We have conducted a set of seven idealized simulations using the Norwegian Earth System Model version 2 (NorESM2-LM, Seland et al., 2020a; Tjiputra et al., 2020). This set of simulations comprises one reference simulation where global warming levels reach approximately 1.7 ∘C in the long term without overshoot and six simulations with different overshoot magnitudes and durations. Our simulations are emission driven; that is, atmospheric CO2 concentrations evolve in response to CO2 emissions and to atmosphere–ocean and atmosphere–land CO2 exchanges. CO2 emissions into the atmosphere are the only forcing applied. Land use and non-CO2 greenhouse gas forcings are kept at pre-industrial levels. We note that NorESM2-LM has a transient climate response (TCR) of 1.48 K, which is at the low end of the CMIP6 model ensemble (Tokarska et al., 2020). Consistent with the relatively low TCR, NorESM2 also shows a low transient climate response to cumulative emissions (TCRE) of 1.32 K Eg C-1 (CMIP6 range 1.32–2.30; Arora et al., 2020) and therefore shows relatively little warming for a given amount of CO2 emissions.

NorESM2 employs version 5 of the Community Land Model (CLM5, Lawrence et al., 2019), which is capable of simulating key thermal, hydrologic, and biogeochemical processes associated with permafrost and their response to climate change. Compared to previous model versions, CLM5 includes several improvements (increased soil depth, improved vertical resolution particularly in the top 3 m, vertically resolved soil biogeochemistry, and changes to modelled snow density, among others) enabling more realistic modelling of permafrost and active layer dynamics (Lawrence et al., 2019). The permafrost region is defined here as the geographic area where the model simulates a maximum active layer thickness shallower than 3 m. We note that CLM5 does not simulate the spatial dynamics of vegetation cover and competition between different plant functional types. For example, northward expansion of plant species due to climate warming is not represented. Changes in carbon stocks associated with vegetation are therefore owed to changes in the plant carbon metabolism response to variations in atmospheric CO2 and temperature only.

Our simulations are subdivided into phases of positive and negative emissions and include a period with zero emissions at the end. The positive emission trajectories follow the Zero Emissions Commitment Model Intercomparison Project (ZECMIP) protocol (Jones et al., 2019): emissions are constructed as bell-shaped curves with 50 years of increasing emissions and 50 years of decreasing emissions (see Fig. 2 of Jones et al., 2019, and Fig. 1a herein). Negative emission trajectories are constructed in the same way but with a negative sign. The reference simulation, referred to as B1500, has cumulative carbon emissions of 1500 Pg during the first 100 years and zero emissions afterwards until the year 400.

Each of the six overshoots is also simulated for 400 years in total. They are branched from simulations that follow the same emission profile as the reference B1500 but with higher emissions (referred to as B1750, B2000, and B2500 – the superscript indicates the amount of cumulative emissions, Table 1). The negative emission phases of the overshoots also last for 100 years, and we apply 250, 500, or 1000 Pg of CDR to simulate overshoots of different magnitudes. To simulate overshoots with a different duration, there is a phase of zero emissions for 100 years between the positive and negative emission phases for three of the six overshoots. All simulations are extended by a phase of zero emissions until the year 400. After negative emissions cease, the amount of cumulative carbon emissions is the same for the reference simulation and all overshoots (1500 Pg C, Fig. 1e).

We refer to the six overshoot simulations as OSyx, where the superscript indicates the cumulative amount of CDR and the subscript indicates the duration of the zero-emission phase between positive and negative emissions. For example, OS0250 refers to an overshoot simulation with CDR of 250 Pg C, which starts immediately after positive emissions cease. In the following text we also refer to these overshoots as “low”, “medium”, and “high” (250, 500, and 1000 Pg CDR, respectively) and as “short” and “long” (0 and 100 years between positive and negative emission phases). The simulation design is summarized in Table 1. For the reference simulation as well as for the low and high overshoot cases, we have run three ensemble members. Although three ensemble members are not enough to derive robust statistics, this provides an idea of the magnitude of interannual to multi-decadal internal variability in our model under the applied forcing.

Our simulations are idealized and are not meant to represent a specific socio-economic scenario or to be realistic in terms of technical and socio-economic feasibility, nor are the qualifiers high, medium, low, long, and short meant to judge their realism or feasibility. The carbon dioxide removal from the atmosphere is also idealized and is not meant to represent a specific CDR method. There is no manipulation of terrestrial or marine carbon sinks that many proposed CDR methods would rely on (e.g. bioenergy with carbon capture and storage or ocean alkalinization). The closest analogue to the CDR applied in our experiments would be the method of direct air capture with carbon capture and storage (e.g. Fuhrman et al., 2021) assuming permanent and perfect storage (no leakage back to the atmosphere). To put the positive and negative cumulative emissions of our simulations into perspective, we mention that 2500 Pg C cumulative (positive) emissions used for the high overshoots are similar to SSP5–8.5 emissions of 2622±144 Pg C (CMIP6 model mean and standard deviation, Liddicoat et al., 2021), and 1500 Pg C used for the reference simulation without overshoot corresponds roughly to SSP4–6.0 emissions (1420±98 Pg C), with the more widely known mitigation scenario SSP2–4.5 having somewhat lower emissions than this (1273±84 Pg C).

