Estimating Remaining Carbon Budgets Using Temperature Responses Informed by CMIP6

A remaining carbon budget (RCB) estimates how much CO2 we can emit and still reach a specific temperature target. The RCB concept is attractive since it easily communicates to the public and policymakers, but RCBs are also subject to uncertainties. The expected warming levels for a given carbon budget has a wide uncertainty range, which increases with less ambitious targets, i.e., with higher CO2 emissions and temperatures. Leading causes of RCB uncertainty are the future non-CO2 emissions, Earth system feedbacks, and the spread in the climate sensitivity among climate models. The latter is investigated in this paper, using a simple carbon cycle model and emulators of the temperature responses of the Earth System Models in the Coupled Model Intercomparison Project Phase 6 (CMIP6) ensemble. Driving 41 CMIP6 emulators with 127 different emission scenarios for the 21st century, we find almost perfect linear relationship between maximum global surface air temperature and cumulative carbon emissions, allowing unambiguous estimates of RCB for each CMIP6 model. The range of these estimates over the model ensemble is a measure of the uncertainty in the RCB arising from the range in climate sensitivity over this ensemble, and it is suggested that observational constraints imposed on the transient climate response in the model ensemble can reduce uncertainty in RCB estimates.


INTRODUCTION
The concept of remaining carbon budgets (RCBs) is appealing and highly applicable to climate mitigation In this paper we compare the cumulative emissions after 2018 in emission scenarios from the Integrated  (Smith et al., 2018;Leach et al., 2020). A linear relationship between GSAT and cumulative emissions computed this way is estimated using linear 48 regression, and the slope of the regression line serves as an estimate of TCRE. We define a climate target 49 as a particular GSAT-value, e.g., 2.0 • C above the pre-industrial baseline, and the estimated RCB for this 50 target is obtained by the estimated linear relationship. 51 The transient climate response obtained by this procedure is the so-called effective transient climate 52 response to cumulative emissions of carbon (ETCRE), since the emission scenarios contain other 53 anthropogenic emissions than CO 2 (Matthews et al., 2017). The ETCRE includes warming from other 54 greenhouse gases than CO 2 , most importantly methane, and for cooling effects due to atmospheric aerosols. 55 In contrast, the CO 2 -only TCRE is defined as the warming attributable to CO 2 forcing alone. One can 56 estimate the CO 2 -only TCRE from ESM experiments, driven by atmospheric CO 2 concentration increases 57 by 1% per year. The CO 2 emissions can be derived from the specified atmospheric CO 2 concentrations and 58 the modeled atmosphere-ocean and atmosphere-land CO 2 -fluxes, and hence the CO 2 -only TCRE can be 59 computed by dividing the GSAT increase by the cumulative emissions. Using 15 CMIP5 models, Gillett  The basis of these estimates are scenarios where atmospheric CO 2 concentration increases by 1% per 64 year, and not scenarios where we reduce emissions to mitigate climate change. The reason why this does not pose a problem is the above mentioned scenario-independence of the relation between the GSAT 66 and the cumulative emissions. The physical mechanism behind this scenario-independence is a subtle 67 Table 2). In these scenarios, the total emissions of various greenhouse gasses and aerosols emissions 91 are known, and we can obtain corresponding temperatures using a simplified version of the FaIR model 92 (Smith et al., 2018;Leach et al., 2020). To assess the uncertainties in RCBs, one should ideally explore 93 an ensemble of realistic mitigation scenarios using the full set of ESMs in the CMIP6 ensemble, which 94 is not feasible due to the computational costs. In this study, we parametrize the temperature response 95 module in our simple model by fitting those model parameters to the temperature response in two standard 96 CO 2 -forcing scenarios in each of the ESMs in the CMIP6 ensemble. Each of these simple response models 97 emulates the corresponding temperature response to total forcing in the ESM. Combining this temperature 98 module with the greenhouse gas and aerosol forcing module in the FaIR model we compute a temperature 99 response to each of the emission scenarios, and the resulting GSAT time series and CO 2 emission time 100 series in each of these model runs allows us to analyze the relationship between cumulative emissions and 101 peak temperatures, and estimate ETCRE and RCBs. Our simple modeling set-up, described in Section 2, is 102 based on generally accepted results from the climate modeling literature, while keeping them operational 103 and straightforward.

MODELLING SET-UP
We use a simple modeling set-up where atmospheric CO 2 concentrations are computed from the emissions, 105 E CO 2 (t), using the approach of Leach et al. (2020): where C CO 2 ,PI = 280 ppm is the pre-industrial concentration, and State-dependence is built into the model by letting α depend on the global temperature T (t) and the 110 cumulative uptake, The time t 0 refers to the year 1750. The model for α is 113 α(T, G u ) = g 0 exp r 0 + r u G u + r T T g 1 .

114
We model the concentrations of methane and nitrous oxide as linear responses of scenario data for 115 emissions:

123
The radiative forcing associated with greenhouse gas concentrations is computed using Eqs. 6-8 in (Smith 124 et al., 2018) with parameters presented in Table 3: where µ = 1 ppm. The number F 2×CO 2 is the forcing associated with a CO 2 -doubling. This number is 132 model-dependent and obtained from the Gregory plots for the abrupt 4×CO 2 experiments in the CMIP6 133 ensemble (Gregory et al., 2004). Aerosol forcing is modeled to be proportional to aerosol emissions: where the additional term All parameter values are listed in Table 3.

141
Our model for the temperature response is 142 To prevent statistical overfitting we use fixed, but well-separated times scales (Fredriksen and Rypdal, 2017) is the unit step function, and F(t) = F 2×CO 2 (log(1.01)/ log(2))t, for the two experiments, respectively.

151
The slow climate response, in this case the parameter d 3 , is not well constrained by 150-yr runs (Sanderson, 152 2020). However, the analyses in presented in this paper only concern GSAT up to the year 2100, and are 153 insensitive to this uncertainty. is not accounted for, induced by potentially changing feedbacks in the dynamics of the Earth system, . Nevertheless, it is still assumed that state-of-the-art models remain too stable (Valdes, 2011).

228
The presence of positive feedbacks and potential tipping points within the Earth system adds a layer of 229 uncertainty to RCBs that is extremely difficult to quantify.

CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted in the absence of any commercial or financial 231 relationships that could be construed as a potential conflict of interest.   Wm −2 /ppb −6.5 × 10 −7

TABLES AND FIGURES
1/(Mt SO 2 yr −1 ) 3.7 × 10 −3 β 3 1/(Mt BC yr −1 ) 13.9 × 10 −3 Table 3. Overview of the model parameters used to compute greenhouse gass concentrations, greenhouse forcing, and aerosol forcing, following the approaches in (Smith et al., 2018;Leach et al., 2020).   Table 1. The regression lines demonstrate approximately linear relationships between total positive CO 2 emissions between 2018 and 2100 and the maximum GSAT for the ensemble of emission scenarios for each of the 41 different climate models in the CMIP6 ensemble. ETCRE estimates are obtained from the slopes of regression lines.  T C R > 2 . 2 K T C R < 2 .2 K Figure 6. Each column of points shows maximum GSAT for a given emission scenario, so their spread indicates the variance over the ensemble of ESMs. The red points are for ESMs with TCR>2.2 K, and the blue points for ESMs with TCR<2.2 K.