the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Climate feedbacks with latitude derived from climatological data and theory
Abstract. Most current methods for evaluating climate feedbacks utilise variation with time in Earth’s energy balance and surface temperatures, either from observations or Earth system model perturbation experiments. This study presents a new semiempirical evaluation of Earth’s climate feedbacks at equilibrium, constrained instead by variation with latitude in recent mean climatology. Latitudinally binned surface temperature and outgoing radiation climatology provides a first order net climate feedback estimate λ= 1.3±0.1 Wm^{2 }K^{1}, but this does not isolate the temperature influence on outgoing radiation from other factors. To isolate the surface temperature influence: First, we derive approximated functional relations for outgoing shortwave and longwave radiation in terms of surface temperature, surface relative humidity, fractional cloud amount, tropopause height and incident solar radiation. Second, we use observations of current zonalmean climatology to constrain the relations and apply calculus to evaluate noncloud climate feedbacks with latitude, including the Planck, water vapourlapse rate and surface albedo. Our novel climatologybased evaluations of climate feedbacks weighted by the recent warming pattern, when combined with a recent estimate of cloud feedback from multiple lines of evidence, implies a global mean total net climate feedback λ= 1.1 (0.8 to 1.4 at 66 % range) Wm^{2 }K^{1} consistent with recent assessments of the literature. Our latitudinal method to constrain noncloud climate feedback is independent of previous temporal approaches, using different observational lines of evidence, and so our method complements existing methods to help constrain climate feedback and climate sensitivity.
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RC1: 'Comment on egusphere20232307', Anonymous Referee #1, 07 Dec 2023
This study presents a new method to estimate climate feedbacks by using zonal variations in radiative fluxes and surface temperatures. In contrast to existing methods, this new method utilizes much larger variations in data (by invoking the use of zonal variations) leading to high signal to noise ratio. To estimate the feedback, the authors derive functional relations relating zonal changes in radiative fluxes to changes in surface temperature.
The estimated feedback terms in this study are in good agreement with the expected range reported by other studies. The calculations performed in the paper and the framework presented are novel and important. I believe this study can be accepted for publication. I do however have some concerns/confusions that I would like to see clarified:
Comments:
C1: The authors describe the outgoing longwave radiation as a function of bulk emissivity and surface temperature such that Rlout is of the form E*sigma*T^4 (equation 5).
C1.1: Does it then assume that all the outgoing longwave radiation at top of atmosphere is emitted from the surface? How does this account for the atmospheric absorption and reemission?
C1.2: Satellite measurements show that Earth’s outgoing longwave radiation is an essentially linear function of surface temperature over a wide range of temperatures (Koll & Cronin, 2018). How does this align with equation 5?
C1.3: Latitudinal variations in Rlout can also carry information about the heat transports from tropic to poles and humid to arid regions (Ghausi et al., 2023). By fitting equation 5, these effects will get embedded in the bulk emissivity term. As a result, the clearsky emissivity alone may vary from dry to humid regions. Can this then influence the feedbacks calculated from equation 9b? Also, the functional form of clearsky emissivity described in equation B6 does not consider these effects.
C2: The interpretation of bulk emissivity should be made clear while presenting plots of clearsky and allsky emissivity in figure 3. Usually the clearsky and allsky emissivity have been defined for the atmosphere such that allsky emissivity would be greater than clearsky emissivity (Brutsaert 1975; Crawford & Duchon 1999; Tian et al., 2023).
C3: The authors use surface temperatures, outgoing longwave/shortwave radiation and cloud area fraction data from different sources. Was it done deliberately? Would not CERES alone provide all these variables?
C4: Are the results presented here sensitive to the bin width of latitudinal bands chosen for analysis?
C5: Perhaps the expression for functional relationships derived and described in Appendix B (equation B6) and Appendix D can be moved to the main text in section 3 and 4 respectively. It will also be useful if the fitted cubic expression in Appendix D between T_cold and T_warm is also described.
Corrections:
There is no equation 10c (mentioned in lines 226,229, 297 etc). Equation 10 had only one form as given in line 218.
