Projections of the Transient State-Dependency of Climate Feedbacks

When the climate system is forced, e.g. by emission of greenhouse gases, it responds on multiple time scales. As temperatures rise, feedback processes might intensify or weaken. Current methods to analyze feedback strength, however, do not take such state dependency into account; they only consider changes in (global mean) temperature and assume all feedbacks are linearly related to that. This makes (transient) changes in feedback strengths almost intangible and generally leads to underestimation of future warming. Here, we present a multivariate (and spatially explicit) framework that facilitates dissection of climate feedbacks over time scales. Using this framework, information on the composition of projected (transient) future climates and feedback strengths can be obtained. Moreover, it can be used to make projections for many emission scenarios through linear response theory. The new framework is illustrated using the Community Earth System Model version 2 (CESM2).


Introduction
Were the climate system free of feedback processes, it would be easy to predict and control the future climate (Arrhenius, 1896). Unfortunately, the climate system is peppered with feedback loops (Heinze et al., 2019;Von der Heydt et al., 2020). Because of these, climate change can get suppressed or enhanced, making future projections hard (Sherwood et al., 2020). Therefore, detailed knowledge of all relevant feedback processes is required to accurately assess potential future climates. However, knowledge of the current strengths of climate feedbacks is not enough. Over time, as the climate state changes, the strengths of climate feedbacks also change (Gregory & Andrews, 2016;Marvel, Pincus, Schmidt, & Miller, 2018;Armour, Bitz, & Roe, 2013); for instance, the albedo-increasing effect of ice is only relevant when there (still) is ice.
As the Earth warms, the strength of positive feedbacks increases (Bony et al., 2006). Therefore, projections based only on current knowledge of climate feedbacks underestimate future climate change -especially the committed warming that is to come even if zero emission is achieved (Marvel et al., 2018;Goodwin, 2018;Senior & Mitchell, 2000). To properly assess different emission scenarios, it is crucial to identify all relevant feedback mechanisms and, additionally, to quantify how their strengths change over time.
For future temperature projections with global climate models, currently the focus lies with the following feedback processes Klocke, Quaas, & Stevens, 2013;Cess, 1975). First, the Planck radiation feedback suppresses warming due to increased outgoing radiation. Second, the lapse rate feedback also suppresses warming due to an increase in long-wave radiation escaping to space (Sinha, 1995). Third is the ice-albedo feedback that enhances warming through a less reflective Earth surface (Curry, Schramm, & Ebert, 1995). Fourth is the water vapour feedback which boosts warming because of increased atmospheric water vapour content (Manabe & Wetherald, 1967). Finally, there is the cloud feedback, describing changes in cloud type and distribution, which could enhance or suppress warming (Cess & Potter, 1987).
The effect of a feedback process is quantified by its contribution to the climate system's radiative response to a certain experienced radiative forcing -in the current Anthropocene primarily caused by greenhouse gas emissions. Specifically, the top-of-atmosphere radiative imbalance ∆N (that equals zero when the climate system is in equilibrium) is given by the sum of the radiative forcing F and the radiative response ∆R, i.e.

