Path Independence of Carbon Budgets When Meeting a Stringent Global Mean Temperature Target After an Overshoot

Emission pathways that are consistent with meeting the Paris Agreement goal of holding global mean temperature rise well below 2 °C often assume a temperature overshoot. In such overshoot scenarios, a given temperature limit is first exceeded and later returned to, under the assumption of large‐scale deliberate carbon dioxide removal from the atmosphere. Here we show that although such strategy might result in a reversal of global mean temperature, the carbon cycle exhibits path dependence. After an overshoot, more carbon is stored in the ocean and less on land compared to a scenario with the same cumulative CO2 emissions but no overshoot. The near‐path independence of surface air temperature arises despite the path dependence in the carbon cycle, as it is offset by path dependence in the thermal response of the ocean. Such behavior has important implications for carbon budgets (i.e. the total amount of CO2 emissions consistent with holding warming to a given level), which do not differ much among scenarios that entail different levels of overshoot. Therefore, the concept of a carbon budget remains robust for scenarios with low levels of overshoot (up to 300 Pg C overshoot considered here) but should be used with caution for higher levels of overshoot, particularly for limiting the environmental change in dimensions other than global mean temperature rise.


Introduction
Following the Paris Agreement's long-term temperature goal of "holding the increase in the global average temperature to well below 2°C above preindustrial levels and pursuing efforts to limit the temperature increase to 1.5°C above preindustrial levels, recognizing that this would significantly reduce the risks and impacts of climate change" (UNFCC, 2015;Rogelj, Popp, et al., 2018;Rogelj, Shindell, et al., 2018), numerous studies published possible emission pathways for limiting warming to below 2 or to 1.5°C. The Paris Agreement's long-term temperature goal does not suggest that a given level of temperature rise may be temporarily exceeded, but studies have made various assumptions leading to pathways that either stay below or temporarily exceed the 1.5 or 2°C limits (Grubler et al., 2018;Luderer et al., 2018;Rogelj, Popp, et al., 2018;Rogelj, Shindell, et al., 2018;Sanderson et al., 2016;Sanderson et al., 2017). The precise interpretation of the Paris Agreement's long-term temperature goal remains a topic of active discussion in policy circles (Schleussner et al., 2016;Rogelj et al., 2017). Meeting the Paris Agreement's long-term temperature goal requires ambitious mitigation efforts on a global scale (Rogelj, den Elzen, et al., 2016). Reducing emissions drastically over the coming decades requires more investment in the near term compared to a situation in which action is postponed and emission reductions are assumed to be implemented later. Because of the way mitigation pathways were designed, there are structural biases toward delaying emission reductions (Rogelj et al., 2019). These biases could be resolved if scenarios would apply a different and more appropriate scenario logic (Rogelj et al., 2019). However, emission pathways often first breach the intended temperature target level with the assumption that temperatures can be returned to the desired temperature target level by the Year 2100 (Rogelj, Popp, et al., 2018;Rogelj, Shindell, et al., 2018;Sanderson et al., 2016Sanderson et al., , 2017Tanaka & O'Neill, 2018). Such pathways are referred to as overshoot pathways because they temporarily exceed the target temperature level and aim at stabilizing temperature rise at a level lower than the target later on (Matthews & Solomon, 2013). Geophysically, an overshoot can only be achieved through the implementation of deliberate carbon dioxide removal from the atmosphere (CDR), which results in net-negative emissions for a given period. (For a comprehensive summary of various CDR technologies and their feasibility see Minx et al., 2018;Fuss et al., 2018). If overshoot is avoided, the reliance on CDR is reduced (Grubler et al., 2018;Strefler et al., 2018;van Vuuren et al., 2018).
While temperature rise has been shown to be reversible in scenarios that implement CDR, other components of the climate system, such as sea level rise (Ehlert & Zickfeld, 2018;Mengel et al., 2018;Palter et al., 2018;Tokarska & Zickfeld, 2015) or ocean acidification (Mathesius et al., 2015) and marine net primary productivity (John et al., 2015) take centuries to millennia to be restored to their initial levels, even under large-scale implementation of CDR. Recent studies examined the carbon-climate responses to CDR (Jones et al., 2016;Schwinger & Tjiputra, 2018;Tokarska & Zickfeld, 2015;Vichi et al., 2013;Zickfeld et al., 2016), indicating that carbon sinks often weaken, as expected, when atmospheric CO 2 is deliberately reduced, or even become sources of carbon (i.e., release CO 2 back to the atmosphere) in response to deliberate net removal of the anthropogenic CO 2 from the atmosphere (Jones et al., 2016;Tokarska & Zickfeld, 2015).
The reversibility of climate change in response to CDR has implications for the concept of a carbon budget (i.e., the total amount of CO 2 that can be emitted, consistent with achieving a given temperature level; Zickfeld et al., 2009), which has shown to be independent of the CO 2 emission pathway, particularly in scenarios with monotonically increasing CO 2 emission rate. However, the question remains whether the concept of a carbon budget is robust in case of overshoot scenarios. This is particularly pertinent because the carbon budgets concept is applied to emissions pathways that stay below a given target level but also to pathways that temporarily overshoot a specific temperature limit (Rogelj, den Elzen, et al., 2016;Rogelj, Schaeffer, et al., 2016;Rogelj, Popp, et al., 2018;Rogelj, Shindell, et al., 2018). An overshoot of a given carbon budget followed by the implementation of CDR resulting in net-negative emissions leads to a temperature overshoot, where a given level of warming is first exceeded and then returned to. Previous work by MacDougall et al. (2015) suggests that carbon budgets after an overshoot are smaller than carbon budgets before the overshoot, for scenarios with large levels of overshoot.
Here we use an Earth System Model of intermediate complexity (UVic ESCM; section 2.2), driven by a set of idealized emission scenarios that exceed a target carbon budget by different amounts and are followed by large-scale global CDR, such that the total amount of CO 2 emitted is the same in each scenario by the Year 2200 (or Year 2250 in the 1.5°C scenario group; section 2.1). We explore the response of land and ocean carbon sinks to overshoot scenarios in which different temperature targets of 1.5, 2.0, and 2.5°C are first exceeded and then restored in the long term. Our main analysis focuses on the 2.0°C scenario group, and the two remaining groups are shown in the supporting information. We compare the carbon cycle response in the stylized overshoot scenarios to that in a reference scenario that limits warming to the given temperature target (such as 2.0°C) without overshoot at the time the same amount of cumulative CO 2 emissions is reached and explore to what extent the difference is dependent on the level of overshoot.