Scenarios that are consistent with reaching 1.5 ∘C towards the end of the century, typically contain up to 330 Pg C negative emissions (IPPC, 2018) until 2100, although the uncertainties surrounding the feasibility of such scenarios are large (e.g. Fuss et al., 2018). Also, the quoted 330 Pg C is gross negative emissions, but, in realistic scenarios, part of the gross carbon dioxide removal is used to compensate for residual, difficult to mitigate positive emission. Thus, our low overshoot cases with 250 Pg of net-negative carbon emissions are consistent with the amount of negative emissions applied in IPCC scenarios. Given the longer time horizon of our simulations compared to IPCC scenarios, we also consider our medium overshoot cases roughly consistent with the order of magnitude of CDR applied in such scenarios. The amount of negative emissions in our high overshoot cases (1000 Pg C) is certainly beyond a realistic range.

Our experiment design aims at simulating, in an idealized fashion, emission pathways that eventually meet the temperature target mentioned in the Paris Agreement of keeping global warming to well below 2 ∘C after a period of overshoot (with global warming > 2 ∘C), and we ask whether the state of the Earth system towards the end of these pathways is similar, or reversible, compared to the reference pathway without overshoot. Our definition of reversibility relies on cumulative carbon emissions: all six overshoot simulations reach the same amount of cumulative emissions as the reference simulation (1500 Pg C) after CDR has been applied. For surface temperature to be reversible, the close-to-linear dependence of global warming on cumulative emissions, known as the transient climate response to cumulative carbon emissions (TCRE e.g. Gillett et al., 2013), needs to remain valid under negative emissions. As mentioned above, irreversibility emerges as a deviation from a linear dependence of climate change on cumulative carbon emissions in this approach.

Many previous studies on Earth system reversibility have been conducted with Earth system models of intermediate complexity, where internal variability is limited (e.g. Tokarska et al., 2019; Jeltsch-Thömmes et al., 2020; Lie et al., 2022). For fully coupled Earth system model simulations we need a thorough definition of reversibility in the presence of internal variability. Similar to the concept of “time of emergence” (e.g. Keller et al., 2014), we consider an aspect of modelled climate reversible if the ensemble mean of the overshoot simulation returns to the reference pathway within the range of internal variability. Since our ensemble is too small (three members) to derive this internal variability from the ensemble spread of single years, we define a measure of internal variability as follows: for a given point in time t we calculate the variance of an annually and spatially averaged time series xi(t) for ensemble member i (i=1 … 3) over 11 years, centred at point t, and define internal variability (IV) as the square root of the mean of the variances of the three ensemble members: 1IV(t)=13∑i=13111∑k=-55xitk-x‾i2. This definition is similar to defining internal variability as the standard deviation (1σ) of a time series over a period of 33 years, except that we compensate for a shorter time interval by using several ensemble members. The 11-year sliding window used here is consistent with the fact that we present most of our results as moving averages with the same window length (11 years). Defining internal variability as 1 standard deviation (1σ) of interannual variations leads to relatively conservative threshold for reversibility, compared to some previous studies, which used a 2σ definition of internal variability (e.g. Keller et al., 2014). Decadal and longer-term variability, which can be larger than interannual variations for some variables, is partly removed in our approach by averaging over three ensemble members.

Reversibility depends on the timescale considered, since most Earth system variables show a considerable hysteresis behaviour when CDR is applied (e.g. Boucher et al., 2012; Jeltsch-Tömmes et al., 2020). In this study, we evaluate reversibility for a centennial time horizon after all emissions cease (i.e. after all simulations have reached cumulative carbon emissions of 1500 Pg). We define REVt as the reversibility after a time span t (in years) from this point in time. When comparing overshoots of different durations, there are two possible choices for assessing reversibility. First, we can compare REVt for fixed t, which are different years of the simulations for the short and the long overshoots. For example, REV95 is the year 295 of the simulation for the short overshoot (emissions cease at the year 200) but the year 395 for the long overshoot (emissions cease at the year 300). Alternatively, we can compare reversibility for a given year of the simulations, hence comparing reversibility at different times after emissions cease. For example, when calculating reversibility at the end of our simulations (based on 11-year means for the years 390 to 400), we actually compare REV195 for the short overshoots to REV95 for the long overshoots. In this study, we focus on the second option, arguing that an advantage of applying CDR sooner rather than later would be that the Earth system has more time to recover from the consequences of a shorter overshoot. Hence, we assess reversibility at the end of our simulations, that is, REV195 for the short overshoots to REV95 for the long overshoots.

