The radiative feedback continuum from Snowball Earth to an ice-free hothouse

Domain: earth_science
Published: 2024-08-03
Editorial outcome: major_revision

Abstract
Paleoclimate records have been used to estimate the modern equilibrium climate sensitivity. However, this requires understanding how the feedbacks governing the climate response vary with the climate itself. Here we warm and cool a state-of-the-art climate model to simulate a continuum of climates ranging from an nearly ice-covered Snowball Earth to an nearly ice-free hothouse. We find that the pre-industrial (PI) climate is near a stability optimum: warming leads to a less-stable (more-sensitive) climate, as does cooling of more than 2 K. Physically interpreting the results, we find that the decrease in stability for climates colder than the PI occurs mainly due to the albedo and lapse-rate feedbacks, and the decrease in stability for warmer climates occurs mainly due to the cloud feedback. These results imply that paleoclimate records provide a stronger constraint than has been calculated in previous studies, suggesting a reduction in the uncertainty range of the climate sensitivity.

Introduction
Recent community assessments1,2 have substantially narrowed the estimated range of Earth’s equilibrium climate sensitivity (ECS) for the first time in decades, leading to better constraints on future warming3. This narrowing of the uncertainty in the ECS (which is defined as the equilibrium global-mean surface temperature response to CO2 doubling from pre-industrial levels) was achieved in large part through the use of paleoclimate records from times when the climate was substantially different from today. In Sherwood et al.1, the ECS likelihoods derived from proxy reconstructions of temperatures and estimates of radiative forcing during the Last Glacial Maximum (LGM) and mid-Pliocene warm period (mPWP) provided the strongest line of evidence against high ECS values. In another assessment2, proxy reconstructions of LGM, mPWP, and Eocene temperatures also informed the strongest line of evidence against high ECS values: so-called “emergent constraints” wherein a relationship between temperature changes and ECS within an ensemble of Earth System Models (ESMs) is combined with observations or paleoproxy reconstructions of those temperature changes to derive a constraint on ECS.

A confounding factor in the use of paleoclimate records to inform the sensitivity of the modern climate to greenhouse gas forcing is that the radiative feedbacks governing the climate response can vary with the underlying climate itself1,2. That is, the use of paleoclimate records to constrain ECS requires understanding how modern radiative feedbacks (which govern ECS) relate to radiative feedbacks operating in climates much colder or much warmer than today.

Following previous work4–6, Sherwood et al.1 represented the dependence of radiative feedbacks on the underlying climate by including a quadratic feedback term in the standard model of global energy balance used to relate reconstructions of temperature and climate forcing to modern-day ECS. This approach typically represents the net radiative feedback as becoming less negative (i.e., a more-sensitive climate) with global warming and more negative with global cooling (e.g., ref. 1). While higher-order terms that are cubic and beyond in surface temperature could be included, they are typically assumed to be small and omitted. This raises key questions regarding the range of temperatures over which this approximation applies, what causes it to fail outside this range, and relatedly how confident we can be in the structure of the radiative feedback dependence on global temperature over a wider range of climate states. The answers to these questions also have implications for emergent constraints, in which the mapping of feedbacks between past and future climate states is implicitly accounted for through the use of ESMs to simulate the paleoclimate states and ECS values on which the constraints rely.

Here we warm and cool a state-of-the-art ESM to simulate a continuum of climates ranging from an nearly ice-covered Snowball Earth to a nearly ice-free hothouse planet. We analyze how the radiative feedbacks depend on the underlying climate, and we physically interpret the results.

Results
Using NCAR’s Community Earth System Model Version 2 (CESM2)7 in its standard workhorse configuration, we ramp CO2 concentrations over a range of 11.5 doublings. Specifically, we start from the end of a 500-year pre-industrial (PI) control simulation, which has a constant CO2 concentration of 284.7 ppm, and we either increase or decrease the atmospheric CO2 concentration at a rate of 1% per year (Fig. 1a). The Warming simulation, which extends a preexisting gradual CO2 quadrupling simulation8, is 279 years long and ends with a CO2 concentration of 4522 ppm, which is 16 times the PI value. The Cooling simulation is 514 years long and ends with a CO2 concentration of 1.6 ppm, which is 1/175 times the PI value. See the Methods for details.

This leads to a 59 K range in simulated annual-mean global-mean surface temperature, with climates ranging from an nearly ice-covered Snowball Earth to an nearly ice-free hothouse planet. Averaged over the last decade of the PI control simulation, the global-mean surface temperature is 15°C, and the ice area is 11.4% of the global surface area. The latter includes sea ice, snow cover on land, and prescribed time-invariant glacial ice cover (see the Methods for details), with sea ice covering 6.1% of the ocean (4.3% of the globe). In the Warming simulation, the annual-mean global-mean surface temperature increases by 33°C (Fig. 1b), and the ice area decreases to 0.0% of the ocean. In the Cooling simulation, the temperature decreases by 41 K to −26°C (Fig. 1b), and the ice area increases to 68.7% of the globe (Fig. 1c), with sea ice covering 70.3% of the ocean.

The surface temperature in the deep tropics (averaged annually and over 10°S–10°N) is 28°C in the PI (Fig. 1e), and it reaches 42°C in the final decade of the Warming run (Fig. 1f) and 5°C in the final decade of the Cooling run (Fig. 1d). In the PI climate, the polar surface temperature (averaged annually and over both hemispheres poleward of 70°) is −24°C, and this region is largely covered with snow and ice (Fig. 1h). In the final decade of the Warming run, the polar temperature reaches 4°C (Fig. 1f), and the remaining ice cover is almost exclusively glacial ice (Fig. 1i), which is a specified surface type in CESM2 with an area that does not evolve during the simulations. In the final decade of the Cooling run, the polar temperature reaches −68°C (Fig. 1d), and the ice cover extends into the tropics (Fig. 1g).

In order to evaluate the net radiative feedback over this continuum of climates, we adopt the standard model of global energy balance and climate feedbacks:

ΔN = ΔFGHG + ΔFnet = ΔFGHG + λnet ΔT, (1)

with

λnet = ΔFnet / ΔT = (ΔN − ΔFGHG) / ΔT, (2)

Here all quantities are averaged annually and globally: N is the top-of-atmosphere (TOA) net energy flux reported by the model (using top-of-model fields), FGHG is an estimate of CO2 radiative forcing relative to PI based on previously published line-by-line radiative transfer calculations9 (see Methods and SI Fig. S1), Fnet ≡ N − FGHG is the net radiative response of the climate system, λnet is the net radiative feedback parameter, and T is the surface temperature. The fluxes are defined to be positive in the downward direction, and the feedback parameter is negative for a stable climate. The modifier Δ is described below.

The radiative forcing FGHG and resulting value of N are plotted in Fig. 2a, b, and the difference Fnet is plotted in Fig. 2c. It can be readily seen that Fnet does not depend linearly on T. Specifically, the slope of the Fnet versus T curve (Fig. 2c) is most negative near the PI climate (black vertical dashed line), being less steep in warmer and colder simulated climates. In extremely cold climates, the slope becomes zero around T = 0°C and then changes sign for climates with T < 0°C, implying that additional incremental levels of cooling lead to less energy coming into the climate system and hence more cooling.

We consider two approaches to define λnet in Eq. (2), following previous work10 (see Methods for details):

(i) The “effective feedback” λeffnet, which describes the radiative feedback processes operating between a given climate state and the PI climate. In this case, we define Δ as the anomaly from the PI climate, and Eq. (2) is calculated from Fnet after applying a polynomial smoothing. Note that this allows λeff to vary smoothly even in the limit ΔT → 0 (see Methods and SI Fig. S2).

(ii) The “differential feedback” λdiffnet, which describes the feedback processes operating within a given climate. Hence λdiffnet is the local tangent value of the slope in Fig. 2a. In this case, we define Δ as the anomaly associated with an incremental change in climate, and Eq. (2) is calculated using a regression of Fnet versus T within a running window (see Methods).

The effective feedback may be seen as most directly relevant to current discussions of ECS, since they often involve estimates of past climates compared with today, rather than estimates of past climate variability (e.g., refs. 1, 2). On the other hand, the differential feedback reflects the radiative response to a temperature perturbation in a given underlying climate, and hence it may be somewhat easier to physically interpret.

The net feedback parameter calculated using each of these approaches is plotted in Fig. 2d, e. A striking result is that the PI climate is near the stability optimum. The differential feedback λdiffnet, which indicates the stability of the climate system to perturbations, is most negative when the global temperature is 2 K cooler than the PI value (Fig. 2d). Starting from the PI, warming leads to less-stabilizing radiative feedbacks and hence a more-sensitive climate, as does cooling of more than 2 K. The effective feedback λeffnet shows similar behavior, being most negative when the global temperature is 5 K cooler than the PI value (Fig. 2e).

The TOA net energy flux when the climate has reached equilibrium is N = 0, as is approximately the case in the simulated PI climate (Fig. 2b). Hence from Eq. (2), the equilibrium warming response to a change in CO2 is ΔT = −ΔFGHG / λnet. This is known as the ECS in the special case of a doubling of CO2 from PI levels, as mentioned above. It is given by ECS = −F2× / λnet,2×, where F2× = 4.2 W/m2 is the value of the radiative forcing ΔFGHG when CO2 is doubled from its PI value of 284.7 ppm, and λnet,2× is the value of the feedback parameter λeffnet operating in this climate state. For other climate states, the effective climate sensitivity (EffCS) is similarly defined using the effective feedback parameter:

EffCS = −F2× / λeffnet, (3)

The EffCS is plotted in Fig. 2f. This shows that the sensitivity is lowest near the PI climate, with more-sensitive climates at warmer and much colder temperatures. The continuum of simulated climates spans a range of EffCS values from 2°C to 15°C. Note that the EffCS (Fig. 2f) scales as the inverse of λeffnet (Fig. 2e).

Under more-extreme cooling, the value of λdiffnet in Fig. 2d becomes positive when the global temperature drops below approximately 0°C, which is 15 K colder than the PI climate. At this point there is a change in the sign of the slope of the Fnet versus T curve in Fig. 2c: as the temperature drops below this point, incremental coolings are accompanied by incremental decreases in the level of heating by the net radiative response of the climate system. This corresponds to the Snowball Earth bifurcation point, beyond which the sea ice in the model expands toward the equator in an irreversible process. Note that this is the point at which the global temperature and ice area begin to change in the Cooling simulation (Fig. 1b, c). The implication is that the climate is transitioning across a range of temperatures for which the only equilibrium climate state is unstable (SI Fig. S3). Previous studies have demonstrated that bifurcations and bistability associated with the Snowball Earth climate occur in climate models of varying levels of complexity in climates with ice cover, and so on, with 11–15. Note that the λeffnet and λeffnet framework can give potentially misleading results about the stability of the underlying climate state because it is based on anomalies from the PI climate.

Under warming, the values of λeffnet and λdiffnet increase monotonically. Notably, the climate remains stable (λdiffnet is negative) even at extreme levels of global warming nearing 15 K above the PI. Notably, previous studies using idealized single-column radiative models have found that the net climate feedback becomes more negative with warming for climates warmer than approximately 25 K above the PI16,17.

Note that when the climate is forced to transiently evolve away from an equilibrated state, it is possible for the climate feedback parameter to become less negative due to the spatial pattern of surface temperature changes18. In SI Section S1, we investigate the extent to which this may explain the results in Fig. 2d, e by using a standard two-layer model of the climate system19 that includes a term to represent the deep ocean heat uptake efficacy. The results show that although deep ocean heat uptake efficacy can cause λdiffnet and λeffnet to become less negative under both warming and cooling as the climate gets farther from its equilibrated state, a moderate (CMIP5-mean) deep ocean heat uptake efficacy leads to far smaller changes in λnet than we find in CESM2 (SI Fig. S4). Furthermore, even with a large ocean heat uptake efficacy, the two-layer model results in a “V”-shaped feedback dependence on temperature that is centered at the equilibrated climate (purple curve in SI Fig. S4), in contrast to the “U” shape centered at a temperature several degrees colder than the PI that we find in CESM2 (Fig. 2d, e). This suggests that the changes in λnet shown in Fig. 2d, e are considerably outside of what would be expected from changing surface temperature patterns associated with deep ocean heat uptake, and that feedback nonlinearities with global temperature are the main cause of the dependence of the net feedback on the underlying climate in CESM2 over the simulated range considered here.