CO 2 Emission Pathways Design
We designed three groups of idealized emission scenarios, where fossil fuel CO 2 emissions peak within the next few decades (in Years 2018-2030) and decline to 0 by the Year 2100 through emission reductions and deliberate removal of CO 2 from the atmosphere (CDR), followed by net-negative emissions in the period 2100-2200, and net-zero emissions in the period 2200-3000. Each scenario group was designed to reach the same level of cumulative CO 2 emissions by the Year 2200 (or 2250 for the 1.5°C scenario group), in order to achieve the same long-term temperature stabilization in each scenario, at the desired level (of 1.5, 2.0, or 2.5°C). Each set of simulations includes a reference nonovershoot pathway ("OS 0"), where the given temperature stabilization level is not exceeded, and emissions are rapidly reduced to net zero, without resorting to CDR (Figure 1a; supporting information Figure S1a). These scenarios are designed to explore the carbon cycle consequences of overshoot and are not intended as interpretations of the Paris Agreement.
We define the overshoot cumulative emissions as the difference in cumulative CO 2 emissions between the peak cumulative emissions (maximum cumulative CO 2 emissions before net zero for each scenario) and the cumulative emissions in the reference nonovershoot scenario "OS 0" in Year 2200, when cumulative CO 2 emissions reach the same level (Figure 1c), or Year 2250 for the 1.5°C scenario group (supporting information Figure S1c). Scenario names in the legend indicate the amount of overshoot cumulative emissions. We focus the main analysis on the 2.0°C scenario group, and additional figures for the 1.5 and 2.5°C scenarios are shown in the supporting information. While the scenarios slightly differ in the design, they are comparable for the purpose of this study.
In all scenarios, land use change (LUC) emissions follow RCP 2.6 (van Vuuren et al., 2011) to the Year 2100. After the Year 2100, LUC emissions are extended linearly to reach zero in the Year 2150. Biophysical effects of LUCs are not considered. In addition to CO 2 emissions from fossil fuels and LUC, all scenarios include forcing from sulfate aerosols and non-CO 2 greenhouse gases, with slight differences in aerosol forcing

