The TCRE measures the dependence of surface warming to cumulative CO₂ emissions and is defined by the change in global-mean, surface air temperature, ΔT(t) in K, relative to the pre industrial divided by the cumulative carbon emission, Iem(t) in EgC, such that 1TCRE≡ΔT(t)Iem(t), where Δ represents the change since the time of the pre industrial. The TCRE is approximately scenario independent and depends only on the cumulative carbon emissions.

The TCRE from Eq. () may be related to an identity involving the product of two terms, the Transient Climate Response (TCR) affected by climate processes and the airborne fraction affected by the carbon cycle , such that 2TCRE=ΔT(t)Iem(t)=ΔT(t)ΔIA(t)︸TCRΔIA(t)Iem(t)︸carbon cycle, where the TCR is defined by the ratio of the surface temperature change, ΔT(t), and change in the atmospheric carbon inventory, ΔIA(t), and the airborne fraction is defined by the ratio of the change in the atmospheric carbon inventory, ΔIA(t), and the cumulative carbon emissions, Iem(t).

The TCRE can also be equivalently defined by separating the TCR term in Eq. () into a product of two terms, involving separate thermal and radiative dependencies, such that 3TCRE=ΔT(t)Iem(t)=ΔT(t)ΔF(t)︸thermalΔF(t)ΔIA(t)︸radiativeΔIA(t)Iem(t)︸carbon cycle, where the thermal dependence is given by the ratio of the surface temperature change ΔT(t) and the change in the radiative forcing, ΔF(t), and the radiative dependence from the ratio of the change in the radiative forcing, ΔF(t), and the change in the atmospheric carbon inventory, ΔIA(t) . The benefit of this additional step is to gain insight into the thermal and radiative effects on the TCRE, by drawing upon the energy balance at the top of the atmosphere and the logarithmic dependence of radiative forcing on atmospheric CO₂.

The ZEC measures the temperature change relative to the pre industrial, ΔT(t), minus the temperature change at the time of net zero, tZE, ΔT(tZE), and is defined by 4ZEC≡ΔT(t)-ΔT(tZE). This definition of the ZEC measures the absolute value of the temperature change and is likely to be sensitive to the warming level experienced from the emission scenario.

Alternatively, a geometric measure of the ZEC may be employed, given by the ratio of the temperature change, ΔT(t), and the temperature change at the time of net zero, ΔT(tZE), which measures the fractional zero emission commitment, 5ΔT(t)ΔT(tZE). A positive ZEC corresponds to this geometric measure, ΔT(t)/ΔT(tZE)>1, and a negative ZEC to ΔT(t)/ΔT(tZE)<1.

The temperature change, ΔT(t), used to define the ZEC may be related to the cumulative carbon emission, Iem(t), by the product of the thermal, radiative and carbon-cycle contributions, 6ΔT(t)=ΔT(t)ΔF(t)ΔF(t)ΔIA(t)ΔIA(t)Iem(t)Iem(t), so that the geometric ZEC from ΔT(t)/ΔT(tZE) may be expressed as a product of normalised thermal, radiative and carbon-cycle contributions, 7ΔT(t)ΔT(tZE)=ΔT(t)ΔF(t)/ΔT(tZE)ΔF(tZE)︸thermal⋅ΔF(t)ΔIA(t)/ΔF(tZE)ΔIA(tZE)︸radiativeΔIA(t)ΔIA(tZE)︸carbon cycle. The dependence of the emissions is removed after the time of net zero as Iem(t) is taken to be fixed for t>tZE.

Our aim is gain insight into the controls of the ZEC and interpret the continued warming or cooling response in terms of its normalised thermal, radiative and carbon-cycle contributions. Next consider the thermal, radiative and carbon-cycle terms in Eq. () that determine the ZEC response.

The responses of 9 full Earth system models are analysed following the ZECMIP protocols , involving an annual 1 % rise in atmospheric CO₂ until a cumulative carbon emission of 1000 Pg C is reached and then there is no further carbon emission (Fig. a). A single realisation is analysed for each model.

