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

The 90 % credible interval (CI) of the exponent and relaxation time were

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.

The Earth is a complex, heterogenous system with turbulent atmospheric and oceanic processes operating over scales ranging from millimetres up to planetary scales. When considered by timescale, there are three main regimes: the weather, macroweather and climate

A major challenge is to determine the Earth's decadal and centennial response to anthropogenic and natural perturbations. At the moment, projection uncertainties – famously exemplified in the range 1.5–4.5 K for a

In order to construct macroweather and climate models, beyond linearity and stochasticity, we require additional model constraints, the classical one being energy balance. Starting with the first energy balance models (EBMs) proposed by

Energy conservation is an important symmetry principle, yet when implemented in box-type models, it violates another symmetry: scale invariance. This is because box models are integer ordered differential equations whose response functions (Green's functions) are exponentials

To make more realistic models, the key issue is energy storage. Storage is a consequence of imbalances in incoming short wave and outgoing long wave radiation and it must be accounted for in applications of the energy balance principle

To understand the FEBE's key new features, recall that linear differential equations can be solved with Green’s functions; in the classical integer-ordered case, these are based on exponentials. However, in the general case where one or more terms are of fractional order, they are instead based on “generalized exponentials”, themselves based on power laws. In the FEBE, there are two distinct power-law regimes with a transition at the relaxation time (estimated to be of the order of a few years; see below). While the low-frequency Green's function can be very close to

The following text introduces the FEBE (Sect.

The zero-dimensional FEBE may be written as

If we solve the FEBE using Green's functions, we obtain

Mathematically, when

To see if this is compatible with the value estimated from the low-frequency response to external forcings, consider the low-frequency behaviour (

We consider natural and anthropogenic sources of external forcing: solar and volcanic, greenhouse gases and aerosols. We use the standard semi-empirical carbon-dioxide-concentration-to-forcing relationship

We follow the CMIP5 recommendations for anthropogenic and solar forcing, while volcanic forcing is unprescribed

The global climate is warming and most of the observed changes are due to increases in the concentration of anthropogenic greenhouse gases (GHGs)

The RCP scenarios are derived from estimates of emissions computed by a set of integrated assessment models (IAMs); these emissions are converted to concentrations using the Model for the Assessment of Greenhouse-gas Induced Climate Change

The wide spread between the scenarios allows for the investigation of the consequences of various future policies, from strong mitigation (RCP2.6, SSP1-26) to no-policy reference (RCP8.5, SSP5-85) shown in Fig.

In this paper, we use the forcing due to carbon dioxide equivalent,

Aerosols are a strong component of radiative forcing associated with anthropogenic emissions, resulting from a combination of direct and indirect aerosol effects. There exists high uncertainty of the aerosol forcing, arising from a poor understanding of how clouds respond to aerosol perturbations

We obtained the CMIP5 aerosol forcing from the total

The total amount of aerosol forcing in 2005 given at the 90 % CI in the IPCC Fifth Assessment Report (AR5) is [

The prescribed CMIP6 SSP aerosol forcing,

The other external forcings considered are solar and volcanic. Although there exist other natural forcings such as mineral dust and sea salt, they are small and will be implicitly included with the internal variability. We use the CMIP5 recommendation for solar forcing,

Volcanic forcing

The volcanic forcing series,

It is well established that volcanic forcing must be scaled down by 40 %–50 % in order to produce a comparable effect on surface temperature, and thus most EBMs linearly scale volcanic forcing

The normalization is such that the mean is unchanged:

The volcanic response appears to be non-linear as the intermittency (“spikiness”, sparseness of the spikes) parameter

We consider the standard assumption about internal variability that it is forced by a Gaussian “delta-correlated” white noise

Using

Working in a linear framework, we write the forcing series,

We used five historical records of surface air temperature for our analysis each spanning the period 1880–2020, with median monthly temperature anomalies in relation to the reference period of 1880–1910: Hadley Centre/Climatic Research Unit Temperature version 4 (HadCrut4,