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Earth's Future among scenario groups (supporting information Figure S3). Both forcings are stabilized after the historical period, and set to a constant, Year 2010 value, for future years. Natural forcing due to volcanic eruptions is set to zero for future scenarios, as future eruptions are unpredictable. Hence, while our scenario design is well suited to explore the relative impact of differing cumulative CO 2 emissions, its aim is not to provide carbon budget estimates for specific warming levels, as non-CO 2 warming plays an important role here as well (Rogelj et al., 2015;Rogelj, den Elzen, et al., 2016;Rogelj, Schaeffer, et al., 2016;Tokarska et al., 2018).

The University of Victoria Earth System Model
The University of Victoria Earth System Climate Model (UVic ESCM) is an intermediate complexity model with a horizontal grid resolution of 1.8°(meridional) × 3.6°(zonal) (Weaver et al., 2001). The physical model consists of an atmosphere model coupled to an ocean general circulation model (including both organic and inorganic carbon cycle), a sea ice model, and a land surface model together with dynamic terrestrial vegetation (Weaver et al., 2001) and is coupled to models of the marine and terrestrial carbon cycle.
The atmosphere is represented by an energy-moisture balance model with winds prescribed from observation-based data (Weaver et al., 2001). The atmospheric model is coupled to a three-dimensional ocean general circulation model, consisting of 19 vertical levels and global resolution of 1.8°(meridional) × 3.6°(zonal) (Weaver et al., 2001). The ocean model is based on the Geophysical Fluid Dynamics Laboratory Modular Ocean Model 2.2, including both organic (Ocean Ecosystem Biogeochemical Model, represented by nutrient, phytoplankton, zooplankton, and particle detritus; Schmittner et al., 2008) and inorganic (Orr et al., 1999) carbon cycle components. The ocean general circulation model is coupled to a sea ice model (Weaver et al., 2001), which includes thermodynamic components (open water sea ice and its changing area) and dynamic components governed by momentum balance, which respond to oceanic and atmospheric (wind) stresses (Weaver et al., 2001). The land surface model, represented by a simplified version of the Hadley Centre Met Office Surface Exchange Scheme, is coupled to a dynamic terrestrial vegetation model (Top-down Representation of Interactive Foliage and Flora Including Dynamic vegetation model; Weaver et al., 2001;Meissner et al., 2003;Eby et al., 2009). Changes in temperature and atmospheric CO 2 concentration drive dynamic changes in vegetation distribution, consisting of five different plant functional types: broadleaf tree, needle-leaf tree, C3 grass, C4 grass, and shrubs (Weaver et al., 2001). The vegetation fraction for each functional plant type is calculated based on Lotka-Volterra competition equations (Meissner et al., 2003). Terrestrial carbon feedbacks due to increased CO 2 levels (CO 2 fertilization) and weakening of the carbon sinks driven by the increase in the temperature are also included (Matthews & Caldeira, 2008). Ice sheets, land-based ice dynamics and carbon release from peatlands and permafrost are not included in this version of the UVic model.
Compared with other Earth System Models of Intermediate Complexity, the UVic model generally ranks high with regards to the complexity of the ocean, land surface and biosphere components and generates responses that fall within the uncertainty envelope of the observed surface air temperature in the historical period . In more comprehensive Earth System Models, the representation of carbon cycle feedbacks spans a range of responses especially regarding uncertain terrestrial carbon uptake. The UVic model carbon cycle responses are generally close to the median of the Earth System Model range (Arora et al., 2013).