Under the ZECMIP protocol, each individual model experiment branches at the time of net zero, one branch continuing with the 1 % rise in atmospheric CO₂ and the other branch continuing with no further emissions. The time of net zero as defined by the branch point varies from 61 to 71 years across the set of Earth system models .

Prior to net zero, the global-mean surface temperature increases nearly linearly with the rise in cumulative carbon emissions (Fig. b). There is a nearly constant slope of the temperature change versus cumulative carbon emissions up until the maximum emission, which defines the climate metric, the TCRE (Fig. c).

After net zero, there are a range of temperature responses from a slight cooling to a slight continued warming (Fig. b), where the temperature change relative to the temperature at net zero defines the ZEC. The continued temperature change is also evident in the positive and negative excursions in temperature at the maximum carbon emissions in Fig. c.

The temperature response after net zero is interpreted in terms of a geometric ZEC involving the continuing temperature change, ΔT(t), divided by the temperature change at net zero, ΔT(tZE). This estimate of the temperature change at net zero is performed using a 20 year averaging window around the time of net zero to reduce the effect of interannual variability. The averaging is performed on the 1 % branch experiment that always includes carbon emissions with an approximately linear rise in temperature, rather than combining a forced response up to net zero and an unforced response after net zero; this choice follows MacDougall et al. (2020) to avoid a possible bias in the estimate of the temperature at net zero, ΔT(tZE). This averaging approach is applied for all the variables evaluated at net zero in our normalised framework.

The temperature response after emissions defining the ZEC involves a variety of competing drivers (Fig. a–f) involving changes in carbon inventories, radiative forcing, radiative response and planetary heat uptake. These changes are next described and our framework applied to quantify the relative importance of these competing drivers.

The ZEC measures the temperature change after net zero. The timing of net zero varies from years 61 to 71 in the set of models and, in our subsequent analysis, we choose to align their time series so that the timing of net zero coincides. The ZEC, defined by ΔT(t)-ΔT(tZE), reaches -0.04 ± 0.14 K for year 25, and -0.11 ± 0.19 and -0.12 ± 0.25 K for years 50 and 90 after net zero (Fig. a, b; Tables 1a and A1 for individual models) .

Alternatively, the geometric ZEC, given by the ratio of the temperature change relative to the pre industrial, ΔT(t), and the change for net zero, ΔT(tZE), varies from 0.97 ± 0.09 for year 25 to 0.93 ± 0.11 and 0.92 ± 0.14 respectively for years 50 and 90 after net zero (Fig. c; Table 1a). The model-mean changes in the geometric ZEC are relatively small, accounting for a temperature anomaly decrease of only 8 % after net zero. However, the individual model responses are much larger, reaching 20 % changes after net zero; as previously highlighted by MacDougall et al. (2020).

The ZEC response is made up of competing responses that are quantified in the normalised framework (Eq. ): (i) the normalised thermal contribution, ΔT(t)/ΔF(t), is large and positive, reaching 1.22±0.11 and 1.33±0.15 for 50 and 90 years after net zero respectively (Fig. d; Table 1b); and (ii) the normalised radiative contribution, ΔF(t)/ΔIA(t), is relatively small, only reaching 1.09 ± 0.02 and 1.11 ± 0.02 after 50 and 90 years respectively (Fig. e); and (iii) the normalised carbon contribution, ΔIA(t), is large and negative, reaching 0.69 ± 0.05 and 0.62 ± 0.06 after 50 and 90 years respectively (Fig. f; Table 1b). Hence, the geometric ZEC is primarily determined by a competition between the normalised thermal and carbon contributions; these contributions are discussed in more detail in the next subsections.

For individual models, there are some large variations, with the normalised thermal contribution exceeding a 30 % increase for CESM2, CNRM-ESM2 and UKESM1, and the normalised carbon contribution reaching a 30 % decrease for CanESM5, CESM2, CNRM-ESM2, GFDL-ESM2, and NorESM2 (Fig. , red and green lines; Table A1).