In the B simulations, global average near-surface temperature (SAT) peaks approximately at the year 80 (i.e. 20 years before positive emissions cease) at 1.86, 2.20, 2.35, and 3.01 ∘C above the pre-industrial level for the simulations with emissions of 1500, 1750, 2000, and 2500 Pg C (Fig. 1b, Table 2). After this peak, SAT slowly decreases by 0.39 to 0.66 ∘C over the next 100–150 years, consistent with the negative zero-emission commitment (ZEC) values found for our model (MacDougall et al., 2020). ZEC is defined as the global mean surface temperature deviation from the point in time when emissions cease (Fig. 1c). Our simulations show a strongly variable ZEC, which is first negative (up to -0.4 ∘C) and then shows positive values (up to 0.5 ∘C) about 100 to 200 years after emissions cease (later for the stronger emission cases). The temperature decrease is mainly seen north of 40∘ N, since there is a strong reduction in the strength of the AMOC (Fig. 1d) during the positive emission phases, which leads to a reduced northward heat transport by the ocean. During the positive emission phases, the AMOC strength at 40∘ N decreases by 11 to 17 Sv (1 Sv: 1 Sverdrup = 1×106 m3 s-1) or 48 %–76 % in our model (Table 2), with a stronger reduction observed in simulations with higher emissions. After positive emissions cease, there is a gradual recovery of AMOC strength, which is slower for higher emissions. The recovery of AMOC leads to increasing surface temperatures and to a second SAT maximum at the years 258, 280, 290, and 388 in the B1500, B1750, B2000, and B2500 simulations, respectively, which coincides well with the point in time when AMOC and northward heat transport in the Atlantic recover to almost pre-industrial strength (vertical lines in Fig. 1c and d). For a more detailed discussion of the evolution of AMOC and high-latitude SAT in our simulations, we refer the reader to Schwinger et al. (2022). The final temperature for the reference simulation without overshoot is 1.70 ∘C above pre-industrial level.

We find that global average SAT is reversible in our simulations, according to our definition, in all overshoots except the high and long overshoot OS1001000 (Fig. 1f and g, Table 3). In all six overshoot simulations, global SAT decreases below the temperatures found in the reference simulation towards the end of the negative emission phases. After this cooling phase, SAT slowly recovers towards the reference temperatures. In our experiments, negative emissions are applied to a model state where AMOC and northward heat transport are strongly reduced compared to the reference case (Fig. 1h). As a consequence, SAT drops more strongly than expected from the reduction in radiative forcing alone, particularly in the region north of 40∘ N. This aspect of our simulations is discussed in Schwinger et al. (2022), and we refer the reader to this study for more details. Here, we focus on the final state of the Earth system after overshoots of different magnitudes and durations, that is, on the last 10 to 50 years of our overshoot simulations. We note that previous studies with intermediate-complexity ESMs (MacDougall, 2013; Jeltsch-Thömmes et al., 2020) have shown that hysteresis and irreversibility generally increase with increasing climate sensitivity. Therefore, owing to its low climate sensitivity, NorESM2 most likely shows a relatively high degree of reversibility compared to higher-sensitivity ESMs.

TCRE derived from our simulations shows a complex behaviour (Fig. 2). During the positive emission phases of the B simulations, the relationship between ΔSAT and cumulative emissions is initially approximately linear and very similar across different rates of emissions. However, already before the peak in SAT (marked by stars in Fig. 2) is reached, there is a deviation from this linear relationship towards smaller values of TCRE. Calculated at the point in time when positive emissions cease (x markers in Fig. 2), TCRE is smaller compared to the values at the SAT maxima by 0.13 to 0.25 K Eg C-1. Furthermore, during the zero-emission phases of the B simulations, SAT decreases and recovers as described above. The final TCRE is between the values at the SAT maximum and at the cessation of emissions, except for the highest-emission simulation B2500, where the final TCRE is substantially higher. We note however, that there is a decreasing temperature trend after the secondary SAT maximum in the B2000 and B2500 simulations. The secondary SAT maximum in B2500 only occurs a few years before the end of our simulations, and therefore the high final TCRE value in B2500 might be caused by the fact that this simulation is less equilibrated at the year 400 than the simulations with lower emissions.