References:
Koll, Daniel DB, and Timothy W. Cronin. "Earth’s outgoing longwave radiation linear due to H2O greenhouse effect." Proceedings of the National Academy of Sciences 115.41 (2018): 1029310298.
Ghausi, S. A., Tian, Y., Zehe, E., & Kleidon, A. (2023). Radiative controls by clouds and thermodynamics shape surface temperatures and turbulent fluxes over land. Proceedings of the National Academy of Sciences, 120(29), e2220400120.
Brutsaert, W. (1975). On a derivable formula for long‐wave radiation from clear skies. Water resources research, 11(5), 742744.
Crawford, T. M., & Duchon, C. E. (1999). An improved parameterization for estimating effective atmospheric emissivity for use in calculating daytime downwelling longwave radiation. Journal of Applied Meteorology and Climatology, 38(4), 474480.
Tian, Y., Zhong, D., Ghausi, S. A., Wang, G., & Kleidon, A. (2023). Understanding variations in downwelling longwave radiation using Brutsaert's equation. EGUsphere, 2023, 117.
Citation: https://doi.org/10.5194/egusphere20232307RC1 
AC3: 'Reply on RC1', P. Goodwin, 26 Apr 2024
Responses to reviewer comments and open comment.
We thank the reviewer for their comments, and editor for their reading of the manuscript and those comments. We are confident that a revised version of our manuscript will answer all points and concerns raised. Notably, in a revised version:
 We will demonstrate the skill of our approach. This will include, for example:
 moving from a 5degree latitude zonally averaged framework to a 1 degree by 1 degree spatial resolution in latitude and longitude;
 separating the effects of local surface elevation and surface pressure on outgoing radiation from the surface temperature effect (in addition to separating the relative humidity and tropopause height effects).
 We will demonstrate the independence of our approach for calculating climate feedbacks from current methods (where climate feedbacks are defined as the sensitivity of outgoing radiation to surface temperature for a change in climate state). This will include, for example:
 showing that our approach calculates climate feedbacks by isolating the sensitivity of outgoing radiation to surface temperature for a change in climate state by isolating the temperature impact on outgoing radiation from other factors in spatial climatology (where existing methods typically consider a change in climate state from temporal variation);
 showing that our approach calculates the climate feedbacks for an infinitesimal change in climate state, and so is not subject to the linearising assumption of methods that use radiative kernels analysed for large finite changes in climate state.
 We will adopt the term ‘effective emissivity’ for the emissivity definition we use, and explain the motivation for this definition and place this definition in context with the literature.
Below, we now detail how we will answer the specific points raised by reviewer 1.
Reviewer 1:
“This study presents a new method to estimate climate feedbacks by using zonal variations in radiative fluxes and surface temperatures. In contrast to existing methods, this new method utilizes much larger variations in data (by invoking the use of zonal variations) leading to high signal to noise ratio. To estimate the feedback, the authors derive functional relations relating zonal changes in radiative fluxes to changes in surface temperature.
The estimated feedback terms in this study are in good agreement with the expected range reported by other studies. The calculations performed in the paper and the framework presented are novel and important. I believe this study can be accepted for publication. I do however have some concerns/confusions that I would like to see clarified:”
We thank the reviewer for their careful assessment of the manuscript, and are pleased that they see the novelty and importance of the calculations performed. Below we detail how we will alter a revised manuscript to answer the concerns raised.
“Comments:
C1: The authors describe the outgoing longwave radiation as a function of bulk emissivity and surface temperature such that Rlout is of the form E*sigma*T^4 (equation 5).
C1.1: Does it then assume that all the outgoing longwave radiation at top of atmosphere is emitted from the surface? How does this account for the atmospheric absorption and reemission?”
The emissivity of any substance is defined as the ratio of the radiation emitted divided by the expected radiation emitted for a black body at the same temperature. In this study, we adopt an ‘effective emissivity’ of the Earth’s surface as observed from space, where we divide the longwave radiation emitted to space by the expected radiation emitted by the Earth’s surface if it were a perfect black body.