∆N (t) = F (t) + ∆R(t).
(1) This response ∆R is the sum of the feedback contributions of all relevant feedbacks. So, writing ∆R j for the feedback contribution of feedback j, where the sum is over all feedback processes. Classically, the feedback strength λ j (also called feedback parameter ) is given by the feedback contribution per unit warming, i.e. λ j := ∆R j (t) ∆T (t) . (3) From this, the top-of-atmosphere radiative imbalance ∆N can be related to the warming ∆T as which can be used for climate projections (Gregory et al., 2004). However, the climate system responds on many, vastly different time scales . In particular, not all feedback processes react similarly on all of these time scales. Consequently, there is no linear relationship between all feedback contributions ∆R j and the global warming ∆T Knutti & Rugenstein, 2015). Hence, computations of feedback strengths λ j often find these to change over time (Senior & Mitchell, 2000;Boer, Hamilton, & Zhu, 2005); they are certainly not constant, as is often implicitly assumed Klocke et al., 2013;Meraner, Mauritsen, & Voigt, 2013).
We propose here to extend the classical framework to include dynamics on multiple time scales. To this end, feedback strengths should not be computed for the total time series, as in equation (3), but they should be separated in (Koopman) modal contributions instead. This makes the time-dependency of the various feedback processes explicit and opens the possibility of projections -both transient and equilibrium -for individual feedback contributions under any forcing scenario.
The rest of this paper is structured as follows. In section 2, we describe the theoretical framework, define new feedback strength metrics, and explain the computations of the feedback contributions. In sections 3.1-3.2, the framework is applied in a study of climate feedbacks in a 1, 000 year long abrupt4xCO2 experiment in the 'Community Earth System Model version 2' (CESM2). In section 3.3, projections of feedback contributions are made for and compared against output of CESM2's 1pctCO2 experiment. Finally, section 4 contains a summary and discussion.
2 Method and Model 2.1 Method: the new climate feedback framework From linear response theory , the change (response) in an observable O due to a certain forcing g is given by where G [O] is the Green function for observable O. The Green functions in the climate system are wellapproximated by a sum of decaying exponentials (the Koopman modes), with only the contribution of the different modes depending on the observable O. That is, where the sum is over all M modes, τ m is the m-th mode's time scale and β [O] m is the contribution of mode m to the observable's Green function. Because of this, following e.g. Proistosescu and Huybers (2017), the response of observable O can also be written as where M g m (t) := m all observable-dependencies. Since all feedback contributions ∆R j and the global warming ∆T adhere to this formulation, it is possible to define the feedback strength of feedback j per mode m as Additionally, the instantaneous feedback strength at time t can be defined as the incremental change in feedback contribution ∆R j per incremental change in temperature ∆T (i.e. the local slope of the graph (∆T, ∆R j ) at time t): Using this framework, it is also possible to estimate properties of the eventual equilibrium state -if such state exists. For instance, if g(t) ≡ḡ, M g m (t) =ḡτ m 1 − e −t/τm , the equilibrium ∆O * is given by The equilibrium feedback strength λ * j for feedback j is then defined as A more elaborate treatment of the mathematics of this framework is given in SI-Text S1. Below, we use this framework on two idealised experiments in CESM2 to understand the temporal evolution of (the strength of) climate feedbacks in these scenarios.

Model: CESM2 abrupt4xCO2 and 1pctCO2 experiments
We focus on forcing experiments conducted with the model CESM2 (Danabasoglu et al., 2020), specifically on the model's abrupt4xCO2 (1, 000y long) and 1pctCO2 (150y long) experiments in CMIP6 (Eyring et al., 2016). The abrupt4xCO2 experiment is used for fitting of the parameters and computations of feedback strengths per mode; the 1pctCO2 experiment is used to test projections made within the presented framework.
As feedback contributions are not outputted directly by any climate model, these have been determined using additional methodology. For this, we have used the radiative kernel approach , in which feedback contributions are computed (per grid point at location x) as where y j is the relevant (derived) state variable for the j-th feedback and ∂N/∂y j (x) is the pre-computed radiative kernel. We have used the publicly available CESM-CAM5 kernels from Pendergrass, Conley, and Vitt (2018); . There are others available, but differences between kernels are typically small. The procedure for the cloud feedback contribution is somewhat different because of its more nonlinear nature, but is nevertheless computed following common practice Shell et al., 2008;Klocke et al., 2013) -see SI-Text S4 for more details.

