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Many attempts have been made to constrain ECS, typically using either the record of historical warming or reconstructions of past climates4. Methods based on historical warming are affected by uncertainties in ocean heat uptake and the contribution of aerosols to net radiative forcing5,6. These methods also diagnose the effective climate sensitivity over the historical period, which may be different to ECS, owing to the strength of climate feedbacks varying with the evolving pattern of surface temperature change4,7,8,9. Although methods based on past climatic periods, such as the Last Glacial Maximum10, are more closely related to the concept of equilibrium, they suffer instead from even larger uncertainties in the reconstruction of net radiative forcing.

As an alternative, the emergent constraint approach uses an ensemble of complex Earth system models to estimate the relationship between a modelled but observable variation in the Earth system and a predicted future change2,11. The model-derived emergent relationship can then be combined with the quantification of the observed variation to produce an emergent constraint on the predicted future change2,11,12. Here we present an emergent constraint on ECS that is based on the variability of global-mean temperature.

To inform our search for an emergent constraint, we consider the simple ‘Hasselmann model’13 for the variation in global mean temperature ΔT in response to a radiative forcing Q:

The constant heat capacity C in this model is a simplification that is known to be a poor representation of ocean heat uptake on longer timescales14,15,16. However, we find that it still offers very useful guidance about global temperature variability on shorter timescales. The climate feedback factor λ determines how the net top-of-atmosphere planetary energy balance N varies with temperature change ΔT in response to a radiative forcing change Q. ECS and λ are inversely related, with a constant of proportionality that is the radiative forcing due to doubling of atmospheric CO2, Q2×CO2 so that ECS = Q2×CO2/λ. Although the diagnosed Q2×CO2 varies across the model ensemble17, the uncertainty in ECS is predominantly due to uncertainty in λ, which varies from 0.6 W m−2 K−1 to 1.8 W m−2 K−1, as shown in Extended Data Table 1.

If Q can be approximated as white-noise forcing with variance , the Hasselmann model can be solved to give expressions for the variance of global temperature and the one-year-lag autocorrelation of the global temperature α1T, which can be combined to yield an equation for ECS (see Methods):

where is our key metric of global temperature variability. This equation is essentially a fluctuation–dissipation relationship18 relating the variability of the climate () to its sensitivity to external forcing (ECS).

Observational records of global mean temperature change3 enable Ψ to be estimated for the real world. The variance of the net radiative forcing is approximately equal to the variance of the top-of-the-atmosphere flux , which can in principle be estimated from satellite measurements. However, the available satellite records are currently too short to provide reliable estimates of σN. In addition, the radiative forcing due to doubling CO2 (Q2×CO2) is not observable in the real world. This means that the right-hand side of equation (2) cannot be directly estimated from observations. Fortunately, we find that the variation in ECS is weakly correlated with across the model ensemble (see Extended Data Table 1). We can therefore approximate the predicted gradient of the ECS versus Ψ emergent relationship using the ensemble mean value of (= 8.7). Our theory therefore predicts a gradient of the ECS versus Ψ emergent relationship of  = 12.2.

Figure 1a shows the simulation of global warming in the historical simulations with the 16 models in the CMIP5 ensemble19,20 used here (see list in Extended Data Table 1). Here and throughout, higher-sensitivity models (λ < 1.0 W m−2 K−1) are shown in magenta and lower-sensitivity models (λ > 1.0 W m−2 K−1) are shown in green. Observations from the HadCRUT4 dataset3 are shown by the black line marked with dots. Figure 1a illustrates that both high- and low-sensitivity models are able to fit the historical record with reasonable fidelity, despite implying very different future climates. Models with higher ECS values also have longer response times, and there are variations across the models in net radiative forcing and in ocean heat uptake—allowing models with both high and low sensitivities to reproduce historical global warming21. As a result, the fit to the global temperature record does not provide a direct constraint on ECS, as shown in Fig. 1b.

Figure 1: Historical global warming.
figure 1

a, Simulated change in global temperature from 16 CMIP5 models (coloured lines), compared to the global temperature anomaly from the HadCRUT4 dataset (black dots). The anomalies are relative to a baseline period of 1961–1990. The model lines are colour-coded, with lower-sensitivity models (λ > 1 W m−2 K−1) shown by green lines and higher-sensitivity models (λ < 1 W m−2 K−1) shown by magenta lines. b, Scatter plot of each model’s ECS against the root-mean-square error in the fit of each model to the observational record. Individual CMIP5 model runs are denoted by the letters listed in Extended Data Table 1.