This ‘effective emissivity’ both characterises the greenhouse effect of the atmosphere, which acts to reduce the radiation emitted to space, and characterises the nonblackbody nature of the Earth’s surface. We agree that this ‘effective emissivity’ is not the same as the emissivity of a layer of atmospheric gas, and nor is it the same as the emissivity of the material the Earth’s surface is made from. In a revised manuscript we will carefully explain the ‘effective emissivity’ we adopt, both how it characterises the greenhouse effect of the atmosphere and how it differs from the emissivity of a particular material or gas. The ‘effective emissivity’ does not assume that the radiation emitted into space is the radiation emitted from the surface, but instead allows the radiation emitted to space to come from the atmosphere.
It has been often noted in Energy Balance Model literature that outgoing clear sky longwave radiation increases nearly linearly with surface temperature (e.g. Budyko, 1969; North, 1975), and this phenomenon has recently been explained as an effect of the greenhouse properties of water vapour (Koll and Kronin, 2018). Goodwin and Williams (2023) demonstrated that this nearlinear increase in clear sky outgoing longwave radiation to surface temperature gives rise to a near linear reduction in ‘effective emissivity’ in clear sky conditions with surface temperature (e.g. see Goodwin and Williams, 2023 – figure 3 therein). Since the definition of ‘effective emissivity’ simplifies the mathematics in our framework, and allows the impact of clouds to be more simply accounted for, we adopt this definition. This motivation for the adoption of the ‘effective emissivity’ will be explained in a revised manuscript.
“C1.2: Satellite measurements show that Earth’s outgoing longwave radiation is an essentially linear function of surface temperature over a wide range of temperatures (Koll & Cronin, 2018). How does this align with equation 5?”
The study here investigates the ‘effective emissivity’ as a function of surface temperature. The recent paper Goodwin and Williams (2023) shows that the outgoing longwave radiation being nearly a linear function of surface temperature (e.g. Koll and Cronin, 2018) is consistent with the ‘effective emissivity’ also being nearly a linear function of surface temperature (equation 5). In a revised version of our manuscript we will both cite this finding of Goodwin and Williams (2023) and show how the result also applies to this study.
“C1.3: Latitudinal variations in Rlout can also carry information about the heat transports from tropic to poles and humid to arid regions (Ghausi et al., 2023). By fitting equation 5, these effects will get embedded in the bulk emissivity term. As a result, the clearsky emissivity alone may vary from dry to humid regions. Can this then influence the feedbacks calculated from equation 9b? Also, the functional form of clearsky emissivity described in equation B6 does not consider these effects.”
In a revised version we will alter our calculation to move to analysing the outgoing radiation and its relationship to surface temperature by bins in longitude and latitude (rather than zonal averages by latitude). This relationship is primarily determined by a vertical radiative balance. In the discussion, we also discuss how horizontal convergence of heat transport affects the surface temperature distribution. This approach considers vertical radiative balance for surface temperature. Whether the surface temperature is maintained by horizontal heat convergence this does not alter the nature of the vertical radiative response to surface temperature. These points will be discussed in a revised version. In a revised manuscript, our definition of clear sky emissivity will include the impacts of surface temperature, relative humidity, the height of the tropopause, and the additional impacts of surface pressure and surface elevation.
“C2: The interpretation of bulk emissivity should be made clear while presenting plots of clearsky and allsky emissivity in figure 3. Usually the clearsky and allsky emissivity have been defined for the atmosphere such that allsky emissivity would be greater than clearsky emissivity (Brutsaert 1975; Crawford & Duchon 1999; Tian et al., 2023).”
Yes, we agree that this point needs to be made clear in a revised manuscript. We will explain how the ‘effective emissivity’ is defined as the longwave radiation emitted to space divided by the expected radiation emitted from the Earth’s surface for a black body at the same temperature. We will explain how the value of the ‘effective emissivity’ reduces if the greenhouse effect strengthens, and that this is the opposite sign to how the emissivity of a layer of gas changes when the greenhouse effect strengthens (e.g. due to the presence of clouds), and we will relate this to the literature pointed out by the reviewer.
“C3: The authors use surface temperatures, outgoing longwave/shortwave radiation and cloud area fraction data from different sources. Was it done deliberately? Would not CERES alone provide all these variables?”