Linear representation of λnet(T)

Many recent studies have suggested that colder climates are more stable than warmer climates, including climates considerably colder than the PI. Specifically, as summarized in a previous assessment2, paleoclimate records20–28 and comprehensive climate models29–36 suggest a general trend toward less-stabilizing radiative feedbacks (hence higher EffCS) with increasing global temperature over a range of climates spanning approximately 6 K colder than today to approximately 10 K warmer than today. However, the results presented here suggest that the PI climate is near a stability optimum, with warming and cooling beyond 2 K both leading to less-stable climates (Fig. 2d). Similarly, warming and substantial cooling both lead to less-negative values of λeffnet and higher EffCS (Fig. 2e, f). While the climate at the temperature characteristic of the LGM (4-6 K colder than the PI) is more stable than the simulated climates that are warmer than the PI, consistent with the studies mentioned above, we find that climates beyond about 6 K colder than the PI can be considerably less stable than climates warmer than the PI.

As noted in the Introduction, previous work has typically represented nonlinearities in the dependence of the net radiative response on the underlying climate by using a quadratic relationship with global temperature (e.g., ref. 1). In this case, Eq. (1) is replaced with

ΔN = ΔFGHG + λ0ΔT + 1/2 αΔT2, (4)

where λ0 is the net feedback near the PI climate and α is a coefficient scaling the nonlinear radiative response. This implies a linear dependence on global temperature for both the effective feedback and the differential feedback:

λeffnet = λ0 + 1/2 αΔT and λdiffnet = λ0 + αΔT: (5)

Note that here we adopt the formalism used in Sherwood et al.1. Sherwood et al.1 use the value α = 0.1 W/m2/K2 (with an uncertainty of ±0.1 W/m2/K2) for the difference in the feedback at the LGM compared with the PI, and they implicitly assume no change in feedback between the PI and warmer climates. We include red dashed lines in Fig. 2d, e to represent a linear dependence of λnet on T that goes through the PI climate (T = 15°C) and the climate with an LGM-like level of cooling (T = 10°C). The slopes of the curves correspond to values of α = −0.01 W/m2/K2 for λdiffnet and α = 0.05 W/m2/K2 for λeffnet. We also include for comparison magenta dashed lines that go through the PI climate and the climate at 5 K of warming (T = 20°C), which have slopes that correspond to values of α = 0.17 W/m2/K2 for λdiffnet and α = 0.10 W/m2/K2 for λeffnet. Note that CESM2 has previously been shown to be among the ESMs with the largest values of α when assessed for temperature changes near the PI climate6.

These results show that the value of α adopted by Sherwood et al.1 for the change in λnet at the LGM is much larger than in the CESM2 results, because the “U” shape in Fig. 2d, e causes the feedback at 5 K of cooling to be similar to the feedback at the PI. This suggests that paleoclimate records provide a stronger constraint on the upper bound of the ECS than has been calculated in previous estimates such as Sherwood et al.1. In other words, if we were to repeat the analysis of Sherwood et al.1 using the value of α that we find here for the difference between the LGM and PI feedbacks, our lower value of α would imply a lower modern-day climate sensitivity than they found. This amounts to a stronger constraint on the upper bound of the ECS than they report and therefore a reduction in the uncertainty range for the ECS. This is because the CESM2 results suggest that the LGM may be a more direct analog to current warming than previously assumed, since the feedbacks are relatively similar. In other words, Sherwood et al.1 took λnet to be more negative at the LGM than the modern value, whereas we find that the feedbacks are similar. So a given paleo estimate of the LGM value of λnet implies a similar modern feedback value according to our results, whereas the analysis of Sherwood et al.1 would take it to imply a less-negative modern feedback and hence a more-sensitive modern climate.

For climates warmer than the PI, these results imply a larger value of α that is actually somewhat similar to the Sherwood et al.1 result. But here the value of α applies to warming rather than cooling. Overall, this suggests that feedback nonlinearities could be large for future warming, consistent with some other studies (e.g., ref. 6), while being relatively small for colder climates similar to the LGM.

These results suggest that the formulation of feedbacks as changing linearly with global temperature applies only over a narrow range of climates, and that because of the “U” shape of the relationship between the feedback and the underlying climate, comparing feedbacks between two climates depends sensitively on the temperatures of the climates.

The range of climates over which the quadratic term in the global energy budget (Eq. (4)) serves as a useful approximation can be seen by comparing the linear fits to the values of λdiffnet and λeffnet (magenta lines in Fig. 2d, e). For the effective feedback λeffnet, the quadratic term captures much of the variation in the feedback parameter for climates with T between about 3 K colder and 8 K warmer than the PI climate, thus serving as a decent approximation to feedback changes over a temperature range spanning the PI climate and CO2 quadrupling but not spanning climates as cold as the LGM. Outside of this temperature range, the quadratic approximation fails spectacularly. For the differential feedback λdiffnet, the quadratic term provides a decent approximation over a similar temperature range. Including additional terms in the Taylor series expansion (i.e., order ΔT3 and higher in Eq. (4)) would be expected to widen the range over which the expansion provides a useful approximation.

Individual radiative feedback parameters

In order to identify what physical processes are responsible for the decrease in stability under both cooling and warming from the PI (Fig. 2b), we begin by using a radiative kernel analysis to assess which individual feedback parameters are driving the changes. We use radiative kernels37 that were generated based on CAM5, which is the previous version of the atmospheric model in CESM2. Using these kernels, we compute the annual-mean global-mean change in the radiative response associated with changes in the surface temperature (FP for Planck feedback), atmospheric lapse rate (FL for lapse-rate feedback), humidity (Fw for water-vapor feedback), and surface albedo (Fα for albedo feedback). We compute the cloud radiative response (Fc) as the difference between the sum of the individual feedbacks and Fnet; hence Fc also includes the residual (Fres) due to inaccuracies in the radiative kernel analysis (see SI Section S2 for details). Each of the resulting radiative responses is shown in SI Fig. S5. Note that the radiative kernel analysis effectively linearizes the simulated response to changing climate fields about a climate near the PI. Although it would be more accurate to use radiative kernels that vary with the climate38, the present analysis could be seen as a preliminary step toward building understanding of feedbacks across a wide continuum of climate changes by using a kernel that does not vary with climate, before considering how the radiative kernels change.

We define each individual feedback parameter as

λi = ΔFi / ΔT, (6)

where the subscript i can indicate any individual feedback and Δ has the same two definitions as in Eq. (2).

The feedback parameters are shown in Fig. 3. We focus on the differential feedback analysis (right column in Fig. 3) and the effective feedback analysis (left column in Fig. 3). Note that although there is some compensation between the lapse-rate feedback (λL) and the water-vapor feedback (λw), as expected, the changes in the combined feedback (λL + λw) are dominated by the lapse-rate feedback (see red dashed lines in third row of Fig. 3).

Physical interpretation of feedback changes

Here we interpret the physical mechanisms driving the changes in the feedback parameters, which are shown in Fig. 3. We focus on the differential feedback analysis because it describes the feedback processes operating within a given climate and hence may be more-readily understood than the effective feedback parameters.

The large range of simulated climate changes may be expected to be annually and zonally uniform to a first approximation. Hence we repeat the analysis in Fig. 3 taking the annual average and the zonal average of each kernel as well as each simulated climate field before multiplying the kernels by the climate fields (see SI Section S3 for details). We find that the result matches closely with the feedback parameters computed using the full 4-dimensional structure of the simulated climate and kernel fields (SI Fig. S6). This suggests that the zonal and seasonal patterns of temperature, surface albedo, humidity, and cloud changes do not play a substantial role in the variations in each feedback parameter shown in Fig. 3, allowing the specific factors driving the variations in each feedback parameter to be more-readily assessed by examining only the meridional and vertical structure of the fields.

The decrease in stability with cooling in cold climates is the main novel result of the present study, since previous work has discussed the decrease in stability with warming. Hence we begin by interpreting the lapse-rate and albedo feedbacks.

Lapse-rate feedback. The lapse-rate feedback describes the impact of changes in the vertical temperature structure. In the tropics today, deep convection occurs, and the temperature profile is close to being moist adiabatic. Warming causes the moist adiabatic lapse rate to decline. This is a negative local feedback, since it means that smaller changes in surface temperature are needed to bring about a given change in outgoing longwave radiation. On the other hand, in the present-day Arctic the planetary boundary layer is often capped by a temperature inversion and hence a very stable stratification, which suppresses vertical mixing and causes temperature changes at the surface not to be propagated aloft, which is a positive local feedback.

The inversion strength can be described by the difference in potential temperature between the 700-hPa level and the surface (cf. ref. 39), which is plotted in Fig. 4a. Across the range of simulated climates, ice-covered regions of the globe tend to have an inversion, as expected because the surface absorbs less solar radiation when it is covered with snow or ice, setting up the potential for a positive lapse-rate feedback in these regions. This leads to a less-negative global lapse-rate feedback as the climate cools and more of the globe resembles the present-day Arctic (Fig. 4a, b).

As the climate warms and sea ice is lost, the erosion of polar inversions lead to less-positive polar lapse-rate feedbacks (Fig. 4a, b). However, the lapse-rate feedback in the tropical region becomes less negative with warming for climates warmer than the PI. An analysis of a previous version of this model lead to fairly similar changes in the spatial pattern of the lapse-rate feedback parameter under varied levels of forced warming40. The mechanisms driving the changes in the tropical temperature profile that cause this are beyond the scope of the current study. The result is that the global lapse-rate feedback becomes somewhat less negative with warming, at least up to temperatures about 10 K warmer than the PI climate, although the cloud feedback dominates the changes in the net feedback parameter for climates warmer than the PI.

The temperature feedback radiative kernel has a spatial structure that varies vertically but is fairly uniform horizontally, suggesting that lapse-rate feedback changes should approximately track changes in the vertical structure of globally-averaged atmospheric warming. Indeed, we get a similar result when we repeat the analysis using the global-mean temperature profile (red line in Fig. 4c), which removes the influence of horizontal variations in the efficiency of radiation to space but still retains vertical variations. This result implies that the changes in the lapse-rate feedback parameter under cooling global temperature are dictated primarily by the global-mean atmospheric temperature profile becoming more similar to the Arctic today, causing the global lapse-rate feedback to approach the positive value in the Arctic today.

Albedo feedback. The albedo feedback occurs because a warmer climate has less ice cover, and ice-free regions absorb more solar radiation rather than reflecting it back to space. We find that the albedo feedback increases approximately monotonically with cooling global temperature across the range of simulated climates. The albedo feedback radiative kernel has a spatial structure with values most negative in the low latitudes, where there is the most incident solar radiation. Nonetheless, we find that the migration of the ice edge into sunnier latitudes has a relatively limited influence on the variations in the albedo feedback parameter: we get a fairly similar result when we repeat the analysis using a spatially-uniform radiative kernel, which removes the influence of spatial variations in incident solar radiation as well as clouds and other factors (red line in Fig. 4e). In this case the albedo feedback parameter is approximated to be proportional to the sensitivity of the ice area to global temperature (i.e., the slope in Fig. 4d). This implies that the albedo feedback becomes more destabilizing primarily because the ice area expands more rapidly with cooling in colder climates.

This behavior continues in climates warmer than the PI, with the change in ice area per change in global temperature continuing to decrease as the climate warms (Fig. 4d), leading to a smaller albedo feedback in warmer climates (Fig. 4e). In the warmest simulated climates there is almost no remaining snow and sea ice (Fig. 1i), and the albedo feedback λdiffα approaches zero (Fig. 3).

Cloud feedback. Clouds cause shortwave cooling and longwave heating, and changes in clouds with climate lead to a feedback that can be either positive or negative. We find that the cloud feedback in CESM2 is approximately zero near the PI climate, but the feedback becomes increasingly destabilizing as the underlying climate warms. Previous work using CESM2 and earlier versions of this model similarly found that cloud feedbacks are more destabilizing in warmer climates29,36,41.

An important caveat associated with the changes in the cloud feedback shown in Fig. 3 is that this term includes the residual due to factors including inaccuracies in the radiative kernel analysis. One measure of this is the residual when the kernel analysis is repeated using clear-sky fields (see “Caveats” section below), which we find contributes about 25% of the diagnosed cloud feedback change between the PI and warmest simulated climates (red dashed line in bottom right panel of Fig. 3).