Temporal evolution, abrupt4xCO2
From CESM2 model output, feedback contributions ∆R j (x, t), top-of-atmosphere radiative imbalance ∆N (x, t) and near-surface temperature increase ∆T (x, t) were computed; changes in observables are determined with respect to the temporal average of a control experiment with pre-industrial CO 2 levels. From these, globally averaged datasets ∆R j (t), ∆N (t) and ∆T (t) were derived. These global time series have been fitted to equation (7), with exception of ∆N (t) that also includes the radiative forcing and is fitted to where the subscript 'abr' indicates that the forcings are from the abrupt4xCO2 experiment. More details of the procedure can be found in SI-Text S2.
First, the amount of modes M needs to be determined. Tests with various values for M have indicated M = 3 is the best choice here: M < 3 does not capture all dynamics and M > 3 has no significant benefits; see SI- Figure  S1. This is also in agreement with other reports that typically find two or three relevant modes (Caldeira & Myhrvold, 2013; Tsutsui, 2017; Proistosescu & Huybers, 2017; Bastiaansen, Dijkstra, & von der Heydt, 2020). Table 1 -Values for the climate feedback parameters λ m j per mode. These have been computed from the fitted parameters (SI-Table S1) via equation (8). Time scales τm have units 'year'; feedback parameters λ m j have units 'W/m 2 /K'. Plusminus values indicate 95% confidence intervals, which were propagated from the 95% confidence intervals for the fitted parameters assuming a normal distribution for all errors and no existing correlations between parameters.
From the fitted parameters, the feedback strengths per mode are computed using equation (8). In Table 1, the results are shown per mode and for the equilibrium values. A plot of the change in (instantaneous) feedback strength over time is given in Figure 2A. This shows behaviour on three distinct time scales: (1) a short time scale τ 1 = 4.5y, (2) an intermediate time scale τ 2 = 127y and (3) a long time scale τ 3 = 889y. The shorter time scale is in agreement with other studies , and long-term behaviour seems very plausible. The Planck feedback does not change much over time. The lapse rate feedback becomes weaker over time since temperatures increase. The surface albedo feedback diminishes as sea-ice is melting. The water vapour feedback, however, strengthens as warmer air can contain more water, which -as demonstrated here -does lead to an almost tripling in this feedback's strength over time. Finally, the cloud feedback becomes more dominant over time. The estimated equilibrium values are a bit higher generally compared to other studies (e.g. Zelinka et al. (2020)) since longer time scales are incorporated in the present study, which leads to higher temperature values and more positive total feedback strength Rugenstein et al., 2020;Marvel et al., 2018;Goodwin, 2018).
Since the individual feedback contributions ∆R j should sum to the total radiative response ∆R, feedback strengths should satisfy λ m = j∈F λ m j . However, because of ignored feedback interactions, imperfect radiative kernels and other limitations of the feedback contribution computations, a mismatch of up to 10% is deemed inconsequential. Looking at the feedback parameters per mode in Table 1, we see a good balance for the intermediate time scale (−0.02 W m −2 K −1 imbalance), but not for the short (+0.54 W m −2 K −1 imbalance) or long time scales (−1.46 W m −2 K −1 imbalance). Taking the large uncertainty into account for the shortest time scale, this imbalance might be insignificant still. However, for the long time scale, the imbalance is large and even the signs are not the same. This points to a missing negative feedback on this time scale (or inaccuracies in the computational methods for the feedback contributions ∆R j ). Because the equilibrium values show a smaller imbalance (−0.43 W m −2 K −1 ), the alleged missing feedback (or feedbacks) seem to be especially important during the transient state.