The HadCRUT4 dataset is a combination of the sea-surface temperature records: HadSST3 was compiled by the Hadley Centre of the UK Met Office along with land surface station records: CRUTEM4 from the Climate Research Unit in East Anglia; the Cowtan and Way dataset uses HadCRUT4 as raw data but interpolates missing data that would lead to bias especially at high latitudes by infilling missing data using an optimal interpolation algorithm (kriging); we use the dataset with land air temperature anomalies interpolated over sea ice. The GISTEMP dataset combines the Global Historical Climate Network version 3 (GHCNv3) land surface air temperature records with the Extended Reconstructed Sea Surface Temperature version 4 (ERSST) along with the temperature dataset from the Scientific Community on Antarctic Research (SCAR) and is compiled by the Goddard Institute for Space Studies; the NOAA National Climate Data Center uses GHCNv3 and ERSST but applies different quality controls and bias adjustments. The final dataset, BEST, makes use of its own land surface air temperature product along with a modified version of HadSST.

The selected CMIP5 models have monthly historical simulation outputs available over the 1860 to 2005 period along with outputs of scenario runs from 2005 to 2100 for RCP2.6, RCP4.5 and RCP8.5, summarized in Table

In this section, we establish a procedure to estimate the probability distribution associated with the climate sensitivity:

Through this framework, each parameter combination (

The residuals are thus equal to the internal temperature variability, i.e. the response to the internal forcing

To calibrate the FEBE, we take the time-dependent forced response calculated for each parameter combination and remove it from the temperature series to obtain a series of residuals which represent an estimator of the historical internal variability. The likelihood function (

Using Bayes' rule, we can obtain the posterior probability density function (PDF) for our parameters using the likelihood function (an a priori probability) and the prior distribution for the parameters,

We use the following Mathematica 12.2

The priors chosen here are intended to reflect knowledge about the historical climate system. Following

Using Bayes, Eq. (

Using Bayes' theorem as described above, we derive PDFs for the model and forcing parameters of the FEBE from the mean likelihood functions of the five observational datasets. The different observational datasets are treated as dependent due to the use of overlapping raw data, with the differences between series coming partly from the different processing of the raw data by different teams. This corresponds to putting the datasets into a Bayesian framework where each has equal a priori probability: HadCRUTv4, C&W, GISTEMP, NOAAGlobalTemp and BEST (

Following IPCC methodologies, we report the “very likely” credible interval at the 90 % credible level throughout this work along with median estimates for the all ensemble spreads. The complete suite of model and forcing parameters and climate sensitivities are summarized in Tables

Model and forcing parameter medians for FEBE calibrated over the historical period (1880–2020) using

The calculated ECS and TCR medians using both parameters corresponding to

Model and forcing parameter medians using

The model is characterized by

For each observational dataset and their average, PDFs are shown for the model parameters: the scaling parameter

The second model parameter is the relaxation time

Presented in Fig.

The aerosol linear scaling factor

The volcanic intermittency correction exponent

For each observational dataset and their average, PDFs are shown for the forcing parameters: the aerosol scaling factor

The total historical (1880–2020) forcing series prescribed by the IPCC using,

In Fig.

The climate sensitivity parameter

For each observational dataset and their average, PDFs are shown for the climate sensitivity parameter

Two standard types of climate sensitivity used for inter-model comparisons: ECS and TCR – our results are summarized in Table

If atmospheric

The PDF for ECS shown in Fig.

Conventionally, TCR quantifies the temperature change that would occur if

The PDFs for ECS

The derived PDFs for TCR are shown in Fig.

The ECS and TCR estimates using the SSP scenarios with the FEBE are lower than those using RCPs due to the overly strong aerosols over the historical period in the SSPs which require a lower aerosol linear factor along with lower ECS to best match the historical temperature record. The difference between the shape of the RCP and SSP aerosol forcing can also account for this.