We also carry out an alternative test of the impact of clouds that does not rely on the radiative kernels. Instead, we redo the net feedback analysis in Fig. 2 using clear-sky fields reported by the model for the change in TOA net energy flux ΔN. The resulting values of λeffnet and λdiffnet are plotted in SI Fig. S8. For both measures of the net feedback in SI Fig. S8, the feedback remains relatively constant in climates warmer than the PI when using clear-sky fields, whereas it becomes steadily less negative with warming when using all-sky fields. This suggests that cloud changes contribute substantially to the trend toward a less-negative net feedback for climates warmer than the PI, consistent with the kernel analysis results in Fig. 3.

This alternative approach also allows us to separate the influence of cloud shortwave effects from cloud longwave effects. We find that using clear-sky fields for only the longwave component of ΔN causes behavior resembling the all-sky results, whereas using clear-sky fields for only the shortwave component of ΔN causes behavior resembling the clear-sky results (SI Fig. S8). This suggests that the increase in the net feedback in warm climates is caused primarily by the cloud shortwave feedback, which is consistent with the results of previous studies29,36,41.

Planck feedback. The Planck feedback describes how warming the surface and atmospheric column above causes more outgoing longwave radiation to space due to the Stefan-Boltzmann law. This feedback remains relatively invariant across the range of simulated climates, although it becomes slightly more negative as the climate cools. Note that because we use a radiative kernel, we account only for changes in the Planck feedback due to the evolving pattern of surface temperature change, and we do not represent how the Planck feedback depends on global temperature. The Planck feedback radiative kernel is most negative in the warmest regions of the control climate (see SI Section S3). The meridional structure of the surface temperature evolution is shown in SI Fig. S7. Simulated surface temperature changes tends to be amplified in ice-covered regions (Fig. 4f), which is expected to occur primarily due to the albedo feedback and lapse-rate feedback. As the ice-covered regions expand equatorward, the amplification moves out of the polar region, which causes the Planck feedback to become slightly more negative (see SI Section S3 for details). Note that Fig. 4f indicates that polar amplification is not a ubiquitous feature of climate change within this wide range of climates.

Water-vapor feedback. The water-vapor feedback occurs because warmer air can hold more water vapor, which is a greenhouse gas. This feedback tends to be more positive in warmer climates, for reasons that can be explained using idealized one-dimensional radiative-convective equilibrium models31. Consistent with this, we find that the strength of the water-vapor feedback varies approximately monotonically with the underlying climate, becoming more positive with warming, although it becomes fairly constant in climates warmer than the PI.

Caveats

The results in Fig. 2 rely on direct model output in addition to the estimated CO2 radiative forcing (FGHG), which is computed using previously published line-by-line radiative transfer calculations9. These instantaneous radiative forcing (IRF) calculations do not account for stratospheric temperature adjustment, although they give similar results for our purposes to other previously published line-by-line radiative model results that do include stratospheric temperature adjustment42 (SI Fig. S1). Neither calculation allows for the rapid adjustments to the tropospheric temperature profile in response to CO2 forcing that are needed to estimate the effective radiative forcing (ERF)43.

We assess the error associated with this approach by comparing with two separate estimates of the ERF associated with CO2 quadrupling from the PI level in CESM2, noting that the error may be larger for climates farther from the PI. First, we use a preexisting CESM2 run44 that has the sea-surface temperature (SST) field fixed at PI values and CO2 increased by 4× in order to estimate the ERF based on the change in TOA net radiation fields. Second, we use the regression method of Gregory45 to estimate the ERF based on the first 20 years of a preexisting CESM2 simulation in which CO2 was instantaneously quadrupled from its PI value46. In the latter analysis, the ERF is obtained by extrapolating the relationship between global TOA net energy flux and surface temperature to zero surface temperature anomaly. The results are 8.90 W/m2 for the fixed-SST ERF estimate and 8.77 W/m2 for the regression method ERF estimate, compared with 8.56 W/m2 in the line-by-line radiative transfer code IRF estimate that we adopt in this analysis. The close agreement between the IRF estimate from the radiative transfer code and the ERF estimates from CESM2 may be coincidental given that CESM2, like most ESMs, shows substantial forcing adjustments from rapid changes in atmospheric temperature and cloud cover in response to CO2 changes (e.g., ref. 47). However, this agreement gives confidence in the use of the IRF estimate (Fig. 2a) as an approximation to the ERF in CESM2 for our calculations.

Another consideration is whether radiative forcing should change with the underlying climate itself. Here we have adopted the standard definition of radiative forcing that assumes that CO2 changes occur within a constant climate (i.e., fixed surface temperature), and hence that all radiatively-important atmospheric and surface field changes beyond rapid adjustments are part of the radiative feedback on surface temperature changes. However, another defensible choice for the differential feedback would be to define radiative forcing relative to the continuously evolving climate, in which case the CO2 forcing would change depending on factors including changes in atmospheric water vapor, cloud cover, and the difference in temperature between the surface and the stratosphere (e.g., refs. 48, 49). Calculating the radiative forcing under this alternative definition, which would require additional simulations, would modify the value of the differential feedback. Note that while this ambiguity in forcing definition is inherent to the differential feedback, the effective feedback only uses the standard radiative forcing definition adopted here because it is defined in terms of anomalies relative to the PI climate43,48.

As noted above, the radiative kernel analysis does not allow the radiative response to perturbations in climate fields to evolve with the underlying climate because it effectively linearizes the simulated response about a climate near the PI. Furthermore, since the radiative kernels are set to zero above a fixed tropopause, radiative responses may not be calculated accurately in climates with a tropopause that is substantially higher than in the PI (e.g., ref. 31).

To assess the accuracy of the kernel analysis, we re-ran the kernel analysis using clear-sky versions of the radiative kernels, which are included with the kernel fields37. The residual between the sum of the clear-sky feedback parameters and the clear-sky TOA net energy flux reported by the model is indicated as a red dashed line in the bottom row of Fig. 3. This provides an estimate of the uncertainty in the analysis. Although not negligible, the values are relatively small. Note that cancelation between feedbacks may play a role in these relatively small residuals (cf. ref. 50), especially for climates far from the PI. Furthermore, for the lapse-rate and albedo feedbacks, which dominate net feedback changes in colder climates, we found that using a horizontally-averaged kernel produced similar results. That is, horizontal variations in the kernel between the warm tropics and cold poles have minimal influence on how feedbacks change across climate states; instead, feedback changes primarily track changes in the global ice extent and the globally-averaged vertical structure of the atmosphere. This insensitivity to capturing differences in the radiative kernels across the range of spatial variations in the control climate (from the tropics to the poles) suggests that changes in the radiative efficiency of the atmosphere across climate states may be of secondary importance, supporting the accuracy of this analysis which uses a kernel that does not vary with climate.

This analysis uses an approximately equilibrated PI climate, whereas the simulated climates that are increasingly warmer or colder than the PI are expected to be increasingly far from equilibrium. Hence it may be seen as a source of concern that the net climate feedback is found to be most negative near the PI and increasingly less negative in climates increasingly warmer or colder than the PI. However, a number of factors suggest that the level of equilibration is not substantially influencing the values of the climate feedback that we calculate. First, we identify simple and fairly basic physical processes that drive the increase in sensitivity with cooling (related to the lapse rate and albedo feedbacks), suggesting that this is likely to be a robust climate response, and the increase in sensitivity with warming has been previously identified as a robust feature of many climate models (e.g., Fig. 7.11 of ref. 2). Second, previous studies have found a loss of stability at the Snowball Earth bifurcation point, implying an increase in sensitivity as λdiffnet approaches zero under extreme cooling. Third, the minimum climate feedback is in a climate that is approximately 2 K colder than the PI, rather than being at the equilibrated PI climate. Fourth, this approach does not depend on the level of equilibration, at least when applied to a simplified representation of the climate system (SI Fig. S3). Fifth, we find that the impact of deep ocean heat uptake efficacy would not produce this shape (SI Fig. S4). And sixth, the clear-sky residual is relatively small (Fig. 3), showing that the alternative approach of using kernels, rather than the TOA balance used to generate the results in Fig. 2d, e, gives a similar result.

Moreover, we compared our results with a previous analysis51 of a CESM2 instantaneous CO2 quadrupling simulation46. Hahn et al.51 used the same radiative kernels as the present study37, and their results include values for the feedback parameters around simulation year 100 (averaged over their simulation years 85-115), at which point the global surface temperature is 6.6°C above the initial PI value. We indicate their feedback parameter values at this level of warming as blue circles in Fig. 3. The agreement with our analysis (blue lines in Fig. 3) adds some confidence to our interpretation that the relationships we find between feedback values and global temperatures do not depend strongly on the degree of equilibration. We similarly included a blue circle indicating their value for λeffnet in Fig. 2e, which agrees with our results (blue line in Fig. 2e).

Finally, the experimental design used here does not allow for slow feedbacks associated with factors including changes in ice sheets, the carbon cycle, and the deep ocean, which could modify the stability of the climate given sufficient time to adjust. These results should thus be interpreted as a measure of how the traditional fast feedbacks (i.e., Planck, water vapor, lapse rate, surface albedo, and clouds) depend on the underlying climate state, and they are relevant to studies that treat ice sheets and other slow feedbacks as external forcings (e.g., the LGM analysis of ref. 1). If ice sheets were allowed to change, it is expected that their distinct spatial structure of ERF would produce different relationships between climate feedbacks and global temperature changes than those under CO2 forcing alone explored here (e.g., refs. 52, 53). It is similarly expected that the results may differ if the model were allowed to approximately equilibrate to each level of CO2, rather than using the 1% per year ramping adopted in the present study.

Methods
Here we describe details regarding the CESM2 simulations, how the CO2 forcing was estimated, and how the feedback parameters were calculated.

Simulation details
We use NCAR CESM2 in its standard workhorse configuration. The atmospheric component is CAM6, and the ocean component is POP2. The atmosphere and ocean both have nominal horizontal resolutions of 1°, and there are 32 vertical levels in the atmosphere and 60 vertical levels in the ocean.

The Warming and Cooling simulations are both branched from the end of year 500 of a previously run pre-industrial (PI) control simulation7 with the forcing fixed at 1850 levels. The atmospheric CO2 concentration is increased or decreased at a rate of 1% per year from the start of each simulation. For the first 150 years of the Warming run, we use the pre-existing CESM2 “1pctCO2” simulation that is part of the CMIP6 archive8, which we extend to simulate further warming by branching to a cloned case. The Cooling run is identical to the Warming run except that the CO2 change has the opposite sign.

Warming run details: This run initially failed during year 151 with the error “bounding bracket for pH solution not found” from co2calc.F90. Adjusting the POP time step from the default value dt_count=48 to dt_count=60 during years 151-152 caused this error to no longer occur. After year 279, there was an error in lnd_import_export.F90 that the coupler was receiving an output of NaN from the land model. We were not able to resolve this error by reducing the CAM time step and ended the run after year 279.

Cooling run details: After year 278, this run failed with the error “bounding bracket for pH solution not found" from co2calc.F90, which was not resolved by increasing dt_count. We then commented out the line in the model code that called this error, which may lead to unreliable simulated pH. After year 332, when the CO2 concentration reached approximately 10 ppm, the land component of the model failed with the error “CO2 is outside of an expected range” in lnd_import_export.F90, and we commented out the line in the model code that called this error. At the end of the 514-year run, there was an error with the iron flux being out of range in marbl_diagnostics_mod.F90, which we were not able to resolve by simply commenting out the line in the model code that called this error.

Quantities analyzed: For CO2, we use the atmospheric field co2vmr, which is the CO2 volume mixing ratio. For surface temperature, we use the atmospheric field TS, which is the radiative surface temperature. For the measure of inversion strength in Fig. 4a, we compute the potential temperature from the atmospheric temperature T at vertical level 23, which is at approximately 700 hPa on the model hybrid vertical coordinate. For ice cover, we take the maximum of the fields FSNO and PCT_GLACIER/100, multiply this value by landfrac, and then add ICEFRAC. Here FSNO is the fraction of ground covered by snow reported by the land model, PCT_GLACIER is the percent of ground covered by glaciers which is included in the surface dataset input used by the land model, landfrac is the fraction of the gridbox covered by land reported by the land model, and ICEFRAC is the fraction of the gridbox covered by sea ice reported by the atmospheric model. We compute the net energy flux N as FSNT − FLNT, with FSNT and FLNT the top-of-model net longwave and solar fluxes reported by the atmospheric model.

CO2 forcing
Byrne and Goldblatt9 used a line-by-line radiative transfer code to calculate forcing from CO2 (as well as other greenhouse gases). The publication includes a supplemental data text file ("text03.txt”) that has radiative forcing associated with CO2 concentrations varying from 1 ppm to 100,000 ppm. Although this is a considerably wider range of CO2 concentrations than mentioned in their actual paper, the supplemental data values are valid output from their radiative model (Brendan Byrne, personal communication, January 2021).

