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Path-dependent reductions in CO2 emission budgets caused by permafrost carbon release

Abstract

Emission budgets are defined as the cumulative amount of anthropogenic CO2 emission compatible with a global temperature-change target. The simplicity of the concept has made it attractive to policy-makers, yet it relies on a linear approximation of the global carbon–climate system’s response to anthropogenic CO2 emissions. Here we investigate how emission budgets are impacted by the inclusion of CO2 and CH4 emissions caused by permafrost thaw, a non-linear and tipping process of the Earth system. We use the compact Earth system model OSCAR v2.2.1, in which parameterizations of permafrost thaw, soil organic matter decomposition and CO2 and CH4 emission were introduced based on four complex land surface models that specifically represent high-latitude processes. We found that permafrost carbon release makes emission budgets path dependent (that is, budgets also depend on the pathway followed to reach the target). The median remaining budget for the 2 °C target reduces by 8% (1–25%) if the target is avoided and net negative emissions prove feasible, by 13% (2–34%) if they do not prove feasible, by 16% (3–44%) if the target is overshot by 0.5 °C and by 25% (5–63%) if it is overshot by 1 °C. (Uncertainties are the minimum-to-maximum range across the permafrost models and scenarios.) For the 1.5 °C target, reductions in the median remaining budget range from ~10% to more than 100%. We conclude that the world is closer to exceeding the budget for the long-term target of the Paris Climate Agreement than previously thought.

Main

Sometimes called ‘carbon budgets’, cumulative anthropogenic CO2 emission budgets compatible with a given global mean warming target have been evaluated in many ways1,2,3,4,5,6,7,8,9,10. Yet, only a handful of the studies11,12,13 made (incomplete) preliminary attempts to account for permafrost thaw. The additional emission of CO2 and CH4 caused by this natural process triggered by warming in the high latitudes13,14 will diminish the budget of CO2 humankind can emit to keep below a certain level of global warming. Permafrost carbon release is also an irreversible process over the course of a few centuries13,14, and may thus be considered a ‘tipping’ element of the Earth’s carbon–climate system15, which puts the linear approximation of the emission budget framework1,4,5,16,17 to the test.

To quantify the impact of permafrost carbon release on emission budgets, we use an Earth system model of reduced complexity whose processes are parameterized to faithfully emulate more complex models. OSCAR v2.2.1 (a minor update of v2.2 (ref. 18)) was run in its default configuration, which is comparable to the median of its probabilistic set-up. Therefore, all our results are for about a 50% chance of meeting the temperature targets. OSCAR is extended here with a new permafrost carbon module that emulates four state-of-the-art land surface models: JSBACH (Methods), ORCHIDEE-MICT (ref. 19), and two versions of JULES (refs 20,21). These complex models were developed specifically to represent high-latitude processes, in particular soil thermic and biogeochemistry mechanisms that control carbon sequestration and emission. In this new emulator, the permafrost carbon in two high-latitude regions is represented as an initially frozen pool that thaws as global temperature increases. Thawed carbon is not immediately emitted: it is split between several pools, each with its specific timescale of emission. We assume that 2.3% of the emission occurs as methane14 (Methods discusses the uncertainty of this value), and this emitted CH4 is fully coupled to the dynamical atmospheric chemistry of OSCAR. More details on the protocol, the emulator and the models are provided in Methods.

We do not assume a priori that reductions in emission budgets can simply be calculated as the cumulative permafrost carbon release in a given scenario. Quite the opposite, we apply three specifically designed approaches to estimate emission budgets. The first one is the ‘exceedance’ approach, in which the budget is a threshold in terms of cumulative anthropogenic CO2 emissions above which the temperature target is exceeded (with a given probability). The second one is the ‘avoidance’ approach, in which the budget is another—typically lower—cumulative emissions threshold below which the target is avoided (also with a given probability). These two approaches were used in the fifth International Panel on Climate Change (IPCC) assessment report6,22. However, neither of these considers the possibility of overshooting the target first, and then returning below it afterwards. To investigate such a case, we adapted the approach of MacDougall et al.12 to create ‘overshoot’ budgets.

Reductions in exceedance and avoidance budgets

With the exceedance approach, budgets are calculated in any given scenario as the maximum cumulative CO2 emissions before the point in time when global temperature reaches the target level for the first time. This is illustrated in Fig. 1 (Methods and example given in Supplementary Fig. 1). Here our exceedance budgets are based on the four extended representative concentration pathways (RCP) emission scenarios23 and two idealized scenarios (Methods and Supplementary Fig. 2).

Fig. 1: Illustration of the three budget-calculation approaches used in this study.
figure1

Exceedance budgets (red) are the amount of CO2 that can be emitted before exceeding a given temperature target. Avoidance budgets (blue) are the amount of CO2 that can be emitted while staying below the target. Capture budgets (yellow) are the amount of CO2 that needs be captured when the target temperature is overshot by a given level. Capture budgets are combined with avoidance budgets to give the net overshoot budgets. Methods and Supplementary Fig. 1 give technical details on how these budgets are calculated. texc is the time at which the target is exceeded (for exceedance budgets), tavo is when temperature peaks at the targeted level (for avoidance budgets), tpeak is when temperature peaks at a given overshooting level above the targeted level (for overshoot budgets) and tcap is when temperature returns below the targeted level (for capture budgets and hence also overshoot budgets).