In the revised manuscript we will use CERES EBAF4.2 to provide all data for outgoing shortwave and longwave radiation (at both clear sky and all sky conditions), and cloud area fraction. Surface temperature will be taken from ERA5 reanalysis as we also need a compatible dew point temperature to calculate relative humidity (and ERA5 provides both surface temperature and dew point temperature). ERA5 will also be used for surface pressure. The data will be analysed to 1 degree resolution over both latitude and longitude.
“C4: Are the results presented here sensitive to the bin width of latitudinal bands chosen for analysis?”
In the first version of the manuscript we binned the data in latitudinal bands of 5 degrees. In the revised manuscript we instead use a 1 degree resolution, and analyse the data to 1 degree both latitudinally and longitudinally (i.e. without taking zonal means). Our results following this move to a 1 degree resolution (in both latitude and longitude) are similar to the results of the initial version of the manuscript (with 5 degree resolution and taking the zonal mean), demonstrating that the results are insensitive to the bin width chosen to analyse the data.
“C5: Perhaps the expression for functional relationships derived and described in Appendix B (equation B6) and Appendix D can be moved to the main text in section 3 and 4 respectively. It will also be useful if the fitted cubic expression in Appendix D between T_cold and T_warm is also described.”
Agreed, we will place the equivalent relationships in the main text in a revised version.
“Corrections:
There is no equation 10c (mentioned in lines 226,229, 297 etc). Equation 10 had only one form as given in line 218”
Thank you, we will correct this in a revised version.
Citation: https://doi.org/10.5194/egusphere20232307AC3  We will demonstrate the skill of our approach. This will include, for example:

AC3: 'Reply on RC1', P. Goodwin, 26 Apr 2024

CC1: 'Comment on egusphere20232307', Timothy Merlis, 16 Dec 2023
Hi all,
The manuscript omits references to Koll et al 2023 (doi:10.1175/JASD220178.1) and Feldl and Merlis 2023 (doi:10.1029/2023GL105796)two published articles that offer approaches with similar aims. These were both presented in conferences in summer 2022 and have now been through peer review.
I'm not a reviewer of this manuscript. My aim is to prompt the authors to engage with existing literature and explain how what they are doing is different than or similar recent work, including but not limited to that which I've contributed to.
Sincerely,
Tim Merlis, Princeton University
Citation: https://doi.org/10.5194/egusphere20232307CC1 
AC1: 'Reply on CC1', P. Goodwin, 26 Apr 2024
We thank Tim Merlis for his comments and for informing us of these existing publications. In a revised version, we will cite both publications and explain how our manuscript fits in context with them.
Briefly, with regards to Koll et al. (2023) that study explores the longwave clear sky feedback from first principles, but does not consider climate feedbacks under all sky conditions or shortwave feedbacks (such as the surface albedo feedback). In comparison, our approach explores the Planck (longwave), water vapourlapse rate (longwave) and surface albedo (shortwave) feedbacks for both clear sky and all sky conditions. Our approach also uses a theoretical framework constrained by observations.
With regards to the Feldl and Merlis (2023) study, they adopt an approach whereby the climate feedback is obtained by multiplying the numerically determined radiative kernels for a particular climate model by theoretical relations for other terms that convert those radiative kernels to the climate feedback. Therefore, their approach relies on comprehensive climate model output to generate the radiative kernels before the climate feedbacks can be determined. Our approach is independent of radiative kernels obtained from comprehensive climate model output, and derives the entire climate feedback from a theoretical framework constrained by observations of the real climate system.
In a revised version of our paper, we will explain this clearly to place our study in context with these previous studies.
Citation: https://doi.org/10.5194/egusphere20232307AC1

AC1: 'Reply on CC1', P. Goodwin, 26 Apr 2024

RC2: 'Comment on egusphere20232307', Anonymous Referee #2, 01 Apr 2024
This study by Goodwin et al. presents an idea of how individual and total feedbacks can be derived from spatial variation of the outgoing radiation, and by extension Earth's equilibrium climate sensitivity (ECS). This is apparently an idea that is brewing in the community, and I looked at some of the papers that are referred to in other comments. These studies should obviously be referenced in an updated manuscript.
