Testing GWP* to quantify non-CO2 contributions in the carbon budget framework in overshoot scenarios

Anthropogenic climate change is caused by emissions of a range of greenhouse gases (GHGs) that widely vary with respect to their radiative efficiency, atmospheric lifetime, and therefore impact on climate system¹. To deal with GHGs of different characteristics on a common scale, the impact of non-CO₂ GHG emissions on climate is sometimes expressed in terms of CO₂ equivalent (CO₂-eq) emissions. In doing so, the emissions of non-CO₂ GHGs are converted to CO₂-eq emissions using GHG emission metrics². The most widely used metric is the 100-year GWP (GWP100), which is defined as the time-integrated radiative forcing over 100 years following a pulse emission of one ton of a GHG divided by the corresponding quantity of one ton of CO₂^3,4. Similarly, other metrics often considered are the GWP20 (with a 20-year reference time period) and the 100-year Global Temperature change Potential (GTP100), which is defined as the global temperature change following a pulse emission of one ton of a GHG after the reference time period of 100 years divided by the corresponding quantity of one ton of CO₂⁵.

International climate treaties, such as the Kyoto Protocol and the Paris Agreement, commonly address multiple GHGs with the use of GWP100 to mitigate climate change cost-effectively^6,7,8,9,10. However, when GWP100 is used to aggregate GHG emissions on a common scale of CO₂, the temperature impacts of different GHGs are not accurately captured owing to their diverse behavior in the atmosphere. GHGs fall into two categories: short-lived climate pollutants (SLCPs) and long-lived climate pollutants (LLCPs)¹¹. There have been decades-long efforts to provide a more accurate emission metric in terms of the consistency with radiative forcing or temperature^{5,12,13,14,15,16}. GWP100 is not a metric that intends to capture the contrasting temperature responses of LLCP and SLCP emissions^2,11,17,18.

CO₂, an LLCP, once emitted, will largely remain in the atmosphere (the so-called airborne fraction) with slightly declining concentrations for centuries to millennia^19,20,21. The slightly declining concentrations lead to near-stable warming due to the thermal inertia of the ocean²². It is thus possible to relate cumulative CO₂ emissions to the temperature rise in an almost linear fashion through the widely used transient climate response to cumulative emissions (TCRE)^23,24. This provides the basis for the ‘remaining carbon budget approach’, which is usually used for CO₂.

In contrast, CH₄, an SLCP, is mainly removed from the atmosphere due to atmospheric chemistry processes on faster time scales and thus has a smaller turnover time in this reservoir. It is currently an active research question how to determine the contribution of CH₄ and other SLCPs in the “carbon budget” framework^{25,26,27,28,29}. Unlike CO₂, a pulse emission of CH₄ or other SLCPs does not result in near-stable warming; a stable warming is the consequence of a sustained or no increase in the rate of CH₄ emissions (or other SLCP emissions). In other words, a one-off emission of CO₂ is approximately comparable to a sustained increase in the emission rate of CH₄ in terms of the temperature outcome^30,31.

A new metric, GWP*, that captures this fundamental relationship has been proposed, allowing an integration of SLCPs into the carbon budget framework¹⁶. Starting from the work of Allen et al.¹⁵, the GWP* metric was built by considering a 20-year time interval to assess the rate of changes in SLCP emissions. Cain et al.¹⁶ further improved GWP* by incorporating the small multi-centennial warming effect arising from SLCP emissions, with an additional term representing the “stock” effect (comparable to the behavior of CO₂) on top of the term for the “flow” effect originally proposed. Therefore, GWP* can be defined by combining the effects from the emission itself (stock) and the change in emissions (flow), with the coefficients s and r, respectively, in Eq. (1). CO₂-eq emissions (({E}_{{CO}2e})), which are referred to as CO₂-warming equivalent emissions when based on GWP* in Cain et al.¹⁶, of SLCP emissions can be calculated as follows:

Here, the term ∆E_SLCP represents the change in the rate of SLCP emissions with respect to the preceding time interval ∆t, whereas E_SLCP is the emission of the SLCP for that year, and GWP_H is the GWP value of the SLCP over the time horizon H. Originally, by fixing the coefficients r and s to 0.75 and 0.25, respectively, it was possible to ensure a good approximation of CH₄ impacts on temperatures over a range of emissions scenarios (RCP2.6, RCP4.5, and RCP6.0) until 2100^16,28. Subsequently, a further modification was put forward to scale the right-hand side of Eq. (1) to ensure the forcing equivalency³². Finally, the GWP* equation adopted in the Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR6)^33,34, which is used in our analysis, is the following form:

({E}_{100}(t)) denotes the emissions of an SLCP multiplied with its GWP100 value. The first and second terms on the right-hand side of Eq. (2) capture the stock and flow effects, respectively, with a much higher weight given on the flow effects. Equation (2) is a special case of Eq. (1) with the time interval (triangle t) of 20 years (also modified with a scaling factor of 1.13 discussed above). Note that this formula is based on GWP100 values in Table 8.A.1 of the IPCC AR5.