The CO2 in our simulations ranges from 1.6 ppm to 3422 ppm. We calculate the associated radiative forcing FGHG using a cubic interpolation of the relationship between the radiative forcing associated with the annual-mean global-mean profile ("GAM” in “text03.txt”) and the logarithm of the CO2 concentration ("CO2” in “text03.txt”), which is shown in SI Fig. S1.

Calculation of effective and differential feedback parameters
For the effective feedback parameters, we smooth each radiative response time series (Fnet or Fi) using a least-squares fit to a 12th-order polynomial in (T − T0) that is constrained to go through (T0, F0), where T0 and F0 are the surface temperature and radiative response (Fnet or Fi) averaged over years 480-499 of the PI simulation. This allows the ratio in Eq. (2) to be smooth even in the limit T → T0. This smoothing of Fnet, and the resulting values of λeffnet and EffCS, are plotted in SI Fig. S2 next to the raw unsmoothed annual-mean simulation output.

For the differential feedback parameters, we regress the radiative response (Fnet or Fi) on the surface temperature T. We use a total-least-squares (TLS) regression, rather than a standard ordinary-least-squares (OLS) regression, because the radiative response (Fnet or Fi) and temperature (T) both play the role of dependent variables. A TLS regression accounts for errors in both variables, whereas an OLS regression accounts for errors in one variable and treats the other as an independent variable. The TLS regression depends on the choice of units, and we normalize each variable by the standard deviation of the residuals of the time trend, following Winton57. We compute the TLS regression in a running window of variable duration that spans temperatures in the range ±3 K.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper performs idealized simulations using an Earth System Model (CESM2) with large increases or decreases of the atmospheric CO2 to map the climate sensitivity across a wide range of warm and cold climates. The authors’ simulations show a U-shaped climate sensitivity with a stability optimum near the PI, meaning climate sensitivity increases for both sufficiently warmer and colder climates. The authors use radiative kernels to understand the behavior and find that the albedo and lapse rate feedbacks are responsible for the increased sensitivity in cold climates and that the cloud and lapse-rate feedbacks explain the increased sensitivity in warm climates. The authors suggest that assumptions of a constant or a linear relationship between climate sensitivity and the global temperature that have been used in previous studies (such as Sherwood et al. 2020), could lead to biased estimation of climate sensitivity.\nThis is a very interesting study using a simple but novel approach to show an important aspect of the climate sensitivity: its state dependence. A better understanding of the state dependence of climate sensitivity has huge implication for the understanding of past warm and cold climates and for the use of paleoclimate to constrain climate sensitivity. This study presents a welcoming step toward mapping climate sensitivity and feedbacks over a wide range of climate.\nDespite the known problems of CESM2 (see my comments below), I think the authors result (a U-shaped climate sensitivity) is robust, mostly because the key processes for the increase of sensitivity with cooling rely on simple and basic physics (ice albedo feedback) and thus should not be qualitatively impacted by CESM2’s problems in parameterizations (e.g., shown in Zhu et al. 2022). In addition, an increase of climate sensitivity with warming seems to be a robust feature of many climate models (e.g., see IPCC AR6 Figure 7.11). Given this and the fact the authors have provide discussion on the caveats of their work, I support the publication of the paper in Nature Communications. Below are a few comments for the authors to consider.\n1. The caveats on CESM2’s high climate sensitivity should be expanded. It has been shown that CESM2 substantially overestimates the global warming during the early Eocene and the global cooling during the LGM (Zhu et al., 2020;2021). How will CESM2’s unrealistically high ECS impact the results in the study? Will the U-shape be broader if a low-ECS model (e.g., CCSM4 or CESM1(CAM5)) is used? Will the heigh of the edge of the “U” be impacted?\n2. The authors could consider expanding the discussion of the state dependence of feedbacks. For example, are the increase of cloud and lapse rate feedbacks with warming robust? What are the physical processes driving the increases? The authors seem to suggest that mechanisms are beyond the scope of the paper, but it would be nice to briefly provide some mechanistic understanding. If these hypotheses are rooted in simple physical processes, the authors findings are more likely to be robust (less likely to be impacted by the uncertain physical parameterizations). Also, this discussion could be done in the context of many previous studies on the state dependence of climate forcing and feedbacks, such as Meraner et al. (2013), Caballero and Huber (2013), and Zhu et al. (2019).\n3. Please consider providing some quantitative estimation on how much the linear relationship between climate sensitivity and the global temperature could impact the estimation of ECS (e.g., in Sherwood et al., 2020). I understand that there are a lot of uncertainties involved, in particular the single-model approach of the present study, but I think some estimation (or speculation) on the magnitude of the impact would be nice for readers to evaluate the importance of the authors finding.\n4. Line 333: “Cloud feedbacks are more destabilizing in warm climates” has been found in multiple versions of CESM, including the CCSM3 in Caballero and Huber (2013), CESM1 in Zhu et al. (2019), and the CCSM4, CESM1, and CESM2 in Zhu and Poulsen (2020).", "reviewer_2": "# General comments\nThe authors estimate the temperature-dependence of the climate feedback parameter—and related climate sensitivity. They do so by forcing a state-of-the-art GCM with a wide range of CO2 concentrations to span surface temperatures from -25 to almost 30 degrees. A regression of energy imbalance against surface temperatures is used to quantify the feedback at different base climates. In addition, the authors try to disentangle the proportion of different feedback components using a radiative kernel technique.\nThe main finding is that the climate sensitivity reaches a local minimum at surface temperatures close to present-day. This finding is in general agreement with idealised modelling studies [Seeley and Jeevanjee (2020), Kluft et al. (2021)]. This study adds two novels aspects: i) the GCM seems to reach the local minimum at lower temperatures than the idealised models, and ii) the authors point out that many studies dealing with the temperature-dependency of the feedback parameters have chosen a linear-approach that is not in agreement with this finding (and the idealised studies).\nIn the second half of the manuscript, the authors try to associate the temperature-dependence of the feedback parameter with individual processes (feedbacks) by using radiative kernels. Although I like the general idea, I find this part of the manuscript not convincing. Radiative kernels are strongly dependent on the base climate they have been calculated in. This dependence can be linearised to a certain degree, but to me the results presented in Fig. 3 indicate that the kernels are not suitable for this large range of surface temperatures. My main concerns are:\n1. The almost constancy of the Planck feedback is in stark contrast to the expected T^3 dependence. This difference is likely caused by the fact that it is not represented in the kernels (L263-264).\n2. The very muted increase of the water-vapour feedback at high temperatures (L313-314)). Many line-by-line modelling studies [Koll and Cronin (2018), Seeley and Jeevanjee (2020), Kluft et al. (2021)] as well as more analytical approaches [Simpson (1928), Nakajima et al., 1992), Ingram (2010)] point towards a rapidly declining water-vapour feedback as soon as the atmospheric emission window shuts due to increased continuum absorption at high surface temperatures.\n3. The imbalance of water-vapor and lapse-rate feedbacks. Usually, one would assume that these two feedbacks cancel each other quite well (as seen for surface temperatures between 5 and 20 degrees). This cancellation vanishes at colder temperatures, which the authors interpret as a more stabilising lapse-rate feedback. An alternative explanation would be that the linear extrapolation of the radiative kernels if not sufficient anymore: an increase/decrease of upper-tropospheric temperatures is thought to be accompanied by a moistening/drying of the same atmosphere layers; an effect that is likely not captured in the kernels.\nThe authors defend the usage of the kernels in two different ways. First, they compare their results to another study which used the same radiative kernels. This gives confidence in the proper application of the kernels, but not in the kernels themselves. Second, they compute the residual between the net clear-sky TOA simulated by the GCM to results obtained using the clear-sky radiative kernels. This ought to simultaneously prove the accuracy of the cloud feedback (L340-384) and the usage of the kernels in general (L399-403). What I don’t understand is, why this residual is almost at its maximum for the PI climate and apparently becomes negligible at temperatures far below the reference climate.\nDespite my concerns about the methodology used for decomposing the feedback parameter, the study's findings regarding the local minimum in ECS challenge some widespread assumptions and offer a valuable extension to the work on more idealised models. However, I think it is necessary to address these concerns and refine the methodology to further strengthen the impact of this research.\n# Minor comments\nL82 28.7C is not 14.8C degrees warmer than the 15.0C in PI.\nL465-466 In my opinion, the relationship between feedback and temperature should depend on physical processes and therefore be similar across models? Could this be rephrased as how varying feedbacks across GCMs could be linked to their different base climates?\nL683-684 Is this sufficient to keep the model close to an equilibrated state? In my understanding this would be necessary to estimate the feedbacks from the TOA imbalance.\nL699 It would be nice to know what the modifications were. Otherwise the model differs from NCAR CESM2 in a way that is not transparent.\nL356-392 and L715-726 I agree that the definition of the instantaneous forcing allows changes in the water-vapour concentration to be considered as feedbacks rather the forcing. However, the state-dependence of the CO2 forcing is rather large at these CO2 and T ranges. Is it possible to estimate how large the additional “feedback” is and in which of the kernel/feedback it would be visible?", "reviewer_3": "Review of “The radiative feedback continuum from Snowball Earth to an ice-free hothouse”\nAuthors: I. Eisenman and K. C. Armour\nIn this study the authors explore how climate sensitivity varies as the earth warms or cools. Past studies indicate that climate sensitivity increases as the earth warms and decreases for a slight cooling. It is also known that were the earth cool enough, climate would become unstable, and snow and ice would cover the entire earth’s surface. Finally, the concept of a runaway greenhouse warming is not new and indicates that the climate of a much warmer earth would also be unstable with the Earth losing all of its water (as did Venus). So, it isn’t particularly surprising that the present climate state is more stable than both warmer and colder climate states. The current study indicates that the present climate is more stable than all other states except those a few degrees cooler. Quantification with the CESM2 model of climate stability as a function of global mean temperature is a new result.\nThe analysis is sound and the exposition is clear. Thank you! The results reported are novel in that they indicate the limits to the approximation that climate sensitivity depends linearly on temperature, which the authors show is valid only in the range 3K cooler to 10K warmer than present day. This makes it more difficult to directly infer from paleoclimates (with temperatures cooler by more than 4K) the climate sensitivity of today’s climate. To the authors’ credit, several caveats are discussed, which help the reader interpret which aspects of the results may not be robust.