The TCR-to-ECS ratio is a non-dimensional measure of the fraction of committed warming already realized after a steady increase in radiative forcing; in this case, with a doubling of

In the next section, we show that with a lower and more constrained climate sensitivity parameter (Figs.

With the above collection of model and forcing parameter probability distributions, the FEBE was used to reconstruct the temperature over the historical period, as well as make projections of the forced temperature response for the coming century using forcings prescribed by the RCP and SSP scenarios.

The CI provided for the MME corresponds to the spread between the different GCMs, “structural uncertainty”, while for the FEBE it is parametric uncertainty

For the FEBE, the spread of the forced projections is purely from the uncertainty in the parameters: the contribution to uncertainty from internal variability has been averaged out (it is effectively the average over an infinite ensemble of realizations of internal variability). In order to make projections, we therefore draw samples of parameters from the (correlated) multidimensional parameter space (approximated by the multivariate normal distribution in Eq.

In this section, we present the full historical reconstruction using the FEBE observation-based projections with those from the GCMs in the CMIP5/6 MME. In order to make a proper comparison with data, we must include both the forced deterministic temperature response, with its purely parametric uncertainty, as well as the internal variability of the mean observational temperature series, estimated to be

An important characteristic of probabilistic forecasts is their reliability that quantifies the difference between the forecast and actual probability distributions. Consider for example, a set of predictions derived from ensemble forecasts. In some realizations, it is predicted that the chance of above-average seasonal-mean temperature for the coming season will be 70 %. If the probabilistic forecast system is reliable, then one can expect that in 70 % of these predictions the actual seasonal-mean temperature will be above average

This is expected for a reliable model and is an analogous validation of probabilistic aspects of the projection as unlike weather forecasts where we have many past test cases; climate change projections cannot be calibrated in the same manner

The small-scale limit of the validity of the FEBE is not known, although it is likely to be

This estimate of the internal variability forcing can be compared with that of

As with GCMs, the FEBE predicts the forced deterministic response as well as the statistical properties of the internally driven stochastic part. We can therefore evaluate the accuracy of the stochastic part by comparing the FEBE temperature statistics with those from observational time series. It was already shown in

Below Milankovitch timescales, there are three main scaling regimes observed in the atmosphere: the weather, macroweather and climate

We have shown that the FEBE hindcasts are reliable (Sect.

Unlike the comparison in Fig.

Throughout the historical period, the hindprojection of the FEBE and the median of the CMIP5/6 MME are close. Between 1915–1960, the CMIP5/6 MME is consistently warmer than the FEBE hindprojection and historical temperature records, although generally by less than 0.05 K. The slowdown in global warming during the first decade of the 21st century, termed as the slowdown (“hiatus”)

The 90 % CI of projected warming relative to the pre-industrial reference period (1880–1910) for the RCP scenarios analysed in this study based on the FEBE and the CMIP5 MME. Summary of Fig.

The 90 % CI of projected warming relative to the pre-industrial reference period (1880–1910) for the SSP scenarios analysed in this study based on the FEBE and the CMIP6 MME. Summary of Fig.

Following the monthly resolution reliability confirmation in Sect.

We now consider the deterministic (infinite ensemble) FEBE projections to 2100. At first, the temperature increase in each case is nearly identical; the future pathways only diverging into their respective scenarios roughly two decades after their beginning (RCPs begin in 2005; SSPs begin in 2014). Further into the future, the warming rate begins to depend more on the specified scenario, the highest being in RCP8.5/SSP5-85 (Fig.

The deterministic forced temperature response projected using the FEBE (blue), with parameters calibrated using

While the forcing of the (perhaps most realistic) middle scenario, RCP4.5/SSP2-45, stabilizes in the mid 2060s, the temperature projections continue rising throughout the 21st century for both FEBE and the CMIP5/6 MME (Fig.

The projections of both the FEBE and the CMIP5 MME for the strong emission scenario, RCP8.5, show alarming warming rates of 3.5 K with 90 % CI [2.9, 4.1] K and 4.8 K with 90 % CI [3.5, 6.0] K in 2100 shown in Fig.

