Beyond Forcing Scenarios: Predicting Climate Change through Response Operators in a Coupled General Circulation Model

Global Climate Models are key tools for predicting the future response of the climate system to a variety of natural and anthropogenic forcings. Here we show how to use statistical mechanics to construct operators able to flexibly predict climate change. We perform our study using a fully coupled model - MPI-ESM v.1.2 - and for the first time we prove the effectiveness of response theory in predicting future climate response to CO2 increase on a vast range of temporal scales, from inter-annual to centennial, and for very diverse climatic variables. We investigate within a unified perspective the transient climate response and the equilibrium climate sensitivity, and assess the role of fast and slow processes. The prediction of the ocean heat uptake highlights the very slow relaxation to a newly established steady state. The change in the Atlantic Meridional Overturning Circulation (AMOC) and of the Antarctic Circumpolar Current (ACC) is accurately predicted. The AMOC strength is initially reduced and then undergoes a slow and partial recovery. The ACC strength initially increases due to changes in the wind stress, then undergoes a slowdown, followed by a recovery leading to a overshoot with respect to the initial value. Finally, we are able to predict accurately the temperature change in the North Atlantic.


predicting climate change using the Ruelle Response theory
In this paper we show for the first time how Ruelle's response theory can be used to perform successful centennial climate predictions in the fully coupled climate model MPI-ESM v.1.2 40 . We consider two ensemble experiments. One experiment features an instantaneous CO 2 doubling ( xCO 2 2) taking place at year 1850, and the outputs of its ensemble members are used to compute the Green functions of several observables of interest, as described in the Methods section (Appendix A). We then perform an ensemble of runs where the CO 2 concentration is increased -starting also at year 1850 -at the rate of 1% per year until doubling ( pctCO 1 2), which takes place after about 70 years (y). We use the Green function computed with xCO 2 2 to reconstruct the response of pctCO 1 2 and compare the prediction with direct numerical simulations. As discussed in the Methods section, the Green function is defined for a period of 2000 y. Therefore, in what follows we are able to perform predictions beyond the time frame simulated in the pctCO 1 2 scenario (which is only 1000 y long), thus showing very useful predictive power.
www.nature.com/scientificreports www.nature.com/scientificreports/ First, we analyse the response of the globally averaged near-surface temperature (T m 2 ) on short and long time scales. We then focus on two key aspects of the large-scale ocean circulation, namely the Atlantic Meridional Overturning Circulation (AMOC) 41,42 and the Antarctic Circumpolar Current (ACC) 43 , and show that we can achieve excellent skill in predicting the the slow modes of the climate response. We also look into the global ocean heat uptake (OHU) 44 . A non-vanishing OHU indicates the presence of a global net energy imbalance. In current conditions, the ocean is well-known to absorb a large fraction of the Earth's energy imbalance due to global warming and to store it through its large thermal inertia, up to time scales defined by the deep ocean circulation 45 . Finally, we prove the validity of our approach for predicting the change in the surface temperature in the North Atlantic, where the ocean deep water formation takes place. This region features a complex interplay between local processes and large-scale meridional energy transport, thus being particularly sensitive to the strength of the forcing and the changes in the large-scale circulation 46 .

Results
Global Mean Surface temperature. Figure 1 shows that the change of T m 2 under the pctCO 1 2 scenario is predicted with very good accuracy through response theory. The prediction is accurate both for the fast (first 70 y) and the subsequent slow response. The time pattern of temperature change indicates that the contribution of the fast feedbacks saturates after few decades, and the slow modes dominate the response for the rest of the period. The warming goes on for multicentennial scales, in a way that is not captured at all by models featuring a non-dynamic ocean 18,27 . The importance of the slow modes of climate response, associated with the oceanic thermal inertia, can be quantified considering the ratio between the transient climate response (TCR) and the equilibrium climate sensitivity (ECS), sometimes referred to as the realised warming fraction 3 ; see the Methods section for the precise definition of these quantities. Here we have TCR ECS / 05 ≈ . (ECS K 3 5 ≈ . ), which is much smaller than what found (≈0.85) by Ragone et al. 27 , indicating a more prominent role of the slow modes of variability in the model investigated here. The prediction obtained via response theory shows the establishment of steady state conditions for times larger than 1000 y.