The resulting geometric ZEC response involves a competition between these normalised thermal and carbon contributions. For example, the positive ZEC response for UKESM1 is due to a strong thermal contribution and only a moderate opposing carbon contribution, while the positive ZEC response for CNRM-ESM2 involves a very strong thermal contribution and an opposing strong carbon contribution. Meanwhile the negative ZEC response for NorESM2 is due to a relatively modest thermal contributions and relatively strong opposing carbon contributions.

The inter-model spread of the geometric ZEC, measured by the coefficient of variation, reaches 0.12 for 50 years after net zero and is made up of contributions of 0.09 for the thermal contribution, 0.01 for the radiative contribution and 0.07 for the carbon contribution (Table 1). Hence, the thermal contribution is the most important contributor to the inter-model spread in the geometric ZEC, closely followed by the carbon contribution and the radiative contribution is least important.

The normalised contributions to ZEC from the thermal, radiative and carbon responses can also be analysed to quantify the contribution of each term to the spread across models. By varying just one input term in Table A1, the thermal and carbon terms can explain 58 % and 40 % respectively of the variance in ZEC, whereas the radiative term explains only 2 % of the variance. This analysis confirms that both the model spread in thermal response and the model spread in carbon sink both contribute significantly to the spread in ZEC, and both remain high priority research areas to understand in order to reduce uncertainties in ZEC.

These competing carbon and thermal contributions for the ZEC are next addressed in more detail in terms of their own dependencies.

The carbon contribution to the geometric ZEC response involves a normalised decrease in the atmospheric carbon inventory, ΔIA(t), which is achieved by an increase in both land and ocean carbon inventories.

In order to compare these different carbon sinks, the carbon changes of each inventory are henceforth normalised by the same cumulative carbon emission at net zero, as given by the airborne, landborne and oceanborne fractions. Each of these fractions are evaluated at a particular time using a 20 year time window centered on that time. The airborne fraction, ΔIA(t)/Iem(tZE), is a maximum at net zero and then declines in time for all models (Fig. , black line) due to the increase in the landborne and oceanborne fractions (Fig. , green and blue lines). The airborne fraction is 0.52±0.03 at net zero and decreases to 0.38±0.05 and 0.34±0.05 at years 50 and 90 after net zero (Table 1d). The landborne fraction, ΔIL(t)/Iem(tZE), increases from 0.26±0.04 at net zero reaching 0.34±0.07 and 0.35±0.08 for 50 and 90 years later respectively; and the oceanborne fraction, ΔIO(t)/Iem(tZE) increases from 0.22±0.03 at net zero reaching 0.28±0.04 and 0.31±0.04 for 50 and 90 years later. Hence, initially after net zero, the carbon uptake by the terrestrial system dominates over that by the ocean for most models, but they become comparable to each other by 90 years.

The landborne fraction is much larger than the oceanborne fractions for CanESM5, CNRM-ESM2 and GFDL-ESM2, while the landborne and oceanborne fractions are comparable for UKESM1 and the landborne fraction is much smaller than the oceanborne fraction for ACCESS-ESM1.5 (Fig. ; Table A1).

These different relative strengths of the land and ocean carbon sinks are likely due to structural differences in the land carbon model, which contributes a much greater model spread than the ocean response . The three models with higher landborne fractions may be overestimating the possible land carbon sink as they neglect the role of nutrient limitations, while the other models include land nitrogen cycling and limitation of carbon allocation. Models without an explicit terrestrial nitrogen cycle can project unrealistic land carbon sinks which could not be supported by available nutrients . In particular, showed explicitly for the ACCESS model, the important role of nutrients – both nitrogen and phosphorus - in reducing land carbon sinks. showed this inclusion or absence of nutrient limitation to be the largest systematic difference in the carbon response of CMIP6 Earth system models, with a distinct split in the land carbon response to climate and CO₂ between models with and without a nitrogen cycle. Consequently, the inclusion of nitrogen limitation on land acts to reduce the increase in the projected land carbon sink and so acts to increase the resulting ZEC.