\nI have two suggestions for improving the manuscript. First, I would eliminate panel f of figure 2, which is a plot of “effective climate sensitivity” as a function of temperature. The curve shown is proportional to the negative inverse of the curve in panel e, and thus it is unnecessary since it provides no additional information. More importantly, the interpretation of “effective climate sensitivity” as defined for this figure is problematic. “Effective climate sensitivity”, which is the equilibrium temperature change expected were atmospheric CO2 concentration doubled, has been seen by some as a more “accessible” measure of climate sensitivity than the more fundamental and useful “feedback parameter”, which also characterizes sensitivity. But in the context of this study, it makes little sense to plot the effective climate sensitivity as a function of temperature. Consider, for example, a climate some 35K cooler than the present (i.e., at ~-20K). At that temperature, the authors show the earth is plunging toward a snowball condition, and if CO2 were doubled, even instantaneously, this would not prevent earth ending up in an even cooler state. Yet, panel f indicates that the earth should after doubling of CO2 end up about 10K warmer than its initial temperature (i.e., reach an equilibrium temperature of about -10K). We know that at -10K the earth will still eventually become a “snowball”, so the contention that panel f is useful in indicating “effective climate sensitivity” is unsupported. Besides deleting panel f, the discussion of it and of “effective climate sensitivity” would also, of course, need to be eliminated.\nThe second suggestion involves the claim in the concluding section that the results are important because they indicate that reliance on paleoclimate estimates of climate sensitivity (for temperatures 2K cooler than pre-industrial) cannot constrain current model estimates of climate sensitivity. More discussion and qualification of this conclusion are needed. In particular, the statement that “net feedback (and climate stability) change relatively rapidly for temperatures more than 2K colder than the PI” isn’t supported by figure 2, which by eye seems to indicate the climate sensitivity remains within about 10% of its present value even for climates as much as 10K cooler than present. Since the range of climate sensitivities estimated by models is much larger than this, I think the paleoclimate data may still be of substantial value. The authors should therefore be more circumspect in their discussion of the implications (and practical significance) of their results.\nBesides the above two suggestions, I have the following comments and suggestions:\nLines 76-86: In the discussion of “ice cover” and its changes, you should mention here that glacial ice amounts were prescribed and held fixed.\nLine 118: You need to indicate how “effective feedback” is estimated for PI conditions (and plotted on fig. 2e) since it is undefined when delta-T is 0. I guess you must interpolate, but you should mention this.\nEq. 4: a plus sign is missing between the first two terms on the RHS.\nLine 219: replace “atmospheric temperature” with “atmospheric lapse rate”.\nLine 237: indicates that “cloud and lapse-rate feedbacks” are primarily responsible for the decreased stability as climate warms. It looks to me like the contribution from the residual might be as significant as lapse-rate/water vapor feedback, and this should probably be noted.\nLine 321: I don’t think subsections are labeled in the SI, so replace “Sec. 5.2” with “Sec. 5”.\nLines 404-416: Yes, the Hahn et al. (2021) results seem consistent with the present study, but I’m not sure this tells us much about whether the kernels can be held fixed, independent of climate state. And the fact that the clear-sky residuals are not too large also isn’t informative in this regard. How do you know that the kernels don’t strongly depend on climate state (especially as climate becomes substantially different from the present one?\nLine 420: replace “Cooling” with “cooling”.\nLine 441: replace “was” with “were”.\nLine 451: replace “lead” with “led” and replace “surface temperature” with “transient surface temperature changes”."}, "author_response": "Responses to Reviewer #1\nWethankthereviewerforthesehelpfulcomments. Wehaveaddressedthemasfollowsintherevised\nmanuscript(reviewer’scommentsinitalics):\nThis paper performs idealized simulations using an Earth System Model (CESM2) with large in-\ncreases or decreases of the atmospheric CO2 to map the climate sensitivity across a wide range\nof warm and cold climates. The authors’ simulations show a U-shaped climate sensitivity with a\nstability optimum near the PI, meaning climate sensitivity increases for both sufficiently warmer\nand colder climates. The authors use radiative kernels to understand the behavior and find that\nthe albedo and lapse rate feedbacks are responsible for the increased sensitivity in cold climates\nand that the cloud and lapse-rate feedbacks explain the increased sensitivity in warm climates. The\nauthors suggest that assumptions of a constant or a linear relationship between climate sensitivity\nand the global temperature that have been used in previous studies (such as Sherwood et al. 2020),\ncouldleadtobiasedestimationofclimatesensitivity.\nThisisaveryinterestingstudyusingasimplebutnovelapproachtoshowanimportantaspectofthe\nclimate sensitivity: its state dependence. A better understanding of the state dependence of\nclimate sensitivity has huge implication for the understanding of past warm and cold climates and for the\nuse of paleoclimate to constrain climate sensitivity. This study presents a welcoming step toward\nmappingclimatesensitivityandfeedbacksoverawiderangeofclimate.\nDespite the known problems of CESM2 (see my comments below), I think the authors result (a U-\nshaped climate sensitivity) is robust, mostly because the key processes for the increase of sensitivity\nwith cooling rely on simple and basic physics (ice albedo feedback) and thus should not be quali-\ntatively impacted by CESM2’s problems in parameterizations (e.g., shown in Zhu et al. 2022). In\naddition, an increase of climate sensitivity with warming seems to be a robust feature of many cli-\nmate models (e.g., see IPCC AR6 Figure 7.11). Given this and the fact the authors have provide\ndiscussion on the caveats of their work, I support the publication of the paper in Nature Communi-\ncations. Belowareafewcommentsfortheauthorstoconsider.\nReply: Weappreciatethisthoughtfulandgeneroussummaryofourmanuscript.\n1. The caveats on CESM2’s high climate sensitivity should be expanded. It has been shown that\nCESM2 substantially overestimates the global warming during the early Eocene and the global\ncooling during the LGM (Zhu et al., 2020;2021). How will CESM2’s unrealistically high ECS\nimpact the results in the study? Will the U-shape be broader if a low-ECS model (e.g., CCSM4 or\nCESM1(CAM5))isused? Willtheheighoftheedgeofthe“U”beimpacted?\nReply: We appreciate this comment and have expanded our discussion of this topic in the revised\nmanuscript. Although we do not think we can confidently assess how the U-shape would differ in\nanother model without carrying out additional simulations, we thank the reviewer for this framing\nofthequestionintermsofthewidthandheightoftheU-shape,whichwehaveincorporatedintothe\nrevisedmanuscript. Theexpandeddiscussionincludesthefollowing(lines555-572):\nSincetherelationshipbetweenthesimulatednetfeedbackandunderlyingclimateisex-\npected to depend on the choice of model, it would be useful to reproduce the present\n2\nanalysis using other ESMs. This would be particularly valuable because paleoclimate\nconstraints on the ECS all rely on mapping feedbacks between different climate states.\nRecent studies using CESM2 identified an apparent cold bias in the simulation of the\nLGM climate (Zhu et al., 2021) and warm bias in the simulation of the early Eocene\n(Zhuet al.,2022), andanew versionof themodelwas developedwithcloud feedbacks\ntuned to be less positive (“CESM2-PaleoCalibr”, Zhu et al., 2022), which reduced the\nLGMbiasandalsoresultedinareducedmodern-dayECS.Comparingthepresentanal-\nysis with a similar analysis that used CESM2-PaleoCalibr rather than CESM2 would\nfurther identify to what extent the tuning caused the dependence of the net feedback on\nthe underlying climate to be shifted or restructured, which may shed further light on\nthe way feedbacks in past climate states serve as analogs for feedbacks in the modern\nclimate. That is, future work could determine whether identified biases in simulations\nof past warm climates using ESMs become reduced by changes in the value of the net\nfeedback applying to all climates states (a vertical shift of the “U” shape in Figs. 2d,e)\norbychangesinthenetfeedbackdependenceontheunderlyingclimatestate(achange\ninthehorizontalwidthofthe“U”shapeinFigs.2d,e).\n2. The authors could consider expanding the discussion of the state dependence of feedbacks. For\nexample, are the increase of cloud and lapse rate feedbacks with warming robust? What are the\nphysical processes driving the increases? The authors seem to suggest that mechanisms are beyond\nthe scope of the paper, but it would be nice to briefly provide some mechanistic understanding. If\nthese hypotheses are rooted in simple physical processes, the authors findings are more likely to be\nrobust(lesslikelytobeimpactedbytheuncertainphysicalparameterizations). Also,thisdiscussion\ncouldbedoneinthecontextofmanypreviousstudiesonthestatedependenceofclimateforcingand\nfeedbacks,suchasMeraneretal. (2013),CaballeroandHuber(2013),andZhuetal. (2019).\nReply: In the revised manuscript, we have focused the discussion more clearly on the increase in\nsensitivity incold climates,in order tokeep theemphasis on themost novelresults. We haveadded\ntexttoclarifythis,including(lines318-320):\nThe decrease in stability with cooling in cold climates is the main novel result of the\npresentstudy,sincepreviousworkhasdiscussedthedecreaseinstabilitywithwarming.\nHencewebeginbyinterpretingthelapse-rateandalbedofeedbacks.\nFollowing the reviewer’s suggestion, we now point readers to other papers in our discussion of the\ncloudfeedback(lines379-382):\nPrevious work using CESM2 and earlier versions of this model similarly found that\ncloud feedbacks are more destabilizing in warmer climates (Caballero & Huber, 2013;\nZhuetal.,2019;Zhu&Poulsen,2020).\nFurthermore, in the revised manuscript we note that our finding that the cloud feedback changes\nareprimarilyduetothecloudshortwavefeedbackis“consistentwiththeresultsofpreviousstudies\n(Caballero&Huber,2013;Zhuetal.,2019;Zhu&Poulsen,2020)”(lines403-405). Wealsodiscuss\nchangesinthelapse-ratefeedbackinclimateswarmerthanthePI,noting(lines340-342):\ntial pattern of the lapse-rate feedback parameter under varied levels of forced warming\n(Merlisetal.,2022).\nWehopetheserevisionsaresufficienttoaddressthereviewer’scomment.\n3. Please consider providing some quantitative estimation on how much the linear relationship\nbetween climate sensitivity and the global temperature could impact the estimation of ECS (e.g., in\nSherwood et al., 2020). I understand that there are a lot of uncertainties involved, in particular\nthe single-model approach of the present study, but I think some estimation (or speculation) on the\nmagnitudeoftheimpactwouldbeniceforreaderstoevaluatetheimportanceoftheauthorsfinding.\nReply: WehaveaddedabriefdiscussionregardingtheimpliedchangeintheSherwoodetal.(2020)\nresult(lines236-249):\nTheseresultsshowthatthevalueofaadoptedbySherwoodetal.(2020)forthechange\nin l at the LGM is much larger than in the CESM2 results, because the “U” shape\nnet\nin Figs. 2d,e causes the feedback at 5K of cooling to be similar to the feedback at the\nPI. If we were to repeat the Sherwood et al. (2020) analysis using the value of a that\nwe find here for the difference between the LGM and PI feedbacks, our lower value\nof a would imply a lower modern-day climate sensitivity than Sherwood et al. (2020)\nfound, which amounts to a stronger constraint on the upper bound of the EffCS than\nthey report. This is because the CESM2 results suggest that the LGM may be a more\ndirect analogue to current warming than previously assumed, since the feedbacks are\nrelatively similar. In other words, Sherwood et al. (2020) took l to be more negative\nnet\nattheLGMthanthemodernvalue,whereaswefindthatthefeedbacksaresimilar. Soa\ngivenpaleoestimateoftheLGMvalueofl impliesasimilarmodernfeedbackvalue\nnet\naccordingtoourresults,whereastheanalysisofSherwoodetal.(2020)wouldtakeitto\nimplyalessnegativemodernfeedbackandhenceamoresensitivemodernclimate.\n4. Line333: “Cloudfeedbacksaremoredestabilizinginwarmclimates”hasbeenfoundinmultiple\nversions of CESM, including the CCSM3 in Caballero and Huber (2013), CESM1 in Zhu et al.\n(2019),andtheCCSM4,CESM1,andCESM2inZhuandPoulsen(2020).\nReply: Weappreciatethereviewerbringingthisup. Asalsonotedabove,wehaveaddedthefollow-\ningsentence(lines379-382):\nPrevious work using CESM2 and earlier versions of this model similarly found that\ncloud feedbacks are more destabilizing in warmer climates (Caballero & Huber, 2013;\nZhuetal.,2019;Zhu&Poulsen,2020).\nWethankthereviewerforthesehelpfulcomments. Wehaveaddressedthemasfollowsintherevised\nmanuscript(reviewer’scommentsinitalics):\n#Generalcomments\nThe authors estimate the temperature-dependence of the climate feedback parameter–and related\nclimate sensitivity. They do so by forcing a state-of-the-art GCM with a wide range of CO2 con-\ncentrations to span surface temperatures from -25 to almost 30 degrees. A regression of energy\nimbalance against surface temperatures is used to quantify the feedback at different base climates.