When permafrost carbon is ignored, we estimated total exceedance budgets of 2,350 (2,290–2,480) Gt CO2 for the 1.5 °C target and 3,260 (3,110–3,550) Gt CO2 for 2 °C, with 1870 as the preindustrial reference year (Supplementary Table 1, which also contains budgets for 2.5 °C and 3 °C). (Uncertainties are the minimum-to-maximum range across the permafrost models and scenarios.) Our results are ~3% different from the IPCC estimates based on complex models6. This confirms that OSCAR’s default configuration gives results consistent with the Earth system models used in previous climate change assessments.

When permafrost carbon processes are included, exceedance budgets are reduced by 30 (10–120) Gt CO2 for 1.5 °C and 60 (10–200) Gt CO2 for 2 °C (Fig. 2a and Supplementary Table 1). This is only a few percentage points of the total budgets, but it corresponds to a more substantial reduction in the remaining budgets (Fig. 2b). It is also smaller in magnitude than previously estimated with a model of intermediate complexity12, which can be explained by the oversensitivity of the permafrost carbon model used in that study (Table 1). An important (known) caveat of the exceedance approach is that it ignores the system’s dynamics after the point in time at which the temperature target is reached22. This is especially important for permafrost carbon, as a significant part of the thawed carbon keeps being emitted long after the target is first reached (Supplementary Fig. 3), which implies the temperature target will actually be surpassed if budgets are based on this approach.

Fig. 2: Change in emission budgets caused by permafrost carbon release.
figure2

The different budget-calculation approaches include different levels of overshoot and the avoidance budgets based on two subgroups of scenarios: NetNegEm0 and NetNegEm+. The uncertainty bars show the full range and the symbols show the average, across all permafrost models and scenarios. a, Absolute reductions in emission budgets. b, Relative reductions in the remaining budgets, calculated by assuming 2,240 Gt CO2 (ref. 44) was emitted between 1870 (the preindustrial reference year) and 2017 (Methods). To better isolate the effect of permafrost carbon in b, we present values under a constant present-day non-CO2 radiative forcing (other non-CO2 backgrounds are available in Supplementary Table 1). The grey hashed area represents a permafrost-induced reduction of 100% or more, which would imply that the budget was already exceeded in 2017.

Table 1 Comparison of cumulative permafrost carbon release estimated in 2100, 2200 and 2300 (Pg C (1 Pg C = \(\frac{{44}}{{12}}\) Gt CO2))

With the avoidance approach, budgets are calculated using a large ensemble of peak-and-decline emission scenarios whose values of peak temperature and maximum cumulative CO2 emissions are used for interpolation (Fig. 1, Methods and Supplementary Fig. 1). This approach accounts for the complete system’s dynamic by ensuring that the temperature target is never exceeded. Its drawback, however, is its intense computing requirement that makes it extremely costly to follow by complex models. Here we create and use an ensemble of 3,120 scenarios, by combining 520 fossil fuel CO2 emission scenarios of our own making (Supplementary Fig. 4) to the land-use and non-CO2 climate forcers from the six scenarios previously used for exceedance budgets (Methods).

Permafrost carbon reduces avoidance budgets by 60 (10–180) Gt CO2 for 1.5 °C and 100 (20–270) Gt CO2 for 2 °C (Fig. 2a). This reduction in avoidance budgets is systematically larger than for exceedance budgets: by 20% to 140% across all the emulated permafrost carbon models (Fig. 3). This confirms that the exceedance approach only partially captures the impact of permafrost carbon release on emission budgets. We conjecture that other slow and strongly non-linear processes, such as forest dieback15,24,25, are also incompletely accounted for with exceedance budgets. As the exceedance approach was the only one used by complex models in the fifth IPCC assessment report6,22, we conclude that future updates of the emission budgets based on such models will remain biased without a change in experimental protocol.

Fig. 3: Path dependency of the permafrost-induced budget reductions.
figure3

Reductions in avoidance and overshoot budgets are compared to reductions in exceedance budgets (reference case of 100%). Each type of budget represents a different archetype of pathway, and they are roughly sorted by the increasing intensity of the permafrost effect. The uncertainty bars show the full range of our results. Values >100% mean that the emission budgets are more largely reduced than in the exceedance case. If the permafrost effect was path independent, all the points would be close to 100%.

Path dependency and overshooting pathways

Our ensemble of 3,120 scenarios for the avoidance budgets covers a large enough spectrum of possible futures that it can be split into two groups (Methods and Supplementary Fig. 4). In the subgroup of scenarios that have no net negative emissions (NetNegEm0), the permafrost-induced reduction in avoidance budgets is 90 (10–230) Gt CO2 for 1.5 °C and 150 (30–340) Gt CO2 for 2 °C (Fig. 2a). This is systematically more than in the subgroup of scenarios in which the net negative emissions are extensively implemented (NetNegEm+) (Fig. 3). The physical reason for this is that extensive net negative emissions artificially make the temperature peak a few years after they are introduced, whereas when net negative emissions are not available the peak of the temperature is caused by natural processes only, and permafrost carbon emissions can delay it for decades (Supplementary Fig. 5). That the effect of permafrost carbon release depends on the emission pathway is proof that inclusion of such a previously unaccounted for tipping process renders emission budgets path dependent. In other words, the emission budget compatible with a given target depends on both the timing and magnitude of the anthropogenic emissions, and not on their magnitude alone.

