The Half-order Energy Balance Equation, Part 1: The homogeneous HEBE and long memories

Hébert

2020

Preprint

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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.

Section: Discussionsupporting

confidence: 71%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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The Fractional Energy Balance Equation for Climate projections through 2100

Procyk

Hébert

2020

Preprint

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“…First, although the domain of the FEBE discussed here – macroweather in time – has low intermittency (it is not far from being quasi‐Gaussian, we model it as such), it is in fact the only dynamical atmospheric regime where the multifractality is low. Yet in the spatial domain, macroweather multifractality is on the contrary very high (Lovejoy, 2018), so that for regional (space–time FEBE: Lovejoy, 2021a; 2021b) multifractality is important. Second, it is worth pointing out that there is also a large mathematical literature on fBm and fGn processes that characterize them parametrically.…”

Section: Estimating H τmentioning

confidence: 99%

The fractional energy balance equation

Quart J Royal Meteoro Soc

Procyk

Hébert

et al. 2021

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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.

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

Amador

2021

Clim Dyn

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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.