\nIn addition, the authors try to disentangle the proportion of different feedback components using a\nradiativekerneltechnique.\nThe main finding is that the climate sensitivity reaches a local minimum at surface temperatures\nclose to present-day. This finding is in general agreement with idealised modelling studies [Seeley\nandJeevanjee(2020),Kluftetal. (2021)]. Thisstudyaddstwonovelsaspects: i)theGCMseemsto\nreach the local minimum at lower temperatures than the idealised models, and ii) the authors point\nout that many studies dealing with the temperature-dependency of the feedback parameters have\nchosenalinear-approachthatisnotinagreementwiththisfinding(andtheidealisedstudies).\nIn the second half of the manuscript, the authors try to associate the temperature-dependence of\nthefeedbackparameterwithindividualprocesses(feedbacks)byusingradiativekernels. AlthoughI\nlikethegeneralidea,Ifindthispartofthemanuscriptnotconvincing. Radiativekernelsarestrongly\ndependentonthebaseclimatetheyhavebeencalculatedin. Thisdependencecanbelinearisedtoa\ncertaindegree,buttometheresultspresentedinFig. 3indicatethatthekernelsarenotsuitablefor\nthislargerangeofsurfacetemperatures.\nReply: Weappreciatethissummaryofourmanuscript.\nSeeourcommentbelowregardingFig.3,whichhadanerrorintheoriginalsubmission.\nMymainconcernsare:\n1. ThealmostconstancyofthePlanckfeedbackisinstarkcontrasttotheexpectedTˆ3dependence.\nThisdifferenceislikelycausedbythefactthatitisnotrepresentedinthekernels(L263-264).\nReply: Therevieweriscorrect. Wepointthisoutonlines409-412:\nNote that because we use a radiative kernel, we account only for changes in the Planck\nfeedback due to the evolving pattern of surface temperature change, and we do not\nrepresenthowthePlanckfeedbackdependsonglobaltemperature.\nSimilarly,wealsoexplain(lines477-479):\nNotethatcancelationbetweenfeedbacksmayplayaroleintheserelativelysmallresid-\nuals(cf. Koll&Cronin,2018),especiallyforclimatesfarfromthePI.\n2. The very muted increase of the water-vapour feedback at high temperatures (L313-314)). Many\n5\nline-by-line modelling studies [Koll and Cronin (2018), Seeley and Jeevanjee (2020), Kluft et al.\n(2021)] as well as more analytical approaches [Simpson (1928), Nakajima et al., 1992), Ingram\n(2010)] point towards a rapidly declining water-vapour feedback as soon as the atmospheric emis-\nsionwindowshutsduetoincreasedcontinuumabsorptionathighsurfacetemperatures.\nReply: We appreciate the reviewer raising this point and have added a brief discussion of it, clar-\nifying that previous studies have identified this effect in climates warmer than we simulate (lines\n176-179):\nNote that previous studies using idealized single-column radiative models have found\nthatthenetclimatefeedbackbecomesmorenegativewithwarmingforclimateswarmer\nthanapproximately40 C(Seeley&Jeevanjee,2021;Kluftetal.,2021).\n \nBy contrast, we have focused the discussion in the revised manuscript more clearly on the increase\ninsensitivityincoldclimates,inordertokeepemphasisonthemostnovelresults,explaining(lines\n318-320):\nThe decrease in stability with cooling in cold climates is the main novel result of the\npresentstudy,sincepreviousworkhasdiscussedthedecreaseinstabilitywithwarming.\nHencewebeginbyinterpretingthelapse-rateandalbedofeedbacks.\n3. The imbalance of water-vapor and lapse-rate feedbacks. Usually, one would assume that these\ntwo feedbacks cancel each other quite well (as seen for surface temperatures between 5 and 20 de-\ngrees). This cancellation vanishes at colder temperatures, which the authors interpret as a more\nstabilisinglapse-ratefeedback. Analternativeexplanationwouldbethatthelinearextrapolationof\theradiativekernelsifnotsufficientanymore: anincrease/decreaseofupper-tropospherictempera-\nturesisthoughttobeaccompaniedbyamoistening/dryingofthesameatmospherelayers;aneffect\nthatislikelynotcapturedinthekernels.\nReply: The reviewer is correct that the water-vapor and lapse-rate feedbacks largely cancel for\nclimates similar to the reference climate on which the kernel is based (red dashed lines in third row\nofFig.3),whereasthereisconsiderablylesscancellationforcolderclimates,whichcouldbeseenas\nacauseforconcern. However,therelativelysmallclear-skyresidual(reddashedlinesinbottomrow\nof Fig. 3) implies that if the kernel is missing a compensating effect between these two feedbacks,\nthenitisdoingthisinawaythatthetotalclear-skyfeedbackisstillrelativelywellrepresented.\nTheauthorsdefendtheusageofthekernelsintwodifferentways. First,theycomparetheirresultsto\nanotherstudywhichusedthesameradiativekernels. Thisgivesconfidenceintheproperapplication\nof the kernels, but not in the kernels themselves. Second, they compute the residual between the\nnet clear-sky TOA simulated by the GCM to results obtained using the clear-sky radiative kernels.\nThis ought to simultaneously prove the accuracy of the cloud feedback (L340-384) and the usage\nof the kernels in general (L399-403). What I don’t understand is, why this residual is almost at\nits maximum for the PI climate and apparently becomes negligible at temperatures far below the\nreferenceclimate.\nReply: It turns out that I inadvertently overwrote a variable in the code to make the final version of\nof Fig. 3, and I somehow failed to notice that the final version of the figure had this error. I am\nextremely grateful to the reviewer for catching this! I have corrected it in the revised manuscript,\nwhichshowsaconsiderablysmallerclear-skyresidualinFig.3.\nDespite my concerns about the methodology used for decomposing the feedback parameter, the\nstudy’s findings regarding the local minimum in ECS challenge some widespread assumptions and\noffer a valuable extension to the work on more idealised models. However, I think it is necessary to\naddresstheseconcernsandrefinethemethodologytofurtherstrengthentheimpactofthisresearch.\nReply: Weappreciatethiscommentandhopethatourrevisions,asdiscussedabove,havesufficiently\naddressedthereviewer’sconcerns.\n#Minorcomments\nL8228.7Cisnot14.8Cdegreeswarmerthanthe15.0CinPI.\nReply: We appreciate the reviewing catching this embarrassing typo! We have corrected it. It\nnow reads that the PI temperature of “15 C” (line 79) “increases by 15K to 30 C” (line 82). (We\n   \nrounded these values to the nearest integer for consistency with other reported temperatures in the\nrevisedmanuscript.)\nL465-466 In my opinion, the relationship between feedback and temperature should depend on\nphysicalprocessesandthereforebesimilaracrossmodels? Couldthisberephrasedashowvarying\nfeedbacksacrossGCMscouldbelinkedtotheirdifferentbaseclimates?\nReply: Thereviewerraisesaninterestingpoint(asweunderstandit): thatiftherearesimplephysical\nprocesses linking feedback parameter values to the mean state climate, as we’re suggesting, then\nthiscouldalsohaveimplicationsforunderstandingfeedbackdifferencesacrossclimatemodelsthat\nsimulate different mean states. We have not looked into this question but see it as a potentially\npromising topic to investigate in a future study. At the same time, though, the level of inter-model\ndifferences in their mean states is relatively small compared with the range of climates that we are\nlooking at. The mean states all essentially hover around the bottom of the “U” that we find, which\ngives us less confidence that the impact of mean state differences in the context of this study would\nexplainmuchoftheinter-modeldifferencesinfeedbackstrengths.\nL683-684 Is this sufficient to keep the model close to an equilibrated state? In my understanding\nthiswouldbenecessarytoestimatethefeedbacksfromtheTOAimbalance.\nReply: No,the1%/yrrampingisnotsufficientlyslow,andthemodelisexpectedtobecomeincreas-\ninglyoutofequilibriumasCO2isrampedfromthePIlevel. Toclarifythis,intherevisedmanuscript\nwenotethat“thesimulatedclimatesthatareincreasinglywarmerorcolderthanthePIareexpected\ntobeincreasinglyfarfromequilibrium”(lines490-492). However,ouranalysisdoesnotassumethat\nthe climate system is near equilibrium. This is a subtle point, and we have added a substantial new\ndiscussiontoclarifythisintherevisedmanuscriptonlines875-913intheSIandFig.S3,including\n(lines904-905): “Thishelpstoillustratehowtheanalysisusedinthisstudydoesnotdependonhow\nequilibrated the climate system is with the evolving value of DF .” We have also explained in\nGHG\nthe main text (lines 504-506): “this approach does not depend on the level of equilibration, at least\n7\nwhenappliedtoasimplifiedrepresentationoftheclimatesystem(SIFig.S3)’.’\nL699Itwouldbenicetoknowwhatthemodificationswere. OtherwisethemodeldiffersfromNCAR\nCESM2inawaythatisnottransparent.\nReply: Themodificationtothecodewassimplytocommentoutaline. Wehaveclarifiedthisinthe\nrevised manuscript as follows (lines 818-819): “we commented out the line in the model code that\ncalledthiserror,whichmayleadtounreliablesimulatedpH.”\nL356-392andL715-726Iagreethatthedefinitionoftheinstantaneousforcingallowschangesinthe\nwater-vapour concentration to be considered as feedbacks rather the forcing. However, the state-\ndependence of the CO2 forcing is rather large at these CO2 and T ranges. Is it possible to estimate\nhowlargetheadditional“feedback”isandinwhichofthekernel/feedbackitwouldbevisible?\nReply: This is an interesting question that we think deserves more study, although we expect that\nwe would need to use fixed-SST simulations to evaluate it, and we see it as outside the scope of the\npresentstudy.\nWethankthereviewerforthesehelpfulcomments. Wehaveaddressedthemasfollowsintherevised\nmanuscript(reviewer’scommentsinitalics):\nReviewof“TheradiativefeedbackcontinuumfromSnowballEarthtoanice-freehothouse”\nAuthors: I.EisenmanandK.C.Armour\nIn this study the authors explore how climate sensitivity varies as the earth warms or cools. Past\nstudies indicate that climate sensitivity increases as the earth warms and decreases for a slight\ncooling. It is also known that were the earth cool enough, climate would become unstable, and\nsnow and ice would cover the entire earth’s surface. Finally, the concept of a runaway greenhouse\nwarming is not new and indicates that the climate of a much warmer earth would also be unstable\nwith the Earth losing all of its water (as did Venus). So, it isn’t particularly surprising that the\npresent climate state is more stable than both warmer and colder climate states. The current study\nindicates that the present climate is more stable than all other states except those a few degrees\ncooler. Quantification with the CESM2 model of climate stability as a function of global mean\ntemperatureisanewresult.\nThe analysis is sound and the exposition is clear. Thank you! The results reported are novel in that\ntheyindicatethelimitstotheapproximationthatclimatesensitivitydependslinearlyontemperature,\nwhich the authors show is valid only in the range 3K cooler to 10K warmer than present day. This\nmakes it more difficult to directly infer from paleoclimates (with temperatures cooler by more than\n4K) the climate sensitivity of today’s climate. To the authors’ credit, several caveats are discussed,\nwhichhelpthereaderinterpretwhichaspectsoftheresultsmaynotberobust.\nReply: Weappreciatethisthoughtfulandgeneroussummaryofthemanuscript.\nI have two suggestions for improving the manuscript. First, I would eliminate panel f of figure 2,\nwhich is a plot of “effective climate sensitivity” as a function of temperature. The curve shown\nis proportional to the negative inverse of the curve in panel e, and thus it is unnecessary since\nit provides no additional information. More importantly, the interpretation of “effective climate\nsensitivity” as defined for this figure is problematic. “Effective climate sensitivity”, which is the\nequilibrium temperature change expected were atmospheric CO2 concentration doubled, has been\nseenbysome asamore“accessible”measureof climate sensitivitythanthemorefundamentaland\nuseful “feedback parameter”, which also characterizes sensitivity. But in the context of this study,\nit makes little sense to plot the effective climate sensitivity as a function of temperature. Consider,\nfor example, a climate some 35K cooler than the present (i.e., at -20K). At that temperature, the\nauthors show the earth is plunging toward a snowball condition, and if CO2 were doubled, even\ninstantaneously,thiswouldnotpreventearthendingupinanevencoolerstate. Yet,panelfindicates\nthat the earth should after doubling of CO2 end up about 10K warmer than its initial temperature\n(i.e., reach an equilibrium temperature of about -10K). We know that at -10K the earth will still\neventually become a “snowball”, so the contention that panel f is useful in indicating “effective\nclimate sensitivity” is unsupported. Besides deleting panel f, the discussion of it and of “effective\nclimatesensitivity”wouldalso,ofcourse,needtobeeliminated.