The Green function -see Fig. 5 -provides information on the time scales of the response. As a result of the presence of slow oceanic time scales, the Green function significantly departs from a simple exponential relaxation behavior, which is sometimes adopted to describe the relaxation of the climate system to forcings 30,47 . The idea of defining a general Green function as a sum of multiple or infinite 29 exponential functions with different timescales, generalising Hasselmann's ideas -the so-called pulse-response method -and along the lines of previous studies 38,48,49 is beyond the scope of our analysis and will be investigated further in a future work. In our case, after a fast decrease for short time scales, the Green function tends to zero at a much slower pace for times longer than 100 y, in agreement with what reported by Held et al. 47 .
Atlantic meridional overturning circulation. The AMOC is strongly influenced by buoyancy perturbations in the Atlantic basin 41 . It is relevant at climatic level because it encompasses about 25% of the total (atmospheric and oceanic) meridional heat transport 42 . The time series of annual mean AMOC strength in the pctCO 1 2 scenario is shown in Fig. 2a. The AMOC strength undergoes a decrease by about 30%, reaching its minimum in about 150 y. Successively, the AMOC slowly recovers.
The prediction of the AMOC change obtained via response theory captures very well the ensemble mean of the time evolution for the pctCO 1 2 in the first 1000 y. The corresponding Green function is shown in the inset of Fig. 6a. On short time scales, we have a reduction of AMOC, as a result of the negative value of the Green function. On longer time scales (>100 y), a negative feedback acts as a a restoring mechanism, associated with a positive sign in the Green function. The presence of fast (meaning here decadal) response associated with the GHG forcing has already been found in other models 50,51 , and is most likely related to the timescales of the sea-ice melting, consistently with paleoclimate simulations of the last interglacial climate with prescribed freshwater influx from reconstructed sea-ice melting 52 . The slow recovery of the AMOC might be understood as a heat [53][54][55] and freshwater 56 advection feedback. www.nature.com/scientificreports www.nature.com/scientificreports/ In the 1001-2000 y period, response theory shows that a steady state is progressively reached over multi-centennial scales. The newly established AMOC is significantly weaker than the unperturbed AMOC, although a large ensemble spread is found. This is consistent with simulations obtained from higher resolution versions of the same model 57 , intermediate-complexity models 51 and other fully coupled models inclusive of an interactive carbon cycle 58 .
the Antarctic circumpolar current. The ACC is by far the strongest large-scale oceanic current and its role in the general circulation is two-fold. On one hand, it isolates Antarctica from the rest of the system, being associated with a very strong zonal circulation in the Southern Ocean. On the other hand, although eminently wind-driven, it marks the area of outcropping of deep water occurring at the southern flank of the subtropical gyre, as part of the global-scale overturning circulation 43 .
The Green function is shown in the inset of Fig. 6b. We find that the initial strengthening of the ACC can be associated with an increase in surface zonal wind stress (not shown here). Such a surface forcing determines  www.nature.com/scientificreports www.nature.com/scientificreports/ an enhanced Eulerian mean ACC transport, consistently with previous low resolution simulations 59 . On decadal scales, we have a loss in the correlation between wind stress and ACC, corresponding to the Green function turning negative after about 30 y. Beyond these time scales, we have time-wise coherent response of the AMOC and ACC, underlying the response of the global ocean circulation. Other models 60,61 also feature such a behavior on intermediate time scales, consistently with the idea that the two circulations are related via the thermal wind balance 62 . Figure 2b shows that the prediction of the ACC strength evolution in the pctCO 1 2 scenario is rather accurate for the first 1000 y, except for an underestimation of the positive short-term response, which is smoothed out. This points to an insufficient ability of response theory in representing the complex coupling between surface wind stress and downward momentum transfer. Furthermore, we observe the presence of a strong variability (on decadal time scales) of the predicted signal. This might result from either the small ensemble size or, more interestingly, could be the signature of the natural variability, encoded by a Ruelle-Pollicott pole 2,28,29 ; see discussion in Sect. IA.
