Summarizing the response of the Earth system to anthropogenic forcers with metrics has long been practised as a way to illustrate uncertainty in Earth system response to greenhouse gases. For example, the concept of the equilibrium climate sensitivity (ECS), the equilibrium global mean temperature increase which would be observed in response to a doubling of atmospheric carbon dioxide concentrations , has existed for over 50 years and a significant amount of literature has been devoted to constraining its value .

The Earth system responds to a step change in forcing on timescales ranging from days to millennia , so an “effective climate sensitivity” (EffCS here on) is often used as a proxy for decadal to centennial feedbacks. EffCS is generally calculated in a coupled atmosphere–ocean model from the output of the “abrupt4xCO2” simulation, a standard experiment in which CO2 concentrations are quadrupled instantaneously from pre-industrial levels and the model is allowed to evolve .

EffCS is calculated by assuming that a model is associated with a single feedback parameter (i.e. a rate of change of top of atmosphere radiative flux per unit surface temperature increase), allowing the equilibrium temperature response to a step change forcing to be predicted by linear extrapolation. Another metric, the transient climate response (TCR) at the time of CO2 doubling or quadrupling (T140) is calculated from an “1pctCO2” idealized experiment in which CO2 concentrations are increased by 1 % each year, starting from a pre-industrial state, resulting in linearly increasing forcing.

Although it was generally assumed that TCR would be a better predictor of transient warming under a high emissions scenario such as Representative Concentration Pathway 8.5 (RCP8.5) , a complication has arisen due to the fact that EffCS seems to be better correlated than TCR with 21st century warming from present-day levels under a business-as-usual scenario . The reason for this is not yet well understood given that the radiative pathway in RCP8.5 leading up to 2100 is relatively similar to that of the 1 % annual increase experiment used to measure T140. Furthermore, neither EffCS nor TCR is well correlated with end-of-century temperatures in a mitigation scenario such as RCP2.6 , which calls into question the relevance of such summary metrics in the discussion of mitigation adaptations.

Similarly, a number of studies have shown that the EffCS approximation does not well describe the true equilibrium behaviour of most models . When general circulation model (GCM) abrupt4xCO2 simulations are continued for thousands of years, many are found to deviate significantly from the linear trend line one would fit to a 150-year simulation .

The conceptual models representing the evolving feedbacks as a function of timescales vary slightly between studies – either modulating the efficacy of deep ocean heat uptake or by representing the climate system as sum of warming patterns which emerge on different adjustment timescales , each associated with their own feedback parameter. However, the analytical set of solutions for the temperature response to a step change in forcing is the same in either case – a superposition of decaying exponential modes with different timescales varying between a few years and a few centuries . It has been shown that the implications of these additional degrees of freedom and ambiguity over contributions from different timescales of response might imply that EffCS may not be strongly constrained by temperature change over the last century , and that the long-term equilibrium (LTE) sensitivity may be greater than that implied by EffCS .

This state of understanding leads to a number of emerging critical questions which we discuss in this paper – can we explain the non-intuitive result that EffCS is a better predictor than T140 of end-of-century temperatures under RCP8.5? Which summary metrics of global sensitivity to greenhouse gas forcing are most useful for effective policy decisions? Finally, do the implicit structural assumptions underpinning the applicability of these metrics to the real world cause us to mis-categorize and potentially underestimate future warming risk?

We begin by considering an idealized ensemble of climate model simulations. We use a two-timescale thermal response model, conceptually representing the deep ocean (with a response timescale of a century or more) and shallow ocean response timescales (with a response timescale of 10 to 50 years). Such a model, although simple, is capable of resolving evolving feedback amplitudes and can emulate the climatological responses of complex Earth system models on two timescales. Such a model makes a structural assumption that the Earth can be modelled as a discrete sum of linear decaying exponential responses to forcing, but this model has been found to well describe GCM evolution on a century timescale and is sufficiently complex to illustrate the limitations of defining system sensitivity through TCR or EffCS.

To efficiently describe the response of the system to a generic forcing, this study employs a linear Green function which describes the forcing by convolution with an impulse response ( – in this case, the step change in CO2 forcing). This approach can be used to approximate and simplify global climate dynamics , and its computational efficiency allows Markov chain Monte Carlo parameter estimation for the physical parameters. Furthermore (and critically for this study), the pulse–response formulation can be used to self-consistently relate different metrics of climate sensitivity on a range of timescales .