We find that TCRE at the SAT maximum during the positive emission phases decreases with increasing emissions; i.e. TCRE is smaller by about 0.1 K Eg C-1 for total emissions of 2500 Pg C compared to emissions of 1500 Pg C (stars in Fig. 2b). This behaviour is consistent with the study by Herrington and Zickfeld (2014). However, our findings are partly different from Krasting et al. (2014), where a decreasing trend was shown for cumulative emissions below approximately 2000 Pg C, and an increasing TCRE trend was shown for simulation with larger emissions. The latter simulations, however, had rather low emission rates and might therefore not be comparable with our simulations. TCRE values calculated at the time when emissions cease in our simulations (x markers in Fig. 2b) show no clear trend.

TCRE at the end of the B1500 reference simulation and all overshoot simulations (Fig. 2c) reflect the final global mean near-surface temperatures shown in Fig. 1g (since cumulative emissions are the same for all overshoots and the reference). The ensemble mean of each overshoot falls within the internal variability of TCRE found in the reference simulation except for the high and long OS1001000 overshoot. Within the groups of short and long overshoots there is a common pattern, where the medium-size overshoot (500 Pg carbon removal) has the lowest final SAT and TCRE. Although there is only one ensemble member available for the medium-sized overshoots, this result is consistent with other findings discussed in the next section.

Towards the end of our simulations, atmospheric CO2 concentrations (Fig. 3a and b) converge to a similar level in all six overshoots and the reference simulation, with differences less than 7 ppm (i.e. less than 15 Pg atmospheric carbon, Table 3) at the year 400. According to our definition, however, the atmospheric CO2 concentration is reversible only in the OS0250 and OS01000 simulations, since the internal variability is small (about 1 ppm only). Nevertheless, given the vastly different pathways of atmospheric CO2 concentrations in our simulations with freely evolving CO2, a difference of less than 7 ppm is remarkable. We note that during the negative emission phases, atmospheric CO2 concentrations decrease below the concentration in the reference simulation in all overshoots. This effect is largest (about 50 ppm) for the large overshoots, which have the fastest rate of CDR.

This similarity is caused by a near-perfect compensation between stronger ocean uptake and weaker land uptake during an overshoot (Fig. 3c and d), consistent with the findings of Tokarska et al. (2019). The cumulative fractions of carbon stored in the atmosphere (CFA), in the ocean (CFO), and on land (CFL) are shown in Fig. 4a–c. In all B simulations, CFO increases continuously during the positive emission phase and the following phase of zero emissions, as the ocean circulation constantly removes carbon from the surface ocean to depth. CFO is lower for high (positive) emissions due to chemical and temperature-mediated feedbacks (e.g. Arora et al., 2020). For land, the cumulative uptake fraction initially increases but shows a decreasing trend in the second half of the simulations, indicating that land becomes a source of carbon to the atmosphere. Same as for CFO, and consistent with current understanding of carbon cycle feedbacks, CFL becomes smaller in the simulations with higher emissions.

During phases with negative emissions in the overshoot simulations, both land and ocean become a source of CO2 such that their carbon stocks are reduced (Fig. 3c and d). For the ocean this happens as soon as the CO2 partial pressure difference between atmosphere and ocean becomes negative. For the land, a reduced CO2 fertilization effect shifts the overall balance between carbon uptake through net primary production and carbon release through heterotrophic respiration towards the latter. However, since these processes are slow and lag the reduction in the cumulative total of emissions, there is a rapid increase in CFO and CFL (Fig. 4b and c). At the end of the simulations, we find a common pattern between the group of short and the group of long overshoots: while CFA is almost equal in the low and high overshoots, it is lower by about 0.5 %–0.6 % in the medium overshoots (Fig. 4d), as reflected in the atmospheric CO2 concentration towards the end of the simulation period (Fig. 3b). This is caused by a non-monotonic behaviour of the land carbon fluxes with increasing overshoot size. While the cumulative ocean uptake increases monotonically for larger overshoots (Fig. 4e), the land carbon fluxes show a more complex behaviour. Relative to the reference simulation, there is a decrease in land carbon stocks in all overshoots except for the OS0500 case where there is a small gain. Likewise, the long medium-size overshoot OS100500 shows the smallest carbon loss among all long overshoot simulations. Hence, somewhat counterintuitively, the overall land carbon storage is closest to the reference case in the medium-sized overshoots (both long and short), which is caused by different levels of compensation between permafrost carbon losses and gains in other carbon reservoirs (vegetation and non-permafrost soils) as discussed further below in Sect. 3.5.

For the high overshoot cases, the compensating effect of land and ocean uptake is clearly exhibited in the cumulative fractions at the end of our simulations. For the OS1001000 simulation, CFO is about 2.5 % (37.5 Pg C) larger, while CFL is 2.2 % (33 Pg C) smaller than in the reference simulation, and these values show only small internal variability.

