The Regional Fractional Energy Balance Equation: Projections to 2100

We present the fractional energy balance equation (FEBE) which is a generalization of the standard EBE. The FEBE can be derived either from Budyko–Sellers models or phenomenologically by applying the scaling symmetry to energy storage processes. It is easily implemented by changing the integer order of the storage (derivative) term in the EBE to a fractional value near 1/2. 

The model used a Bayesian framework based on historical temperatures and natural and anthropogenic forcing series for parameter estimation. Significantly, the error model was not ad hoc, rather predicted by the model itself: the internal variability response to white noise internal forcing, a fraction Relaxation noise (fRn). Due to computational constraints, we employ a block bootstrapping method to calculate the likelihoods of our parameters in the Bayesian scheme. Notably we estimate the regional relaxation time directly from empirical data, generally it is calculated for various discrete surface types using heat capacities or globally from fitting a two-box model to GCM outputs, which to the authors knowledge has not been estimated prior to this study. 

The FEBE historical reconstructions (1880–2020) closely follow observations (notably during the “slowdown”, 1998–2015). We also reproduce the internal variability with the FEBE and statistically validate this against centennial scale temperature observations. We show the FEBE to plausibly reproduce the annual cycle at monthly resolution, in particular to explain the lag between the temperature maximum and the maximum in the radiative forcing. 

Using the calibrated FEBE we made temperature projections to 2100 using both the Representative Carbon Pathways (RCP) and Shared Socioeconomic Pathways (SSP) scenarios, shown alongside the Coupled Model Intercomparison Project Phase (CMIP) 5 and 6 multi-model ensemble (MME) at global and regional scales.