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To test whether variability is a better constraint on ECS, we de-trend the global mean temperature records from the models and the observations. Our approach to de-trending is informed by techniques designed to detect precursors of potential tipping points22 such as ‘critical slowing down’23. The method applied in that case is to use a moving window, to linearly de-trend within that window, and then to calculate statistics of the de-trended residuals. For tipping point detection, the favoured variable is often the autocorrelation, which measures the memory in fluctuations of the analysed variable23. We use a similar approach, although here we apply it to analyse the relationship between Ψ and ECS across the ensemble of models, rather than to detect declining system resilience in a single realization of the system.

We analyse the annual-mean global-mean temperature time series from 16 CMIP5 historical simulations and compare to the HadCRUT4 observational dataset. Although there were another 23 historical runs available in the CMIP5 archive, we chose to use just one model variant from each climate centre, to avoid biasing the emergent constraint towards the centres with the most model runs in the archive. Where there was more than one model variant from a modelling centre, we took the model variant from that centre that had the smallest root-mean-square (r.m.s.) error in the fit to the record of observed global warming from 1880 to 2016. The remaining 23 model runs (which included some initial condition ensembles) were subsequently used to test the robustness of the emergent constraint (see Extended Data Fig. 1).

Figure 2a shows the resulting variation in Ψ for each of the models and the observations, using a window width of 55 yr, and data from 1880 to 2016 to match the available observational datasets. Although Ψ varies in time, the different models are clearly distinguished, in contrast to the simulations of historical global warming (Fig. 1a). In particular, the Ψ values separate higher-sensitivity models (magenta lines) from lower-sensitivity models (green lines), with higher-sensitivity models producing larger Ψ values. It is also worth noting that Ψ from the observational data are within the range of the lower-sensitivity models but clearly outside the range of the higher-sensitivity models. Figure 2b shows the emergent relationship between ECS and the time-mean Ψ values across the model ensemble, with a best-fit gradient that is very close to our theoretical value. The vertical blue lines show the observational constraint on Ψ from the HadCRUT4 dataset, but similar observational constraints are also derived from other datasets of global mean temperature (see Extended Data Table 2).

Figure 2: Metric of global mean temperature variability.
figure 2

a, Ψ metric of variability versus time, from the CMIP5 models (coloured lines), and the HadCRUT4 observational data (black circles). The Ψ values are calculated for windows of width 55 yr, after linear de-trending in each window. These 55-yr windows are shown for different end times. As in Fig. 1, lower-sensitivity models (λ > 1 W m−2 K−1) are shown by green lines and higher-sensitivity models (λ < 1 W m−2 K−1) are shown by magenta lines. b, Emergent relationship between ECS and the Ψ metric. The black dot-dashed line shows the best-fit linear regression across the model ensemble, with the prediction error for the fit given by the black dashed lines (see Methods). The vertical blue lines show the observational constraint from the HadCRUT4 observations: the mean (dot-dashed line) and the mean plus and minus one standard deviation (dashed lines).

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As in previous studies11,12 the emergent relationship from the historical runs and observational constraint can be combined to provide an emergent constraint on ECS. This involves convolving the prediction error implied by the fit of the scatter plot to the emergent relationship, with the uncertainty in the observations, to produce a probability density function (PDF) for the y-axis variable (see Methods). Figure 3a shows the resulting PDF for ECS (black curve). For comparison, the prior PDF implied by the equal-weighted model ensemble is shown by the orange histogram. The emergent constraint PDF is sharply peaked around a best estimate of ECS = 2.8 K, which is slightly smaller than the centre of the IPCC range of 1.5–4.5 K. Our best estimate of ECS is considerably larger than the values derived from raw energy budget constraints8,24,25 but similar to some recent estimates that account for time-dependent and forcing-dependent feedbacks9,26.

Figure 3: Emergent constraint on ECS.
figure 3

a, The PDF for ECS. b, The related CDF. The horizontal dot-dashed lines show the 66% confidence limits on the CDF plot. The orange histograms (both panels) show the prior distributions that arise from equal weighting of the CMIP5 models in 0.5 K bins.

