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The fractional energy balance equation

Classical Energy Balance Equations (EBEs) are differential equations of integer order (h = 1), here we generalize this to fractional orders: the Fractional EBE (FEBE, 0 < h ≤ 1). In the FEBE, when the Earth is perturbed by a forcing, the temperature relaxes to equilibrium via a slow power‐law process: h = 1 is the exceptional (but standard) exponential case. Our FEBE derivation is phenomenological, it complements derivations based on the classical continuum mechanics heat equation (that imply h = 1/2 for the surface temperature) and of the more general Fractional Heat Equation which allows for 0 < h < 2. Unlike some of the earlier “scale free” models based purely on scaling, the FEBE has an extra blackbody radiation term that allows for energy balance. It therefore has two scaling regimes (not one), it has the advantage of being stable to infinitesimal step‐function perturbations and it has a finite Equilibrium Climate Sensitivity. We solve the FEBE using Green's functions, whose high‐ and low‐frequency limits are power laws with a relaxation scale transition (several years). When stochastically forced, the high‐frequency parts of the internal variability are fractional Gaussian noises that can be used for monthly and seasonal forecasts; when deterministically forced, the low‐frequency response describes the consequences of anthropogenic forcing, it has been used for climate projections. The FEBE introduces complex climate sensitivities that are convenient for handling periodic (especially annual) forcing. The FEBE obeys Newton's law of cooling, but the heat flux crossing a surface nonetheless depends on the fractional time derivative of the temperature. The FEBE's ratio of transient to equilibrium climate sensitivity is compatible with GCM estimates. A simple ramp forcing model of the industrial‐epoch warming combining deterministic (external) with stochastic (internal) forcing is statistically validated against centennial‐scale temperature series.