Spatial distribution, abrupt4xCO2
With a slight modification of (8), the spatial structure of the (Koopman) modes can be included. Following a similar procedure in Proistosescu and Huybers (2017), we take and similar formulas for ∆N (x, t) and ∆T (x, t). More details can be found in SI-Text S2.2. The space-dependent β [Rj ] m (x) encompasses the spatial structure of the observable ∆R j in the m-th mode. Again, it is also possible to estimate equilibrium properties, leading to an estimation of the spatial distribution of feedback contributions and warming in equilibrium. Using the time scales τ m from the fits of the global observables, the spatial coefficients β [Rj ] m (x) can be computed efficiently using linear regression methods.
Water Vapour feedback (LW) 0 1,000 2,000 3,000 0 Water Vapour feedback (SW) 0 1,000 2,000 3,000 0 Cloud feedback (SW+LW) Figure 1 -Evolution of temperature increase ∆T , radiative imbalance ∆N and feedback contributions ∆Rj for the CESM2 abrupt4xCO2 experiment. Red circles denote data points; blue lines the obtained fits. The dashed lines indicate the estimated equilibrium values (or initial forcing in case of the radiative imbalance plot). SI- Table S1 contains the values for the fitted parameters.    j (x) in equilibrium, and initial (effective) radiative forcing F (x) for the CESM2 abrupt4xCO2 experiment. Equilibrium distributions are derived from the fitted spatial modes; initial forcing F (x) is obtained directly from fits.
In SI-Figures S3-S10, the spatial modes of all feedback contributions, warming and radiative imbalance are given. The estimated equilibrium distributions are shown in Figure 3. The patterns of feedback contributions are in agreement with previous studies Dong et al., 2020;Armour et al., 2013;Boeke, Taylor, & Sejas, 2020;Andrews & Webb, 2018;Dessler, 2013;Soden et al., 2008;Andrews et al., 2015), although they seem more pronounced here -probably due to a better incorporation of long-term behaviour. Near the poles, temperature increase gets close to +30 • C (compared to +12 • C near the equator) and large changes (up to +135 W m −2 ) in surface albedo are found related to sea-ice melting. Moreover, the lapse rate feedback contribution increases by about 10 W m −2 in these polar regions. Near the equator, the water vapour feedback gets much stronger (more than +50W m −2 ) over the oceans -especially in the Pacific cold tongue where changes up to +120W m −2 are found. Cloud feedback changes mostly over southern hemisphere oceans (up to +69 W m −2 ), and northern hemisphere changes are smaller (on average only +2.7 W m −2 ) compared to previous studies (Andrews & Webb, 2018;Dong et al., 2020;Armour et al., 2013).

Projections for 1pctCO2 experiment
In the 1pctCO2 experiment, the forcing (g grad ) is given by a yearly 1% increase in atmospheric CO 2 . Replacing M g abr m by M g grad m in (8), while using the values for τ m and β m as fitted with the abrupt4xCO2 experiment, gives a projection for feedback contributions and warming in the 1pctCO2 experiment. This procedure, following linear response theory Lucarini & Sarno, 2011;, has been used for warming projections already in many models, including global climate models (Lembo, Lucarini, & Ragone, 2020;Aengenheyster et al., 2018;Maier-Reimer & Hasselmann, 1987;Hasselmann, Sausen, Maier-Reimer, & Voss, 1993). Although typically used on an ensemble of model runs and for smaller forcings, we use it here nevertheless to illustrate its potency. More details can be found in SI-Text S3.
In Figure 4, projections for global feedback contributions are shown along with data coming directly from the CESM2 1pctCO2 experiment. Qualitatively, the projections are able to reproduce the actual trends in the globally averaged observables. The quantitative values overestimate feedback contributions by 10% − 20% (less for the lapse rate feedback; more for cloud and long-wave water vapour feedbacks up to about t = 120y), which seems reasonable for this set-up. Projections for the temporal evolution of instantaneous feedback strengths in this scenario are given in Figure 2B. These show lasting changes over centuries -primarily in (long-wave) water vapour feedback, surface albedo feedback and cloud feedback -and a less negative total feedback parameter, indicating the climate system becomes more sensitive to radiative forcing over time.
Using the same approach, it is also possible to make projections for the evolution of the spatial distributions. In SI-Figures S11-S18 the projections for the feedback contributions ∆R j , the radiative imbalance ∆N and the temperature increase ∆T at t = 140y are shown. Videos showing these results for other times are also available (SI-videos). From all of these, it can be seen that the spatial projections reproduce the general larger spatial patterns and even some of the smaller features, although the internal variability is of course not present in the projections. The spatial patterns are similar to those found in the abrupt4xCO2 experiment ( Figure 3) and show the same kind of response in polar and equatorial regions. In general, the projections seem to overestimate the spatial feedback contributions, similar to the overestimation of the global feedback contributions.