\nReply: We appreciatethe reviewerbringing upthis issue. We thinkthe reviewerraises a reasonable\n9\npoint, but after thinking this over and discussing it extensively between the authors, we have ulti-\nmatelydecidedtoretaintheinclusionofleff andEffCS,becausetheEffCSiswidelyusedsowesee\nthisasanimportantpointofcontactwithotherliterature. However,wehaveaddedsomediscussion\\oftheshortcomingsofthisframework,alongthelinesofwhatthereviewerbringsup,includingthe\nfollowing(lines170-173):\nNote that leff remains negative for all climates, in contrast with ldiff, which illustrates\nnet net\nhow the EffCS and leff framework can give potentially misleading results about the\nnet\nstability of the underlying climate state because it is based on anomalies from the PI\nclimate.\nThe second suggestion involves the claim in the concluding section that the results are important\nbecausetheyindicatethatrelianceonpaleoclimateestimatesofclimatesensitivity(fortemperatures\n2Kcoolerthanpre-industrial)cannotconstraincurrentmodelestimatesofclimatesensitivity. More\ndiscussion and qualification of this conclusion are needed. In particular, the statement that “net\nfeedback(andclimatestability)changerelativelyrapidlyfortemperaturesmorethan2Kcolderthan\nthe PI” isn’t supported by figure 2, which by eye seems to indicate the climate sensitivity remains\nwithinabout10%ofitspresentvalueevenforclimatesasmuchas10Kcoolerthanpresent. Sincethe\nrange of climate sensitivities estimated by models is much larger than this, I think the paleoclimate\ndata may still be of substantial value. The authors should therefore be more circumspect in their\ndiscussionoftheimplications(andpracticalsignificance)oftheirresults.\nReply: Weagree that thestatement intheoriginal submissionthatvalues “changerelatively rapidly\nfor temperatures more than 2K colder than the PI” was poorly chosen, and we have removed this\nsentencefromtherevisedmanuscript.\nWe have also revised the discussion in an attempt to be to be more circumspect and to emphasize\nthevalueofpaleoclimatedataforconstrainingECS,including(lines236-249):\nTheseresultsshowthatthevalueofaadoptedbySherwoodetal.(2020)forthechange\nin l at the LGM is much larger than in the CESM2 results, because the “U” shape\nnet\nin Figs. 2d,e causes the feedback at 5K of cooling to be similar to the feedback at the\nPI. If we were to repeat the Sherwood et al. (2020) analysis using the value of a that\nwe find here for the difference between the LGM and PI feedbacks, our lower value\nof a would imply a lower modern-day climate sensitivity than Sherwood et al. (2020)\nfound, which amounts to a stronger constraint on the upper bound of the EffCS than\nthey report. This is because the CESM2 results suggest that the LGM may be a more\ndirect analogue to current warming than previously assumed, since the feedbacks are\nrelatively similar. In other words, Sherwood et al. (2020) took l to be more negative\nnet\nattheLGMthanthemodernvalue,whereaswefindthatthefeedbacksaresimilar. Soa\ngivenpaleoestimateoftheLGMvalueofl impliesasimilarmodernfeedbackvalue\nnet\naccordingtoourresults,whereastheanalysisofSherwoodetal.(2020)wouldtakeitto\nimplyalessnegativemodernfeedbackandhenceamoresensitivemodernclimate.\nBesidestheabovetwosuggestions,Ihavethefollowingcommentsandsuggestions:\niceamountswereprescribedandheldfixed.\nReply: We have revised this text to mentioned that it is prescribed, explaining that the ice area\n“includes sea ice, snow cover on land, and prescribed glacial ice cover” (lines 79-80). Note that in\nthe following paragraph we explain that glacial ice “is a specified surface type in CESM2 with an\nareathatdoesnotevolveduringthesimulations”(line94-95).\nLine118: Youneedtoindicatehow“effectivefeedback”isestimatedforPIconditions(andplotted\non fig. 2e) since it is undefined when delta-T is 0. I guess you must interpolate, but you should\nmentionthis.\nReply: Wehaveclarifiedthisintherevisedmanuscript(lines120-122):\nEq. 2 is calculated from F after applying a polynomial smoothing. Note that this\nnet\nallows leff to vary smoothly even in the limit DT 0, as described in SI Sec. S3 and\nnet\n!\nshowninFig.S2.\nWehavealsorevisedSec.S3toinclude(lines853-857):\nFor the effective feedback parameters, we smooth each radiative response time series\n(F or F) using a least-squares fit to a 12th-order polynomial in (T T ) that is con-\nnet i 0\n \nstrainedtogothrough(T ,F ),whereT andF arethesurfacetemperatureandradiative\n0 0 0 0\nresponse (F or F) averaged over years 480-499 of the PI simulation. This allows the\nnet i\nratioinEq.2tobesmootheveninthelimitT T .\n0\n!\nEq. 4: aplussignismissingbetweenthefirsttwotermsontheRHS.\nReply: Wethankthereviewerforcatchingthistypo! Wehavecorrecteditintherevisedmanuscript.\nLine219: replace“atmospherictemperature”with“atmosphericlapserate”.\nReply: Wehavemadethischangeintherevisedmanuscript(lines279-280).\nLine 237: indicates that “cloud and lapse-rate feedbacks” are primarily responsible for the de-\ncreasedstabilityasclimatewarms. Itlookstomelikethecontributionfromtheresidualmightbeas\nsignificantaslapse-rate/watervaporfeedback,andthisshouldprobablybenoted.\nReply: I inadvertently overwrote a variable in the code to make the final version of Fig. 3, which\ncaused the wrong quantity to be plotted for the clear-sky residual in the bottom row of Fig. 3, and I\nsomehow failed to notice that the final version of the figure had this error. I have corrected it in the\nrevisedmanuscript,whichshowsaconsiderablysmallerclear-skyresidualinFig.3.\nLine321: Idon’tthinksubsectionsarelabeledintheSI,soreplace“Sec. 5.2”with“Sec. 5”.\nReply: Wethankthereviewerforcatchingthis! Wehavecorrecteditintherevisedmanuscript.\nLines 404-416: Yes, the Hahn et al. (2021) results seem consistent with the present study, but I’m\n11\nnotsurethistellsusmuchaboutwhetherthekernelscanbeheldfixed,independentofclimatestate.\nAnd the fact that the clear-sky residuals are not too large also isn’t informative in this regard. How\ndo you know that the kernels don’t strongly depend on climate state (especially as climate becomes\nsubstantiallydifferentfromthepresentone?\nReply: The reviewer raises fair points. We agree that our analysis does not quantify the full-sky\nresidual (since it’s wrapped into the cloud feedback). Regarding the Hahn et al. (2021) result,\nwe agree that this does not assess the accuracy of the kernel analysis, as the original submission\nimplied. We have revised the manuscript to clarify that this result provides an assessment of the\nextenttowhichequilibrationmatters,movingtheparagraphtoourdiscussionofequilibration(lines\n510-520)andrevisingitaccordingly.\nLine420: replace“Cooling”with“cooling”.\nReply: Wehavereplacedthiswith“incolderclimates”(line481)\nLine441: replace“was”with“were”.\nReply: Weappreciatethiscorrectionandhavemadethesuggestedchangeintherevisedmanuscript.\nLine 451: replace “lead” with “led” and replace “surface temperature” with “transient surface\ntemperaturechanges”.\nReply: We appreciate this correction and suggestion and have made both of these changes in the\nrevisedmanuscript.\nReviewer #1:\nRemarks to the Author:\nThe authors have revised the manuscript according to the reviewers’ comments. I enjoyed reading\nthe revised manuscript very much.\nOne more minor comment for the authors to consider: One advantage of the authors’ approach is\nthe simplicity and effectiveness of the experiment, i.e., running two very simple sensitivity\nsimulations of 200–500 years each that can continuously cover a temperature range of 55K and\nthe associated fast feedback processes. I think this strength could be better stressed, e.g., in the\nlast paragraph of the manuscript.\nReviewer #2:\nRemarks to the Author:\nI have reevaluated the manuscript and am pleased to note that the authors have effectively\naddressed all of my concerns. While I maintain reservations about certain quantitative aspects of\nthe methodology, these concerns have been explicitly discussed in the revised manuscript,\nenhancing overall clarity.\nGiven the significant findings, particularly the identification of a U-shaped temperature\ndependence, I believe the manuscript offers valuable contributions to the scientific community.\nTherefore, based on the authors' responsiveness to feedback and the noteworthy findings\npresented, I recommend accepting the paper for publication.\nReviewer #3:\nRemarks to the Author:\nThe authors have adequately addressed most of my concerns, but on rereading the article I\nstumbled across what might be a real problem I missed before:\nAccording to Smith et al. (https://doi.org/10.5194/acp-20-9591-2020), the CMIP6 atmospheric\nmodel fast adjustments (stratospheric primarily) increase radiative forcing by about 50% over the\ninstantaneous radiative forcing. Since it is this adjusted forcing that should appear in eq. 1, the\nsubsequent analysis needs to take that into account. Under “Caveats” the potential impact of\naccounting for stratospheric and tropospheric adjustments is considered, but it is difficult for me to\nunderstand (given the above result) why the adjustment is only about 3%. There needs to be\nsome discussion about this. I would note that the lambda curves that are shown in fig. 2 would be\nsubstantially altered if in eq. 1, an effective radiative forcing were 50% larger than the\ninstantaneous radiative forcing. It shouldn’t be too difficult to consider how that might affect the\nconclusions.\nI also suggest the following:\nLine 133: replace “both” with “each”.\nLine 80: replace “prescribed glacial ice cover” with “prescribed fixed glacial ice cover”.\nLines 141-153: I still think these two paragraphs and panel f of figure 2 are of no value. Why is\nEffCS of interest? To my knowledge, this is the first time EffCS has been defined, at least in this\nway. It clearly has no relationship whatsoever to equilibrium climate sensitivity because\nlambda_2x in the definition of ECS is not the same as lambda^eff_net. ECS allows us to estimate\nwhat the equilibrium temperature will be to a doubling of CO2 assuming lambda is independent of\nclimate state. I have no idea how to interpret EffCS. It’s simply inversely proportional to\nlambda^eff_net, but as a temperature it seems meaningless to me. If you keep these two\nparagraphs, perhaps you could say why they can help us understand the climate system. Why is\nEffCS a useful measure, given that we have lambda^eff_net to gauge climate sensitivity.\nLine 279: The Planck feedback reflects not only changes in surface temperature, but also\natmospheric temperature, usually assumed to undergo the same change as the surface. This\nsentence should be revised to correctly characterize the Planck feedback.\nLine 284: replace “are shown” with “is shown”.\nResponses to Reviewer #1\nWe thank the reviewer for this comment. We have addressed it as follows in the revised manuscript\n(reviewer’scommentinitalics):\nThe authors have revised the manuscript according to the reviewers’ comments. I enjoyed reading\ntherevisedmanuscriptverymuch.\nOnemoreminorcommentfortheauthorstoconsider: Oneadvantageoftheauthors’approachisthe\nsimplicity and effectiveness of the experiment, i.e., running two very simple sensitivity simulations\nof 200-500 years each that can continuously cover a temperature range of 55K and the associated\nfast feedback processes. I think this strength could be better stressed, e.g., in the last paragraph of\nthemanuscript.\nReply: Weappreciatethissuggestion,whichwehaveadoptedbyaddingthefollowingtothepenul-\ntimateparagraphofthemanuscript(lines563-565): “Itisnoteworthythatthe279-yearand514-year\nCO rampingsimulationsgeneratedforthisanalysiscouldbefairlystraightforwardlyrepeatedwith\n2\nadifferentESM.”\nWearegladthereviewerfindsthatwehavesufficientlyaddressedtheircomments(reviewer’scom-\nmentinitalics):\nIhavereevaluatedthemanuscriptandampleasedtonotethattheauthorshaveeffectivelyaddressed\nall of my concerns. While I maintain reservations about certain quantitative aspects of the method-\nology, these concerns have been explicitly discussed in the revised manuscript, enhancing overall\nclarity.\nGiventhesignificantfindings,particularlytheidentificationofaU-shapedtemperaturedependence,\nI believe the manuscript offers valuable contributions to the scientific community. Therefore, based\non the authors’ responsiveness to feedback and the noteworthy findings presented, I recommend\nacceptingthepaperforpublication.\nWe thank the reviewer for these comments. We have addressed them as follows in the revised\nmanuscript(reviewer’scommentsinitalics):\nTheauthorshaveadequatelyaddressedmostofmyconcerns,butonrereadingthearticleIstumbled\nacrosswhatmightbearealproblemImissedbefore:\nAccording to Smith et al. (https://doi.org/10.5194/acp-20-9591-2020), the CMIP6 atmospheric\nmodel fast adjustments (stratospheric primarily) increase radiative forcing by about 50% over the\ninstantaneous radiative forcing. Since it is this adjusted forcing that should appear in eq. 1, the\nsubsequent analysis needs to take that into account. Under “Caveats” the potential impact of ac-\ncounting for stratospheric and tropospheric adjustments is considered, but it is difficult for me to\nunderstand (given the above result) why the adjustment is only about 3%. There needs to be some\ndiscussion about this. I would note that the lambda curves that are shown in fig. 2 would be sub-\nstantially altered if in eq. 1, an effective radiative forcing were 50% larger than the instantaneous\nradiativeforcing. Itshouldn’tbetoodifficulttoconsiderhowthatmightaffecttheconclusions.