Although GWP* is often referred to as a metric, GWP* is not a single value: it is rather a more elaborated way of applying existing GWP100 values. In the traditional approach of using GWP100, the value of GWP100 for the non-CO₂ GHG of interest (for both SLCPs and LLCPs) is simply multiplied with the non-CO₂ GHG emissions to calculate CO₂-eq emissions. In the GWP* approach, however, the value of GWP100 is multiplied with the emissions and the change in the rate of emissions for a non-CO₂ GHG (for SLCPs only) as in Eq. (1). This approach allows for a more accurate translation of the temperature response of SLCP emissions to that of CO₂ emissions when calculating CO₂-eq emissions.

Scenarios that exceed the 1.5 °C or 2 °C warming target level temporarily (so-called overshoot scenarios), in which temperatures rise and then fall, are becoming increasingly relevant in the face of inadequate emission reduction commitments in the policy arena. Although GWP* was proposed with the aim of obtaining a more accurate estimate of temperature changes from cumulative CO₂-eq emissions, the performance of GWP* with declining emissions in overshoot scenarios was not fully assessed in IPCC AR6³⁵ and, at the moment of writing this manuscript, has yet to be systematically tested in any peer-reviewed papers.

Previous studies have relied either on specifying the non-CO₂ warming response for a given scenario and determining the CO₂ contribution as a residual^27,36 or on assuming and exploiting a scenario-dependent relationship between cumulative CO₂ emissions and non-CO₂ climate forcers³⁷. Another recent approach relies on the concept of CO₂ forcing equivalent (CO₂-fe) emissions to consider the contributions of non-CO₂ climate forcers^29,38. However, implications of using GWP* to determine non-CO₂ contributions to the carbon budget for overshoot scenarios have not been systematically explored.

This is particularly relevant to pathways achieving the 1.5 °C warming target, as around 90% of the scenarios considered in the IPCC Special Report on 1.5 °C (SR15) are either overshoot scenarios for the 1.5 °C target or, more generally, peak and decline scenarios³⁹. Similarly, a large majority of low emissions scenarios developed from Integrated Assessment Models have been peak and decline scenarios²², with more recently developed stabilization scenarios with net-zero CO₂ emissions being an exception^40,41. Lastly, given the current emission trajectory and the recent anomalous warming, overshooting the 1.5 °C target level is projected to be imminent^42,43.

Moreover, there is a growing interest in the Earth system modeling community to simulate very long-term overshoot scenarios over a few to several centuries (up to 2300); such very long-term overshoot scenarios are under development in two Horizon Europe projects, RESCUE and OptimESM. GWP* could be considered as a convenient tool to assess the temperature implication of CH₄ emissions of such scenarios. Whereas GWP* may not be originally intended for such use, considering its calibration range to 2100, users of GWP* are not necessarily aware of the limitations associated with the calibration range clearly. It is therefore useful to assess how well GWP* serves its purpose over multi-century time scales with different lengths and magnitudes of overshoot.

Therefore, in this paper, we aim to tackle the following scientific questions: Does the GWP* metric adequately ensure equivalent temperature responses when it is used to convert SLCP emissions to CO₂-eq emissions under multi-century scenarios with different lengths and magnitudes of temperature overshoot? Can this metric be used to include non-CO₂ contributions in the carbon budget approach for overshoot scenarios?

Testing the temperature equivalency of GWP* under SSPs

We first used a simple Impulse Response Function (IRF) model in IPCC AR5¹ to simulate the temperature effect of CH₄ emissions under the SSP1-1.9, SSP1-2.6, SSP4-3.4, SSP4-6.0, SSP5-3.4-OS, and SSP5-8.5 scenarios, up to the year 2300. We also calculated the temperature effect of the CO₂-eq emissions that were converted from the CH₄ emissions using different emission metrics, such as GWP100 and GWP*. By comparing the two results, we assessed the extent to which the temperature pathways are maintained or altered with the use of emission metrics in temperature calculations under different SSPs (i.e., “temperature equivalency”).

Our results in Fig. 1 show that GWP100 ensures an imperfect but reasonably good temperature equivalency approximately up to the point of peak CH₄ emissions; however, its accuracy decreases substantially when CH₄ emissions decline, a well-known issue with GWP100. It is also shown that GWP100 performs better than GWP20 and GTP100 in terms of the temperature equivalency. GWP20 and GTP100 are illustrative of emphasizing and de-emphasizing, respectively, the temperature effect of CH₄ emissions relative to GWP100^44,45,46. Thus, the results from GWP20 and GTP100 are opposite each other, relative to those from GWP100.

a–f represents different SSP scenarios. Dotted black lines indicate CH₄ emissions of each SSP. Thick black lines give the temperature response calculated with the IRF for CH4 emissions without using any emission metric. Thin solid lines show the temperature response calculated with the IRF for CO₂-eq emissions converted from the CH₄ emissions using different metrics such as GWP100 and GWP*. Only the CH₄ emissions or the CO₂-eq emissions converted from CH₄ emissions are considered for temperature calculations here. The temperature calculation is based on the IRF. This figure mimics but extends the presentation of Figure 7.22 of IPCC AR6 WGI, which shows the outcome only till 2100.