Summary and discussion
In this study, we have introduced a new multivariate feedback framework that enables analysis and projections of individual climate feedbacks, including their spatial distribution, over different time scales. This framework deviates from the common practice of linearly regressing feedback contributions to global warming Marvel et al., 2018;Klocke et al., 2013;Meraner et al., 2013) -which is not valid, especially when considering long-term effects Knutti & Rugenstein, 2015). Instead, these contributions are here taken into account directly, and temporal evolution is explicitly considered via a nonlinear regression to obtain (Koopman) modes for each feedback. These modes, indicating behaviour over different time scales, encapsulate the climate system's dynamics and can thus be used both for analysis and projections of climate feedbacks -even for other forcing scenarios.
Using the presented framework, we have analyzed an abrupt4xCO2 experiment in CESM2. This clearly showed evolution of climate feedbacks over time, such as a diminishing of surface albedo feedback in the polar regions as sea-ice melts. Strikingly, the water vapour and cloud feedbacks showed a large increase over time: water vapour feedback, primarily over oceans around the equator, almost tripled in strength over a 1,000 year period and cloud feedback mainly changed over southern hemisphere oceans. Analysis also showed that the (commonly) considered feedback processes do not sum up to the total feedback for long time scales, pointing to a missing negative feedback on these time scales. Since estimated equilibrium feedback strengths do balance, the alleged missing feedback (or feedbacks) is expected to play a role primarily during the transient. As such, the ocean heat uptake is a good candidate to fill this gap. However, as of now, no radiative kernels exist for the ocean heat uptake's effect on the top of atmosphere radiative balance, making it impossible to test this hypothesis. Nevertheless, tracking its effect over time seems necessary to accurately understand the long-term transient behaviour.
Another benefit of tracking the temporal evolution of climate feedbacks is that this makes it possible to project the climatic change for all sorts of emission scenarios without the need to have dedicated model experiments for them all. In this paper, we have shown the capabilities of such projections by comparing them to a (single) 1pctCO2 experiment with the same CESM2 model. The projections did capture the qualitative trends, but also showed an overestimation for almost all global and spatial climate feedback contributions. However, this possibly is due to internal variability associated with a single run. Hence, results are expected to be even better quantitatively when a (potentially small) ensemble of runs is used, as application of similar linear response techniques on e.g. temperature response have shown previously (Lembo et al., 2020;Aengenheyster et al., 2018).
As climate models become more and more detailed, these models also become more computationally expensive. Therefore, it is practically impossible to compute climate responses for many forcing scenarios and/or on very long time scales. As climate feedbacks become more positive with rising temperatures, it is necessary to track these changes to prevent underestimation of climate change and adequately account for committed (long-term) warming. Hence, both extrapolation techniques and response theory can play a significant role to alleviate the gaps. The multivariate feedback framework in this study can contribute to this, allowing for long-term estimation and climate projections for different forcing scenarios. Moreover, as these projections are multivariate, they can inform us on more than global mean warming alone, and also indicate how a potential future climate state may behave differently compared to our current one.
CESM2 data has been downloaded on the fly from Google's cloud storage mirror of CMIP6 data using the 'intake-esm' utility package in Python (https://doi.org/10.5281/zenodo.4243421).

2
The net amount of the top-of-atmosphere radiative imbalance is typically denoted by ∆N and equals the amount 3 of incoming shortwave solar radiation minus the outgoing short-and longwave radiation (i.e. ∆N = N SW,↓ − 4 N SW,↑ − N LW,↑ ). The precise amounts of these outgoing radiative fluxes is influenced by any forcing (e.g. due to 5 CO 2 emissions) and by the many feedback processes of the Earth system (e.g., sea-ice melting, changes in surface 6 albedo and cloud formation). Hence, the radiative imbalance ∆N can be seen as a function of any applied forcing 7 µ and the system state y; that is, 8 ∆N = ∆N ( y; µ) . (S1.1) 9 The Earth system is said to be in (radiative) equilibrium when ∆N = 0. So, for a given (constant) forcing µ = µ A , 10 an equilibrium state y = y A has to satisfy 11 ∆N y A ; µ A = 0. (S1.2)