To investigate this path dependency further, we looked into overshoot budgets using the same ensemble of scenarios as for those avoidance budgets (Methods). The net overshoot budgets were calculated as the sum of two gross budgets: a ‘peak’ budget that is exactly the same as an avoidance budget for a given peak temperature above the long-term target, and a ‘capture’ budget that corresponds to the amount of net negative emission required to return below the long-term temperature target (Fig. 1, Methods and Supplementary Fig. 1). Capture budgets have a mathematical definition analogous to exceedance budgets, and so these budgets have the same caveat of overlooking the system’s evolution after the target is met. Longer-term requirements in CO2 capture to compensate for lasting permafrost emissions26,27 are therefore ignored in our capture budgets (provided in Supplementary Table 2). Also, only net negative emission requirements can be estimated in this way—gross negative emissions could be much larger if the decrease in fossil fuel consumption was not rapid enough28.

In the case of an overshoot amplitude of 0.5 °C, emissions from permafrost thaw reduce the net emission budgets by 130 (30–300) Gt CO2 for the 1.5 °C long-term target (that is for a peak temperature of 2 °C, a case that corresponds to the Paris Climate Agreement), and by 190 (50–400) Gt CO2 for the 2 °C target (Fig. 2a). For an overshoot amplitude of 1 °C, permafrost-induced reductions reach 210 (50–430) Gt CO2 for the 1.5 °C target, and 270 (70–530) Gt CO2 for 2 °C target. (Budgets for other targets and other levels of overshoot are provided in Fig. 2 and Supplementary Table 1.) The permafrost-induced reduction is systematically more pronounced in these cases than in non-overshooting scenarios (Fig. 3) because of the additional capture required to counteract the extra emission from the thawed permafrost that occurs during the overshoot period. It is already known that the rest of the carbon–climate system (that is, excluding permafrost) exhibits a path-dependent behaviour under overshooting scenarios29 (Supplementary Fig. 6), but our results show that permafrost carbon release strongly reinforces this rupture of the linear approximation of the emission budget framework.

Discussion and policy implications

A permafrost-induced path dependency of emission budgets was implied by MacDougall et al.12, although their quantification of the effect was biased by the high sensitivity of their model’s permafrost carbon release in response to high-latitude warming (Table 1; note that an update of their model showed a lower bias26). Their study also focused on exceedance budgets and a handful of overshooting scenarios that did not correspond to political commitments. The Paris Climate Agreement aims to avoid 2 °C, which implies avoidance budgets are needed. It also recognizes an overshooting trajectory by setting the long-term target to 1.5 °C, which means overshoot budgets are also needed.

A few earlier attempts to quantify the permafrost-induced reduction in emission budgets were also made11,13, albeit without applying any of the budget-calculation approaches we use. They simply subtracted cumulative emissions caused by permafrost thaw from cumulative anthropogenic emissions at an arbitrary point in time. Such an approach is not suitable to estimate budget reductions accurately (Supplementary Fig. 7) because it overlooks the dynamical response of the coupled system. It was not retained in the fifth IPCC assessment report6. Additionally, these earlier studies did not find path dependency, either because only one scenario was investigated11 or because path independency was assumed13.

The OSCAR v2.2.1 model, with its new permafrost carbon emulator, estimates future carbon release from thawing permafrost within the range of existing studies (Table 1). A cumulative 60 (11–144) Pg C is projected to be released by 2100 under RCP8.5, slightly lower than the 37–174 Pg C reviewed by Schuur et al.14, and close to the 28–113 Pg C obtained with a data-constrained model by Koven et al.30. Uncertainties in permafrost-related processes and their response to climate change remain very high, however, and there are elements that suggest our results are conservative. Deep (for example, Yedoma) and seabed permafrost thaw is not modelled. Should these carbon stocks be mobilized, budgets would be further reduced. Changes in the nitrogen cycling caused by permafrost thaw are also ignored. These could lead to the emission of N2O (ref. 31), but also to changes in the ecosystems’ net carbon balance32.

We also assume a constant fraction of permafrost carbon is emitted as CH4, but the value and future evolution of this fraction are uncertain14,33,34. With this constant value, we simulate an emission of 3.7 (0.7–10.5) Tg CH4 per year over the 1980–2012 period, in line with a recent review35. This methane contributes a non-negligible fraction of the reduction in emission budgets (Fig. 4). This contribution is also path dependent, contrary to that obtained in previous studies13,33 by using a fixed global warming potential. It is, however, a well-known caveat of global warming potentials (or any other emission metric) that they are linear and constant, whereas the actual Earth system behaves in a complex, dynamical and non-linear fashion36,37,38,39, and so they cannot be naively used in combination with emission budgets40.

Fig. 4: Contribution of CH4 released by permafrost thaw to the budget reductions.
figure4

a,b, Absolute (a) and relative (b) terms with regard to the values shown in Fig. 2a. Uncertainty bars show the full range and symbols show the average, across all permafrost models and scenarios.

These uncertainty sources mean we assume that no probabilistic distribution of the permafrost-induced effect can yet be drawn from our results, and so we provide its full range. To reduce this uncertainty, it is key to foster observation- and model-based research on the permafrost and other tipping processes of the Earth system so as to know if and when the world will enter an overshooting climatic regime. Meanwhile, permafrost adds to the uncertain context under which climate policy decisions must be taken41,42,43. Careful policy-making might entail taking the pessimistic end of our estimates.

Nevertheless, we showed here that accounting for the tipping elements of the Earth system breaks the apparent linear behaviour of the carbon–climate system, which equates to making emission budgets path dependent. This renders manipulating budgets more delicate than previously thought, as budget users have to make assumptions regarding the long-term target, but also the shorter-term target (for example, the risk of overshooting) and even the reliance on certain technologies (as we demonstrate for net negative emissions).