The probability for the global mean surface temperature of exceeding a 1.5 K threshold

Whereas the CMIP5 projections differ from the CMIP6 projections due to both model and forcing series changes, the FEBE projections differ only because of the changes in the forcing series. By comparing the left and right columns of Fig.

Although the FEBE projections are consistently about 15 % cooler than the CMIP5 MME, due to the its smaller uncertainty the FEBE 90 % CI lies entirely within the corresponding CMIP5 CI. Both projection methods support each other and are thus complementary. When compared to CMIP6 projections, although most of 90 % CIs overlap, the median CMIP6 temperatures are nearly 65 % warmer than the corresponding FEBE median, mainly caused by their overpowered aerosols

We can also use the FEBE to estimate the probability of exceeding various warming thresholds. Important tipping points have been established which could lead to irreversible changes in major ecosystems and the planetary climate if certain thresholds in warming are exceeded

According to the FEBE for the low-emission scenario, RCP2.6, it is unlikely to exceed the 1.5 K threshold in 2100 (

In the RCP4.5 scenario, the probability of the FEBE exceeding the 1.5 K threshold is extremely likely (

List of RCP and SSP scenarios analysed in this study and the probabilities of exceeding 1.5 or 2 K based on the FEBE and the CMIP5/6 MME. Summary of Fig.

For the final high-emission, business-as-usual, RCP8.5 and SSP5-85 scenarios; both the FEBE and CMIP5/6 MME project that exceeding the 1.5 K threshold is virtually inevitable by 2100, although in the FEBE projection, it is extremely likely that this threshold is exceeded nearly 15 years after the CMIP5/6 MME projections of 2040. The same is found for the 2 K threshold, with both the FEBE and CMIP5/6 MME exceeding the threshold about 15 years after the 1.5 K threshold. These results are all summarized in Table

In the following section, we summarize the key results presented earlier in the paper: model and forcing parameters (see Table

The two parameters that characterize the model,

The FEBE model also adjusts the deterministic forcings, notably the aerosol and volcanic forcing series which must be scaled (the former linearly and the latter non-linearly) for the temperature response to best match historical temperature records. From our analysis, we find a more constrained aerosol forcing. For the

In comparison to IPCC AR5 and to the CMIP6 MME, we find lower likely ranges for the climate sensitivity parameter, ECS and TCR when using the FEBE with

With all necessary parameters of the FEBE calibrated on observational temperature series, we evaluated the FEBE reliability, showing that it is able to reconstruct the historical temperatures (Sect.

In the low-emission scenario, RCP2.6 (SSP1-26), the FEBE projects the 90 % CI of the temperature in 2100 to be [1.2, 1.6] K ([1.3, 1.8] K) as compared to the CMIP5 (CMIP6) MME of [1.1, 2.4] K ([1.4, 2.8] K). In the middle scenario, RCP4.5 (SSP2-45), the FEBE projects warming reaching [1.6, 2.3] K ([1.9, 2.6] K), narrower than the CMIP5 (CMIP6) MME warming of [1.8, 3.2] K ([2.2, 3.8] K), while in the high-emission scenario, RCP8.5 (SSP5-85), both the FEBE and CMIP5 (CMIP6) MME project extreme temperature increases of [2.7, 3.6] K ([3.0, 4.2] K) and [3.3, 5.3] K ([3.6, 6.0] K), highlighting the need for strong emission mitigation.