Global Mean Responses to Overshoot Scenarios
By design, CO 2 emission pathways ( Figure 1a) differ in the peak emissions rate and the amount of CDR implemented in the period 2100-2200, resulting in net-negative emissions during that period, in order to reach the same level of cumulative CO 2 emissions (Figure 1c) by the Year 2200 (or Year 2250 in the 1.5°C scenario group, supporting information Figure S1; section 2). As a result, the 2°C temperature level is temporarily exceeded in all but the "OS 0" scenario ( Figure 1d), which we will refer to as the reference nonovershoot scenario and to which the overshoot scenarios will be compared. All scenarios reach the same temperature level in the long term, around the Year 2200 ( Figure 1d) because each of them emits the same amount of cumulative CO 2 emissions (Figure 1c). Global mean warming has been shown to be proportional to the cumulative CO 2 emissions (with the proportionality constant referred to as the transient climate

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Earth's Future response to cumulative CO 2 emissions or TCRE; Matthews et al., 2009) and is independent of CO 2 emission pathway for a range of emissions scenarios (Herrington & Zickfeld, 2014;Zickfeld et al., 2012), particularly for monotonically increasing CO 2 emission rates. Overshoot scenarios considered here exhibit the same proportionality of global mean warming to cumulative CO 2 emissions, despite their nonmonotonic nature of CO 2 emission rate, which peaks and declines. Such behavior of overshoot scenarios has important implications in the context of carbon budgets, which we discuss in section 3.5.

Carbon Reservoirs Response
Land carbon storage declines in all overshoot scenarios during the net-negative emissions phase (2100-2200), while the ocean continues to take up carbon for most scenarios albeit at a lower rate than before CDR was implemented (Figure 2). Scenarios with the highest level of overshoot (OS 300, and OS 150) experience a slowdown (or even a sink-to-source transition) in ocean carbon uptake during the CDR phase, which, however, is followed by an increase in the rate of ocean carbon uptake beyond the Year 2200, once the netnegative emission phase stops. After CO 2 emissions return to zero in 2200, land carbon storage continues to decline, whereas ocean carbon storage slowly increases for centuries onward (Figures 2b and 2c). After Year 2200 (when CDR ends, and all scenarios reach the same level of cumulative CO 2 emissions) land carbon

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Earth's Future storage in all overshoot scenarios is lower than in the reference nonovershoot scenario "OS 0," while ocean carbon storage is higher in all overshoot scenarios than in the "OS 0" reference scenario (Figures 2b and 2c).
Such behavior occurs due to the interactions between the atmosphere, ocean and land carbon pools. Atmospheric carbon in the Year 2200 is lower in the overshoot scenarios than in the reference nonovershoot scenario ("OS 0"; Figure 2 a) due to the slow response of ocean CO 2 uptake to CDR. Lower atmospheric CO 2 , in turn, leads to a greater loss of land carbon in overshoot scenarios, compared with the reference scenario, as vegetation productivity decreases due to a diminished CO 2 fertilization effect when atmospheric CO 2 levels decline (Jones et al., 2016). However, ocean carbon uptake continues in the long term (after the CDR phase) and is higher in overshoot scenarios than in the reference nonovershoot scenario, due to its long response timescale. The overall carbon cycle response is qualitatively similar in the 1.5 and 2.5°C scenario groups (supporting information Figures S1 and S2). We acknowledge that the net amount of these changes is small (Figure 2), and internal variability is not represented in the UVic model. However, ongoing work with large initial conditions ensembles of comprehensive Earth System Models suggests that the effects of internal variability on cumulative land and carbon uptake are small and likely insignificant.

Restoration of Carbon Reservoirs to Reference Levels
To further investigate and quantify the differences in carbon reservoirs in response to overshoot scenarios, we calculated the change in carbon storage relative to the nonovershoot reference scenario "OS 0" in the year 2200 when CDR ends and cumulative CO 2 emissions reach the same level for all scenarios in the 2.0°C scenario group (Figure 2d). The difference between the given scenario and the reference scenario increases with increasing overshoot level, with the scenario "OS 300" having the largest difference (~20 Pg C) relative to the nonovershoot reference case "OS 0," despite cumulative CO 2 emissions being the same in all scenarios in the Year 2200 (Figure 2d). Similar results are found for the 1.5 and 2.5°C scenario groups. Time series of this imbalance (i.e., normalized changes of the carbon storage time series relative to the nonovershoot "OS 0" scenario) are shown in supporting information Figure S4.
Differences between each overshoot scenario and the nonovershoot reference scenario "OS 0" in the year when CDR ends and cumulative CO 2 emissions reach the same level in all scenarios (i.e., in the Year 2200 for the 2 and 2.5°C scenario groups, and in the Year 2250 for the 1.5°C scenario group; Figure 1 and supporting information Figure S1) are plotted as a function the cumulative CO 2 emissions overshoot in supporting information Figure S8. There is an approximately linear relationship between the amount of overshoot and the departure from the nonovershoot level for atmosphere, land, and ocean carbon storage, which holds across scenario groups. For temperature ( Figure S8b) the linear relationship holds only within each scenario group, due to slightly different scenario design (section 2; supporting information Figure S3).