Furthermore, we quantified a substantial permafrost-induced reduction in the remaining budgets for low-warming targets. This ranges from ~5% to as much as ~40% for a 2 °C target, and from ~10% to more than 100% for a 1.5 °C one, under present-day non-CO2 forcing and for an about 50% chance of meeting the temperature targets. Whether the world has already breached the budget for 1.5 °C remains elusive, however. It depends on many factors which include the uncertainty on past anthropogenic emissions44,45, the amount of forcing by non-CO2 species that will be mitigated in the near future12,46,47 and a possible bias in the model’s simulated present-day global temperature7,8,9,10 (not accounted for in this study). Irrespective of these uncertainties, it appears that the attainability of the Paris Climate Agreement is more compromised than suggested by the existing literature that largely ignores tipping or irreversible feedbacks of the Earth system.

Methods

OSCAR v2.2.1

OSCAR is a compact Earth system model whose modules are calibrated to emulate the behaviour of more complex models18. Of particular interest to this study, OSCAR features a module for the terrestrial carbon cycle calibrated on TRENDY and CMIP5 data50,51, a module for the oceanic carbon cycle adapted from the literature52 to embed CMIP5 data51, a climate-response module calibrated on CMIP5 models53 and an atmospheric chemistry module for the CH4 tropospheric lifetime54.

We used OSCAR v2.2.1, which is a minor update of v2.2. The only change between the two versions that affects this study is a minor correction of the carbonate chemistry in the surface ocean. This correction implies a better behaviour of the model for high-warming scenarios. All the equations remain the same as in the description paper18.

We used the global RCP data23 to drive the model over the data set’s historical period (1765–2005) and following the four extended RCP scenarios (2006–2500). Concentrations of all the greenhouse gases but CO2 and CH4 are prescribed to the model. Radiative forcings (RFs) of all the near-term climate forcers (ozone and aerosols) and albedo effects (black carbon on snow and land-cover change) are also prescribed. Therefore, the model was run in an emission-driven fashion only for fossil CO2 and CH4 emissions. However, to ensure that we obtained the same atmospheric concentrations of CO2 and CH4 as those of the RCPs when permafrost thaw is turned off, we first ran a concentration-driven simulation which we used to back-calculate the anthropogenic emissions of CO2 and CH4 that are compatible with these atmospheric concentrations28,55. These compatible anthropogenic emissions were then used to drive the model, instead of the original RCP emissions. Land-use and land-cover change data come from the LUH1.1 data set56 up to 2100. After that year, land-cover change is assumed to be zero, and land uses (wood harvest and shifting cultivation) are assumed to be constant.

We also introduced the CST and STOP scenarios. In CST, concentrations of all the greenhouse gases but CO2, radiative forcings of all the near-term climate forcers and albedo effects, and fossil CO2 emissions are kept constant after 2005. In STOP, all these values are set to their preindustrial value after 2005. In both CST and STOP, land-cover change is assumed to be zero after 2005. Land-uses are assumed to be constant after 2005 in CST, and to be zero in STOP.

The above protocol was further adjusted so that, when atmospheric CH4 concentration deviates from that of the original RCP because of CH4 emission from permafrost thaw, OSCAR also calculates the associated change in radiative forcing from stratospheric H2O and tropospheric O3 (refs 18,36).

In this study, OSCAR was not run in a probabilistic fashion—we used the default configuration of the model to save computing time. This implies that the full uncertainty of the Earth system is not sampled in this study, just that of the permafrost system under a close-to-median configuration of the rest of the model. The default configuration has an equilibrium climate sensitivity for CO2 doubling of ~3.2 °C. A comparison of the default and median results for key variables of the model is provided in Supplementary Fig. 8 for our six scenarios and in the case without permafrost thaw. The median results were obtained by running an unconstrained Monte Carlo ensemble of 2,000 elements, as in a previous work18. Supplementary Fig. 8 shows that the default and median atmospheric CO2 and global temperature simulated variables remain close, with a normalized root-mean square error <5%. Two noticeable biases were identified, however. First, the default configuration gives a lower atmospheric CH4 than the median, which suggests that our results underestimate the additional effect of CH4 emission caused by permafrost thaw. Second, for RCP2.6 (a peak-and-decline scenario) and to a lesser extent for RCP4.5 (a stabilizing scenario), the default configuration warms more than the median, which indicates that our capture budgets are probably overestimated (which may partly compensate for the protocol-induced underestimate described in the main text).

Permafrost carbon emulator

We coupled a permafrost emulator to OSCAR v2.2.1, calibrated on four land surface models: JSBACH (see below), ORCHIDEE-MICT19 and JULES20 following the two different versions DeepResp and SuppressResp21. The calibration of the parameters defined hereafter is done using outputs of the complex models for integrations over 1850–2300 of the RCPs 8.5, 4.5 and 2.6. In this emulator, we calibrated and separately ran the permafrost system of two aggregated regions of the globe: North America and Eurasia. In these models, we call ‘permafrost carbon’ the carbon that was frozen (and therefore inactive) during preindustrial times. All the parameter values are given in Supplementary Table 4.