During the Paris Conference in 2015 (COP21), nations of the world strengthened the United Nations Framework Convention on Climate Change by agreeing to hold the increase in the global average temperature to well below 2 K above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5 K. According to our projections, crossing either of these thresholds is delayed with respect to the CMIP5/6 MME projections but will eventually happen if strong mitigation is not implemented. To avert a 1.5 K warming, drastic cuts would have to be made to global greenhouse emissions, similar to that in RCP2.6 (and SSP1-26), for which we found

Ever since the first climate models in the 1970s, multidecadal projections have had large uncertainties with the wide ECS uncertainty limits of 1.5–4.5 K essentially unchanged. For policymakers, the most deleterious consequence of large uncertainties is that projections emanating from quite diverse future scenarios have significant overlap. For example, up until 2050, the RCP2.6 and 8.5 scenarios can both claim to respect the 2 K threshold – albeit with rather different probabilities (Fig.

One way of reducing this uncertainty is by developing complementary types of models. In this paper, we directly constructed such a model in the macroweather regime (roughly 1 month and up) based on the physically principles of energy conservation and scaling: the fractional energy balance equation (FEBE). Although originally derived phenomenologically, it was recently discovered

The FEBE is a parsimonious model with only two shape parameters: an exponent

Bayesian inference allows for a robust probabilistic parameter characterization. The basic external forcings were those prescribed for the historical part of the CMIP5/6 GCMs and these were constrained by five monthly, global resolution empirical temperature series (since 1880). The internal forcing was assumed to be a Gaussian white noise and, since to a good approximation, the FEBE white noise response is a fractional Gaussian noise (fGn), the latter was taken as the Bayesian inference error model.

In order to estimate the parameters, the forcing series required two adjustments. The most important was the aerosol recalibration parameter

The forcings and parameters combined with the RCP and SSP scenarios allow us to make projections through to 2100; we did this for RCP2.6 (SSP1-26), 4.5 (SSP2-45) and 8.5 (SSP5-85). Overall, the observational-based FEBE projections had uncertainties that are smaller by more than a factor of 2 in comparison to the CMIP5/6 MME uncertainties. However, the two modelling approaches have quite different sources of uncertainty. Whereas the CMIP5/6 uncertainty is purely due to differences in the climates of the GCMs (“structural uncertainties”), the FEBE uncertainty is “parametric” and it depends largely on the uncertainty of the historical forcings and temperatures, in particular those associated with aerosols. In fact, a byproduct of the model and Bayesian framework is that we are able to more tightly constrain aerosol forcing, supporting recent literature findings of weaker historical aerosol cooling. As a consequence, the FEBE projections are consistently a little cooler than those of the CMIP5/6 MME, but with uncertainties about half of those of the MME, it still lies within the MME uncertainty bounds. By comparing the FEBE with the CMIP5 and CMIP6 MMEs, we were also able to separately quantify the contribution of changing the RCP to SSP forcing scenarios from that of the difference in the models. The qualitatively different FEBE thus effectively complements the GCMs.

There is a long history – starting with the four-thirds law of turbulent diffusion

In addition, whereas the regional (horizontally varying) FEBE has already been derived from the heat equation

In the future, time-varying parameters may also be considered: for example, the climate sensitivity multiplied by the forcing constitutes an “effective forcing” so that time-varying sensitivities are trivial to include – they essentially just change the forcing. Less trivial is the inclusion of time-varying relaxation times, and at the moment it is not obvious that it is possible to even mathematically define a time-varying order of temporal differentiation (i.e. a time-varying

The FEBE, which is an observational model based on energy and scaling symmetries, and its projections to 2100, are complementary to the GCMs. Future work will explore the full (regional, 2-D) FEBE model

RCP concentrations can be found at

SL was responsible for the conceptualization of the study. RH was responsible for the design of methods for model calibration. RP was responsible for the development the model code and prepared the manuscript with contributions from all co-authors.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Shaun Lovejoy acknowledges some support from the National Science and Engineering research Council (Canada). We thank Dave Clarke and Lenin Del Rio Amador for helpful discussions, and Maya Willard-Stepan for help in editing the manuscript. The work profited from discussions at the CVAS working group of the Past Global Changes (PAGES) programme.

This paper was edited by Valerio Lucarini and reviewed by four anonymous referees.

Raphael Hébert has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement nos. 716092 and 772852.