Quarterly Journal of the Royal Meteorological Society | 2021
The fractional energy balance equation
Abstract
Energy conservation is a useful symmetry for understanding the climate: Energy Balance Models (EBMs) of Budyko-Sellers or box type. Extending these to time varying anomalies involves energy storage; possible mechanisms can be constrained by a second symmetry: scale invariance. Temporal storage scaling scale follows from the spatial scale invariance of the dynamics. Physically, it corresponds to a hierarchy of storage processes. The 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, it relaxes to thermodynamic equilibrium via a slow power law process: H = 1 is the exceptional 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 < 1. We solve the FEBE using Green’s functions, whose high and low frequency limits are power laws with a transition at the relaxation scale (several years). When stochastically forced the high frequency part of the internal variability are fractional Gaussian noises; when deterministically forced, the low frequency responses have been used for climate projections: the FEBE is linked to the Fluctuation Dissipation Theorem. 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 temperature. The FEBE transient to equilibrium climate sensitivity ratio is compatible with GCM estimates. A simple ramp forcing model of the industrial epoch warming combining deterministic (external forcing) with stochastic (internal forcing), is statistically validated against centennial scale temperature series.