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We further found that GWP* gives a nearly perfect temperature equivalency approximately until the end of the century under the range of emission scenarios. This result is rather expected, as GWP* is calibrated to do so under the three RCP scenarios. A notable exception is, however, that when CH₄ emissions drop sharply, GWP* overestimates the cooling effect, as shown most evidently under SSP5-3.4-OS (Fig. 1e). This behavior reflects the fact that none of the RCP scenarios used to calibrate GWP* contain a large overshoot as in SSP5-3.4-OS. It should also be emphasized that, compared to GWP100, GWP* clearly shows superior performances for reproducing the temperature (Table S1). Over the longer run, however, the temperature equivalency provided by GWP* no longer holds. The cooling effect of CH₄ is underestimated or shown even as a slight warming with the use of this metric when CH₄ emissions decrease (in all SSPs). From the physical perspective, this is due to the fact that the “stock” parameter, as defined in Cain et al., (2019), emphasize too much the century-timescale response to earlier methane emission increases, leading to an overestimation of temperatures beyond 2100. These findings are robust since the deviations discussed here are generally confirmed by two other modeling approaches (Fig. S1). Our findings raise the question regarding whether the use of GWP* could be adequate for accounting for the contribution of CH₄ emissions in the carbon budget framework when it is applied for overshoot scenarios, in particular beyond 2100.

We then explored the sensitivity of the foregoing results to the parameters used to define GWP*. We first investigated the sensitivity with respect to the time interval considered between two pulse emissions in the flow term of Eq. 2 (20 years by default, Fig. S2a). By minimizing the Sum of Squared Errors (SSE) between the temperatures calculated from original CH₄ emissions and that from CO₂-eq emissions based on the modified GWP* for all the scenarios in the period 1750-2100, we found the optimal time interval to be 22 years, which is close to the commonly used value of 20 years. Furthermore, we investigated the sensitivity with respect to the formulation of the flow term in the GWP* equation by substituting the default flow term taking the change in emissions 20 years apart (Eq. 2) with an alternative flow term taking the average change in emissions over the previous 5, 10, 20, 30, 40 or 50 years (Fig. S2b). Our results show that the use of average change in emissions over the previous 40 years or longer leads to the best temperature equivalency. We further found that the use of average change in emissions over the previous 40 years yielded a slightly better temperature equivalency than the use of emissions 20 years apart. We can explain this outcome by the fact that the use of average emissions over 40 years, compared to the use of two end-point emissions 20 years apart, more closely resembles the exact CO₂-eq emissions that precisely reproduce the warming from original CH₄ emissions (i.e., the Linear Warming Equivalence emissions)⁴⁷. However, even when the GWP* equation is modified in several different ways, the temperature misfit after the rapid drop in CH₄ emissions persists (Fig. S3).

Using GWP* to quantify the overshoot CO₂-eq budget

Our analysis based on the SSP scenarios has shown that the temperature equivalency of GWP* can be influenced by the rate of CH₄ emission reductions. To gain more insights, we performed a further analysis based on scenarios with different overshoot lengths and magnitudes. To generate a range of overshoot scenarios, we used the Aggregated Carbon Cycle, Atmospheric Chemistry, and Climate (ACC2) model^9,22,48. ACC2 is a simple climate model⁴⁹ that resolves many GHGs and aerosols, accounting for primarily important nonlinearities in the global earth system such as CO₂ fertilization of the land biosphere, saturation of ocean CO₂ uptake under rising atmospheric CO₂ concentrations, and climate-carbon cycle feedback. The interaction between CH₄ and tropospheric ozone through the OH chemistry is parameterized in ACC2. The temperature is calculated by using a land-ocean energy balance model coupled with a heat diffusion model (see “Methods” for details). The simple climate model ACC2 can be further coupled with a climate mitigation module consisting of a set of marginal abatement cost functions^9,50 to calculate least-cost emission pathways for a specified climate target. ACC2 then works as an optimizing climate-economy model, and we took advantage of this model feature to systematically generate overshoot pathways for our analysis. We also took advantage of the more detailed process representations of ACC2, relative to the IRF model, while simplicity is maintained in ACC2 for performing sensitivity simulations.

We use ACC2 to generate a range of pathways that reach the 1.5 or 2 °C target level after overshoot of varying lengths and magnitudes. We calculate least-cost emission scenarios to keep the temperature below a target level throughout the simulation period (temperature stabilization pathways). If we apply the temperature target only after a certain year, the model yields least-cost emission scenarios that lead to temperature overshoot pathways. In this case, the temperature exceeds the target level before that year, since it is less costly to invest in mitigation efforts later in the cost-effectiveness framework as also pointed out by previous studies^51,52. The length and the magnitude of the temperature overshoot are determined by ACC2’s internal cost minimization reflecting assumptions on the maximum rates of emission abatements and the discounting rate. We varied the target years between 2060 and 2180 with an interval of 10 years and selected appropriate target years to be tested for each temperature goal (1.5 °C and 2 °C). For each target year, we further varied the discount rate to 2% and 6%, compared to the original 4% by default, to generate different overshoot magnitudes, while keeping the same overshoot length. Emissions of all climate forcers other than CO₂ and CH₄ are assumed to follow SSP1-1.9 and SSP1-2.6 in the case of the 1.5 and 2 °C targets, respectively.