My main issue is that it is not demonstrated that the method has skill in predicting total feedback and ECS. We are simply presented with an argument in Section 1.2 that the results are close to that obtained using other methods. I find that argument difficult to accept, and it certainly does not qualify the method as an independent line of evidence.
The method would be straightforward to test: simply perform the analysis on CMIP6 models and compare against their true feedback and ECS. There was some attempt to do this in Feldl and Merlis (2023, GRL), albeit unfortunately only on a few individual feedbacks, but the performance is not at a quantitatively accurate level. They find an ok correlation of around 0.6, but the slopes are biased, i.e. not 1:1. Hence attempts to measure feedbacks using these methods are going to yield biased results. At the very least the authors could perform the analysis shown in Figure 1 and demonstrate that the resulting feedback estimate is statistically related to total feedback in the model, though ideally they should run each model through the full analysis, as if they were observations, and compare estimates to the true values.
Along the same lines, I think the authors are overconfident in their results. The statistical uncertainties that are presented do not take into account structural uncertainties that arise from the method as well as internal variability. They seem to just be grounded on the least squares fits. An analysis like that suggested above, which could be extended using some of the large initial value ensembles that are out there, would provide a more faithful estimate of true uncertainty.
Finally, I wonder why the authors chose to treat each 5 degree latitude band the same, despite the fact that they cover very different area? When looking at Figure 1, it seems that most of the weak negative feedback comes from the cold regions which cover a small area of the Earth. Judging by eye, it would seem one could fit a line with a slope somewhere in the 1.5 to 2 Wm2K1 range on the data points between 270 and 300 K. The dropoff at higher temperatures are probably due to tropical anvil clouds, that may or may not exert a feedback under climate change. The colder data points, in particular those below 250 K are situated on Antarctica and hence represent a small area and at the same time have elevated Lout+Sout due to the reflective and high elevation surface. These four data points seem to have a large influence on the fit. From the description in Section 2.1 it is unclear whether the authors apply the area weighting to the linear fits, or just to the warming patterns.
A minor thing is that the AR6 and Sherwood et al. feedback estimates (Table 1) relate only the process understanding line of evidence. When taking the inverse of the respective ECS estimates, which are based on all lines of evidence, different total feedback is obtained.
All in all, I think this paper has some interesting ideas, but the results regarding climate change feedbacks and climate sensitivity are a giant leap of faith. Can the temporal dimension simply be replaced with the zonal direction given the circulation and distribution of clouds, etc? Therefore, also considering the not cited papers that already presented the same idea (Koll et al. 2023, Feldl and Merlis 2023), I cannot recommend publication of the manuscript before more evidence of the quantitative skill of the method has been presented.
Citation: https://doi.org/10.5194/egusphere20232307RC2 
AC2: 'Reply on RC2', P. Goodwin, 26 Apr 2024
We thank reviewer 2 for the their comments, and editor for their reading of the manuscript and those comments. We are confident that a revised version of our manuscript will answer all points and concerns raised. Notably, in a revised version:
 We will demonstrate the skill of our approach. This will include, for example:
 moving from a 5degree latitude zonally averaged framework to a 1 degree by 1 degree spatial resolution in latitude and longitude;
 separating the effects of local surface elevation and surface pressure on outgoing radiation from the surface temperature effect (in addition to separating the relative humidity and tropopause height effects).
 We will demonstrate the independence of our approach for calculating climate feedbacks from current methods (where climate feedbacks are defined as the sensitivity of outgoing radiation to surface temperature for a change in climate state). This will include, for example:
 showing that our approach calculates climate feedbacks by isolating the sensitivity of outgoing radiation to surface temperature for a change in climate state by isolating the temperature impact on outgoing radiation from other factors in spatial climatology (where existing methods typically consider a change in climate state from temporal variation);
 showing that our approach calculates the climate feedbacks for an infinitesimal change in climate state, and so is not subject to the linearising assumption of methods that use radiative kernels analysed for large finite changes in climate state.