\nReply: We thank the reviewer for raising this point. We should have been clearer that the close\nagreementbetweenIRFcalculatedbyradiativetransfercodeandERFfromCESM2doesnotimply\nthatstratosphericandtroposphericadjustmentsaresmall. ItsimplymeansthatCESM2’sERFvalue\n(after adjustments) ends up similar to the forcing values we are using, giving us confidence in our\ncalculation. We clarify this interpretation in the Caveats section of the manuscript, where we have\naddedthefollowing(lines450-456):\nThe close agreement between the IRF estimate from the radiative transfer code and the\nERF estimate from CESM2 may be coincidental given that CESM2, like most ESMs,\nshows substantial forcing adjustments from rapid changes in atmospheric temperature\nand cloud cover in response to CO changes (e.g., Smith et al., 2020). However, this\n2\nagreementgivesconfidenceintheuseoftheIRFestimate(Fig.2a)asanapproximation\ntotheERFinCESM2forourcalculations.\nIalsosuggestthefollowing:\nLine133: replace“both”with“each”.\nReply: Agreed,andthanksforthesuggestion,whichwehaveadopted.\nLine80: replace“prescribedglacialicecover”with“prescribedfixedglacialicecover”.\nReply: We clarified this point by revising the quoted text to “prescribed time-invariant glacial ice\ncover”.\nLines 141-153: I still think these two paragraphs and panel f of figure 2 are of no value. Why is\nEffCS of interest? To my knowledge, this is the first time EffCS has been defined, at least in this\nway. Itclearlyhasnorelationshipwhatsoevertoequilibriumclimatesensitivitybecauselambda 2x\nin the definition of ECS is not the same as lambdaeˆff net. ECS allows us to estimate what the\nequilibrium temperature will be to a doubling of CO2 assuming lambda is independent of climate\n4\nstate. I have no idea how to interpret EffCS. It’s simplyinversely proportional to lambdaeˆff net, but\nas a temperature it seems meaningless to me. If you keep these two paragraphs, perhaps you could\nsay why they can help us understand the climate system. Why is EffCS a useful measure, given that\nwehavelambdaeˆff nettogaugeclimatesensitivity.\nReply: The effective climate sensitivity (EffCS) is a commonly used metric that allows us to illus-\ntrate how the effective radiative feedback relates to the sensitivity of the climate to CO2 forcing.\nOur definition of EffCS, as the forcing from CO2 doubling divided by the effective feedback, ex-\nactly follows many previous studies (e.g., Murphy, 1995; Winton et al., 2010; Armour et al., 2013;\nSherwood et al., 2020; Mitevski et al., 2021; Andrews et al., 2022). See Rugenstein and Armour\n(2021)foradiscussionofreasonswhyEffCScanbeconsideredausefulmetric.\nLine 279: The Planck feedback reflects not only changes in surface temperature, but also atmo-\nspheric temperature, usually assumed to undergo the same change as the surface. This sentence\nshouldberevisedtocorrectlycharacterizethePlanckfeedback.\nReply: That is a fair point. We replaced “surface” with “surface and atmospheric column above” in\ntherevisedmanuscript.\nLine284: replace“areshown”with“isshown”.\nReply: Weappreciatethereviewercatchingthisgrammarmistakeandhavecorrecteditassuggested.\nReferences\nAndrews,T.,etal.,2022: OntheeffectofhistoricalSSTpatternsonradiativefeedback.J.Geophys.\nResearch-atmospheres,127(18),e2022JD036675,doi:10.1029/2022JD036675.\nArmour, K. C., C. M. Bitz, and G. H. Roe, 2013: Time-varying climate sensitivity from regional\nfeedbacks.J.Climate,26(13),4518–4534,doi:10.1175/JCLI-D-12-00544.1.\nMitevski,I.,C.Orbe,R.Chemke,L.Nazarenko,andL.M.Polvani,2021: Non-monotonicresponse\nof the climate system to abrupt CO2 forcing. Geophys. Res. Lett., 48 (6), e2020GL090861, doi:\n10.1029/2020GL090861.\nMurphy, J. M., 1995: Transient-response of the Hadley Center coupled ocean-atmosphere model\nto increasing carbon-dioxide. Part III: Analysis of global-mean response using simple models. J.\nClimate,8(3),496–514,doi:10.1175/1520-0442(1995)008¡0496:TROTHC¿2.0.CO;2.\nRugenstein, M. A. A. and K. C. Armour, 2021: Three flavors of radiative feedbacks and\ntheir implications for estimating equilibrium climate sensitivity. Geophys. Res. Lett., 48 (15),\ne2021GL092983,doi:10.1029/2021GL092983.\nSherwood, S. C., et al., 2020: An assessment of earth’s climate sensitivity using multiple lines of\nevidence.Rev.Geophysics,58(4),e2019RG000678,doi:10.1029/2019RG000678.\nWinton, M., K. Takahashi, and I. M. Held, 2010: Importance of ocean heat uptake efficacy to\ntransientclimatechange.J.Climate,23(9),2333–2344,doi:10.1175/2009JCLI3139.1.\nReviewer #3:\nRemarks to the Author:\nThe authors have adequately addressed the concerns I raised earlier, so I think the article can be\npublished.\nI continue to be uncomfortable with how the “transient” response is relied on to infer the\ndependence of climate sensitivity on climate state and will take the liberty to suggest some\nobvious follow-up work. Perhaps the authors have already considered this.\nIt would be interesting to perform an “abrupt quartering” CO2 experiment to complement the\nabrupt quadrupling experiment. From these two experiments one could fit results to obtain the\nappropriate parameters for Geoffroy’s 2-layer EBM representation of CESM2 under warming\nconditions and under cooling conditions. We might expect the parameters to differ since ocean\nheat uptake would likely have a strong dependence on whether ocean stratification was becoming\nincreasingly or decreasingly stratified. With these parameters, figures could be constructed similar\nto S4 in the supplemental material. The red and purple curves shown there would show a\ndiscontinuity at the initial temperature (control climate temperature), with the curve to the left of\nit dipping more steeply and to a more negative value. When the composite results of the heating\nand cooling run are smoothed, one would get a curve that reaches a minimum at a temperature\ncooler than the control climate, and the sharp “V” point would be rounded into a “U” shape, in a\nposition that might be similar to the blue curve in fig. S4. This would provide a very simple\nexplanation of the results of the CESM2 runs that would not require a strong dependence of\nfeedback strength (and climate sensitivity) on climate state.\nThe above would not rule out a dependence of feedback strength on climate state but is appealing\nbecause it simply explains why the minimum sensitivity is offset from the control climate\ntemperature and shows a general increase (that slows with change) over other temperatures. I\nfind this so appealing that I worry that in fig. S4 the parameters used to construct the red and\npurples curves may be incorrect. If they are indeed correct, then the contribution of the “transient”\nimpact on the estimate of lambda-eff seems to be much to small to explain but a small fraction of\nthe overall behavior. Still, if this were my research, I would double check whether really accurate\nvalues for the 2-layer EBM model might give something close to the behavior of CESM2 under the\nassumption that the parameters are constants but may differ between a warming and cooling\nclimate.\nIf the transient behavior is responsible for a non-negligible portion of the apparent dependence of\nlambda-eff on climate state, one could use the warming/cooling EBM model runs to subtract these\ncontributions from the total lambda-eff obtained from the CESM3 simulations. This would isolate\nthe contribution to variations in climate sensitivity due to variations in the climate state alone.\nOne final detail: in the caption to figure 4 (in the main text), two references are made to Fig. 2, I\nthink these should refer to Fig. 3.\nResponses to Reviewer #3\nWe appreciate the reviewer taking the time to further reflect on our manuscript. Our responses are\nbelow(reviewer’scommentinitalics):\nThe authors have adequately addressed the concerns I raised earlier, so I think the article can be\npublished.\nReply: Wearegladthereviewerfindsthatwehavesufficientlyaddressedtheircomments.\nIcontinuetobeuncomfortablewithhowthe“transient”responseisreliedontoinferthedependence\nof climate sensitivity on climate state and will take the liberty to suggest some obvious follow-up\nwork. Perhapstheauthorshavealreadyconsideredthis.\nIt would be interesting to perform an “abrupt quartering” CO2 experiment to complement the\nabrupt quadrupling experiment. From these two experiments one could fit results to obtain the\nappropriate parameters for Geoffroy’s 2-layer EBM representation of CESM2 under warming con-\nditions and under cooling conditions. We might expect the parameters to differ since ocean heat\nuptake would likely have a strong dependence on whether ocean stratification was becoming in-\ncreasingly ordecreasingly stratified. Withthese parameters, figures could beconstructed similarto\nS4inthesupplementalmaterial. Theredandpurplecurvesshowntherewouldshowadiscontinuity\nat the initial temperature (control climate temperature), with the curve to the left of it dipping more\nsteeplyandtoamorenegativevalue.\nWhenthecompositeresultsoftheheatingandcoolingrunaresmoothed,onewouldgetacurvethat\nreachesaminimumatatemperaturecoolerthanthecontrolclimate,andthesharp“V”pointwould\nbe rounded into a “U” shape, in a position that might be similar to the blue curve in fig. S4. This\nwould provide a very simple explanation of the results of the CESM2 runs that would not require a\nstrongdependenceoffeedbackstrength(andclimatesensitivity)onclimatestate.\nReply: We agree that the ocean heat uptake could depend on whether the climate is warming or\ncooling (as investigated, for example, in Stouffer, 2004: “Time Scales of Climate Response”, J.\nClimate 17, 209-217), and that this can be modeled by varying the parameters in the two-layer\nmodel results shown in our Fig. S4. To illustrate this, we have repeated the result, using the large\nvalueofeasinthepurplecurveinourFig.S4b,butthistimedoublingthevalueofgforthecooling\nsimulation (left half of figure). The result is shown below (purple curve). Next, we smoothed this\ntwo-layer model result using the same procedure that we used for the CESM results, which is also\nincludedbelow(greencurve).\n-1\n-1.5\n-2\n-2.5\n10 20\nT (oC)\n)K/2m/W(\nffid ten\n-1\n-1.5\n-2\n-2.5\n10 20\nT (oC)\n)K/2m/W(\nffe ten\nTwo-layer model (large and varied )\nTwo-layer model (smoothed)\nCESM2\n“U” to be offset to a temperature colder than the starting value. And consistent with this, the green\ncurve in the figure above does have a minimum temperature that is 1.4 C colder than the starting\n \ntemperature(comparedwith4.8 CcolderintheCESM2results).\n \nNonetheless, it can readily be seen from the results in the manuscript that this hypothesis does not\nexplain the CESM2 results. Specifically, Fig. S2b of the manuscript shows the raw unsmoothed\nCESM2 results (blue dots). The blue dots representing the cold climates nearest to the reference\nclimate (blue dots immediately to left of vertical dashed line) do not indicate increasingly negative\nvalues (as in the purple curve above), instead showing the beginnings of an upturn toward less\nnegativevalues.\nIn other words, the reviewer suggests an interesting alternative hypothesis for our results which\nwould suggest that the key features we focus on are actually an artifact of the smoothing technique\nthat we employed, but the figure we include to evaluate our smoothing technique (Fig. S2b) shows\nthatourresultsarenotactuallyconsistentwiththereviewer’shypothesis.\nThe above would not rule out a dependence of feedback strength on climate state but is appealing\nbecause it simply explains why the minimum sensitivity is offset from the control climate tempera-\nture and shows a general increase (that slows with change) over other temperatures. I find this so\nappealing that I worry that in fig. S4 the parameters used to construct the red and purples curves\nmay be incorrect. If they are indeed correct, then the contribution of the “transient” impact on\nthe estimate of lambda-eff seems to be much to small to explain but a small fraction of the overall\nbehavior. Still, if this were my research, I would double check whether really accurate values for\nthe2-layerEBMmodelmightgivesomethingclosetothebehaviorofCESM2undertheassumption\nthattheparametersareconstantsbutmaydifferbetweenawarmingandcoolingclimate.\nIf the transient behavior is responsible for a non-negligible portion of the apparent dependence of\nlambda-eff on climate state, one could use the warming/cooling EBM model runs to subtract these\ncontributions from the total lambda-eff obtained from the CESM3 simulations. This would isolate\nthecontributiontovariationsinclimatesensitivityduetovariationsintheclimatestatealone.\nReply: We appreciate the reviewer raising this alternative hypothesis, which we have considered as\ndescribedabove.\nOne final detail: in the caption to figure 4 (in the main text), two references are made to Fig. 2, I\nthinktheseshouldrefertoFig. 3.\nReply: Weappreciatethereviewercatchingthistypo! Wehavecorrecteditintherevisedmanuscript.\n3"}]

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