First, we modelled the regional air surface temperature change (Ti) in each region i with a linear dependency on global temperature change (T):

$$T^i = \omega ^iT$$
(1)

The parameters ωi are calibrated with a linear fit between Ti and T (Supplementary Figs. 9 and 10, first row). Note that this parameter represents a feature of the climate system. It does not actually come from the emulated land surface model, but rather from the climate model it uses as the input. In the case of JSBACH, this is the MPI-ESM-LR model’s results for CMIP557. In the case of ORCHIDEE and JULES, the detailed protocol of the simulations used is from Burke et al.21. For JULES, we took the average of all the realizations made with IMOGEN, and for ORCHIDEE we took only one realization made with IMOGEN emulating HadCM3.

Second, we calibrated the temperature dependency of the heterotrophic respiration rate of non-permafrost carbon (r) following a Gaussian law58:

$$r^i = r_0^i\exp \left( {\gamma _1^iT^i - \gamma _2^iT^{i\,2}} \right)$$
(2)

γ1 and γ2 are the sensitivity parameters calibrated with forced positive values (Supplementary Figs. 9 and 10, second row), and r0 is the preindustrial heterotrophic respiration rate taken as the average over 1850–1859 in the case of JSBACH, and 1850 in the case of ORCHIDEE and JULES (as IMOGEN features no interannual variability).

Third, we introduced the theoretical thawed fraction (\(\bar p\)) that can take values from –pmin to 1, with a preindustrial value of 0. This corresponds to the fraction of thawed permafrost carbon for a given regional temperature change, but neglects dynamic considerations. It is fitted by an S-shaped function:

$$\bar p^i = - p_{{\rm{min}}}^i + \frac{{1 + p_{{\rm{min}}}^i}}{{\left( {1 + \left( {\left( {\frac{1}{{p_{{\rm{min}}}^i}} + 1} \right)^{\kappa _{\rm{p}}^i} - 1} \right)\exp \left( { - \gamma _{\rm{p}}^i\kappa _{\rm{p}}^iT^i} \right)} \right)^{\frac{1}{{\kappa _{\rm{p}}^i}}}}}$$
(3)

pmin represents a hypothetical (that is, never reached) case of fully frozen soils, κp is a shape parameter and γp the sensitivity parameter. The three parameters are calibrated with the same fit (Supplementary Figs. 9 and 10, third row), with the additional constraint that pmin cannot be greater than the ratio of the model’s non-frozen soil carbon over frozen soil carbon in preindustrial times. In the case of JSBACH, because there is no refreezing in the model, we calibrated this relationship on the scenario with the fastest warming only (that is, RCP8.5). For ORCHIDEE and JULES, the calibration was made with all scenarios. Therefore, the exact physical meaning of \(\bar p\) depends on the emulated model.

Fourth, we introduced an asymmetric dynamic behaviour in the thawing/freezing process by defining the actual thawed fraction (p), which lags behind the theoretical thawed fraction \(\bar p\):

$$\frac{{{\rm{d}}p^i}}{{{\rm{d}}t}} = \nu ^i\left( {\bar p^i - p^i} \right)$$
(4)

with:

$$\nu ^i = \left\{ {\begin{array}{*{20}{c}} {\nu _{\mathrm{thaw}}^i,{\mathrm{if}}\,\bar p^i \ge p^i} \cr {\nu _{\mathrm{froz}}^i,{\mathrm{if}}\,\bar p^{\mathrm{i}} < p^i} \end{array}} \right.$$
(5)

νthaw and νfroz are the speeds of thawing and freezing, respectively. They are calibrated simultaneously with the transient simulations (Supplementary Figs. 9 and 10, fourth row), using equations (3)–(5) driven only by the regional temperature change taken from the emulated model, that is not using equation (1).

Fifth, a frozen carbon pool (Cfroz) changes with time following the thawing carbon flux (Fthaw) calculated as the product of the frozen pool size during preindustrial times (Cfroz,0) by the speed of change in (that is, the time derivative of) the actual thawed fraction:

$$- \frac{{{\rm{d}}C_{{\rm{froz}}}^i}}{{{\rm{d}}t}} = F_{{\rm{thaw}}}^i = \frac{{{\rm{d}}p^i}}{{{\rm{d}}t}}C_{{\rm{froz}},0}^i$$
(6)

Inspired by Koven et al.30, we then split the thawing flux between three thawed carbon pools (Cthaw1, Cthaw2, Cthaw3) following the partitioning coefficients (πthaw1, πthaw2, πthaw3). Note, however, that for some models we reduced the number of thawed carbon pools to avoid overfitting (Supplementary Table 4). Each thawed carbon pool was then subjected to heterotrophic respiration with its own turnover time (τthaw1, τthaw2, τthaw3). The respiration rate is affected by regional temperature change following the same law as in equation (2), except that the sensitivities are modified by a factor κthaw. This gives:

$$\left\{ {\begin{array}{*{20}{c}} {\frac{{{\rm{d}}C_{{\rm{thaw}}1}^i}}{{{\rm{d}}t}} = \pi _{{\rm{thaw}}1}^i F_{{\rm{thaw}}}^i - \frac{1}{{\tau _{{\rm{thaw}}1}^i}}\left( {\frac{{r^i}}{{r_0^i}}} \right)^{\kappa _{\rm{thaw}}^i} C_{{\rm{thaw}}1}^i} \cr {\frac{{{\rm{d}}C_{{\rm{thaw}}2}^i}}{{{\rm{d}}t}} = \pi _{{\rm{thaw}}2}^i F_{{\rm{thaw}}}^i - \frac{1}{{\tau _{{\rm{thaw}}2}^i}}\left( {\frac{{r^i}}{{r_0^i}}} \right)^{\kappa _{\rm{thaw}}^i} C_{{\rm{thaw}}2}^i} \cr {\frac{{{\rm{d}}C_{{\rm{thaw}}3}^i}}{{{\rm{d}}t}} = \pi _{{\rm{thaw}}3}^i F_{{\rm{thaw}}}^i - \frac{1}{{\tau _{{\rm{thaw}}3}^i}}\left( {\frac{{r^i}}{{r_0^i}}} \right)^{\kappa _{\rm{thaw}}^i} C_{{\rm{thaw}}3}^i} \end{array}} \right.$$
(7)