The inter-model spread is much larger for the landborne fraction than the oceanborne fraction with coefficients of variation of 0.21 and 0.13 respectively after 50 years (Table 1d). These inter-model differences in the landborne and oceanborne fractions are partly compensating, since both coefficients of variation are larger than that for the airborne fraction reaching 0.12.

In common with the analysis of , by varying just one model input term at a time in Table A1, the land carbon sink is again found to dominate the spread in the carbon sink contribution to ZEC, accounting for 78 % of the variance in the carbon sink compared with the ocean sink explaining 22 % of the variance at 50 years after net zero. The magnitude of land and ocean sinks are similar on this timescale, but the model spread is greater for the land sink. On longer timescales beyond a century, we expect the land carbon sink to saturate more rapidly and the ocean carbon sink to play a progressively more important role.

The thermal contribution to the ZEC may be understood in terms of the top of the atmosphere energy balance (Eq. ). The radiative forcing, ΔF(t), peaks close to the time of net zero and then declines for each model (Fig. , black line).

The radiative response, ΔR(t), is negative and so represents the part of the radiative forcing that is returned to space. The radiative response varies between models, most involve a peak in magnitude at the time of net zero and then a slight decline in magnitude (such as CESM2 and NorESM2-LM), while in some models (such as CNRM-ESM2 and UKESM1) the radiative response remains relatively constant in time (Fig. , red line). The planetary heat uptake, ΔN(t), represents the mismatch between the radiative forcing and radiative response. The planetary heat uptake is a maximum at the time of net zero and declines in time for all models (Fig. , blue line). For these thermal quantities there is significant interannual variability.

The thermal contribution to the geometric ZEC response, the normalised ΔT(t)/ΔF(t), may be derived from the top of the atmosphere energy balance (Eq. ). This thermal contribution increases after net zero (Fig. , black line) and is made up itself by the product of contributions from the fraction of the radiative forcing escaping to space, the normalised ΔR(t)/ΔF(t), and the inverse of the climate feedback parameter, the normalised λ(t)-1 (Fig. , blue and red lines respectively).

The normalised fraction of the radiative forcing escaping to space, ΔR(t)/ΔF(t), increases after net zero and reaches 1.25±0.11 and 1.30±0.16 for 50 and 90 years later respectively (Fig. , blue line; Table 1c). The normalised inverse of the climate feedback parameter, λ(t)-1, is close to 1, reaching 0.98±0.06 and 1.02±0.05 for 50 and 90 years later respectively. Thus, the dominant contribution to the increase in the thermal contribution to the geometric ZEC response is from an increase in the fraction of the radiative forcing escaping to space, which is equivalent to a decrease in the fraction of radiative forcing used to increase planetary heat.

For most individual models, the thermal contribution to the ZEC response, the normalised ΔT(t)/ΔF(t), is broadly the same as the fraction of radiative forcing escaping to space, ΔR(t)/ΔF(t) (Fig. , blue line; Table A1). However, there is an enhancement of the thermal contribution from the increase in the radiative forcing escaping to space by a time-varying amplification from the climate feedback parameter for UKESM1 and for the latter parts of the temporal record for GFDL-ESM2 and NorESM2-LM (Fig. , red line).

Inter-model differences in the thermal contribution are dominated by differences in the fraction of radiative forcing returned to space, rather than from differences in the inverse of the climate feedback parameter, since their coefficients of variation are 0.09 and 0.06 respectively after 50 years (Table 1c). In addition, by varying just one model input at a time in Table A1, the fraction of radiative forcing returning to space explains 67 % of the variance in the thermal contribution to the ZEC compared with 33 % from the variance in the inverse climate feedback parameter.