Spatial Responses to Overshoot Scenarios
The lag in the ocean thermal response is evident in Figure 3 (left; panels a, c, and e), where scenarios with high level of overshoot (e.g., "OS 300"; Figure 3e) store more heat in the upper 1,000-2,000 m, particularly between 50°S and 50°N and at northern high latitudes, relative to the nonovershoot reference scenario in the Year 2200 when CDR ends and cumulative carbon emissions in all scenarios reach the same level. Conversely, the low-overshoot scenarios (e.g., "OS 150"; Figure 3a) show a negligible lag in thermal response and return to the nonovershoot temperature level almost immediately after the same level of cumulative emissions is reached. Ocean carbon storage in overshoot scenarios is lower at the surface of the tropical and midlatitude oceans and larger at the subsurface and in the Arctic relative to the reference scenario (Figures 3b, 3d, and 3f). The magnitude of those changes becomes more pronounced as the overshoot level increases. Lower carbon storage at the surface is consistent with the lower atmospheric CO 2 concentration in the Year 2200 in overshoot scenarios. Heat and carbon storage at the subsurface in the tropical ocean may be associated with the slow mixing of heat and carbon into the surface, possibly amplified by higher stratification, which prevents heat and carbon from returning to the atmosphere (Mathesius et al., 2015) stratification. The lag in the ocean carbon uptake persists on long timescales, with the high overshoot scenarios returning to the nonovershoot reference level beyond the Year 3000 (i.e., 800 years after the end of CDR; supporting information Figure S7). Regionally, differences in land carbon uptake relative to the reference nonovershoot "OS 0" scenario can be substantial (up to 20% in the "OS 300" scenario; Figure 4, right panels). However, since the changes are of 10.1029/2019EF001312 Earth's Future different signs (negative in the tropics, positive at high latitudes), the globally averaged change is small. Terrestrial carbon storage in the Tropics is lower in the high overshoot "OS 300" scenario than in the reference nonovershoot scenario "OS 0" due to lower net primary productivity in those regions in scenarios that entail high amounts of CDR (such as in the "OS 300" scenario). These changes are likely driven by a diminished CO 2 fertilization effect: As the atmospheric CO 2 concentration drops below the level in the reference scenario, the net primary productivity of tropical forests declines. In contrast, northern high-latitude regions show slightly enhanced terrestrial carbon uptake (north Europe and North America), likely due to warmer conditions at the time of overshoot, which could promote increased productivity for boreal forest (Arora & Boer, 2014) that continues to grow after emissions start to decline as a lagged response to earlier favorable conditions.

Implications for Carbon Budgets in Overshoot Scenarios
The near-path independence of surface air temperature (Figure 1d) arises despite the path dependence in the carbon cycle, as it is offset by path dependence in the thermal response of the ocean (i.e., lower levels of atmospheric CO 2 in high overshoot scenarios are compensated by a lagged ocean thermal response; Figure 3). This has important implications for carbon budgets, which have not been analyzed in depth under overshoot scenarios.

Earth's Future
Carbon budgets before and after an overshoot can be calculated using the relationship between warming and total cumulative CO 2 emissions (TCRE; Figure 5a) and are illustrated in Figure 5b for the 2.0°C warming target, and in supporting information Figures S5 and S6 for the two remaining scenario groups (for reaching the 1.5 and 2.5°C warming levels). The preovershoot carbon budgets were calculated in the year prior to exceeding the given warming target (i.e., 2.0°C in Figure 5) for the first time, while the postovershoot carbon budgets were calculated in the year when temperature falls below that target level of warming (i.e., 2.0°C in Figure 4. Differences in spatial distribution of surface air temperature (a, c, and e) and total carbon (b, d, and f) relative to the reference nonovershoot scenario "OS 0" in Years 2190-2200, in the 2°C scenario group, in scenarios "OS 70," "OS 150," and "OS 300"; (as labeled). Note: CDR ends in the Year 2200 when all scenarios reach the same level of cumulative CO 2 emissions. Total carbon (right panels) is calculated as a sum of soil and vegetation carbon storage on land and vertically integrated dissolved inorganic carbon in the ocean.