What impact on global average SAT could the small differences of up to 7 ppm in atmospheric CO2 concentration due to the different emission pathways during the overshoots have? For a back-of-the-envelope calculation, we assume that the model has already reached an equilibrium towards the end of the simulations, such that ΔSATeq=-FCO2/λ, where λ=-1.34 is the feedback factor for our model (Zelinka et al., 2020) and FCO2 is the radiative forcing of CO2 (Myhre et al., 1998). This relationship shows that a difference of 7 ppm atmospheric CO2 concentration would result in an equilibrium SAT difference of about 0.06 ∘C. Hence, this can potentially explain a part of the relatively low final TCRE value (Fig. 2c) in the OS0500 and OS100500 simulations compared to the other overshoots.

The Northern Hemisphere high-latitude permafrost extent follows the evolution of surface temperature over this area closely (Fig. 5). We define permafrost where the annual maximum active layer depth is shallower than 3 m. During the positive emission phases, permafrost extent declines by between 5.80 to 9.65×106 km2 from its pre-industrial value of 14.7×106 km2. During the negative emission phases, northern high-latitude SAT cools temporarily below the reference level (Fig. 5c, see Schwinger et al., 2022, for details), and the permafrost area follows this pattern of warming–cooling–warming. Towards the end of our simulations, permafrost area is reversible according to our definition (Fig. 5b, Table 3), consistent with previous studies, which show that the physical extent of permafrost area mainly follows the SAT trajectory and tends to recover under temperature reduction (Boucher et al., 2012; MacDougall, 2013; Lee et al., 2021). It is, however, worth mentioning that landscape changes and hydrological responses to permafrost thaw, such as coastal erosion, excess ice melting, and formation of thermokarst lakes, are highly heterogeneous and depend on small-scale processes that are neither resolved nor parameterized in our model. Therefore, irreversible changes at the (unresolved) landscape scale would occur even if the modelled large-scale physical state of the soil is found to be reversible according to our definition.

Vegetation carbon increases in all B simulations during the positive emission phases due to the CO2 fertilization effect (Fig. 6a–d). This effect remains dominant at high latitudes (north of approximately 45∘ N) throughout the 400 years of the B simulations, while at lower latitudes the vegetation carbon stock declines again during the zero-emission phases. Vegetation carbon is generally not reversible according to our definition, but this is mainly due to a low internal variability, and differences relative to the reference simulation remain small (below 8 Pg C globally; see further discussion below). Note that the distribution of vegetation is prescribed in our model, such that (potentially irreversible) changes in vegetation carbon as well as biophysical feedbacks (changes in albedo and roughness length) caused by shifts in vegetation composition are most likely underestimated in our simulations. For example, a northward tree-line expansion or shrubification in high latitudes due to climate warming cannot be captured by our model.

Soil carbon stocks (Fig. 6e–h) increase during the B simulations at all times and all latitudes, except in permafrost regions, where the release of carbon from thawing permafrost dominates. In non-permafrost regions, soil carbon stocks are generally larger after an overshoot compared to the reference pathway without overshoot at the end of the simulations (with the exception of the high-latitude non-permafrost soil carbon in the OS0250 simulation, where soil carbon stock is slightly smaller). In the permafrost region (Fig. 6h), the evolution of permafrost carbon stock is influenced by the temporary cooling during the negative emission phases. Due to the fact that SAT is colder in all overshoot simulations than in the reference pathway without overshoot for some period of time (Fig. 5c), the area affected by permafrost thaw is smaller for some period of time (approximately 100 years) than in the reference simulation (Fig. 5b). As a consequence, the loss of permafrost carbon is smaller in the OS0500 simulation than in the reference pathway B1500. Also, the OS100250 and OS100500 simulations show a very similar carbon loss that is only marginally larger than in the reference simulation.

Figure 7 summarizes the changes in stock size of the main carbon reservoirs at the end of the six overshoot simulations (average over years 390–400) relative to the reference simulation B1500 without overshoot. As pointed out above, there is a near-perfect compensation between land and ocean carbon uptake in our model; that is, while land loses carbon relative to the reference simulation during an overshoot, the ocean gains additional carbon. This compensation leads to a very similar atmospheric CO2 concentration in all overshoots compared to the reference simulation (Fig. 3). The main carbon reservoirs considered here are vegetation carbon, permafrost carbon, and non-permafrost soil carbon for land, as well as remineralized and preformed dissolved inorganic carbon (DIC) for the ocean. Changes in permafrost carbon are obtained as cumulative carbon fluxes summed over all permafrost grid cells, i.e. those grid cells that are defined as permafrost in the pre-industrial control simulation. Preformed DIC originates from atmospheric CO2 that dissolves in the surface ocean and is transported into the interior by ocean circulation. In contrast, remineralized DIC has been transported into the interior ocean through the biological carbon pump: biological uptake by planktonic organisms near the ocean surface, sinking to depth as particulate organic matter, and subsequent remineralization by bacterial activity. We note that the remineralization of organic carbon consumes oxygen (if present in sufficient quantity), such that oxic remineralization can be measured by apparent oxygen utilization (AOU), defined as the oxygen deficit in a water parcel relative to its saturated oxygen content.