For each pathway, we calculated the “overshoot CO₂-eq budget” which is defined as the cumulative net CO₂-eq emissions (GtCeq) from 2020 till the year when the temperature stabilizes at or returns to the 1.5 or 2 °C warming target level after temperature overshoot. The overshoot CO₂-eq budget discussed in this paper implicitly assumes that positive and negative emissions can be equally balanced, although the validity of this assumption will be explored later in this paper. The carbon budget for this time frame is referred to as the threshold return budget elsewhere^25,53, but we include non-CO₂ contributions in the threshold return budget using the emission metric approach: GWP* for SLCPs and GWP100 for LLCPs. For the sake of analysis, our overshoot CO₂-eq budget considers only CO₂ and CH₄, unless stated otherwise, which are our focus but are largely representative of the entire CO₂-eq budget because these two gases are usually most important for climate change mitigation. We present the results calculated from ACC2 emulating IPSL-CM6A-LR in the main paper. Qualitatively similar results can be found from ACC2 emulating other ESMs, as in Figs. S4 and S5 for CNRM-ESM2-1 and MIROC-ES2L.

Multi-gas optimization experiments

We first performed multi-gas optimization experiments (Fig. 2), where CO₂ and CH₄ emission pathways are optimized simultaneously using the ACC2 setup to calculate the least-cost pathways, while all other gases and pollutants are assumed to follow SSP1-1.9 and SSP1-2.6 in the case of the 1.5 °C (Fig. 2a, c) and 2 °C target (Fig. 2b, d), respectively. Panels (a) and (b) of Fig. 2 show the overshoot CO₂-eq budget (with CH₄ converted via GWP*) for different 1.5 °C and 2 °C stabilization years, respectively, with ranges of overshoot magnitude and length. For both the 1.5 °C and 2 °C target year simulations, we found that the longer the temperature overshoot is, the larger the overshoot CO₂-eq budget is. Furthermore, for the same overshoot length, the higher the overshoot is, the larger the overshoot CO₂-eq budget is (the 2 °C case with the 2060 target year is an exception because the target lengths and magnitudes are not clearly separated) (Table 1). The increasing total overshoot CO₂-eq budget with increasing overshoot lengths and magnitudes reflects both the model behavior of ACC2 and the conversion of CH₄ emissions to CO₂-eq emissions via GWP*, which will be disentangled in the analysis presented in the next section.

a, c shows the overshoot CO₂-eq budget and temperature pathways for the 1.5 °C scenarios, whereas (b, d) shows those of the 2 °C scenarios. Both CO₂ and CH₄ emission pathways were optimized using ACC2 (emulating IPSL-CM6A-LR) for each target level and year with different discounting, with all other gases and pollutants following respective SSPs (SSP1-1.9 and SSP1-2.6 for the 1.5 and 2 °C target level, respectively). The figure shows the results of a combination of four different overshoot lengths and three different overshoot magnitudes for each temperature target level. The overshoot CO₂-eq budgets presented in this figure account for contributions from CO₂ and CH₄. The contributions of CH₄ in the overshoot CO₂-eq budget are determined using GWP*. Legends in panels c) and d) represent the duration (years) and magnitude (°C) of temperature overshoot relative to the 1.5 °C or 2 °C target level, respectively.

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The contribution of CH₄ in the overshoot CO₂-eq budget becomes particularly important for 1.5 °C pathways, in which the total overshoot CO₂-eq budget is so small that the contribution from CH₄ can even determine the sign of the total overshoot CO₂-eq budget. This indicates that how to account for CH₄ contributions to the overshoot CO₂-eq budget is crucial for mitigation pathways aiming at the 1.5 °C target level. The overshoot CO₂-eq budgets calculated with our default version of ACC2 are generally low because this version of ACC2 was configured to IPSL-CM6A-LR, which exhibits a high climate sensitivity (Table S4 and Figs. S10 and S11). The overshoot CO₂-eq budget is in fact already negative in about a half of the cases for the 1.5 °C target level, specifically for scenarios with target years in 2090 and 2120. These negative overshoot CO₂-eq budgets reflect deep CO₂ mitigation that is required to compensate for the limited CH₄ mitigation (Fig. S6 for simulations for the target year of 2120 with three different overshoot profiles), as well as the unmasked warming from decreasing emissions of air pollutants (following SSP1-1.9 for 1.5 °C target scenarios). In the remaining cases with high and long overshoots, on the other hand, some more emissions are still left in the budget.