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Figure 3b shows the resulting cumulative density function (CDF), which gives the probability of ECS taking a value lower than the value shown on the x axis. The black horizontal lines in Fig. 3b show the 66% confidence limits (2.2 K to 3.4 K), or approximately 2.8 ± 0.6 K. Relative to the IPCC range of 1.5–4.5 K, this constraint on ECS therefore reduces the uncertainty by about 60%. Indeed, even the 95% confidence limits from the emergent constraint (1.6 K to 4.0 K) fit well within the IPCC ‘likely’ range for ECS. Our constraint is therefore at odds with a suggestion that the lower 66% confidence limit for ECS could be as high as 3 K (ref. 27). If we instead use all 39 historical runs in the CMIP5 archive, we find a slightly weaker emergent relationship, but derive a very similar emergent constraint on ECS (Extended Data Table 2). The constraint is also robust to the choice of observational dataset, and to whether or not the model global temperature is calculated just across the points where there were observations28 (Extended Data Table 2 and Extended Data Fig. 2).

Our choice of window width was informed by sensitivity studies in which the emergent constraint was calculated for a range of this parameter. Figure 4a shows the best estimate and 66% confidence limits on ECS as a function of the width of the de-trending window. Our best estimate is relatively insensitive to the chosen window width, but the 66% confidence limits show a greater sensitivity, with the minimum in uncertainty at a window width of about 55 yr (as used in the analysis above). As Extended Data Fig. 3 shows, at this optimum window width the best-fit gradient of the emergent relationship between ECS and Ψ (= 12.1) is also very close to our theory-predicted value of (= 12.2). This might be expected if this window length optimally separates forced trend from variability.

Figure 4: Sensitivity of the emergent constraint on ECS to window width.
figure 4

a, Central estimate and 66% confidence limits. The thick black bar shows the minimum uncertainty at a window width of 55 yr and the red bar shows the equivalent ‘likely’ IPCC range of 1.5–4.5 K. b, Probabilities of ECS > 4 K (red line and symbols) and ECS < 1.5 K (blue line and symbols).

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Figure 4b shows the probability of ECS > 4 K and ECS < 1.5 K as a function of window width. For comparison, the IPCC ‘likely’ range of 1.5–4.5 K implies a 25% probability of ECS > 4 K, and a 16% probability of ECS < 1.5 K. At the optimum window width of 55 yr, both probabilities are close to their minimum values of less than 2.5%. Our emergent constraint therefore greatly reduces the uncertainty in the ECS value of Earth’s climate, implying a less than 1 in 40 chance of ECS > 4 K, and renewing hope that we may yet be able to avoid global warming exceeding 2 K.

Methods

Theoretical basis for the emergent relationship

We hypothesize that equation (1) (the ‘Hasselmann model’) is a reasonable approximation to the short-term variability of the global mean temperature anomaly ΔT:

If trends arising from net radiative forcing and ocean heat uptake can be successfully removed, the net radiative forcing term Q can be approximated by white noise. Under these circumstances, equation (1) is essentially the Ornstein–Uhlenbeck equation, which describes Brownian motion, and has standard solutions (for example, see https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_process) for the lag-one-year autocorrelation of the temperature:

and the ratio of the variances of T and Q:

These two equations can be combined to eliminate the unknown heat capacity C and therefore to provide an expression for the climate feedback factor λ:

The ECS and λ are inversely related by a constant of proportionality, which is the radiative forcing due to doubling of atmospheric CO2 (Q2×CO2), so that ECS = Q2×CO2/λ. Thus, we can also derive an expression for ECS in terms of the variability of T and Q:

Least-squares linear regression

Least-squares linear regressions were calculated using well established formulae (see for example http://mathworld.wolfram.com/LeastSquaresFitting.html). The linear regression fn between a time series given by yn and a time series given by xn is defined by a gradient b and intercept a:

Minimizing the least-squares error for yn involves minimizing:

where N is the number of data points in each time series. In this case, the best-fit gradient is given by:

Here is the variance of xn and  × is the covariance of the xn and yn time series, with means of and , respectively. The standard error of b is given by:

which defines a Gaussian probability density for b:

Finally, the ‘prediction error’ of the regression is the following function of x:

This expression defines contours of equal probability density around the best-fit linear regression, which represent the probability density of y given x:

where , as described above.

Calculation of the PDF for ECS

The emergent constraint derived in this study is a linear regression across the CMIP5 models between ECS and the Ψ statistic of the de-trended global temperature. In the context of the least-squares linear regression presented above, ECS is equivalent to y, and Ψ is equivalent to x. The linear regression therefore provides an equation for the probability of ECS given Ψ (that is, the equation for P{y|x} above). In addition, the Ψ statistic calculated from the de-trended observational dataset provides an observation-based PDF for Ψ. Given these two PDFs, P{ECS|Ψ} and P(Ψ), the PDF for ECS is calculated by numerically integrating:

Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Code availability

The Python code used to produce the figures in this paper is available from the corresponding author on reasonable request.