Note that in the 1001-2000 yrs period the ACC reaches an approximate steady state a bit later than the AMOC, possibly as a result of having a larger inertia, consistently with the different depth scales of the two currents 62,63 . The AMOC maximum overturning depth scale is indeed located at about 1000 m, whereas the outcropping in the Southern Ocean is related to isopycnal surfaces reaching much deeper. This has profound implications for setting the time scales of the ACC and AMOC response. The propagation of deep water formation anomalies in the Northern Hemisphere is in fact mediated by Kelvin waves in the Northern Atlantic, whereas much slower interior adjustment through Rossby waves communicates the anomaly to the Southern Ocean 64 . ocean Heat Uptake. Looking at the 2000 y of prediction in Fig. 3, we notice that response theory accurately predicts the response at all time scales. The linearity of the OHU increase in the 70 y of integration comes from the convolution of the singular component of the corresponding Green function with the ramp, see the Methods section. After the CO 2 concentration stabilizes, the OHU decreases towards vanishing values. In the last 1000 y, response theory predicts a further decrease in the OHU down to a value of the order of W 5 10 13 ≈ × . As the cli-  www.nature.com/scientificreports www.nature.com/scientificreports/ mate system as a whole relaxes towards the newly established energetic steady state through the negative Planck feedback, each of its subcomponents go through a process of relaxation. What we portray in Fig. 3 after year 1920 is the relaxation of the slowest climatic component, namely the global ocean. The remaining imbalance at the end of the prediction can also be interpreted as either resulting from the ultra-long time scales required for reaching rigorous steady state conditions, or as the signature of a model energy bias, associated with non-vanishing energy budget at steady state; on this aspect, see discussion and the north Atlantic cold Blob. Finally, we study the surface temperature response for a domain covering the North Atlantic (between lon 26° W and lon 53° W and between lat 53° N and lat 69° N), thus including the areas where the deep water formation occurs. The region is identified as a peculiar spot for the effects of the GHG forcing, since the sea-ice melting has been hypothesized to delay the surface warming of this area compared to surrounding regions, through a weakening of the overturning circulation 46 . Indeed -see Fig. 4 -the surface warming over the North Atlantic region is remarkably different from the behavior of the rest of the extratropics, which features a time dependent response (not shown here) similar in shape but somewhat amplified with respect to the global mean depicted in Fig. 1. Indeed, a long-lasting plateau -a hiatus in the temperature increase of more than 100 y -is observed around the end of the CO 2 increase ramp in the North Atlantic. The plateau is well captured by response theory, and comes in agreement with the AMOC weakening predicted in Fig. 2a. This result is non-trivial, given that such local response results from an interplay of local factors and, as mentioned, the response of the large-scale circulation. This hints at the potential of using response theory to identify global quantities that can be used as predictors for the response of local observables 29 .

Discussion and conclusions
We have shown here that response theory is a valuable tool for flexibly predicting crucial features of the climate change signal. All relies on obtaining the observable-dependent Green function from model simulations. The Green function allows one to deal with a continuum of time-dependent forcings, beyond the standard use of reference scenarios. Our findings provide guidance to the climate modellers' community on how to set up climate www.nature.com/scientificreports www.nature.com/scientificreports/ change experimental protocols, minimising the need for computational resources. This is especially relevant when one wants to study the overall impact and individual effects of multiple climatic forcings or investigate geoengineering options.
We have made here use of a fully coupled climate model, highlighting the slow components of the response associated with the oceanic modes of variability. The presence of a vast range of active time scales in the system makes the prediction of the response theoretically challenging and practically extremely relevant. Compared with previous contributions to the literature that have attempted with a good degree of success the prediction of changes in slow climatic fields using some form of linear regression, our approach is mathematically more robust as it derives directly from basic results in non-equilibrium statistical mechanics.
We stress that response formulas based on Ruelle's theory are rigorously valid, but only when considering ensemble averages. On one side, this clarifies the limits of validity of previous attempts to apply similar formulas to individual model runs. On the other side, it indicates the importance of considering large ensemble strategies when planning major computational efforts for climate change projections.
The predicted changes in the AMOC and ACC feature clearly distinguishable fast and slow regimes of response. The former is essentially different in the two branches of the global ocean circulation, being the ACC subject to the effect of surface wind stress anomalies (substantially underestimated, compared to actual simulations). The latter is found to be well correlated between AMOC and ACC, as a signature of the forced response of the global ocean circulation circulation, which they are both part of. Coherently with previous findings 62, 63 , the ACC reaches a steady state much later than the AMOC. The plateau in near-surface warming over the North Atlantic is also related to the initial slowdown of the AMOC, and contrasts with the regular increase of the surface temperature we find globally (and predict accurately).  www.nature.com/scientificreports www.nature.com/scientificreports/ We remark that the time-dependent transient and long-term evolutions resulting from the CO 2 increase are qualitatively different for the various climatic observables investigated here. Nonetheless, in all considered cases response theory successfully predicts the time-dependent change.