12
If the amount of forcing were to change, say µ(t) = µ A + ∆µ(t) (not necessarily as a function of time), the system 13 state y(t) and the radiative imbalance N (t) will respond and change over time. Denoting the new system state 14 by y(t) = y A + ∆ y(t), a substitution in (S1.1) gives the new imbalance as 15 ∆N (t) = ∆N y A + ∆ y(t); µ A + ∆µ(t) . (S1.3)

16
If the perturbation ∆µ(t) is small enough, the response ∆y(t) will also typically be small. In this case, this 17 expression can be approximated via a Taylor expansion as a (multi-)linear function. Specifically,

∆N (t) = ∆N y
In this expression, ∆N y A ; µ A is the radiative imbalance in the unperturbed state, which equals zero per (S1.2).

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The term ∂∆N ∂µ y A ; µ A ∆µ(t) denotes the direct effect of the forcing and is typically called the radiative response 21 F (t). The resulting radiative response ∆R(t) to this forcing is given by ∂∆N ∂ y interaction terms that are assumed to be small .

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The radiative response ∆R(t) is formed by a combination of all relevant climate feedbacks. which -if all feedback processes are taken into account -leads to λ(t) = j∈F λ j (t) (with possibly a small 40 inconsistency because of the neglected higher order and interaction terms in (S1.4)). Here, we have explicitly 41 denoted the time-dependency of λ j (t). However, often, the feedback parameters are interpreted as constants in 42 time. In practice, their values are even inferred from transient simulation data via linear regression of ∆R j (t) = 43 λ j ∆T (t) [e.g. Klocke et al., 2013, Marvel et al., 2018, Meraner et al., 2013.

44
However, such linearity assumption does not hold -especially when incorporating climate dynamics over longer 45 time scales , Knutti and Rugenstein, 2015. In general, a linear dynamical 46 system that is perturbed by some change in forcing g(t) (e.g., g(t) = ∆µ(t)), follows relatively simple dynamics: and (S1.8) could alternatively be written as 59 (S1.11)

60
As ∆R j and ∆T (and ∆R) follow (S1.8), λ j in (S1.6) cannot be a constant (unless M = 1 or in the case the 61 Green functions for ∆R j and ∆T are identical up to a multiplicative factor).

62
At the same time, (S1.8) invites the study of the system's dynamics per mode. Hence, following (S1.8) we write Here, R m j denotes the feedback contribution for mode m (which thus acts on a timescale τ m ), R m the complete 68 radiative response at the m-th mode and T m the (global mean surface) temperature at this mode. 69 We propose here to quantify climate feedback strengths per mode. That is, we define the climate feedback 70 parameters λ m j , which denote the strengths of the j-th feedback at mode m as , (S1.15) 72 and the total feedback parameters λ m for the full radiative response at mode m as (S1.16)

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Similar to the classical definitions of the feedback parameters, per mode the individual feedback parameters should 75 again sum up to the total feedback parameter, i.e. λ m = j∈F λ m j (again small inconsistencies are possible due 76 to ignored higher order and/or interaction terms in (S1.4)).

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It is also possible to estimate the instantaneous feedback strength of a feedback at time t. For this, the local slope 78 of the graph (∆T (t), ∆R j (t)) at time t can be used, which is computed as the fraction of the derivatives, i.e.
where the primes denote derivatives with respect to time.

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We stress that these equilibrium properties have no straightforward connection to the transient feedback parameters 87 λ m j and λ m .

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S2 Climate Feedbacks in CESM2 abrupt4xCO2 experiment The above described framework has been applied to CESM2's CMIP6 abrupt4xCO2 model run [Eyring et al., [Danabasoglu et al., 2020]. To compute the feedback contributions ∆R j (t), radiative kernels made by Pendergrass 92 et al. [2018] with CESM-CAM5 were employed [Pendergrass, 2017a,b]. Computations followed the standard 93 process : radiative kernels were multiplied (per grid cell) with time series for 94 the relevant variable field to obtain location-dependent time series ∆R j (x, t), from which global mean time series 95 ∆R j (t) were derived. In this way, we have investigated the following climate feedbacks: (1) Planck feedback,