Increased biological pump efficiency is the main driver of increased ocean carbon storage during and after an overshoot, which leads to a legacy of increased remineralized carbon in the ocean interior (see also Fig. 10). Changes in preformed DIC play a significant role only if the overshoot is long and intense (overshoots OS100500 and OS1001000) because in contrast to remineralized DIC, preformed DIC is lost from the ocean surface during the negative emission phases of an overshoot when the partial pressure difference with the atmosphere becomes negative. Only during the long and high overshoots is there sufficient time with a large positive CO2 partial pressure difference to increase preformed DIC by significantly more than what is removed later during the negative emission phases.

Changes in soil carbon (permafrost and non-permafrost) dominate the legacy of the overshoots in terrestrial carbon stocks, with permafrost carbon loss being far greater than non-permafrost soil carbon gain in the high overshoot simulations. We note that permafrost carbon is particularly sensitive to the large SAT fluctuations in northern high latitudes seen in our model (see Fig. 5c for SAT over the permafrost area). This is because permafrost is exclusively located in the area where the high-latitude cooling (north of 40∘ N) is large during the negative emission phases in our model. As mentioned above, during the negative emission phases, SAT drops below the reference level temperature for a significant amount of time, such that permafrost carbon is actually better preserved during parts of the overshoots compared to the reference simulation with no overshoot. This is why the additional permafrost carbon loss remains relatively limited in the low and medium-size overshoots (there is even a small gain in the short medium-size overshoot). Permafrost carbon losses were larger if our model showed a weaker AMOC decline and associated high-latitude cooling. In this case, the global budget of carbon stock changes could be dominated by losses from land. This could shift the overall carbon balance after an overshoot from lower to higher atmospheric CO2 concentrations compared to a pathway without overshoot.

There are only relatively small differences in vegetation carbon after the overshoots compared to the reference simulation with no overshoot. This is to be expected since the vegetation carbon reservoir reacts with little time lags to changes in environmental conditions and CO2. Generally, there is less carbon stored in vegetation after an overshoot, except for the OS01000 simulation, which shows a small gain. We note that the inclusion of vegetation dynamics in our model would most likely affect these results, since changes in biogeography would lead to larger changes in land carbon pools and larger time lags between drivers and response.

Ocean heat content (OHC), like ocean carbon content, increases steadily during phases with positive or zero emissions (Fig. 8a). Unlike carbon content, however, the ocean heat content does not decrease significantly during phases of negative emissions (compare Figs. 3c and 8a; note there is a small decrease in OHC in the OS1001000 simulation around the year 250). As a result, OHC after an overshoot deviates much more from the reference simulation with no overshoot, than the accumulated carbon fluxes. While the CO2 partial pressure difference between atmosphere and ocean becomes negative during the negative emission phases (i.e. driving a carbon flux out of the ocean in the global mean), SAT does not decrease enough to cool the ocean significantly in the global mean. Also, the relatively strong AMOC reduction in our model increases the oceanic heat uptake by reducing heat losses to the atmosphere in the North Atlantic (Drijfhout, 2015).

These results imply that steric sea level rise (Fig. 8b, Table 3) is irreversible during the 400 years of all overshoot simulations and higher than in the reference simulation by up to 15 cm for the most extreme overshoot OS1001000 (Fig. 8b, Table 3). This is consistent with a multitude of previous studies (e.g. Boucher et al., 2012; MacDougall, 2013; Tokarska and Zickfeld, 2015; Ehlert and Zickfeld, 2018), which show that sea level rise is largely irreversible on centennial to millennial timescales. We note, however, that negative emissions are indeed effective in reducing the rate of sea level rise after an overshoot to a value similar to or lower than that of the reference simulation (Fig. 8c). The additional steric sea level rise due to an overshoot remains relatively small compared to the sea level rise committed to in the reference simulation in our model (<20 % at the year 400 except for the most extreme overshoot OS1001000). Thus, the rate of sea level rise determines the pace and cost of necessary adaptation for the decades to centuries after an overshoot. Therefore, limiting the rate of sea level rise after an overshoot might arguably be more policy relevant in the context of negative emissions than a relatively limited contribution of the overshoot to the sea level rise itself.