Ruelle's response theory provides a relatively simple yet robust and powerful set of diagnostic and prognostic tools to study the response of climatic observables to external forcings. The availability of a large number of ensemble members allows for constructing more accurate Green functions and for studying effectively the response of a broader class of climatic observables. The approach proposed here could be extremely useful to inform the planning of major computational efforts for climate projections, putting such endeavours on firm theoretical grounds and optimizing the use of computational resources.
We have dealt here with a forcing due only to changes in the CO 2 concentration. This means that the pattern of the forcing is determined by heating rate associated with a spatially homogeneous CO 2 mixing rate. Within the linear regime, different sources of forcings can be treated independently. The next step is to investigate other nontrivial forcings (e.g. aerosol forcings, land-use change, land glaciers location and extension). As an example, response theory has been proposed as a tool for framing geoengineering strategies and understanding its limitations 66 .
Response theory provides a powerful formalism to tackle different problems related to the concepts of feedback and sensitivity. A promising application is the definition of functional relations between the response of different observables of a system to forcings, in the spirit of some recent investigations (see, e.g. Zappa et al. 38 ). This would allow to treat comprehensively the concept of feedback across different time scales and define causal links between variables 29 . On a different line, one of the most promising future applications will be given by the synergy between the formalism of response theory and the recently proposed theory of emergent constraints 13,67 . The combination of the two approaches could lead to much needed insights on the climate response to forcings.
Additionally, along the lines of the pulse-response approach, it is in principle possible to try to extract the characteristic time scales of the response of the system by fitting the Green functions of the considered observables as a weighted sum of (in principle, infinitely many) exponential functions. As explained in Refs. 29,68 , the time scales of the exponential functions are the same for all observables. Instead, the weight of each exponential contribution does depend on the observable of interest, with the response of rapidly equilibrating observables being dominated by the fast time scales, while the opposite holds for observables associated with the slow components of the system. The optimal fit can be obtained through a global optimisation procedure, where the response of various different observables is simultaneously fitted to a sum of exponentials. We will delve into this interesting problem of inverse modelling in a separate publication.
Clearly, in some applications one may want to test accurately to what extent nonlinear effects are relevant, as the theory is also applicable to higher orders 16 . Some insights into the non linear component of the response could also be obtained by appropriately combining forcings differing in sign and magnitude 16,17,66 . Nonetheless, being based on a perturbative approach, response theory (linear and nonlinear) has, by definition, only a limited range of applicability (e.g. one cannot use it to treat arbitrarily strong forcings). Still, the non-applicability of response theory has itself fundamental implications for the knowledge of the dynamics of the system one is studying. At a tipping point [69][70][71][72] (or critical transition) the negative feedbacks of a system are overcome by the positive ones and any linear Green function diverges as a result of the increase in the time correlations of the system due to a critical Ruelle-Pollicott pole 2,28 , signalling the crisis of the chaotic attractor 68 . Instead, near a critical transition, response operators do not converge unless one considers very weak forcings 28,37 . The experimental design provided here is thus also a clear and mathematically sound strategy for the study of conditions leading to tipping points and their role for the climate response 2,71 in state-of-the-art climate models.

Simulations. The analysis is based on two ensembles of simulations with Max Planck Institute Earth System
Model (MPI-ESM) v.1.2 40 , using its coarse resolution (CR) version. It features, for the atmospheric module ECHAM6 73 , a T31 spectral resolution (amounting to 96 gridpoints in longitude and 48 in latitude) and 31 vertical levels, for the oceanic module MPI-OM 74 , a curvilinear orthogonal bipolar grid (GR30) (122 longitudinal and 101 latitudinal gridpoints) with 40 vertical levels. The two ensembles, each including 20 runs, are based on two different scenarios. The first one features an instantaneous doubling in CO 2 concentrations (from a reference value of 280 ppm, characteristic of pre-industrial conditions) at the beginning of the simulations ( xCO 2 2), the other one an increase in the CO 2 concentration at the constant rate of by 1% per year, until the xCO 2 2 level is reached after about 70 years; afterwards, the CO 2 concentration is kept constant ( pctCO 1 2). The procedure for the construction of the ensemble is analogous to the protocol for CMIP5 75 and Grand Ensemble 76 experiments. A control run is performed for 2000 y with pre-industrial conditions. Each of the ensemble members is initialized from a state of the control run. The initial conditions are sampled from the control run every 100 y, in order to ensure sufficient decorrelation among the respective oceanic states (at least in the mixed layer 77,78 ). The xCO 2 2 simulations are run for 2000 y, while the pctCO 1 2 simulations are run for 1000 y with the same 20 initial conditions. As an additional check, one of the xCO 2 2 members is prolonged for 2000 additional y, in order to investigate whether the model converges to the steady state or there is an intrinsic model drift 79 .