 We will place our method in the context of recent publications. This will include:
 showing that our approach does not rely on radiative kernels analysed from comprehensive climate models, thus differing from the recent Feldl and Merlis (2023) that requires radiative kernels to calculate climate feedbacks;
 showing that our approach calculates a range of climate feedback terms for longwave and shortwave feedbacks in clear sky and all sky conditions, where the recent Koll et al. (2023) study calculates only the clear sky longwave feedback.
Below, we now detail how we will answer the specific points raised by reviewer 2.
Reviewer 2:
“This study by Goodwin et al. presents an idea of how individual and total feedbacks can be derived from spatial variation of the outgoing radiation, and by extension Earth's equilibrium climate sensitivity (ECS). This is apparently an idea that is brewing in the community, and I looked at some of the papers that are referred to in other comments. These studies should obviously be referenced in an updated manuscript.”
We thank the reviewer for their reading of the manuscript. We also thank the open comments for bringing these papers to our attention and we will be happy to reference them in a revised manuscript.
“My main issue is that it is not demonstrated that the method has skill in predicting total feedback and ECS. We are simply presented with an argument in Section 1.2 that the results are close to that obtained using other methods. I find that argument difficult to accept, and it certainly does not qualify the method as an independent line of evidence.”
In a revised manuscript we will clearly demonstrate both the skill of our method in analysing climate feedbacks and the independence of our method. The climate feedback is the sensitivity of outgoing radiation at the top of the atmosphere to surface temperature change following a change in climate state. Often in the literature this change in climate state is achieved through a change in climate over time, analysed at the same location in space. Here, we analyse the change in climate state by sampling through local different climate states across space and account for nontemperature factors that alter outgoing radiation relative to surface temperature spatially. We adopt a number of changes to further demonstrate that our method is both independent and has skill in prediction in a revised manuscript, including:
 We move to 1x1 degree horizontal resolution (both latitudinally and longitudinally), in place of the 5degree zonal bands. This change ensures that our method represents changes in outgoing radiation for many more data points, and avoids zonal averaging to lead to aliasing and degrading the skill of our relationships.
 We now explore how outgoing radiation is a function of surface temperature accounting for spatial variation in: relative humidity, the height of the tropopause above the local surface, surface pressure, whether the local surface is land or sea, and the cloudiness of the local surface. In the previous version, we had not considered the land/sea difference, local surface pressure, the local elevation of the surface (which affects the height of the tropopause above the local surface). Where we had considered relative humidity and cloudiness, we had considered these only as zonal average quantities and in a revised version we consider them as local properties in a 1x1 degree resolution.
 We also apply our method to CMIP6 models, and demonstrate that the quantities our relationships utilise (for example the cloud emissivity coefficient and the cloud albedo) are also representative when similarly analysed for CMIP6 models.
These steps will further demonstrate the skill and independence of our method.
“The method would be straightforward to test: simply perform the analysis on CMIP6 models and compare against their true feedback and ECS. There was some attempt to do this in Feldl and Merlis (2023, GRL), albeit unfortunately only on a few individual feedbacks, but the performance is not at a quantitatively accurate level. They find an ok correlation of around 0.6, but the slopes are biased, i.e. not 1:1. Hence attempts to measure feedbacks using these methods are going to yield biased results. At the very least the authors could perform the analysis shown in Figure 1 and demonstrate that the resulting feedback estimate is statistically related to total feedback in the model, though ideally they should run each model through the full analysis, as if they were observations, and compare estimates to the true values.”
Our approach emphases the importance of developing an independent estimate of climate feedbacks, drawing upon remotelysensed observational data and theoretical closures, rather than applying a radiative kernel method.
The reviewer proposes that we apply our method to CMIP6 model output, and compare the climate feedbacks evaluated to the ‘true’ values for these models as analysed through existing methods (e.g. a radiative kernel approach following a 4xCO2 perturbation). The reviewer comments that this approach has been conducted in Feldl and Merlis (2023, GRL).
We disagree with the reviewer that this is a useful strategy for our method, given (1) the differences between our method and the Feldl and Merlis (2023) method, and (2) on the basis that we dispute that the climate feedback values as analysed for the CMIP6 models via radiative kernels following a 4xCO2 experiment are the ‘true’ values that our method should be compared to.