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Earth's Future Figure 5) for the first time after the overshoot. Our results show that the postovershoot carbon budgets are largely scenario independent for the low levels of overshoot considered here (up to 300 Pg C of overshoot).
This near-path independence of preovershoot and postovershoot carbon budgets occurs despite the hysteresis behavior in TCRE after an overshoot (Figure 5a), with the postovershoot TCRE curve above the preovershoot TCRE (Figure 5a). However, this hysteresis is a result of the scenario design rather than a property of the Earth system. In all scenarios considered here, the non-CO 2 forcing is held at a constant level. The high overshoot scenarios reach the 2°C target (for the first time, before the overshoot) early on, when the additional warming contribution from non-CO 2 forcing is smaller, and hence, the preovershoot carbon budget is greater than their postovershoot carbon budget. Conversely, low-overshoot scenarios reach the 2°C target (for the first time, before the overshoot) later on and thus experience a larger contribution of additional warming from non-CO 2 forcing, making the preovershoot carbon budget smaller. Also, the additional warming from non-CO 2 forcings is larger in the Year 2200 than at the time when the 2°C target is first reached (before the overshoot). Hence, the postovershoot carbon budgets can be expected to be smaller than the preovershoot budgets, as shown in Figure 5b. The postovershoot budgets differ little between scenarios (Figure 5b), reflecting the reversibility of global mean temperature at the time the same cumulative emissions are reached, and its pathway independence to CO 2 -only scenarios. In the absence of non-CO 2 Figure 5. Global mean temperature change as a function of cumulative CO 2 emissions (TCRE) (a), and the resulting carbon budgets for temperature stabilization at 2.0°C level (b). Gray bars in panel (b) indicate carbon budgets calculated before the overshoot, while colored bars indicate carbon budgets after the overshoot for each emission pathway, as labeled. Note: The gray vertical line in panel (a) is due to a slight decline in temperature once emissions are stopped (as in Figure 1d). Similar behavior occurs for the remaining scenarios once emissions are stopped, and equivalent vertical lines are colinear with the gray line. See supporting information Figures S5 and S6 for the other two scenario groups for 1.5 and 2.5°C warming levels.

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Earth's Future forcing, the preovershoot and postovershoot TCRE curves would likely be colinear, for the levels of overshoot considered here (up to 300 Pg C of overshoot).
These results are in contrast to the results from earlier studies (MacDougall et al., 2015;Zickfeld et al., 2016), that found considerable differences between the preovershoot and postovershoot carbon budgets in scenarios that entail much larger amount of overshoot (e.g., with reductions of the 2.0°C carbon budget by 785 Pg C when estimated from an RCP8.5 CO 2 -only scenario simulation that included an overshoot of approximately 4,000 Pg C; MacDougall et al., 2015). These studies, however, consider much larger levels of overshoot, for which surface air temperature is not fully restored at the time the reference CO 2 concentration is reached. Furthermore, plausible amounts of CO 2 emission reductions through implementation of CDR after an overshoot are estimated to range from about 10 to 1,000 Gt CO 2 (27-270 Pg C; Rogelj, Popp, et al., 2018;Rogelj, Shindell, et al., 2018) over a time frame until the Year 2100. Our study shows that for low levels of overshoot (up to 300 Pg C overshoot considered here), the postovershoot carbon budgets are nearly independent of the CO 2 emission pathway or the level of overshoot (Figure 5b). Consequently, the concept of a carbon budget remains a robust way for expressing emissions limits consistent with limiting the global mean temperature rise to the desired level in such scenarios. This path independence of overshoot carbon budgets for specific temperature limits when low levels of overshoot are considered arises despite path dependence of carbon cycle responses, which is compensated by the delayed response of the ocean resulting from inertia in the ocean heat uptake, discussed in the previous section.