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(2) lapse rate feedback, (3) surface albedo feedback, (4) water vapour feedback and (5) cloud feedback. The latter 97 (cloud feedback) has not been computed directly with a radiative kernel, but is indirectly inferred from differences 98 between clearsky and fullsky feedback contributions of the other feedbacks, following Pendergrass et al. [2018], 99 Soden et al. [2008]; more details are given in section S4. In addition, also (global mean) time series ∆T (t) for 100 surface temperature and ∆N (t) for net top-of-atsmophere radiative imbalance have been computed.

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can be computed exactly as Hence, time series for ∆T (t), N (t) and all feedback contributions R j (t) should satisfy When fitting (S2.2)-(S2.4) to the (global) data, some value for the number of modes M needs to be specified.

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Since the true amount of modes is unknown, we have tested values up to M = 5 -see Figure S1. As M = 3 116 seemed to produce consistent fits with low residuals we have chosen M = 3 for the fits in this study.

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The thus obtained fits are plotted in Main Text Figure 1 along the data. In Table S1, the values for all fitted 118 parameters are given. From these, the feedback parameters per mode λ m j , as well as equilibrium values λ * j , have 119 been computed; these are reported on in Main Text Table 1.

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Here, we do not have such ensemble, but nevertheless apply this principle to illustrate its potency in projections 146 of feedback contributions. Specifically, we use the previously found fits for the CESM2 abrupt-4xCO2 experiment 147 to construct projections for its accompanying 1pctCO2 experiment -an experiment in which the atmospheric 148 CO 2 concentrations are increased by 1% every year. As that experiment has been performed, we can compare 149 the projections to the actual model output. To do so, we have computed feedback contributions ∆R j and other 150 observables using the same techniques as before with the abrupt-4xCO2 experiment.

151
To relate the response in the abrupt-4xCO2 experiment to the response in the 1pctCO2 experiment, M g m need 152 to be computed for both experiments. For this, we denote the forcing in the abrupt4xCO2 setting by g abr (t) and 153 in the 1pctCO2 setting by g grad (t) -where 'abr' stands for 'abrupt' and 'grad' for 'gradual'.

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The radiative forcing due to changes in atmospheric CO 2 content is related logarithmically . Hence, experienced 155 (change in) forcing g is given by where A is some constant (according to Myhre et al. [2014], A = 5.35 W m −2 for the effect of CO 2 alone, but higher 158 values are also used sometimes to mimic enslaved increase in other greenhouse gases ; 159 in any case, the precise value is completely irrelevant for the computations here as it appears as multiplicative 160 constant for all forcings) and where c(t) denotes the concentration of CO 2 in the atmosphere over time. We note 161 that the formalism presented here also would hold if g was just the atmospheric CO 2 concentration, but then 162 the 'smallness of forcing' requirement is much stricter, making projections useless in practice (this can be seen 163 'intuitively' by making a Taylor approximation of (S3.1) for small changes in atmospheric CO 2 ).

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Now, M g m can be computed straightforwardly. Since c abr (t) = 4c(0), g abr (t) = A log (4)  Figures S10-S17 the spatial projections at t = 140y (at that moment, the amount of CO 2 has been quadruppled 175 compared to t = 0) are given along with the actual values and the errors (the spatial distribution for surface 176 albedo feedback also appears as Main Text Figure 4). Movies of the spatial projections over the years are also 177 available.

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As before, the instantaneous feedback strengths λ inst j (t) can be computed. For the 1pctCO2 experiment these are

S4 Cloud feedback
Following Soden et al. [2008], the cloud feedbacks have been derived from differences between clearsky and fullsky 186 feedback contributions of the other feedbacks. As the total radiative response is the sum over all feedback  Because the radiative forcings F (t) and F cs (t) cannot be obtained from model output like ∆N (t), ∆N cs (t),

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∆R j (t) and ∆R cs j (t), their values need to be derived from fits instead. Hence, in the fits we use the following (S4.6)