The Fractional Energy Balance Equation for Climate projections through 2100

Procyk, Roman; Lovejoy, S.; Hébert, Raphaël

doi:10.5194/esd-2020-48

Cited by 9 publications

(36 citation statements)

References 52 publications

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“…This conclusion is therefore important not only for scientists but also for policy makers, stressing the need for mitigation and adaptation measures. The SCRF is a the first of a new family of approaches that directly model the temperature at monthly scales and longer, see for example Procyk et al (2020) that, in addition to scaling, directly exploits the principles of energy-balance (the fractional energy-balance equation) and produces projections and climate sensitivity estimates in a similar range, although with a better estimation of the scaling exponent H by also utilizing the macroweather high-frequency response.…”

Section: Resultsmentioning

confidence: 99%

“…For comparison, the key exponent H (denoted by q in his notation) was estimated as −0.15 compared to our value of −0.5 +0.4 −0.5 (below). An approach related to the scaling CRF we develop below is that of Procyk et al (2020) which is based on the Earth energy balance. A fractional generalization of conventional box models was considered and solved using Mittag-Leffler functions, often called generalized exponentials, also characterized by a scaling exponent H which was estimated as H ∈ [0.33, 0.44] and is equivalent to the negative of the SCRF's H (below).…”

Section: The Linear Response Frameworkmentioning

confidence: 99%

“…an infinite ECS. An interesting alternative to model high-and lowfrequency regimes with scaling regimes can be obtained by the fractional generalization of the energy-balance equation, leading to a CRF with a similar (convergent) low-frequency behaviour as the SCRF (Procyk et al 2020).…”

Section: Scaling Relaxation Processesmentioning

confidence: 99%

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An observation-based scaling model for climate sensitivity estimates and global projections to 2100

2020

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We directly exploit the stochasticity of the internal variability, and the linearity of the forced response to make global temperature projections based on historical data and a Green’s function, or Climate Response Function (CRF). To make the problem tractable, we take advantage of the temporal scaling symmetry to define a scaling CRF characterized by the scaling exponent H, which controls the long-range memory of the climate, i.e. how fast the system tends toward a steady-state, and an inner scale $$\tau \approx 2$$ τ ≈ 2 years below which the higher-frequency response is smoothed out. An aerosol scaling factor and a non-linear volcanic damping exponent were introduced to account for the large uncertainty in these forcings. We estimate the model and forcing parameters by Bayesian inference which allows us to analytically calculate the transient climate response and the equilibrium climate sensitivity as: $$1.7^{+0.3} _{-0.2}$$ 1 . 7 - 0.2 + 0.3 K and $$2.4^{+1.3} _{-0.6}$$ 2 . 4 - 0.6 + 1.3 K respectively (likely range). Projections to 2100 according to the RCP 2.6, 4.5 and 8.5 scenarios yield warmings with respect to 1880–1910 of: $$1.5^{+0.4}_{-0.2}K$$ 1 . 5 - 0.2 + 0.4 K , $$2.3^{+0.7}_{-0.5}$$ 2 . 3 - 0.5 + 0.7 K and $$4.2^{+1.3}_{-0.9}$$ 4 . 2 - 0.9 + 1.3 K. These projection estimates are lower than the ones based on a Coupled Model Intercomparison Project phase 5 multi-model ensemble; more importantly, their uncertainties are smaller and only depend on historical temperature and forcing series. The key uncertainty is due to aerosol forcings; we find a modern (2005) forcing value of $$[-1.0, -0.3]\, \,\,\mathrm{Wm} ^{-2}$$ [ - 1.0 , - 0.3 ] Wm - 2 (90 % confidence interval) with median at $$-0.7 \,\,\mathrm{Wm} ^{-2}$$ - 0.7 Wm - 2 . Projecting to 2100, we find that to keep the warming below 1.5 K, future emissions must undergo cuts similar to RCP 2.6 for which the probability to remain under 1.5 K is 48 %. RCP 4.5 and RCP 8.5-like futures overshoot with very high probability.

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Section: Resultsmentioning

confidence: 99%

Section: The Linear Response Frameworkmentioning

confidence: 99%

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An observation-based scaling model for climate sensitivity estimates and global projections to 2100

2020

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“…Using the empirical value H F ≈ −0.5 thatHébert et al (2021) obtained by (Bayesian) fitting the global temperature series since 1880 to the IPCC CO 2eq forcing, we find h ≈ 0.5. Similarly,Procyk et al (2020) fitted the FEBE (using the exact G 0,h Green's function with the same centennial-scale climate forcing) to estimate h = 0.38 ± 0.06. In the next section we consider the high-frequency behaviour and use the internal variability (assumed to be forced by a Gaussian white noise) that gives the more precise estimate h ≈ 0.42 ± 0.02 (del Rio Amador andLovejoy, 2019).…”

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confidence: 99%

The fractional energy balance equation

Lovejoy

Procyk

Hébert

et al. 2021

Quart J Royal Meteoro Soc

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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.

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“…It is also accurate enough for obtaining the detrended temperature anomalies, whose characterization, modelling and prediction are the focus of the following sections. A more accurate method that takes into account the physics of the system adding memory effects to the heat balance equation, was presented in (Procyk et al 2020).…”

Section: Data Preprocessingmentioning

confidence: 99%

Using regional scaling for temperature forecasts with the Stochastic Seasonal to Interannual Prediction System (StocSIPS).

Amador

Lovejoy

2021

Preprint

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Over time scales between 10 days and 10–20 years – the macroweather regime – atmospheric fields, including the temperature, respect statistical scale symmetries, such as power-law correlations, that imply the existence of a huge memory in the system that can be exploited for long-term forecasts. The Stochastic Seasonal to Interannual Prediction System (StocSIPS) is a stochastic model that exploits these symmetries to perform long-term forecasts. It models the temperature as the high-frequency limit of the (fractional) energy balance equation (fractional Gaussian noise) which governs radiative equilibrium processes when the relevant equilibrium relaxation processes are power law, rather than exponential. They are obtained when the order of the relaxation equation is fractional rather than integer and they are solved as past value problems rather than initial value problems. StocSIPS was first developed for monthly and seasonal forecast of globally averaged temperature. In this paper, we extend it to the prediction of the spatially resolved temperature field by treating each grid point as an independent time series. Compared to traditional global circulation models (GCMs), StocSIPS has the advantage of forcing predictions to converge to the real-world climate. It extracts the internal variability (weather noise) directly from past data and does not suffer from model drift. Here we apply StocSIPS to obtain monthly and seasonal predictions of the surface temperature and show some preliminary comparison with multi-model ensemble (MME) GCM results. For one month lead time, our simple stochastic model shows similar values of the skill scores than the much more complex deterministic models.

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The Fractional Energy Balance Equation for Climate projections through 2100

Cited by 9 publications

References 52 publications

An observation-based scaling model for climate sensitivity estimates and global projections to 2100

An observation-based scaling model for climate sensitivity estimates and global projections to 2100

The fractional energy balance equation

Using regional scaling for temperature forecasts with the Stochastic Seasonal to Interannual Prediction System (StocSIPS).

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