The evolution of global average marine net primary production (PP) in the B and overshoot simulations is shown in Fig. 9a and b. During the positive emission phases, PP decreases by between 2.0 and 2.8 Pg C yr-1 from its pre-industrial value of 33.6 Pg C yr-1. During the zero-emission phases of the B simulations, there is a gradual recovery of PP except for the highest-emission simulation B2500. In general, the recovery of PP is weak compared with, for example, the recovery of AMOC strength (Fig. 1d). Global PP is reversible according to our definition for all overshoot simulations (Fig. 9b, Table 3). We note that in our model the export of particulate organic carbon (POC, not shown) from the surface ocean shows the same qualitative behaviour as PP. Consistent with previous studies (Schwinger et al., 2014; Arora et al., 2020), the reduced PP and carbon export are overcompensated for by a reduction in ocean circulation and upwelling of nutrients and carbon from the deep ocean under climate change, such that the remineralized component of DIC and nutrients in the interior ocean increases steadily in all simulations. This process is irreversible at the 400-year timescale of our simulations such that the excess of remineralized phosphate over the pre-industrial values is larger by up to 30 % in the overshoot simulations compared to the reference simulation (Fig. 9c).

The global average surface oxygen concentration (not shown) is mostly driven by surface temperature and is, consistent with our results for SAT, reversible for all but the OS1001000 overshoot. Surface pH (not shown) is not formally reversible according to our definition with the exception of the short and low overshoot OS0250. This is, however, due to the very low internal variability of global mean surface pH, and we note that surface pH is actually higher (i.e. slightly closer to pre-industrial values) towards the end of all overshoot simulations. This behaviour has been observed in a previous study (Li et al., 2020) and can be explained by the fact that atmospheric CO2 has been lower than in the reference simulation for a period of time before the reference level of atmospheric CO2 is approached (Fig. 3a).

Global ocean oxygen content is not reversible in any of the overshoot simulations (Fig. 9d, Table 3). The non-reversibility in oxygen content is governed in approximately equal parts by ocean warming and changes in the biological pump in our model. This can be seen from apparent oxygen utilization (AOU, Fig. 9e), which is roughly 2500 Tmol larger at the end of the OS1001000 overshoot than in the reference simulation, explaining about half of the 5000 Tmol non-reversibility of global oxygen content. The other half can be attributed to ocean warming, which reduces the solubility of oxygen, and thus the interior oxygen content once surface waters are transported into the interior ocean.

Global average pH (Fig. 9f) is not reversible either, although the differences relative to the reference simulation are small except for the most extreme overshoot OS1001000. That is, the negative emissions are generally effective in bringing pH close to the reference simulations, and the non-reversibility (according to our definition) is due to the very small internal variability of global average pH.

To understand the spatial structure of interior changes in oxygen, remineralized DIC (and phosphate), and pH, it is instructive to look at changes in the simulated ideal age of water masses (Fig. 10a–c for the long overshoots in the Pacific; see Figs. A1–A3 for corresponding figures for the Atlantic Basin and the short overshoots). In the reference simulation B1500, the simulated ideal age relative to the pre-industrial state increases by more than 100–200 years everywhere below approximately 2000 m depth due to increasing ocean stratification. Consequently, the water masses that are upwelled in the Southern Ocean and reach the surface around 60 to 70∘ S are getting older, too. Above these ageing water masses, there is a region where water mass ages get younger under ongoing climate change. The core of this region extends from approximately 30∘ S to 30∘ N, and the increased ventilation here is caused by a reduction in vertical transport of older waters from below (Gnanadesikan et al., 2007), a pattern that is commonly found in ESMs (Cabré et al., 2015). During the overshoot simulations these patterns are amplified, such that most of the old waters in the deep ocean get older while the ventilation of intermediate water masses generally increases.

The influence of seawater temperature is mostly limited to the upper 1000 to 2000 m in the Pacific but reaches down to about 3000 m with Atlantic deep waters (Fig. A1). Temperature changes in the upper ocean (surface down to 500–1000 m) are largely reversible, although, for the most extreme overshoot OS1001000, the volume of water with reversible temperature changes becomes small, particularly in the Atlantic Ocean. Below 1000–2000 m depth temperature changes (but also changes in the other variables discussed here) are largely irreversible. This is partly because the internal variability in the deep ocean is small, and this favours the emergence of non-reversibility according to our definition. However, for changes mediated by the biological pump (oxygen, remineralized DIC) this is not necessarily true. Here, non-reversible changes in the deep ocean can be as large as or even larger than in the upper ocean (Fig. 10g–l). Expectedly, the changes in interior oxygen content (Fig. 10g–i) show a negative correlation with the changes in ideal age, as do the changes in the remineralized component of DIC (Fig. 10j–l) and nutrients (not shown). Note that remineralized DIC is based on a preformed phosphate tracer in our model, such that it is not derived from oxygen via apparent oxygen consumption. Where water masses get older, oxygen content decreases because more oxygen is used for remineralization of organic matter. Consequently, in the Pacific, the loss of oxygen is largest in the tropical deep ocean and in the Southern Ocean upwelling, where the legacy of the overshoots is largest (Fig. 10g–l).

