Using the fractional energy balance equation for accurate temperature projections through 2100
Authors: Procyk Roman , Lovejoy Shaun , Del Rio Amador Lenin 
Published: March 2020

&lt;p&gt;The conventional energy balance equation (EBE) is a first order linear differential equation driven by solar, volcanic and anthropogenic forcings. &amp;#160;The differential term accounts for energy storage usually modelled as one or two &amp;#8220;boxes&amp;#8221;. &amp;#160;Each box obeys Newton&amp;#8217;s law of cooling, so that when perturbed, the Earth&amp;#8217;s temperature relaxes exponentially to a thermodynamic equilibrium.&lt;/p&gt;&lt;p&gt;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.&lt;/p&gt;&lt;p&gt;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 &lt; H &amp;#160;&lt; 1 rather than the EBE value H = 1. &amp;#160;Mathematically the FEBE is a past value problem, not an initial value problem. &amp;#160;&amp;#160;&amp;#160;Recent support for the FEBE comes from [Lovejoy, 2019a] who shows that the special H = 1/2 case (close to observations), the &amp;#8220;Half-order EBE&amp;#8221; (HEBE), can be analytically obtained from classical Budyko-Sellers energy balance models by improving the boundary conditions.&lt;/p&gt;&lt;p&gt;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]. &amp;#160;We directly exploit both aspects to make projections based on historical data estimating the parameters using Bayesian inference.&amp;#160; Using global instrumental temperature series, alongside CMIP5 and CMIP6 standard forcings, the basic FEBE parameters are H &amp;#8776; 0.4 with a relaxation time &amp;#8776; 4 years. &amp;#160;&lt;/p&gt;&lt;p&gt;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.&lt;/p&gt;&lt;p&gt;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.&amp;#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. &amp;#160;Our global temperature projections are therefore within the&amp;#160; CMIP5 90% confidence limits and thus give them strong, independent support.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Lovejoy, S., The half-order energy balance equation, J. Geophys. Res. (Atmos.), (submitted, Nov. 2019), 2019a.&lt;/p&gt;&lt;p&gt;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.&lt;/p&gt;