Single-gas optimization experiments

We further performed single-gas optimization experiments (Fig. 3). In this case, only the emission pathway of one of the two gases (CO₂ or CH₄) is optimized to reach the target temperature with lowest cost, while the emission pathways of all other gases and pollutants are kept at the respective SSP scenario (SSP1-1.9 or SSP1-2.6) corresponding to the target temperature (1.5 or 2 °C target level). This experiment is useful to isolate the relative contribution of CO₂ and CH₄ to the overshoot CO₂-eq budget. That is, unlike the previous experiments, in which emission reductions of CO₂ and CH₄ can be traded to yield a least-cost solution, emission reductions of only one gas can change with varying overshoot lengths and magnitudes in these experiments.

a, b show the overshoot CO₂-eq budget for CO₂ and CH₄ optimization simulations, for different scenarios. The overshoot CO₂-eq budgets in panel a) represent only the contributions from CO₂. Those in (b) represent contributions from CH₄. c, d describe the emissions pathways for the 1.5 °C and 2 °C for CO₂ simulations. e, f describe the emissions pathways for the 1.5 °C and 2 °C for CH₄ simulations. Either CO₂ or CH₄ emission pathways were optimized using ACC2 (emulating IPSL-CM6A-LR) for each target level and year with different discounting, with all other gases and pollutants following respective SSPs (SSP1-1.9 and SSP1-2.6 for the 1.5 and 2 °C target level, respectively). The contributions of CH₄ in the overshoot CO₂-eq budget were determined using GWP*. Legends in (c–f) represent the duration (years) and magnitude (°C) of temperature overshoot relative to the 1.5 °C and or 2 °C target level, respectively.

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As for Fig. 2, panels a) and b) of Fig. 3 report the overshoot CO₂-eq budget calculated for each simulation of the single-gas optimization experiments, for CO₂ cumulative emissions and CO₂-eq cumulative emissions (CH₄ converted with GWP*), respectively. In the case of CO₂-only optimization experiment (Fig. 3a), the longer the temperature overshoot is, the larger the overshoot CO₂-eq budget is. This finding is in contrast to the general belief that the carbon budget is reduced with increasing overshoot lengths^54,55. We found qualitatively similar results for the CH₄-only optimization experiment as well (Fig. 3b). Caution must be taken in drawing conclusions from this, as it could be interpreted that a late mitigation is preferred compared to early mitigation actions. It should be noted that this finding is dependent on model assumptions, such as CO₂ fertilization effect (Figure S8), and model characteristics as discussed in the next section.

The results for overshoot magnitudes provide a further insight: the higher the overshoot is for the same overshoot length, the larger the overshoot CO₂-eq budget is from the CO₂-only optimization experiment, but the smaller the overshoot CO₂-eq budget is from the CH₄-only optimization experiment (in most cases). The results of the CO₂-only optimization experiment reflect the dynamic behavior of ACC2 (analyzed in the next section); those of the CH₄-only optimization experiment, on the other hand, not only reflect the model’s dynamic behavior but also the conversion of CH₄ emissions to CO₂-eq emissions via GWP*. In the analysis that follows, we further delve into this finding separately in the ramp-up and ramp-down periods of overshoot scenarios. Note that CH₄ contributions to the overshoot CO₂-eq budget of the CH₄-only optimization experiment are higher in the 1.5 °C target case than those in the 2 °C target case, which appears counterintuitive first since a lower temperature target usually implies stricter emission limits. This phenomenon is due to the higher relative contributions of the prescribed CO₂ baseline scenario to the total radiative forcing in SSP1-2.6 (used for the 2 °C target simulations) compared to those in SSP1-1.9 (used for the 1.5 °C target simulations).

TCRE of the ramp-up and ramp-down phases of overshoot scenarios

To gain insight into how the model’s dynamic behavior influences the overshoot CO₂-eq budget in different phases of overshoot scenarios, we looked into the TCRE separately for the ramp-up and ramp-down phases. We computed TCRE, for each single-gas optimization experiment, as the ratio of the temperature change over the cumulative CO₂-eq emissions for the sum of three gases (CO₂, GWP*-weighted CH₄, and GWP100-weighted N₂O). We distinguish TCRE between the ramp-up (from year 1750 to the year of peak emissions) and the ramp-down phases (from the year of peak emissions to the year when the temperature target is met after overshoot) as TCRE+ and TCRE−, respectively. The current literature does not provide sufficient clarity regarding the extent to which the linear relationship between cumulative CO₂ emissions and temperature changes holds in case of negative emissions⁵⁶. As reported in Chapter 5 of IPCC AR6 WGI, there is no clear agreement among models for the relationship between TCRE+ and TCRE−³⁵. Figure 4 shows the results from both the CO₂-only optimization and CH₄-only optimization experiments for the 1.5 °C target. In Fig. 4a, c, cumulative emissions consider only one gas (CO₂ or GWP*-weighted CH₄), but the warming is calculated from the emissions of all gases, indicating that the cumulative emissions are not necessarily consistent with the simulated warming. In Fig. 4b, d, on the other hand, cumulative emissions consider the three major gases, which is thus approximately consistent with the simulated warming, but the effects from individual gases cannot be isolated. The values of TCRE+ and TCRE− (°C/1,000 GtCeq) reported in panels b) and d) represent the mean of all TCRE+ and TCRE− values from the scenarios presented in the panels. The general indication is that TCRE− is larger than TCRE+, which is consistent with the finding from Fig. 2 and Fig. 3: the longer or higher the temperature overshoot is, the larger the overshoot CO₂-eq budget is. An exception is the decreasing TCRE− with increasing overshoot magnitudes from the CH₄-only optimization experiments (Fig. 4d). By taking the average of TCRE− across the scenarios with low, medium, and high overshoot (i.e., for discount rates of 2%, 4%, and 6%), the TCRE− values are indeed decreasing with increasing overshoot magnitudes, which are 3.1, 2.9, and 2.7, respectively (mean value of 2.9, as reported in Fig. 4d). This can be confirmed with our earlier finding that the overshoot CO₂-eq budget decreases with increasing overshoot magnitudes in the CH₄-only optimization experiment owing to the use of GWP*.