To ensure mass balance, we must have πt3 = 1 – πt1 – πt2. The other six parameters were calibrated simultaneously with transient simulations by fitting the respiration flux simulated by our emulator to the actual complex model’s flux (Supplementary Figs. 9 and 10, fifth row). To do so, we used only equation (7), driven by the complex model’s thawing carbon fluxes and heterotrophic respiration rates.

Finally, the permafrost carbon emissions (Epf) were deduced as:

$$E_{{\rm{pf}}}^i = - \frac{{{\rm{d}} C_{{\rm{thaw}}1}^i}}{{{\rm{d}}t}}- \frac{{{\rm{d}} C_{{\rm{thaw}}2}^i}}{{{\rm{d}}t}} - \frac{{{\rm{d}} C_{{\rm{thaw}}3}^i}}{{{\rm{d}}t}} - \frac{{{\rm{d}} C_{{\rm{froz}} }^i}}{{{\rm{d}}t}}$$
(8)

The overall performance of the emulator is shown in Supplementary Figs. 9 and 10 (sixth row), with the emulator driven only by the emulated model’s global temperature change (the only driver of our permafrost module). Overall, the performance of the emulator is very satisfying, with a normalized root-mean square error for global cumulative permafrost carbon emissions of 2.8%, 5.3%, 5.7% and 7.1% for JULES-DeepResp, JULES-SuppressResp, JSBACH and ORCHIDEE-MICT, respectively.

Effect of methane emissions

A fraction of 2.3% (ref. 14) of permafrost carbon emission is assumed to be CH4. This value is assumed to remain constant throughout the simulations because the future response of this fraction to environmental changes (for example, climate or CO2) is unclear33. In OSCAR, the atmospheric evolution of these CH4 molecules is tracked in a separate manner, so that, when the permafrost-induced CH4 is oxidized in the atmosphere, we add the newly formed CO2 to the atmospheric CO2 pool. Therefore, the long-term addition of CO2 to the atmosphere caused directly by permafrost thaw does not depend on the CH4 fraction. The transient warming and ensuing feedbacks in the system, however, are a function of this fraction.

To investigate this effect, two additional series of simulations were performed: one without and one with a doubled methane emission (that is, fractions of 0% and 4.6%, respectively). The methane effect shown in Fig. 4 is equal to the difference between the budgets obtained in the main simulations with 2.3% of methane and those obtained in the simulations with 0%. We also found that the difference between the 4.6% and 2.3% simulations is approximately the same as that between 2.3% and 0% (not shown), which suggests the absolute contribution of methane is roughly linear in this domain.

Exceedance budgets protocol

To obtain the exceedance budgets, we ran our six scenarios with the permafrost module turned off and with its four alternative configurations. This is a total of 6 × 5 = 30 simulations. By definition, for each of these simulations, the exceedance budget is the maximum cumulative amount of anthropogenic CO2 that is emitted up to the time when the given temperature target is exceeded22. So, the exceedance budgets are calculated as:

$$B_{{\rm{exc}}} = \mathop {{\max }}\limits_\tau \mathop {\smallint }\limits_{t_0}^\tau E_{{\rm{FF}}}\left( t \right) + E_{{\rm{LUC}}}\left( t \right){\rm{d}}t$$
(9)

for \(T\left( \tau \right) \le T_{{\rm{target}}}\) and where Bexc is the exceedance budget, EFF the yearly fossil fuel CO2 emission, ELUC is the yearly CO2 emission from land-use change, t0 the year the simulation starts, T the simulated temperature change and Ttarget the target temperature change.

Fossil fuel CO2 emission pathways

For the avoidance budgets, we require a set of varied CO2 emission pathways that cover a wide range of possible futures. We created these emission pathways as the sum of one positive emission pathway and one negative: \(E_{{\rm{FF}}} = E_{{\rm{FF}} + } + E_{{\rm{FF}} - }\).

The pathway of positive emission is defined using a parameterized analytical formula of the peak-and-decline form on a semi-infinite interval59:

$$E_{\mathrm{FF}} {+ } = \left\{ {\begin{array}{*{20}{c}} {E_{\mathrm{FF}}{ + }\left( t \right),{\mathrm{if}}\,t \le t_1} \cr {E_{\mathrm{FF}}{ + }\left( {t_1} \right)\exp \left( {r\left( {t - t_1} \right)} \right),{\mathrm{if}}\,t_1 < t \le t_{{\rm m}}} \cr {F_{{\rm m}}\left( {1 + \left( {r + m} \right)\left( {t - t_{{\rm m}}} \right)} \right)\exp \left( { - m\left( {t - t_{{\rm m}}} \right)} \right), {\rm{if}} \,t > t_{{\rm m}}} \end{array}} \right.$$
(10)

where \(F_{{\rm m}} = E_{{\rm{FF}} + }\left( {t_1} \right)\exp (r(t_1 - t_{{\rm m}}))\). We also defined the total cumulative positive emission (Q+) of this pathway:

$$Q_ + = \mathop {\smallint }\limits_{t_0}^\infty E_{{\rm{FF}} + }\left( t \right){\rm{d}}t = Q_{t_1} + \mathop {\smallint }\limits_{t_1}^\infty E_{{\rm{FF}} + }\left( t \right){\rm{d}}t$$
(11)