Changes in pH (Fig. 7m–o) mainly depend on the surface history of a water parcel and ventilation pathways, although changes in organic matter export and remineralization as well as changes in CaCO3 export and redissolution contribute to pH changes at depth (Lauvset et al., 2020). Close to the surface, pH is mostly reversible, since surface pH follows atmospheric CO2 concentrations closely. At locations in the surface ocean where pH is not reversible, the pH value is actually higher than in the reference simulation, reflecting the fact that atmospheric CO2 has been lower during the negative emission phases of the overshoots for some time (Fig. 3a). Likewise, in the deep-water formation regions of the Southern Ocean, we find pH values that are higher after an overshoot. At intermediate depth, there is a legacy of lower pH values in water masses that have ventilated these depths during the overshoot periods. In the deep ocean, mainly the Pacific, we find irreversibly lower pH values that are most likely caused by an increase in remineralized carbon (these water masses have not been ventilated during the course of the overshoot simulations). We note that our model simulates too large marine primary production in the equatorial Pacific and excessive oxygen minimum zones below, such that the effect of changes in remineralized carbon in the deep Pacific on pH might be overestimated.

So far, we have chosen to assess reversibility for the same simulation year, that is, we compare REV95 for the long overshoot simulations (95 years after negative emissions ceased, Fig. 10) to REV195 for the short overshoot simulations (195 years after negative emissions cease, Figs. A2 and A3). To allow for a clean comparison of the effect of overshoot length on reversibility, Fig. A4 shows REV95 for the short overshoots (i.e. reversibility derived from simulation years 290–300). Comparing Figs. A4 and 10 reveals that the volume of water masses showing irreversible change tends to be larger in the long overshoots, although the spatial patterns are broadly similar. The main difference between the short and long overshoots is that the irreversible changes in the deeper ocean are much more pronounced for the long overshoot duration, indicating a clear benefit of limiting the duration of an overshoot.

It has been suggested that tipping points exist in the Earth system, which would, once a critical threshold is crossed, trigger self-amplifying feedbacks and “tipping cascades” that could lead to irreversible changes and a “hothouse Earth” climate state (Steffen et al., 2018; Lenton et al., 2019). Such tipping mechanisms could exacerbate the risk of relying on CDR to return to a less dangerous climate state after an overshoot. Our Earth system model simulations represent some of the proposed cascading tipping mechanisms, while others are not modelled. The release of carbon from permafrost soils is one of the sources that could increase atmospheric CO2 concentrations and thus reinforce global warming. In our simulations, permafrost carbon release is indeed the largest irreversible contribution to carbon losses from land during most of the overshoot simulations. However, continuous and century-long losses from land carbon stocks are compensated for by increased ocean carbon uptake in our model, such that the atmospheric CO2 concentration is not larger after an overshoot compared to the scenario with no overshoot.

Compared to other ESMs, our model simulates a relatively strong reduction, followed by a gradual recovery, of AMOC strength (Schwinger et al., 2022). Therefore, tipping points and cascades related to a strong weakening of AMOC should be represented in our simulations. We note that, due to the northern high-latitude multi-decadal cooling trend simulated after emissions cease and the amplification of this cooling during phases of CDR, the AMOC slowdown actually helps to limit the impact of an overshoot on Arctic sea ice, permafrost, and presumably the Greenland ice sheet (the latter is not represented in our simulations). We also do not see boreal forest dieback, even in the most extreme overshoot simulation considered here. In the high latitudes, vegetation carbon increases with increasing temperature in our model and tends to be slightly higher after an overshoot compared to the reference simulation. For the Amazon rainforest, although vegetation carbon tends to be slightly lower after an overshoot, the system is stable, also after the most extreme overshoot, and there are no abrupt shifts. Although the ocean primary productivity decreases with progressing climate change, the efficiency of the biological pump (measured as the stock of remineralized carbon in the ocean) increases in our simulations, consistent with results from CMIP5 and CMIP6 ESMs (Schwinger et al., 2014; Arora et al., 2020). This increase is due to an overcompensation of reduced PP by reduced ocean circulation and upwelling. In general, we do not find abrupt large-scale shifts that would be indicative of tipping points. This result is broadly consistent with previous model studies using intermediate-complexity ESMs (e.g. Steinacher and Joos, 2016; Jeltsch-Thömmes et al., 2020) or CMIP6 ESMs (e.g. Koven et al., 2022). We have, however, not screened our model results for abrupt regional shifts as done, for example, in Drijfhout et al. (2015).