a, b show the relationship between the temperature change and the cumulative emissions considering only CO₂ and the sum of CO₂, CH₄, and N₂O, respectively, for the period 2020-2200 from the CO₂-only optimization experiment. c, d show the relationship between the temperature change and the cumulative emissions considering only CH₄ and the sum of CO₂, CH₄, and N₂O, respectively, also for the same period but from the CH₄-only optimization experiment. CH₄ emissions were converted to CO₂-eq emissions with GWP*, whereas N₂O emissions were converted to CO₂-eq emissions with GWP100. Arrows indicate the direction of time.

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The relatively large TCRE− from the CO₂-only optimization experiment can be further confirmed with the Zero Emission Commitment (ZEC) parameters of this model. Following the ZEC Model Intercomparison Project (ZECMIP) framework described in Jones et al.⁵⁷, we performed the B.1, B.2, and B.3 experiments as in MacDougall et al.⁵⁸ to compute the ZEC parameters (Fig. S9). These three experiments use 100-year bell-shaped CO₂ emissions of 1000 GtC, 750 GtC, and 2000 GtC, respectively, starting from pre-industrial conditions, aiming to assess the model-specific climate inertia in the decades after a complete cessation of emissions. We report the estimates of ZEC25, ZEC50, and ZEC90, representing the temperature change respectively 25, 50, and 90 years after the start of zero emissions. Table 2 shows that all three ZEC parameters are negative for ACC2, which supports the large TCRE− relative to TCRE+. We further note that the ZEC parameters for different configurations of ACC2 (emulated to CNRM-ESM2-1 and MIROC-ES2L used for sensitivity analyses) also present negative values (Fig. S9).

It should be emphasized that there is generally no consensus for the relationship between TCRE+ and TCRE−, among models, as demonstrated by previous studies^{54,55,59,60,61,62}. The lack of consensus is also evident from the analysis of 1pct-CO₂-cdr idealized experiment⁵⁶, where MIROC-ES2L and CanESM5 report larger TCRE− than TCRE+ whereas UKESM1-0-LL and ACCESS-ESM1-5 show smaller TCRE+ than TCRE− (Fig. S12).

This study examined how the emissions of non-CO₂ GHGs, in particular CH₄, can be accounted for in the overshoot CO₂-eq budget framework. A key requirement for including non-CO₂ contributions in the CO₂-eq budget framework is to ensure the temperature equivalency. That is, non-CO₂ emissions should be converted to CO₂-eq emissions in such a way that the implied temperature outcome remains the same. Recently, the GWP* approach has received attention; however, this approach has not been extensively evaluated for overshoot scenarios or, more generally, peak and decline scenarios in a long time scale.

Our analysis using the SSP scenarios confirmed that GWP* does a reasonably good job of ensuring a temperature equivalency for a range of pathways up to 2100 when it is used to convert CH₄ emissions to CO₂-eq emissions. An exception is that when the temperature drops sharply in the ramp-down phase of overshoot, the use of GWP* leads to an overestimation of cooling. Over a longer time scale up to 2300, when the temperature decreases moderately, the use of GWP* can lead to an underestimation of cooling or indicate a slight warming instead. These findings are related to the fact that GWP* was tuned to a certain set of pathways up to 2100. Our results indeed showed that the deviation tends to become more evident after 2100. These findings suggest that GWP* should be cautiously used for scenarios with strong decline in temperatures and/or longer-term peak and decline scenarios beyond 2100.

Nevertheless, GWP* is still far superior to the standard and most widely used metric GWP100 in terms of the temperature equivalency under all scenarios. We have attempted to improve GWP* by modifying the parameters in the GWP* equation, but this did not result in any substantial improvement in temperature equivalency.

It should be noted that the IRF was used for temperature calculations here in order to be consistent with the underlying methods used to derive metric values in AR5 and the GWP* formula in AR6. When ACC2 and FaIR are used for temperature calculations, our general findings still hold; however, the quantitative results become different particularly from ACC2, reiterating the fact that the GWP* formula is model-dependent or reflects certain model assumptions, a point that deserves more attention.

Our study further investigated how the overshoot CO₂-eq budget for the 1.5 and 2 °C target levels, including the contribution of CH₄ based on GWP*, can be affected by the temperature overshoot of varying lengths and magnitudes. We generated a set of scenarios to reach the representative temperature target levels with overshoot of different lengths and magnitudes through cost-effective calculations of ACC2 configured to several different state-of-the-art ESMs. Our results generally showed that the overshoot CO₂-eq budget increases with the length and magnitude of overshoot, except under CH₄-varying scenarios with different overshoot magnitudes. This finding is mainly related to the model property of ACC2 that it takes larger cumulative CO₂ emissions in absolute terms to produce a given amount of warming than it takes to produce the same amount of cooling in absolute terms. In other words, TCRE+ is smaller than TCRE− in ACC2, which is also consistent with the negative ZEC, found in our analysis for ACC2. Importantly, this finding is modified by the conversion of CH₄ emissions into CO₂-eq emissions via GWP*. The application of GWP* on CH₄ has an effect of decreasing the apparent overshoot CO₂-eq budget with increasing overshoot magnitudes.