Sources of Uncertainty
In our experimental design, aerosols and non-CO 2 greenhouse gases follow the historical trajectory and are held constant (at Year 2010 value) in the future years (section 2). In reality, non-CO 2 forcing would likely change along with CO 2 emissions, which would introduce nonlinearities to the TCRE framework, which is applicable predominantly for CO 2 -only emissions. The preovershoot and postovershoot budgets are likely to differ among scenarios if time-varying non-CO 2 forcings were considered.
While the primary effect of non-CO 2 forcing is their direct temperature effect (warming or cooling), non-CO 2 agents are also shown to affect the carbon sinks uptake rate through the warming effect (Gillett & Matthews, 2010;MacDougall & Knutti, 2016;Tokarska et al., 2018). Changes in temperature and carbon reservoirs are sensitive to the choice of non-CO 2 emission trajectory, which makes the application of a carbon budget for policy dependent on several aspects other than CO 2 emissions (MacDougall et al., 2015;Mahowald et al., 2017;Mengis et al., 2018;Tokarska et al., 2018). Aerosols also impact the carbon cycle through deposition (Mahowald et al., 2017).
Another source of uncertainty arises from permafrost carbon cycle feedbacks (not included in this version of the model), which are shown to be also dependent on the CO 2 emission pathway (MacDougall et al., 2015;Gasser et al., 2018) and may result in additional loss of soil carbon in the permafrost region, especially under scenarios that do not entail ambitious mitigation (Comyn-Platt et al., 2018;McGuire et al., 2018). Especially in the case of large overshoot of carbon budgets and thus a reliance on large amounts of CDR, additional release of permafrost carbon may be higher than in the case of low or nonovershoot scenarios (Gasser et al., 2018).
Our simplified scenario design is based on the assumption that CO 2 is removed from the climate system without considering a specific CDR technology, though, in principle, direct capture of CO 2 from the air would have a similar effect. Costs of implementation of CDR at a large scale and challenges and risks related to CO 2 storage or its utilization once it is captured are not considered here.

Discussion and Conclusions
Including the possibility of deliberate carbon dioxide removal in the carbon budgets framework, resulting in net-negative emissions, is often based on an assumption that exceeding the carbon budget by one ton of CO 2 implies that exactly one ton of CO 2 needs to be removed through CDR in order to achieve a given temperature target level. Based on such reasoning, delays in reducing emissions over the next decades could be compensated by CDR implementation in the future to remove the budget overshoot.

Earth's Future
Our results show that surface air temperature is reversible after an overshoot on global scales. However, the state of the carbon cycle is sensitive to whether a given temperature level is reached for the first time, or whether it is returned to after previously being exceeded (in case of overshoot). Compared to a reference nonovershoot scenario, more carbon is stored in the ocean (with implications for ocean acidification and ocean pH changes) and less carbon is stored on land when the same amount of cumulative emissions is reached. Thus, although the temperature response is largely path independent, carbon cycle responses do show a path dependency. Finally, if permafrost carbon cycle feedbacks were included, this path dependence would likely be more profound, due to pathway dependence of carbon budgets on CO 2 emissions from permafrost thaw (Gasser et al., 2018;MacDougall et al., 2015). Regional responses also show path dependence, with high overshoot scenarios taking a long time to return to the nonovershoot level after the net-negative CDR implementation. The ocean heat and carbon reservoir return to the nonovershoot level only centuries after the overshoot. It is worth noting that if CDR were accomplished partly or largely by enhanced rock weathering, there would be a cobenefit in mitigating ocean acidification impacts (Taylor et al., 2015), which would result in different ocean changes than discussed in this study.
The near-path independence of surface air temperature arises despite the path dependence in the carbon cycle, as it is offset by path dependence in the thermal response of the ocean. Hence, our results suggest that the concept of carbon budgets (i.e., a total amount carbon that can be emitted to meet a given temperature target) remains a valid metric for global temperature rise when considering overshoot scenarios with low levels of overshoot (up to 300 Pg C of overshoot considered here). Therefore, for stringent temperature target levels (such as the 1.5 and 2°C targets) and low-overshoot scenarios, carbon budgets are very similar between scenarios with and without overshoot of a temperature target. However, noticeable and potentially important differences arise in the state of the carbon cycle, especially for higher levels of overshoot, with implications for climate change impacts that are directly related to atmospheric CO 2 (e.g., ocean acidification).

Data Availability Statement
The UVic ESCM model is available online at: http://terra.seos.uvic.ca/model/. Model data used in this study is available at: http://data.iac.ethz.ch/Tokarska_et_al_2019_Overshoot_UVicESCM/.