patterns (see, for instance, negative anomalies in Δδ13CDIC in the North Atlantic deep water (NADW) region of Fig. b). Generally, though, a smooth pattern emerges for changes in Δland. Similarly, the doubling of the organic weathering flux during the glacial results in a strong uniform change in Δδ13CDIC (Figs. h and a). Increased rates of coral reef regrowth have little impact on Δδ13CDIC as the isotopic signature is similar for DIC and deposited CaCO3 (Fig. f). The slight increase in Δδ13CDIC in response to enhanced coral reef regrowth has been attributed to changes in fractionation during marine photosynthesis .

Changes in the other four mechanisms affect the spatial pattern of Δδ13CDIC significantly (Fig. c, d, e, g). We note that these patterns result from adjustment processes acting on different timescales. We recall the forcing history for changes in Southern Ocean wind stress, Southern Ocean gas transfer rate, the rain ratio, the remineralization profile, and the organic weathering rate; PI values are set to LGM values at 40 ka, kept constant from 40 to 18 ka, scaled back to the PI value over the termination (18 to 11 ka), and kept constant thereafter (see also Fig.  and Table ). Results are therefore not directly comparable to the step-change experiments described by . As discussed in detail by and , a more poorly ventilated ocean due to reduced wind stress leads to an increase in δ13Catm. Changes in temperature and circulation in response to decreased Southern Ocean wind stress at the beginning of our simulation (Fig. f) lead to lower marine oxygen concentrations and an increase in POC sedimentation. Subsequent removal of 13C-depleted carbon from the ocean increases δ13CDIC. Restoring wind stress to its initial value over the deglacial reverses the changes. As the sediment feedbacks act on longer timescales, propagation of the signal to the whole ocean is not yet achieved. A complex interplay of processes, such as changes in circulation and subsequent changes in export fluxes, oxygen concentrations, and remineralization of POC, as well as changes in the lysocline and CaCO3 cycling spatially, overlay each other, yielding the Δδ13CDIC distribution seen in Fig. C.

Ocean–sediment interactions also play an important role for δ13CDIC in the case of changes in the Southern Ocean gas transfer rate. Decreasing the gas transfer rate at the beginning of the 40 kyr simulation (Fig. f) leads to a strong increase in δ13Catm. The ocean–sediment interactions are very similar to those described for SO wind stress changes above. The sedimentation fluxes of POC and CaCO3 increase in response to reduced oxygen concentrations from a lower gas transfer rate and changes in the CO32- concentration, as does the δ13C signature of the flux, leading to a removal of 13C from the ocean. From LGM to PI the opposite process takes places (increasing gas transfer back to its initial value); however, due to the long timescales of ocean–sediment interactions, the signal has not propagated to the whole ocean, and in the Atlantic and Pacific negative Δδ13CDIC still prevails (Fig. d).

As discussed in detail in , responses in δ13C from reducing the export rain ratio are small in both the ocean and atmosphere (Fig. a–b) and arise from weathering–sedimentation imbalances and changes in the isotopic value of CaCO3 and POC. This negative perturbation in δ13C is not removed entirely over the deglacial and Holocene due to the long ocean–sediment response timescale, leaving the generally negative Δδ13CDIC as seen in Fig. e.

Finally, a change to a linear remineralization profile in the upper 2 km during the glacial leads to a dipole pattern in δ13CDIC with enriched values in the upper 1 km of the ocean and depleted values underneath (not shown). Across the deglacial, the process is reversed, yielding the pattern in Δδ13CDIC seen in Fig. g from the overlap of different response timescales. Applied changes in the remineralization profile lead to loss of carbon from the atmosphere–ocean system from LGM to PI.

In summary, the carbon and carbon isotope balance in our model simulations can change as a result of altered input fluxes, changes in the surface ocean δ13C signatures, which are then reflected in POC and CaCO3 export fluxes and coral reef regrowth, and altered sedimentation fluxes as a result of altered bottom water CO32- and O2 concentrations. For all seven mechanisms, weathering–sedimentation imbalances and feedbacks exert an impact on the carbon and carbon isotope budget and need to be taken into account on glacial–interglacial timescales. Also, the overlap of different response timescales for atmosphere–ocean and ocean–sediment interactions shapes the patterns in Δδ13CDIC.