The efficient Earth system model (WASP) includes air-sea exchange of CO₂ with a full carbonate chemistry solver for the surface ocean . Sub-surface ocean boxes then exchange carbon with the surface ocean with each sub-surface box having an e-folding timescale prescribed over which the sub-surface box becomes chemically equilibrated with the surface ocean. The land carbon cycle in WASP is separated into a vegetation carbon pool and a soil carbon pool. The net primary production removes carbon from the atmosphere into the vegetation pool. Net primary production is dependent upon atmospheric CO₂ via a logarithmic relationship using a CO₂-fertilisation coefficient, and net primary production is sensitive to global mean temperature via a net primary production-temperature coefficient. The carbon flux from the vegetation to soil carbon pools is via leaf litter, which is only dependent upon the size of the vegetation pool. The soil carbon pool returns carbon to the atmosphere with an e-folding timescale, which is temperature dependent via a third coefficient. WASP includes time-varying climate feedbacks, to represent time-varying changes in the pattern effect .

In these experiments, 10 million prior simulations are integrated using historical forcing and following the SSP245 experiment from year 2014 with varied model parameters (Table S1; ). In an initial prior ensemble the coefficients are varied independently. This prior ensemble is historically forced and compared to observational reconstructions. Only ensemble members with land and ocean carbon uptake that are in accord with historic observational reconstructions are retained in the final WASP ensemble (<1 % of prior ensemble members). Of these simulations, 1138 posterior solutions are identified that satisfy observable quantities .

These 1138 posterior ensemble members are then integrated forward following two different experiments: (i) an annual 1 % increase in atmospheric CO₂ with emissions ceasing at 1000 Pg C (referred to as the 1pctCO2 case as for ZECMIP) or (ii) a constant emission rate of 10 Pg C yr⁻¹ for 100 years until there is 1000 Pg C emitted (referred to as the flat10 case) . This comparison is included as flat10 is a scenario choice for CMIP7 and has the benefit of a more constant forcing regime.

In the 10 million historically forced prior simulations used to determine observational consistency the WASP model simulations include an imposed internal variability . This internal variability is turned off when the posterior simulations are then forced with idealised experiments.

The ZEC responses reveal a slight decrease in surface temperature after net zero for the median of the ensembles for both the 1pctCO2 and flat10 experiments (Fig. a, b, blue line).

For the 1pctCO2 experiment, the median ZEC and the 5 % to 95 % ensemble range in brackets are -0.10 K (-0.47 to 0.43 K) after 50 years, increasing in magnitude to -0.09 K (-0.56 to 0.82 K) after 100 years (Fig. a; Table S2). There is close agreement in these ZEC estimates with the ZECMIP model mean of -0.10 K at 50 years (Table 1) and the ZECMIP range is comparable to the 1 standard deviation range from WASP (Fig. , orange line and dark blue shading).

For the flat10 experiment, the median ZEC and the 5 % to 95 % ensemble range in brackets are slightly smaller: -0.06 K (-0.25 to 0.41 K) after 50 years, increasing in magnitude to -0.07 K (-0.31 to 0.73 K) after 100 years and further to -0.19 K (-0.46 to 1.02 K) after 400 years (Fig. b, Table S3). This slightly smaller magnitude response for flat10 is due to there being a stronger radiative forcing for the 1pctCO2 case as the forcing is more exponential.

There is some slight curvature in the ZEC responses between the 1pctCO2 and flat10 experiments with a greater range for the 1pctCO2 ensembles (Fig. c).

The changes in carbon inventories, radiative forcing, radiative feedback and planetary heat uptake (Figs. S1, S2; Tables S2 and S3) vary in a broadly similar manner as for the ZECMIP diagnostics over 100 years (Figs.  and ), although include a much greater ensemble spread and extend for much longer to 400 years.

The geometric ZEC from Eq. () slightly decreases after net zero for the ensemble median to 0.92, 0.92 and 0.82 for years 50, 100 and 400 for the 1pctCO2 experiment, and 0.95, 0.94 and 0.83 for years 50, 100 and 400 after net zero for the flat10 experiment (Tables S2 and S3). There is a wide inter-ensemble spread with the geometric ZEC varying from 0.57 to 1.70 for 1pctCO2 and 0.71 to 1.50 for flat10 at year 100 including a tail of ensembles with much higher geometric ZEC (Fig. a).

