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.

Abstract. On scales of ≈ 10 days (the lifetime of planetary-scale structures), there is a drastic transition from high-frequency weather to low-frequency macroweather. This scale is close to the predictability limits of deterministic atmospheric models; thus, in GCM (general circulation model) macroweather forecasts, the weather is a high-frequency noise. However, neither the GCM noise nor the GCM climate is fully realistic. In this paper we show how simple stochastic models can be developed that use empirical data to force the statistics and climate to be realistic so that even a two-parameter model can perform as well as GCMs for annual global temperature forecasts.The key is to exploit the scaling of the dynamics and the large stochastic memories that we quantify. Since macroweather temporal (but not spatial) intermittency is low, we propose using the simplest model based on fractional Gaussian noise (fGn): the ScaLIng Macroweather Model (SLIMM). SLIMM is based on a stochastic ordinary differential equation, differing from usual linear stochastic models (such as the linear inverse modelling -LIM) in that it is of fractional rather than integer order. Whereas LIM implicitly assumes that there is no lowfrequency memory, SLIMM has a huge memory that can be exploited. Although the basic mathematical forecast problem for fGn has been solved, we approach the problem in an original manner, notably using the method of innovations to obtain simpler results on forecast skill and on the size of the effective system memory.A key to successful stochastic forecasts of natural macroweather variability is to first remove the lowfrequency anthropogenic component. A previous attempt to use fGn for forecasts had disappointing results because this was not done. We validate our theory using hindcasts of global and Northern Hemisphere temperatures at monthly and annual resolutions. Several nondimensional measures of forecast skill -with no adjustable parameters -show excellent agreement with hindcasts, and these show some skill even on decadal scales. We also compare our forecast errors with those of several GCM experiments (with and without initialization) and with other stochastic forecasts, showing that even this simplest two parameter SLIMM is somewhat superior. In future, using a space-time (regionalized) generalization of SLIMM, we expect to be able to exploit the system memory more extensively and obtain even more realistic forecasts.

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.

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.

Reliable climate projections at the regional scale are needed in order to evaluate climate change impacts and inform policy. We develop an alternative method for projections based on the transient climate sensitivity (TCS), which relies on a linear relationship between the forced temperature response and the strongly increasing anthropogenic forcing. The TCS is evaluated at the regional scale (5° by 5°), and projections are made accordingly to 2100 using the high and low Representative Concentration Pathways emission scenarios. We find that there are large spatial discrepancies between the regional TCS from 5 historical data sets and 32 global climate model (GCM) historical runs and furthermore that the global mean GCM TCS is about 15% too high. Given that the GCM Representative Concentration Pathway scenario runs are mostly linear with respect to their (inadequate) TCS, we conclude that historical methods of regional projection are better suited given that they are directly calibrated on the real world (historical) climate.

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Abstract. We produce climate projections through the 21st century using the fractional energy balance equation (FEBE): a generalization of the standard energy balance equation (EBE). The FEBE can be derived from Budyko–Sellers models or phenomenologically through the application of the scaling symmetry to energy storage processes, easily implemented by changing the integer order of the storage (derivative) term in the EBE to a fractional value. The FEBE is defined by three parameters: a fundamental shape parameter, a timescale and an amplitude, corresponding to, respectively, the scaling exponent h, the relaxation time τ and the equilibrium climate sensitivity (ECS). Two additional parameters were needed for the forcing: an aerosol recalibration 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 rather predicted by the model itself: the internal variability response to white noise internal forcing. The 90 % credible interval (CI) of the exponent and relaxation time were h=[0.33, 0.44] (median = 0.38) and τ=[2.4, 7.0] (median = 4.7) years compared to the usual EBE h=1, and literature values of τ typically in the range 2–8 years. Aerosol forcings were too strong, requiring a decrease by an average factor α=[0.2, 1.0] (median = 0.6); 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. The key parameter ECS in comparison to IPCC AR5 (and to the CMIP6 MME), the 90 % CI range is reduced from [1.5, 4.5] K ([2.0, 5.5] K) to [1.6, 2.4] K ([1.5, 2.2] K), with median value lowered from 3.0 K (3.7 K) to 2.0 K (1.8 K). Similarly we found for the transient climate response (TCR), the 90 % CI range shrinks from [1.0, 2.5] K ([1.2, 2.8] K) to [1.2, 1.8] K ([1.1, 1.6] K) and the median estimate decreases from 1.8 K (2.0 K) to 1.5 K (1.4 K). As often seen in other observational-based studies, the FEBE values for climate sensitivities are therefore somewhat lower but still consistent with those in IPCC AR5 and the CMIP6 MME. Using these parameters, we made projections to 2100 using both the Representative Concentration Pathway (RCP) and Shared Socioeconomic Pathway (SSP) scenarios, and compared them to the corresponding CMIP5 and CMIP6 multi-model ensembles (MMEs). The FEBE historical reconstructions (1880–2020) closely follow observations, notably during the 1998–2014 slowdown (“hiatus”). We also reproduce the internal variability with the FEBE and statistically validate this against centennial-scale temperature observations. Overall, the FEBE projections were 10 %–15 % lower but due to their smaller uncertainties, their 90 % CIs lie completely within the GCM 90 % CIs. This agreement means that the FEBE validates the MME, and vice versa.

Abstract. It is virtually certain that the mean surface temperature of the Earth will continue to increase under realistic emission scenarios, yet comparatively little is known about future changes in climate variability. This study explores changes in climate variability over the large range of climates simulated by the Coupled Model Intercomparison Project Phase 5 and 6 (CMIP5/6) and the Paleoclimate Modeling Intercomparison Project Phase 3 (PMIP3), including time slices of the Last Glacial Maximum, the mid-Holocene, and idealized experiments (1 % CO2 and abrupt4×CO2). These states encompass climates within a range of 12 ∘C in global mean temperature change. We examine climate variability from the perspectives of local interannual change, coherent climate modes, and through compositing extremes. The change in the interannual variability of precipitation is strongly dependent upon the local change in the total amount of precipitation. At the global scale, temperature variability is inversely related to mean temperature change on intra-seasonal to multidecadal timescales. This decrease is stronger over the oceans, while there is increased temperature variability over subtropical land areas (40∘ S–40∘ N) in warmer simulations. We systematically investigate changes in the standard deviation of modes of climate variability, including the North Atlantic Oscillation, the El Niño–Southern Oscillation, and the Southern Annular Mode, with global mean temperature change. While several climate modes do show consistent relationships (most notably the Atlantic Zonal Mode), no generalizable pattern emerges. By compositing extreme precipitation years across the ensemble, we demonstrate that the same large-scale modes influencing rainfall variability in Mediterranean climates persist throughout paleoclimate and future simulations. The robust nature of the response of climate variability, between cold and warm climates as well as across multiple timescales, suggests that observations and proxy reconstructions could provide a meaningful constraint on climate variability in future projections.