<p>The conventional energy balance equation (EBE) is a first order linear differential equation driven by solar, volcanic and anthropogenic forcings. &#160;The differential term accounts for energy storage usually modelled as one or two &#8220;boxes&#8221;. &#160;Each box obeys Newton&#8217;s law of cooling, so that when perturbed, the Earth&#8217;s temperature relaxes exponentially to a thermodynamic equilibrium.</p><p>However, the spatial scaling obeyed by the atmosphere and its numerical models implies that the energy storage process is a scaling, power law process, a consequence largely of turbulent, hierarchically organized oceans currents but also hierarchies of land ice, soil moisture and other processes whose rates depend on size.</p><p>Scaling storage leads to power law relaxation and can be modelled via a seemingly trivial change - from integer to fractional order derivatives - the Fractional EBE (FEBE): with temperature derivatives order 0 < H &#160;< 1 rather than the EBE value H = 1. &#160;Mathematically the FEBE is a past value problem, not an initial value problem. &#160;&#160;&#160;Recent support for the FEBE comes from [Lovejoy, 2019a] who shows that the special H = 1/2 case (close to observations), the &#8220;Half-order EBE&#8221; (HEBE), can be analytically obtained from classical Budyko-Sellers energy balance models by improving the boundary conditions.</p><p>The FEBE simultaneously models the deterministic forced response to external (e.g. anthropogenic) forcing as well as the stochastic response to internal forcing (variability) [Lovejoy, 2019b]. &#160;We directly exploit both aspects to make projections based on historical data estimating the parameters using Bayesian inference.&#160; Using global instrumental temperature series, alongside CMIP5 and CMIP6 standard forcings, the basic FEBE parameters are H &#8776; 0.4 with a relaxation time &#8776; 4 years. &#160;</p><p>This observation-based model also produces projections for the coming century with forcings prescribed by the CMIP5 Representative Concentration Pathways scenarios and the CMIP6 Shared Socioeconomic Pathways.</p><p>We compare both generations of General Circulation Models (GCMs) outputs from CMIP5/6 alongside with the projections produced by the FEBE model which are entirely independent from GCMs, contributing to our understanding of forced climate variability in the past, present and future.&#160; When comparing to CMIP5 projections, we find that the mean projections are about 10- 15% lower while the uncertainties are roughly half as large. &#160;Our global temperature projections are therefore within the&#160; CMIP5 90% confidence limits and thus give them strong, independent support.</p><p>&#160;</p><p><strong>References</strong></p><p>Lovejoy, S., The half-order energy balance equation, J. Geophys. Res. (Atmos.), (submitted, Nov. 2019), 2019a.</p><p>Lovejoy, S., Fractional Relaxation noises, motions and the stochastic fractional relxation equation Nonlinear Proc. in Geophys. Disc., https://doi.org/10.5194/npg-2019-39, 2019b.</p>