Retrieval of AMoc and Acc.
Typically, the large-scale circulation in the ocean is measured in terms of the mass transport across a suitably chosen section of a basin. The strength of the AMOC is computed as the vertically integrated mass weighted meridional mass streamfunction across lat 26.5° N 80 . This is a standard diagnostics of the MPI-ESM model. The ensemble average of the AMOC volume transport amounts to 17.3 Sv (1 Sv = 10 6 m 3 s -1 ), which is consistent with recent available measurements from the RAPID monitoring array 81 . The ACC is roughly zonally symmetric, and its location is closely related to the isopycnal slopes in the Southern Ocean.

Scientific RepoRtS |
(2020) 10:8668 | https://doi.org/10.1038/s41598-020-65297-2 www.nature.com/scientificreports www.nature.com/scientificreports/ Traditionally 60 , it has been measured in terms of the strength of the mass transport across the Drake passage. Similarly, we take the vertically integrated barotropic streamfunction difference between the 2° × 2° boxes centered around the lon 68° W, lat 54° S and lon 60° W, lat 65° S locations. The ensemble average of the ACC is 138 Sv, which is consistent with the multi-model mean estimate found in Mejers et al. 60 , amounting to 155 ± 51 Sv. It is also not far from the value commonly used as benchmark for the assessment of climate models (173 Sv 82 ).
Linear Response theory. Response theories allow one to predict how the statistical properties of a system changes as a result of acting modulations in its external or internal parameters. The validity of the corresponding response formulas is heavily dependent on the hypothesis that the unperturbed system is structurally stable, i.e., roughly speaking, far from bifurcations, or, in terms of geophysical systems, from tipping points (see related discussions in a climate context 18,[26][27][28]. Rigorous derivations of response theories have been provided for the case of deterministic 15,22,23 and stochastic 21 dynamics. We only remark here that statistical mechanical arguments encoded by the chaotic hypothesis 83 (a non-equilibrium analogue of the ergodic hypothesis) indicate the feasibility of the methodology proposed here.
In this paper we follow to a large extent the approach presented by Lucarini et al. 18 and by Ragone et al. 27 (see also Refs. 16,26 ) for the study of a large ensemble of intermediate-complexity atmospheric model runs and follow the deterministic route for response theory 15,22,23 . Let us consider a dynamical system described by the state vector x N ∈  , whose dynamics is described by the set of differential equations =  x F x ( ), where ∈ F N  is a smooth vector field. We add a perturbation to the dynamics in the form Ψ = x t X x f t ( , ) ( ) ( ), where ∈  X N is a smooth vector field that gives the structure of the forcing in the phase space, whilst f is its time modulation. The expectation value of any observable x ( ) Φ = Φ can be written as: where ⟨ ⟩ Φ 0 is the expectation value in the unperturbed state, and the term Φ t ( ) gives the n th order perturbative contribution. We consider here only the first order contribution 〈Φ〉 t ( ) f (1) . The linear correction is given by the convolution of the linear Green function with the time modulation of the perturbation: is the linear Green function of the generic observable Φ. Because of causality, the linear Green function vanishes for negative times. For ease of notation we have not indicated in Eq. A1 the dependence of the response on X, as in the applications considered in this paper X is fixed and only the time modulation f is varied. Note that for a time modulation f such that , as observed in this paper for all observables except the OHU. In this latter case, one has t lim ( ) 0 t f 0 〈Φ〉 ≠ → because the Green function has a singularity (in the form of a Dirac's δ contribution) for = t 0 29 . We remark that in previous works [33][34][35][36]38 , the linear prediction of the desired climate observable is instead obtained by convolving time pattern of another climatic observable -assumed to be the driver -rather than the actual external forcing -with an effective transfer function -rather than the true Green function. The conditions under which climate observables can be used as both predictands and predictors have been discussed by Lucarini 29 .
