2 Methods and data
2.1 Impulse-response step
model
We use a step model (Good et al. 2011) to provide a benchmark of EBM
emulator performance for temperature projections. The step-response
function for each AOGCM was derived by dividing the projected
temperature changes from a single realization of a CMIP6 abrupt-4xCO2
simulation by the radiative forcing for 4xCO2 (Byrne & Goldblatt 2013).
The step-response function was smoothed using cubic splines, and linear
regession (years 121-150) was used for extrapolation beyond the 150
years of the abrupt-4xCO2 simulations. Temperature projections from the
step model were produced by convolution of annual changes in ERF and the
step-response functions.
2.2 Two-layer EBM
In the two-layer EBM (EBM2) (Held et al. 2010; Geoffroy et al. 2013a)
the upper layer represents the Earth’s atmosphere, land surface and
ocean mixed layer, and the lower layer represents the deep ocean. The
rate of temperature change in each model layer is determined from:
\(C_{1}\frac{dT_{1}}{\text{dt}}=F+\lambda T_{1}-\varepsilon\gamma(T_{1}-T_{0})\)(1)
\(C_{0}\frac{dT_{0}}{\text{dt}}=\gamma(T_{1}-T_{0})\) (2)
Where C representations heat capacity, T temperature, F ERF, λ the
climate feedback parameter and γ the heat transfer coefficient between
the upper layer (layer 1) and the lower layer (layer 0). We follow the
formulation of Geoffroy et al. (2013b) which includes an efficacy
parameter for deep ocean heat uptake (ε) to account for the forced
pattern effect in surface temperature (Stevens et al. 2016). As is
commonplace (Geoffroy et al. 2013a, b; Gregory et al. 2015; Cummins et
al., 2020), the EBM2 parameters (Table S1) were calibrated for each
AOGCM using a single realization of a CMIP6 abrupt-4xCO2 simulation
(Table S1).
2.3 Calibration of EBM2 using linear
optimization
As an alternative to abrupt-4xCO2 calibration, we use a linear
optimization algorithm (scipy.optimize.minimize v1.6.2) to optimize the
λ and ε parameters by minimizing the root mean squared error (RMSE) of
the emulated temperatures compared to the AOGCM. The temperature
projections are less sensitive to changes in the other EBM2 parameters
(i.e., C0, C1, and γ), so these
parameters are unchanged from their abrupt-4xCO2 calibrations. We also
applied the linear optimization methodology to the abrupt-4xCO2
simulations and affirmed the calibrated parameter values of Geoffroy et
al. (2013b).
2.4 Three-layer EBM
We use a three-layer EBM (EBM3) (Cummins et al. 2020) as a second
benchmark for EBM2 performance. We follow the method of Cummins et al.
(2020) to calibrate EBM3 parameters for each AOGCM using a single
realization of a CMIP6 abrupt-4xCO2 simulation.
2.5 Data
We use projections of global annual mean near-surface temperature and
radiative fluxes at the top of atmosphere (TOA) from the CMIP6 archive.
We emulate temperatures for eight AOGCMs selected because data was
available for the CMIP6 experiments of interest. For projections of
recent and future climate change, the Historical and SSP experiments
were used. The Detection and Attribution Model Intercomparison Project
(DAMIP) experiments (Gillett et al. 2016) are used for projections of
temperature change attributed to different sources of ERF. RFMIP
experiments (Pincus et al. 2016; Smith et al. 2021) are used for
estimates of ERF during the historical period and ERF projected to 2100
under SSP2-4.5. Following Forster et al. (2013), unforced drift is
removed from the AOGCM projections using the preindustrial control
simulation.