For each ensemble, the geometric ZEC value (Fig. a) equates to the product of the ensemble values for each of the ZEC contributions (Fig. b–d). The thermal contribution to the geometric ZEC, the normalised ΔT/ΔF, increases in time for all ensembles to a median of 1.37 (5 % and 95 % range of 0.92 to 2.18) for 1pctCO2 and 1.24 (5 % and 95 % range of 0.97 to 1.75) for flat10 after 100 years (Fig. b). The radiative contribution to the geometric ZEC, the normalised ΔF/ΔIA, only slightly increases in time for all ensembles to 1.07 (1.04 to 1.09) after 100 years (Fig. c). The carbon contribution to the geometric ZEC given by the normalised change in atmospheric carbon, ΔIA, decreases for all ensembles to 0.71 (0.67 to 0.86) after 100 years (Fig. d).

The thermal contribution to the geometric ZEC, the normalised ΔT(t)/ΔF(t) and its contributions, reveals similar changes as for median ensemble response as in ZECMIP (Fig. a, b, full lines): there is a strengthening in the normalised ΔT(t)/ΔF(t) in time, which is primarily due to the strengthening in the normalised fraction of radiative forcing returned to space, 1-ΔN(t)/ΔF(t) augmented by a strengthening in the normalised inverse climate feedback, λ(t)-1.

This contribution is made up of the product of two terms, the normalised radiative response divided by the radiative forcing, the normalised ΔR(t′)/ΔF(t′), and the normalised inverse of the climate feedback parameter, the normalised λ(t′)-1. There is a consistent increase in the fraction of the radiative forcing returned to space or equivalently a decrease in the fraction of radiative forcing used for planetary heat uptake with a relatively tight ensemble spread and the median increasing to 1.16 (5 % and 95 % range of 0.98 and 1.33) for 1pctCO2 and 1.12 (1.00 and 1.19) for flat10 after 100 years (Fig. a, b, blue line and shading; Tables S2 and S3). The normalised inverse of the climate feedback parameter only slightly increases in time for the median to 1.19 and 1.10 after 100 years for 1pctCO2 and flat10 respectively (Fig. 10a, b, orange line and pale shading), but with a wide, asymmetrical spread (5 % to 95 % ranges extending from 0.72 to 2.23 and 0.84 to 1.73). Hence, including ensembles with much larger λ(t)-1 providing a greater amplification by climate feedbacks compared with the ZECMIP diagnostics.

The carbon contribution to the geometric ZEC, the normalised atmospheric carbon, ΔIA(t), may be understood by the changes in airborne fraction, ΔIA(t)/Iem(t), which progressively decreases in time (Fig. c, d, black line and grey shading; Tables S2 and S3). The dominant contribution to the changes in airborne fraction alters from being from the landborne fraction close to the time of net zero, ΔIL(t)/Iem(t), to the oceanborne fraction on timescales greater than 50 years after net zero, ΔIO(t)/Iem(t) (Fig. c, d, green and blue lines and shading respectively).

There is a much larger ensemble spread around the landborne and ocean borne responses than for the airborne fraction, which implies that the changes in land and ocean carbon sinks partly compensate for each other. This partial compensation in carbon sinks is consistent with the coefficient of variation being larger for the landborne and oceanborne fractions than the airborne fraction as diagnosed for ZECMIP (Table 1). This reduced spread of the airborne fraction in WASP is also partly due to how the WASP ensembles are constructed with historic constraints on atmospheric carbon being much narrower than the historic constraints for land and ocean carbon.

In summary, the ZEC responses and their normalised contributions are broadly similar in the diagnostics of the large ensemble WASP and the smaller set of 9 Earth system models in ZECMIP. The WASP assessment reveals partial compensation between changes in landborne and oceanborne fractions, and a larger spread in the effect of the climate feedback and the possibility of climate amplification of the ZEC.