Here t1 is the last year of the historical data, tm the time at which mitigation begins, r the historical growth rate of fossil CO2 emissions and m the mitigation rate. The value of r is taken as the mean of the growth rate over the last ten years of the historical period (r= 0.022623 yr–1). The mitigation rate m is deduced from the other parameters:

$$m = \frac{{F_{{\rm m}}}}{{A_r}}\left( {1 + \sqrt {1 + \frac{{A_r}}{{rF_{{\rm m}}}}} } \right)$$
(12)

with \(A_r = Q_ + - Q_{t_1 + } - \frac{{F_{{\rm m}}}}{r}\left( {1 - \exp \left( r(t_1 - t_{{\rm m}}) \right)} \right)\). Each positive emission pathway is uniquely defined by the tuple (t1, tm, Q+). In a similar manner, the negative emission pathways are defined following a logit-normal law on a finite interval:

$$E_{{\rm{FF}} - } = \frac{{Q_ - }}{{x\left( {1 - x} \right)}}\frac{1}{{\sigma \sqrt {2{\mathrm{\pi}} } }}{\mathrm{exp}}\left( { - \frac{{\left( {\log \left( {\frac{x}{{1 - x}}} \right) - \mu } \right)^2}}{{2\sigma ^2}}} \right)$$
(13)

with:

$$x = \frac{{t - (t_{{\rm m}} + t_{{\rm{lag}}})}}{{t_{\rm{f}} - (t_{{\rm m}} + t_{{\rm{lag}}})}}$$
(14)

where tf is the last year of the simulation, μ and σ are two shape parameters, tlag is the time between the start of the mitigation of the positive emissions and that of the negative emissions and Q the cumulative amount of negative emissions. Each negative emission pathway is uniquely defined by the tuple (tlag, μ, σ, Q).

Using the above equations, we created 520 fossil fuel CO2 emission pathways by combining different values for the positive emission tuple (t1, tm, Q+) and the negative emission one (tlag, μ, σ, Q). A full list of these 520 combinations of parameters is provided in Supplementary Table 5. The obtained emission pathways are also represented in Supplementary Fig. 4.

Avoidance budgets protocol

To calculate the avoidance budgets, we ran simulations with the 520 fossil fuel CO2 emission pathways combined with the six scenarios we already had for all the other drivers of the model (that is, non-CO2 forcings and land-use drivers), with the permafrost module turned off and with its four alternative configurations. This led to a total of 520 × 6 × 5 = 15,600 simulations. We note that our approach of combining two independent sets of scenarios probably leads to an overestimation of the scenario-related uncertainty, as we implicitly combine inconsistent sources of CO2 and non-CO2 emissions46. It allows, however, for a systematic analysis of the effect of non-CO2 forcing (using, for example, Supplementary Table 1).

Then, for each of these simulations (superscript i), we calculated the maximum temperature of the simulation:

$$T_\mathrm{max}^i = \mathop {{\max }}\limits_{\tau < t_f} T^i(\tau )$$
(15)

and its maximum cumulative CO2 emissions:

$$B_{\mathrm{max}}^i = \mathop {{\max }}\limits_{\tau < t_{\mathrm{f}}} \mathop {\smallint }\limits_{t_0}^\tau E_{\mathrm{FF}}^i\left( t \right) + E_{\mathrm{LUC}}^i\left( t \right){\mathrm{d}}t$$
(16)

If any of these two maxima occurred at the last time step of the simulation (tf), the simulation was discarded. With this approach, we were certain that an emission budget of Bmax ensures that the global temperature does not go above Tmax, given the non-CO2 and land-use forcings of the ith scenario.

However, we have no control over the individual values of Tmax. Therefore, to deduce the avoidance budgets (Bavo) for an exact temperature target, we interpolate linearly in the \(\left( {B_{{\rm{max}}}^i,T_{{\rm{max}}}^i} \right)\) phase space, within the Tmax value interval of ±0.2 °C around Ttarget. We acknowledge this is not exactly the approach followed Rogelj et al.46. However, our approach does respect the philosophy of the ‘avoidance’ budget in ensuring that the temperature target is, indeed, avoided. Obviously, for a given non-CO2 and land-use scenario, any budget lower than the deduced Bavo also implies avoiding the temperature target.

Net overshoot budgets protocol

Net overshoot budgets (Bnet) are the combination of two budgets: an emission budget to reach the peak temperature (Bpeak) and a capture budget that consists of the cumulative amount of negative emission required to go back to the targeted temperature (Bcap < 0), \(B_{{\rm{net}}} = B_{{\rm{peak}}} + B_{{\rm{cap}}}\). Therefore, net overshoot budgets are defined for a given temperature target and a given level of overshoot (Tover), with the peak temperature then being given by \(T_{{\rm{peak}}} = T_{{\rm{target}}} + T_{{\rm{over}}}\).