But how do the findings in the simple model framework reconcile with findings in the CMIP5 and CMIP6 multi-model ensembles? Firstly, it is plausible that there is some commonality in the lack of skill of TCR (the transient response after 70 years) in our simple model ensemble and in the CMIP ensembles. In our simple model case, the ensemble members were explicitly calibrated to reproduce the 20th and early 21st century warming – which is a very strong constraint on the value of TCR in this idealized setup.

Earth system model calibration is conducted in a much larger parameter space by groups with a wide range of objectives which complicate interpretation , but simulations are generally only published using models which are able to adequately describe the 20th century and thus might be subject to a similar effective constraint on TCR which renders the metric ineffective for describing variance in the future evolution of the model. But there remains a direct contradiction for T140, where the simple model suggests T140 should be a better predictor than EffCS for non-mitigation warming in the 21st century, whereas the opposite was found in the CMIP correlations (see Fig. S2 and ).

To understand this, we need to consider how the properties of the simple model ensemble differ from the CMIP archive. Although the thermal response of the simple model is broadly able to represent the climatological response of CMIP models to step forcing and transient forcing in CO2 over a century timescale , it contains no internal climate variability, and all experiments in Sect.  are conducted from an idealized, perfectly spun-up state.

Both of these assumptions are not true for CMIP5 or CMIP6. Measurements of EffCS and TCR are complicated by internal variability , and many models still exhibit some temperature drift in the control simulation from which the 1pctCO2 simulations and abrupt4xCO2 simulations are branched (Fig. ). This creates uncertainty from two sources – firstly, it is not always apparent at what point during the control simulations the 1pctCO2 simulation has been branched; thus, there is uncertainty in how the anomaly should be measured. Secondly, there is the potential for an unknown contribution of control drift to be erroneously included in the temperature evolution of the 1pctCO2 and abrupt4xCO2 simulations.

To assess the contribution of control drift bias in sensitivity metrics, we implement idealized representations of non-equilibration into our simple model from Sect. . We then create an idealized distribution of drift similar to that seen in the CMIP ensembles in the simple model ensemble by initializing the model 500 years before the experiment begins, defining an effective “baseline” period from which anomalies are measured to be the average temperature between the years 400 and 500. Climate internal variability is represented by a second-order autoregressive model, which is fitted to each CMIP model in turn. The ensemble-mean autoregressive parameters are used to create artificial “noisy” simulations by linearly adding noise generated from the autoregressive model to the output of the simple model.

We consider the range of control drifts observed in the CMIP5 and CMIP6 ensembles (illustrated in Fig. l) which range from -0.3 to +0.6 K per century in the CMIP5 and CMIP6 models considered in this study. An idealized distribution of drift in the simple model ensemble is created by initializing the model 500 years before the abrupt4xCO2 or 1pctCO2 simulation with a non-zero, constant forcing drawn from a flat distribution ranging from -1 to +1 W m-2, which results in a distribution of control drift of -0.4 to +0.4 K per century (i.e. broadly comparable to the CMIP case). For each simulation, we consider a baseline for temperature to be defined by the average global mean temperature in years 400–500.

To represent the first-order effect of climate noise, we fit a second-order autoregressive model to the detrended global mean temperature time series in each available model in the CMIP5/6 ensemble. Taking CMIP mean parameters for the variance and autoregressive parameters, we generate noise for each realization of the simple model (though we note, in practice, that the noise characteristics vary by CMIP model).

The results are illustrated in Fig. a, where the simple model ensemble is initialized in a non-equilibrium state with additive Gaussian noise. With these additional sources of error, both EffCS and A140 are not strongly impacted when measured in the noisy/non-equilibrated model variants (Fig. b and c), but the T140 measurement is strongly degraded (Fig. refdriftd). Indeed, in this ensemble, the biased measurements of EffCS or A140 are slightly better correlated with true T140 than the biased measurement of T140 itself. This provides a possible explanation for why T140 may be a poor predictor of RCP8.5 warming in CMIP.

In our simple framework, the reasons for the more accurate measurement of EffCS are primarily associated with the lack of equilibration. Simply adding noise from the autoregressive model has little effect on the accuracy of EffCS, T140 or A140 (where both T140 and A140 are estimated using the average of years 131 to 150 in the simulation; see Table ).