To summarize, the use of GWP* to include CH₄ in the CO₂-eq budget framework in case of overshoot scenarios works in most scenario until 2100, except for cases with a rapid decline in CH₄ emissions. Over a longer run beyond 2100, the use of GWP* can overestimate the warming effect of CH₄ emissions and consequently may lead to an overestimation of CO₂-eq budget. It should however be noted that these results depend on the model configuration used for the analysis, requiring further studies based on different modeling approaches, including uncertainty analyses of Earth system feedbacks.

Description of three reduced-complexity models

Our analysis used three different reduced-complexity models as follows in the order of increasing complexity: the Impulse Response Function (IRF) model of IPCC AR5¹, the Finite Amplitude Impulse Response (FaIR) model^63,64, and the Aggregated Carbon Cycle, Atmospheric Chemistry, and Climate (ACC2) Model^9,22,48.

First, the IRF model is a simplified representation of the atmospheric response to a pulse emission of CO₂. It approximates the fraction of emitted carbon that remains in the atmosphere over time²¹, to estimate the temperature impact of GHG emissions according to the radiative efficiencies of the single GHG. The IRF model used in our study is derived from the latest estimates summarized in the IPCC AR5. It is employed to estimate the temperature response of CO₂-eq emissions derived from CH₄ using various conversion metrics.

Second, the FaIR v1.3 has been extensively used for assessing the temperature impacts of different emissions pathways and mitigation scenarios^16,27,38,65. FaIR has a simple representation of the carbon cycle and accounts for nonlinear feedback such as the temperature and saturation dependency of land and ocean carbon sinks. The model calculates both CO₂ and non-CO₂ GHG atmospheric concentrations by tracking their time-integrated airborne fraction from their initial emissions. From concentrations, Effective Radiative Forcing (ERF) for thirteen forcings, including land-use change, tropospheric and stratospheric O₃ as well as other GHG are computed; finally the temperature change is calculated as the sum of its slow and fast components, representing temperature changes from a response to forcing from the upper ocean and the deep ocean, respectively.

Third, ACC2 is a simple climate-economy model composed of four modules: i) carbon cycle, ii) atmospheric chemistry, iii) physical climate, and iv) economy. The carbon cycle module is made of three ocean boxes, a coupled atmosphere-mixed layer box, and four land boxes. The physical climate module is the land-ocean energy balance model coupled with the heat diffusion model DOECLIM, which is used to calculate the temperature response to radiative forcing⁴⁸. The radiative forcing is calculated for the following direct and indirect climate forcers: CO₂, CH₄, N₂O, O₃, SF₆, 29 halocarbon species, OH, VOC, CO, NO_x, stratospheric H₂O, sulfate aerosols, and carbonaceous aerosols (both direct and indirect effects). The changes in CH₄ and N₂O concentrations are given as a function of their atmospheric concentrations, the sum of anthropogenic and natural emissions, and their lifetime. ACC2 accounts for nonlinear feedbacks such as the saturation of ocean CO₂ uptake under rising CO₂ concentration, CO₂ fertilization of the land biosphere, and increasing heterotrophic respiration of the land biosphere with rising temperatures. Furthermore, uncertain parameters in ACC2 are calibrated with a Bayesian approach^13,22. The model is developed in the GAMS programming language and numerically solved using CONOPT3 and CONOPT4, two nonlinear optimization solvers included in the GAMS software package. When ACC2 is used as a simulator, the first three modules are used to calculate temperature changes based on prescribed emissions scenarios. On the other hand, when ACC2 is used as an optimizer, all four modules, including the economy module consisting of the marginal abatement cost curves⁵⁰ for global CO₂, CH₄, and N₂O emissions, are used to derive least-cost emission pathways for a given climate target. ACC2 has participated in model intercomparison projects^21,49 and has been subject to comparison with other models in other studies⁶⁶. This model has been applied to analyze pathways to achieve the targets of the Paris Agreement^9,22,67 including GHG removal options⁶⁸. The performance of ACC2 configured for this paper with three EMSs was benchmarked against other simple climate models under commonly used scenarios RCP2.6 and RCP4.5 in the emission-driven mode⁴⁹ (Figs. S10 and S11).

Carbon cycle parameters in ACC2

As in many simple climate and carbon cycle models, a steady state is assumed for the carbon cycle in the preindustrial period⁶⁹. In 1750, the preindustrial start year of ACC2, the Net Primary Production (NPP) was assumed to be constant and in balance with the heterotrophic respiration. The perturbation of the NPP and the resulting changes in heterotrophic respiration are described by the four-reservoir land box model. A change in atmospheric CO₂ concentration logarithmically influences the NPP (i.e., CO₂ fertilization), the strength of which is scaled with the BETA factor. On the other hand, the heterotrophic respiration is exponentially dependent on the temperature as parameterized through a Q10 factor, at the rate of which the heterotrophic respiration increases with a temperature increase of 10 °C.