(1) The differences between our method and the approach of Feldl and Merlis (2023).
Feldl and Merlis (2023) utilise semianalytical theory to calculate climate feedbacks spatially in comprehensive climate models, where the semianalytical equations rely on the radiative kernels analysed numerically from finite perturbation (e.g. 4xCO2) experiments conducted using a comprehensive climate model. The semianalytical equations solve for the rest of the terms that modify the radiative kernel to obtain the climate feedbacks, but they do not themselves solve for the radiative kernel itself. Thus, it is fair to compare the Feldl and Merlis (2023) approach to the climate feedback analysed numerically for the comprehensive climate models for the same finite perturbation experiments that the radiative kernel is derived from (e.g. a 4xCO2 experiment).
In contrast, our approach calculates climate feedbacks independently from the radiative kernels analysed for comprehensive climate models. Our approach uses data from the real world to constrain functional equations that isolate the temperature sensitivity of outgoing radiation from other factors. These functional relations are then differentiated with respect to surface temperature to calculate climate feedbacks. Our climate feedbacks thus apply to an infinitesimal perturbation to the climate state, not a 4xCO2 perturbation, and are independent of radiative kernels analysed for comprehensive climate models.
(2) The mismatch between our climate feedbacks and the ‘true’ values analysed from finite perturbation in CMIP models.
When radiative kernels applied to a 4xCO2 experiment are used to calculate climate feedbacks in CMIP5 and CMIP6 models (e.g. Zelinka et al., 2020), it is assumed that individual climate feedbacks from different processes add linearly to produce the total net climate feedback from all processes. However, this assumption of linear combination is not true given that the feedbacks are analysed from large finite perturbations, and so the analysed feedbacks are not the ‘true’ feedbacks but are estimates of the ‘true’ feedbacks in the comprehensive climate models. For example, in Zelinka et al. 2020 there are significant addition errors when combining all the different climate feedbacks to produce the net climate feedback: the rms error in net climate feedback is 17.2 % for the 27 CMIP6 models analysed by Zelinka et al. 2020 and 17.6 % for the 28 CMIP5 models. We do not know which individual climate feedbacks these errors arise from, nor whether some terms have errors of one sign and other terms have errors of another sign for any given model.
In contrast, our climate feedbacks are analysed for infinitesimal perturbation (since they are from differentiated functional relationships), and so do not suffer from any errors of linear addition. Thus, if we apply our approach to the CMIP6 models we would not expect to perfectly reproduce the climate feedback estimates from radiative kernels analysed for finite perturbation in those models and we do not know where the errors of linear addition are in the existing finite perturbation estimates of CMIP5 and CMIP6 model climate feedback estimates.
Nonetheless, we appreciate the spirit of the reviewer’s suggestion and agree that further validation of our method against CMIP6 models would bolster confidence in our findings. In a revised manuscript we will include comparisons to CMIP6 models in terms of our dependences that are important in our theoretical closures. We will utilise our approach to analyse properties from our equations in CMIP6 models, for example analysing c_ep and the albedo of cloud, as well as the values of ∂epsilon/∂T and ∂albedo/∂T, to assess whether our approach produces representative dependences not only for real data but for the CMIP6 models. We will also explore the inter annual variability from our approach, by comparing shorter timescale estimates of our climate feedbacks to longer timescale climatology.
Our study aims to calculate climate feedbacks from observations, independent from radiative kernels analysed from comprehensive climate models. As such, detailed calculation of climate feedbacks within comprehensive climate models is outside the scope of our study.
“Along the same lines, I think the authors are overconfident in their results. The statistical uncertainties that are presented do not take into account structural uncertainties that arise from the method as well as internal variability. They seem to just be grounded on the least squares fits. An analysis like that suggested above, which could be extended using some of the large initial value ensembles that are out there, would provide a more faithful estimate of true uncertainty.”
We acknowledge that there are structural uncertainty and internal variability in both the model method using radiative kernels and in our method using remotelysensed data and theoretical closures. One of the main benefits of our independent approach is that the structural uncertainties are likely to be different to the existing radiative kernel methods as there are different underlying assumptions.