Both A140 and EffCS are less sensitive to non-equilibrated initial states than T140. The former experiences the same variance due to the uncertain climate drift, but the absolute value of A140 tends to be larger than T140; thus, there is less relative error in its estimation. The effect on the drift on EffCS is muted because the near-linear climate drift primarily biases the estimation of slow rather than fast feedbacks (see Fig. S1 in the Supplement). Because EffCS is primarily a measure of fast-mode feedback strength (see Fig. f), its value is less impacted if experiments are started from a non-equilibrium state.

There is some evidence that the lack of equilibration has an outsized effect on the estimation of TCR in the CMIP models. In Fig. , we attempt to unbias the estimate of TCR in two ways. Firstly, we estimate the baseline temperature by regressing the temperatures in the first 20 years of the 1 % CO2 ramp experiment as a function of time (see Fig. S4). Anomalies in temperature (and TOA fluxes for ECS) are measured relative to the corrected baselines derived from the 1pctCO2 simulation, and estimated linear pre-industrial trends are subtracted from the 1pctCO2 and abrupt4xCO2 time series. This pre-processing of the temperature time series improves the correlation between TCR and 21st century warming under RCP8.5 from 0.86 to 0.89. It also improves the correlation between EffCS and 21st century warming slightly from 0.94 to 0.95 (and A140 from 0.89 to 0.91).

These “corrected” values (listed in Table ) are estimates only, given that we would expect the regression estimate based on a short 20-year period to be itself subject to internal variability noise, and we are assuming that the abrupt4xCO2 simulation and 1pctCO2 simulation have the same baselines. However, the improvement in correlation with future warming seen over the case with the pre-industrial average baseline supports the hypothesis that control drift adds uncertainty to the estimation of all quantities (and particularly TCR). However, it is not a complete explanation –and even after this adjustment, EffCS remains better correlated to RCP8.5 transient warming than TCR in the multi-model ensemble.

The question of which metric of climate sensitivity is most useful for summarizing uncertainty in future projections is conditional on a number of factors. Any single metric of sensitivity, even if known perfectly, cannot constrain Earth system response on all timescales and scenarios. We have shown here that one can produce a number of model variants which can exhibit the same value of EffCS or TCR but with a range of responses, especially in a mitigation scenario such as RCP2.6.

In an idealized environment where models can be brought to a complete equilibrium control state, and ensemble sizes for “1pctCO2” simulations are large enough to avoid the effects of internal variability, the T140 metric would be the best idealized warming measure for century-scale warming under a high emissions scenario. However, the presence of even moderate control drift can act as a significant source of error in the measurement of T140, and so here we find that EffCS is likely to be a more accurate practical sensitivity metric in Earth system model applications where full equilibration is difficult to achieve.

EffCS itself has limitations; it is relatively insensitive to slow timescale feedbacks, which means that it poorly correlated with century-scale warming under RCP2.6 (where a large fraction of warming occurs due to slow feedback response to historical emissions) and for warming on multi-century timescales under a high emissions scenario (where concentrations stabilize post-2100). We find that a simple but useful alternative is to simply use the mean warming from years 131 to 150 of the abrupt4xCO2 simulation – which is skilled comparably to EffCS in predicting RCP8.5 warming in 2100 but more sensitive to century timescale feedbacks than EffCS – therefore, it is better correlated with RCP2.6 end-of-century warming.

It is notable that the most common metrics of sensitivity (EffCS, T140 and TCR) provide very little guidance on peak warming expected under climate mitigation. The focus on these metrics has also given rise to the issue that slow feedbacks in Earth system models are not well constrained by the set of experiments currently conducted by default in CMIP. The standard 150-year simulation used to calculate effective climate sensitivity does not constrain true equilibrium climate sensitivity, and only a limited set of CMIP-class models have run models for long enough to be informative about equilibrium response .

It should be noted that these conclusions are derived from the consideration of a relatively simple two-timescale pulse response model. In this model, we can show that certain sensitivity metrics are insufficient to constrain future projections, and that non-equilibration may confound measurement. However, the constrained distributions for the metrics are subject to the structural assumptions of the model. The real world may have more than two response timescales or may be better described as a continuous sum . Further work should identify how such complexity impacts uncertainty in relevant climate metrics.

The diversity of simulated global mean dynamical response to greenhouse gas forcing over the coming centuries can be represented in simple models with a relatively small number of parameters , but we cannot reduce uncertainty in climate projections on all timescales to a single degree of freedom. Summary metrics of climate response have value if the context of those metrics (and their range of applicability in relation to projection uncertainty) is well understood, but their limitations should be kept in mind.