By taking the Fourier transform of the Green function G (1) Φ , one obtains the linear susceptibility of the observable ( ) (1) χ ω Φ , where ω is the frequency. The susceptibility gives the frequency response to a forcing f t ( ) as ω  , where with ⋅ we indicate the Fourier transform. The susceptibility gives a spectroscopic description of the properties of the response of the observable, and its analysis can give interesting information on the most relevant time scales and related processes that determine the response of the observable. procedure for the Retrieval of the Green function. The strategy for testing the prediction of the mentioned key variables with the coupled model ensembles is as follows. First, we compute the Green function from the xCO 2 2 experiment. The time variable is defined in such a way that the instantaneous doubling occurs at = t 0. Hence, the time modulation of the forcing is given in this case by f t is a constant depending on the amplitude of the forcing. Since the radiative forcing is approximately proportional to the logarithm of the CO2 concentration, such a constant is given by f log(2) if the final CO 2 concentration were p times as large as the initial one). Equation A2 can be thus rewritten as: (1) The outputs of the xCO 2 2 experiments and the corresponding Green functions for the observables described above are presented in Figs. 5, 6, 7 and 8.
In particular, the time evolution of OHU for this scenario is shown in Fig. 7. The positive forcing due to the instantaneous CO 2 doubling leads to an instantaneous jump in the OHU, leading to an annual average value of more than 1 PW in the first year. Equation A3 then suggests that the Green function G OHU (1) has a singular behav- www.nature.com/scientificreports www.nature.com/scientificreports/ iour at = t 0 (cfr. Ref. 29 ), while a regular behaviour is found for t 0 > , corresponding to the negative radiative Planck feedback.
For all observables, we then use the Green functions above to perform predictions for the pctCO 1 2 scenario using Eq. A2. Thanks to the proportionality of the radiative forcing to the logarithm of the CO 2 concentration, the time modulation of such forcing can be expressed as = f f r t ( ) pctCO 1 2 , where r t ( ) is a ramp function (cfr. Refs. 18 and 27 ): where the time scale 70 τ ≈ y denotes the time needed to reach the doubling in the CO 2 concentration and where = f f pctCO x CO 1 2 2 2 because at the end of the ramp the CO 2 concentration is doubled. From the Green function, one could in principle compute the susceptibility and perform a spectral analysis of the properties of the response. However, the correct identification of spectral peaks in the susceptibility requires a much richer statistics than what we have available here. The reason is that, while the Green function is an integral kernel whose specific values at each t are not of crucial importance per se, as it is their integrated contribution through convolution with the cosidered time pattern that determines the response, in the case of the susceptibility it is extremely important to make sure that the signal to noise ratio is very large at each individual value of ω. This translates into the fact that, despite the Green function and the susceptibility being strictly connected, to obtain a satisfactory estimate for the latter require a statistics orders of magnitude larger than for the former, and possibly different and dedicated numerical estimation approaches 16,17,26 . An analysis of the susceptibility in experiments similar to what done in this work was attempted in Ragone et al. 27 , but using ten times more ensemble members. We therefore do not present an analysis of the susceptibility. While the analysis of the detailed frequency response of a climate model remains a very interesting and promising topic, it has to likely wait until experiments with at least several hundreds ensemble members will be available. equilibrium climate Response and transient climate Sensitivity. Response theory allows to place on solid formal ground operational definitions of the sensitivity of the climate system 2,18,27 . One of the most important indicators of the global properties of the response of the system to climate change is the equilibrium climate sensitivity (ECS), which is the long term ( → ∞ t ) response of the observable T m 2 to an abrupt doubling of CO2 concentration 3 . Another common measure of the response is the transient climate response (TCR), which is the change in T m 2 realised in the pctCO 1 2 scenario at the end of the ramp of CO 2 increase 84 . Using the formalism discussed in this paper, the ECS can be straightforwardly linked to the susceptibility, because 18,27 . Additionally, the TCR can be computed as the result at time t τ = of the convolution of the Green function of T m 2 with the forcing given in Eq. A4. Indeed, more generally, the susceptibility can be interpreted as a generalised sensitivity function. In particular, as explained in Ragone et al. 27 , one can find an explicit functional relation for the realised warming fraction 3 , given by the ratio between TCR and ECS: where = sinc x x x ( ) sin( )/ . The integrand in the second term on the right hand side of Eq. A5 gives the contribution of each time scale to the inertia of the system. Note that this approach allows for treating seamlessly -by changing the value of -the case of transient response to steeper or gentler increases of CO 2 . However, a detailed analysis of the relationship between TCR and ECS requires an accurate estimate of the susceptibility, that as explained above is beyond the scope of this work.
The theory of emergent constraint has been used to study the ECS 85 and the TCR 86 in climate models. The relationship between ECS and TCR proposed in Eq. A5 might be helpful elucidating and better understanding such results.