Statistical estimation of global surface temperature response to forcing under the assumption of temporal scaling

Quart J Royal Meteoro Soc

Procyk

Hébert

et al. 2021

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.

Section: Scaling Storagementioning

confidence: 99%

The fractional energy balance equation

Quart J Royal Meteoro Soc

Procyk

Hébert

et al. 2021

“…However the use of pure power law CRFs (e.g. (Rypdal, 2012), (Myrvoll-Nilsen et al, 2020) leads to divergences: the "runaway Green's function effect" (Hébert and Lovejoy, 2015) so that if the Earth is perturbed by even an infinitesimal step function forcing, its temperature monotonically increases without ever attaining thermodynamic equilibrium: its equilibrium climate sensitivity (ECS) is infinite. Whereas the classical EBMs conserve energy but violate scaling, the pure scaling CRF models are scaling but violate energy conservation.…”

Section: Introductionmentioning

confidence: 99%

The Fractional Energy Balance Equation for Climate projections through 2100

Procyk

Hébert

2020

Preprint

Abstract. We produce climate projections through the 21st century using 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 FEBE has two shape parameters: a scaling exponent H and relaxation time τ; its amplitude parameter is the equilibrium climate sensitivity (ECS). Two additional parameters were needed for the forcing: an aerosol re-calibration factor α to account for the large aerosol uncertainty, and a volcanic intermittency correction exponent ν. A Bayesian framework based on historical temperatures and natural and anthropogenic forcing series was used for parameter estimation. Significantly, the error model was not ad hoc, but was predicted by the model itself: the internal variability response to white noise internal forcing. The 90 % Confidence Interval (CI) of the shape parameters were H = [0.33, 0.44] (median = 0.38), τ = [2.4, 7.0] (median = 4.7) years compared to the usual EBE H = 1, and literature values τ typically in the range 2–8 years. We found that aerosols were too strong by an average factor α = [0.2, 1.0] (median = 0.6) and the volcanic intermittency correction exponent was ν = [0.15, 0.41] (median = 0.28) compared to standard values α = ν = 1. The overpowered aerosols support a revision of the global modern (2005) aerosol forcing 90 % CI to a narrower range [−1.0, −0.2] W m−2 compared with the IPCC AR5 range [1.5, 4.5] K (median = 3.2 K). Similarly, we found the transient climate sensitivity (TCR) = [1.2, 1.8] K (median = 1.5 K) compared to the AR5 range TCR = [1.0, 2.5] K (median = 1.8 K). As commonly seen in other observational-based studies, the FEBE values are therefore somewhat lower but still consistent with those in IPCC AR5. Using these parameters we made projections to 2100 using both the Representative Carbon Pathways (RCP) and Shared Socioeconomic Pathways (SSP) scenarios and shown alongside the CMIP5/6 MME. The FEBE hindprojections (1880–2019) closely follow observations (notably during the hiatus, 1998–2015). Overall the FEBE were 10–15 % lower but due to their smaller uncertainties, their 90 % CIs lie completely within the GCM 90 % CIs. The FEBE thus complements and supports the GCMs.

“…Q is the mean top-of-the-atmosphere flux (≈ 341 W m 2 ), S(x) is the dimensionless local solar constant with local coalbedo a(x) (in the notation of North and Kim, 2017), and the time-dependent part of the radiative balance is specified by the additional incoming energy flux, the "forcing" F (x, t). Although in this paper I mostly ignore temporal albedo variations (see, however, Sect.…”

Section: Radiative Heat Fluxesmentioning

confidence: 99%

“…There may also be a small advection velocity v; it is not considered to be a true physical velocity but only an ad hoc parameter needed to prevent κ h from being negative (Sellers, 1969). The standard presentation (North et al, 1981;North and Kim, 2017) avoids the problem by using the diffusivity as in Sect. 3.1.…”

Section: The Climatological Temperature Distribution and Budyko-sellers Modelsmentioning

confidence: 99%

“…Ever since Budyko (1969) and Sellers (1969) proposed a simple model describing the exchange of energy between the Earth and outer space, energy balance models (EBMs) have provided a straightforward way of understanding past, present, and possible future climates. The models usually have either zero or one spatial dimension, respectively representing the globally or latitudinally averaged meridional temperature distribution (for a review, see McGuffie andHenderson-Sellers, 2005, andKim, 2017).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The half-order energy balance equation – Part 1: The homogeneous HEBE and long memories

2021

Earth Syst. Dynam.

Abstract. The original Budyko–Sellers type of 1D energy balance models (EBMs) consider the Earth system averaged over long times and apply the continuum mechanics heat equation. When these and the more phenomenological box models are extended to include time-varying anomalies, they have a key weakness: neither model explicitly nor realistically treats the conductive–radiative surface boundary condition that is necessary for a correct treatment of energy storage. In this first of a two-part series, I apply standard Laplace and Fourier techniques to the continuum mechanics heat equation, solving it with the correct radiative–conductive boundary conditions and obtaining an equation directly for the surface temperature anomalies in terms of the anomalous forcing. Although classical, this equation is half-ordered and not integer-ordered: the half-order energy balance equation (HEBE). A quite general consequence is that although Newton's law of cooling holds, the heat flux across surfaces is proportional to a half-ordered (not first-ordered) time derivative of the surface temperature. This implies that the surface heat flux has a long memory, that it depends on the entire previous history of the forcing, and that the temperature–heat flux relationship is no longer instantaneous. I then consider the case in which the Earth is periodically forced. The classical case is diurnal heat forcing; I extend this to annual conductive–radiative forcing and show that the surface thermal impedance is a complex valued quantity equal to the (complex) climate sensitivity. Using a simple semi-empirical model of the forcing, I show how the HEBE can account for the phase lag between the summer maximum forcing and maximum surface temperature Earth response. In Part 2, I extend all these results to spatially inhomogeneous forcing and to the full horizontally inhomogeneous problem with spatially varying specific heats, diffusivities, advection velocities, and climate sensitivities. I consider the consequences for macroweather (monthly, seasonal, interannual) forecasting and climate projections.