The governing equation for each of the four land reservoirs is:

Here, the first term on the right-hand side of Eq. (3) represents added carbon to the reservoir l: the increase in NPP of year t ((delta {f}_{{NPP}}left(tright))) combined with the preindustrial constant NPP (({bar{f}}_{{NPP}}^{{pre}})). ({A}_{{ter},l}) and ({tau }_{{ter},l}) represents the coefficients and the overturning time constants of the IRF describing the temporal evolution of biomass for each reservoir. The second term represents released carbon from the reservoir l: the release of carbon from both preindustrial (({bar{c}}_{{ter},l}^{{pre}})) and fertilized storage (({c}_{{ter},l}left(tright))), which can be enhanced under warming through (qleft(tright)), the Q10 factor, accounting for climate-carbon cycle feedbacks.

Furthermore, the BETA (({beta }_{{NPP}})) and Q10 factors used to parameterize the CO₂ fertilization effect on photosynthesis and temperature control on heterotrophic respiration, respectively, are expressed in the two following equations:

Note that the BETA factor is a commonly used parameter representing the CO₂ fertilization effect (more technically, a scaling parameter expressing the sensitivity of the change in NPP to the logarithmic change in atmospheric CO₂ concentrations) but importantly differs from another commonly used parameter β also representing a quantity including the CO₂ fertilization effect (a scaling parameter expressing the sensitivity of the carbon storage to the change in atmospheric CO₂ concentrations, expressed in GtC ppm⁻¹)^62,70,71,72.

Emulating ACC2 to CMIP6 ESMs

We emulated ACC2 to five state-of-the-art ESMs (CanESM5, CNRM-ESM2-1, IPSL-CM6A-LR, MIROC-ES2L, and UKESM1-0-LL, see Table S2 for an overview of the ESMs) developed under the Coupled Model Intercomparison Project 6, CMIP6⁷³. The emulation was performed using SSP5-3.4-OS, a high-temperature overshoot scenario combining the high emission SSP5-8.5 scenario until 2040 with strong mitigation thereafter⁷⁴. We used the data from the fully-coupled (COU) configuration of ESMs. Our emulation method focuses only on three key parameters: namely, the equilibrium climate sensitivity (ECS), BETA, and Q10. Overall, three configurations of ACC2 emulating IPSL-CM6A-LR, CNRM-ESM2-1, and MIROC-ES2L have been used for the analysis because relatively large number of scenarios were obtained from these three configurations for the range of temperature targets and overshoot profiles considered in this study.

We first calibrated the ACC2 temperature response with that of each ESM. By tuning the ECS parameter in ACC2 (Table S3), we determined, for each ESM, the value of ECS which yielded the closest projection to each ESM (Fig. S13 and Table S4).

Then, we tuned BETA and Q10 values to emulate NBP (Table 2). We considered NBP to capture carbon concentration and carbon-climate feedback effects. We adjusted NBP estimates of ACC2 to correct for a lack of representation of land-use change (LUC) processes in ACC2. That is, LUC emissions data from the Global Carbon Budget⁷⁵ from 1850 to 2014 and integrated assessment models (IAM) output from 2015 to 2100 for the SSP5-3.4-OS^76,77 were used to make the ACC2 output comparable with ESMs. We identified the best combination of BETA and Q10 values for each ESM, for which the RMSE of the NBP estimates from ACC2 relative to the NBP from the ESM is minimized.

The results of the emulation process are shown in Table S4 and Figs. S14–S17. Our emulation for ACC2 gives similar or systematically lower ECS values compared to those reported elsewhere⁷⁸, with an average difference of 0.64 °C.

In Fig. S14, NBP from ACC2 with the best combination of BETA and Q10 values for each ESM is compared with NBP from the respective ESM. Apart from the inter-annual variability, the emulation shows overall a good performance for both the ramp-up and ramp-down phases. We further evaluated the emulation of IPSL-CM6A-LR with ACC2 by using the optimal combination of ECS, BETA, and Q10 parameters to simulate seven additional scenarios. Figures S15–S17 show the land carbon uptake, ocean carbon uptake, and temperature evolution from the emulated ACC2 and IPSL-CM6A-LR. The temperature change was reproduced very well overall for all the SSP scenarios considered. Some over-estimation of ocean carbon uptake, on the other hand, occurred in ACC2 when the ocean carbon uptake declined. This inconsistency may be related to a faster mixing of carbon from the surface to the deep ocean in simple climate models⁶⁶. Land carbon uptake showed a significant misfit in the cases of medium to high forcing scenarios (SSP2-4.5, SSP4-6.0, SSP3-7.0, and SSP5-8.5), which gives >3 °C warming at the end of this century. The largest warming level considered in our analysis using the ranges of overshoot scenarios (Figs. 2, 3) is approximately 3 °C (or 1 °C overshoot of the 2 °C target level). Thus, the misfit found here does not pose a problem for our application, given the range of our model application.