To calculate Bpeak and Bcap, we used the same ensemble of scenarios as for the avoidance budgets. In each case, we took only the subset of scenarios whose maximum temperature was ±0.2 °C of the chosen Tpeak and then declined by at least Tover. For each of these scenarios (superscript j), we calculated \(T_{{\rm{max}}}^j\) and \(B_{{\rm{max}}}^j\) exactly as we do for the avoidance budgets in equations (15) and (16). We also calculated \(B_{{\rm{neg}}}^j\):

$$B_{{\rm{neg}}}^j = \mathop {{{\mathrm{min}}}}\limits_\tau \mathop {\smallint }\limits_{t_1}^\tau \left( {E_{{\rm{FF}}}^j\left( t \right) + E_{{\rm{LUC}}}^j\left( t \right)} \right)\left[ {E_{{\rm{FF}}}^j\left( t \right) + E_{{\rm{LUC}}}^j\left( t \right) \le 0} \right]{\rm{d}}t$$
(17)

for \(T^j\left( \tau \right) \ge T_{{\rm{max}}}^j - T_{{\rm{over}}}\) and using Iverson brackets in the notation (which take a value of 1 if and only if the logical test in the brackets is true, and 0 otherwise.)

Then, just as with the avoidance budgets, we linearly interpolated Bpeak and Bcap in the \(\left( {B_{{\rm{max}}}^j,T_{{\rm{max}}}^j} \right)\) and \(\left( {B_{{\rm{neg}}}^j,T_{{\rm{max}}}^j - T_{{\rm{over}}}} \right)\) phase spaces, respectively. The net overshoot budget Bnet is deduced by summation of Bpeak and Bcap. We note again that this protocol, being somewhat similar to the exceedance protocol in its formulation, ignores everything that may occur after the temperature returns below the targeted value. It therefore provides a lower-bound estimate of future capture requirements.

Extra data processing

To be consistent with IPCC6, we adjusted our budgets for a preindustrial year of 1870. To do so, before actually calculating any budget, global temperatures (T) were offset by a value equal to the average over 1861–1880, and cumulative CO2 emissions (B) were reduced by the cumulative amount of CO2 emitted over 1765–1870. Budgets are rounded to the nearest 10 Gt CO2 in the tables and main text. We also discarded estimates of Bavo and Bpeak for which the coefficient of determination of the linear fit was less than 0.50.

Remaining budgets calculation

The remaining budgets (ΔB) were calculated as \({\mathrm{\Delta }}B = B - B_{{\rm{hist}}}\), where B can be Bexc, Bavo or Bnet, and Bhist is the historical cumulative CO2 emission from anthropogenic activities (from all sources: fossil fuel burning, industry and land-use changes). We take Bhist = 2,240 Gt CO2 (ref. 44). In Fig. 2b, we show the relative reduction in remaining budgets caused by permafrost carbon release, that is, \({\mathrm{\Delta }}B_1/{\mathrm{\Delta }}B_0 - 1\), where the subscript ‘1’ is for a case with permafrost carbon processes, and ‘0’ for one without. In Fig. 3, we show reductions in budgets relative to those in the exceedance case, \({\mathrm{\Delta }}B/{\mathrm{\Delta }}B_{{\rm{exc}}}\).

Permafrost in JSBACH

The earlier CMIP5 version of the Max Planck Institute Earth system model land surface scheme JSBACH60,61 is extended with a multilayer hydrology scheme62, a representation of permafrost physical processes63 as well as the improved soil carbon model YASSO64. For permafrost carbon stocks, we represented carbon cycling in the active layer by the YASSO model, and we prescribed frozen carbon stocks below the active layer from the Northern Circumpolar Soil Carbon Database (NCSCD) version 2 (ref. 65). When the active layer thickness changed, we transferred carbon from the prescribed frozen carbon stocks to the active YASSO carbon pools.

Code availability

The source code of OSCAR is available at https://github.com/tgasser/OSCAR. The code used to generate all the results of this study is available from the corresponding author upon request.

Data availability

RCP scenarios are available at http://www.pik-potsdam.de/~mmalte/rcps/. The data that support the findings of this study are available from the corresponding author upon request.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Change history

  • 21 November 2018

    In the version of this Article originally published, data given for total exceedance budgets of CO2 for 1.5 °C and 2 °C targets were incorrect in the main text, although the correct values were given in Supplementary Table 1. These errors also resulted in an incorrect estimation of the percentage difference between the authors’ results and estimates by the IPCC. These errors have now been corrected in the online versions.

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Acknowledgements

We thank A. H. MacDougall for sharing data and O. Boucher for data used in Supplementary Fig. 6. This work is part of the European Research Council Synergy project ‘Imbalance-P’ (grant no. ERC-2013-SyG-610028). Simulations with OSCAR were carried out on the IPSL Prodiguer-Ciclad facility, which is supported by CNRS, UPMC and Labex L-IPSL, and funded by the ANR (grant no. ANR-10-LABX-0018) and the European FP7 IS-ENES2 project (grant no. 312979). E.J.B. was supported by PAGE21 (EU project no. GA282700), CRESCENDO (EU project no. 641816) and the Joint UK DECC/Defra Met Office Hadley Centre Climate Programme (GA01101). A.E. was also supported by PAGE21.

Author information

T.G. designed the study. T.G. developed the permafrost emulator with inputs from P.C. and M.K. T.K. provided JSBACH data. Y.H., D.Z. and P.C. provided ORCHIDEE data. E.J.B. and A.E. provided JULES data. T.G. and M.K. set up the simulations with OSCAR, processed the outputs and created the figures. T.G., M.K., P.C. and M.O. discussed the preliminary results. T.G. wrote the manuscript with contributions from all the authors.

Competing interests

The authors declare no competing interests.

Correspondence to T. Gasser.

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