The fractional energy balance equation for climate projections through 2100
Authors: Procyk Roman , Lovejoy Shaun , Hébert Raphael 
Published: January 2022

Abstract. We produce climate projections through the 21st century using the fractional energy balance equation (FEBE): a generalization of the standard energy balance equation (EBE). The FEBE can be derived from Budyko–Sellers models or phenomenologically through the application of the scaling symmetry to energy storage processes, easily implemented by changing the integer order of the storage (derivative) term in the EBE to a fractional value. The FEBE is defined by three parameters: a fundamental shape parameter, a timescale and an amplitude, corresponding to, respectively, the scaling exponent h, the relaxation time τ and the equilibrium climate sensitivity (ECS). Two additional parameters were needed for the forcing: an aerosol recalibration factor α to account for the large aerosol uncertainty and a volcanic intermittency correction exponent ν. A Bayesian framework based on historical temperatures and natural and anthropogenic forcing series was used for parameter estimation. Significantly, the error model was not ad hoc but rather predicted by the model itself: the internal variability response to white noise internal forcing. The 90 % credible interval (CI) of the exponent and relaxation time were h=[0.33, 0.44] (median = 0.38) and τ=[2.4, 7.0] (median = 4.7) years compared to the usual EBE h=1, and literature values of τ typically in the range 2–8 years. Aerosol forcings were too strong, requiring a decrease by an average factor α=[0.2, 1.0] (median = 0.6); the volcanic intermittency correction exponent was
ν=[0.15, 0.41] (median = 0.28) compared to standard values α=ν=1. The overpowered aerosols support a revision of the global modern (2005) aerosol forcing 90 % CI to a narrower range [−1.0, −0.2] W m−2. The key parameter ECS in comparison to IPCC AR5 (and to the CMIP6 MME), the 90 % CI range is reduced from [1.5, 4.5] K ([2.0, 5.5] K) to [1.6, 2.4] K ([1.5, 2.2] K), with median value lowered from 3.0 K (3.7 K) to 2.0 K (1.8 K). Similarly we found for the transient climate response (TCR), the 90 % CI range shrinks from [1.0, 2.5] K ([1.2, 2.8] K) to [1.2, 1.8] K ([1.1, 1.6] K) and the median estimate decreases from 1.8 K (2.0 K) to 1.5 K (1.4 K). As often seen in other observational-based studies, the FEBE values for climate sensitivities are therefore somewhat lower but still consistent with those in IPCC AR5 and the CMIP6 MME. Using these parameters, we made projections to 2100 using both the Representative Concentration Pathway (RCP) and Shared Socioeconomic Pathway (SSP) scenarios, and compared them to the corresponding CMIP5 and CMIP6 multi-model ensembles (MMEs). The FEBE historical reconstructions (1880–2020) closely follow observations, notably during the 1998–2014 slowdown (“hiatus”). We also reproduce the internal variability with the FEBE and statistically validate this against centennial-scale temperature observations. Overall, the FEBE projections were 10 %–15 % lower but due to their smaller uncertainties, their 90 % CIs lie completely within the GCM 90 % CIs. This agreement means that the FEBE validates the MME, and vice versa.