We also agree that the structural uncertainty should be quantified. In a revised manuscript, we will make alterations that account for structural uncertainty and internal variability. A revised manuscript moves to a 1x1 degree horizontal resolution (latitude and longitude), as opposed to 5 degree zonal averaging as in the initial version. We will quantify structural uncertainty by arising from the form of the underlying functional relationships we adopt between the properties (e.g. temperature) and outgoing radiation. We will also explore internal variability. In our main analysis, we are taking a 21year climatology from 2003 to 2023. We will compare this 21year climatology to shorterterm periods, and thus evaluate the impact of internal variability on the climate feedbacks we analyse. There is no reason why further studies cannot apply the approach to different time periods altogether.
“Finally, I wonder why the authors chose to treat each 5 degree latitude band the same, despite the fact that they cover very different area? When looking at Figure 1, it seems that most of the weak negative feedback comes from the cold regions which cover a small area of the Earth. Judging by eye, it would seem one could fit a line with a slope somewhere in the 1.5 to 2 Wm2K1 range on the data points between 270 and 300 K. The dropoff at higher temperatures are probably due to tropical anvil clouds, that may or may not exert a feedback under climate change. The colder data points, in particular those below 250 K are situated on Antarctica and hence represent a small area and at the same time have elevated Lout+Sout due to the reflective and high elevation surface. These four data points seem to have a large influence on the fit. From the description in Section 2.1 it is unclear whether the authors apply the area weighting to the linear fits, or just to the warming patterns.”
The climate feedbacks we determine are areaweighted in terms of converting the local value of lambda into a global mean. In the revised version we will area weight each 1x1 degree gridpoint when calculating the global climate feedback. The dependencies cover all of temperature space so we do not want to average out endpoints.
We do note that the fit in figure 1 is not used in the calculating climate feedbacks we present in the study – only in providing a zerothorder estimate. Agreed that surface pressure differences (due to different elevation) will have a significant impact on effective emissivity that is independent of temperature, and we agree that this covariation needs to be accounted for. In a revised version, we will explicitly account for how altitude affects local outgoing radiation separately to surface temperature, and so the surface temperature impact on outgoing radiation will no longer include the altitude effect.
The move to a spatial resolution of 1 degree in both latitude and longitude in a revised manuscript (from zonal averaging the longitude) will mean that in a revised manuscript the spatial analysis of both alpha_cloud (shortwave impact of clouds) and c_epsilon (longwave impact of clouds) will capture the spatial variation in these values reflecting different cloud types.
As we are attempting to isolate the temperature sensitivity of outgoing radiation, accounting for all other factors, we wish to sample over the available range of surface temperatures on Earth. Averaging by area will remove much of the temperature variation in the system, whereas averaging bins on a 1 degree by 1 degree horizontal resolution will preserve temperature variations at high latitudes that would otherwise be averaged within the same equalarea bin.
“A minor thing is that the AR6 and Sherwood et al. feedback estimates (Table 1) relate only the process understanding line of evidence. When taking the inverse of the respective ECS estimates, which are based on all lines of evidence, different total feedback is obtained.”
Agreed, we will make this clear in a revised version.
“All in all, I think this paper has some interesting ideas, but the results regarding climate change feedbacks and climate sensitivity are a giant leap of faith. Can the temporal dimension simply be replaced with the zonal direction given the circulation and distribution of clouds, etc? Therefore, also considering the not cited papers that already presented the same idea (Koll et al. 2023, Feldl and Merlis 2023), I cannot recommend publication of the manuscript before more evidence of the quantitative skill of the method has been presented.”
We are pleased the reviewer finds the initial version of the paper has interesting ideas. A revised version will demonstrate the skill of the method in analysing climate feedbacks from observations and theory, independently from comprehensive climate models. Notably, we will:
 Extend the method to calculate climate feedback for a 1 degree latitude by 1 degree longitude resolution, instead of simply averaging in the zonal direction. Thus, we will capture the zonal distribution of clouds and other factors.
 Show that our approach can be applied to both data and comprehensive climate models by analysing the relevant terms in our framework and showing they behave internally consistent in both observations and in models.
 Place our approach in context of previous studies mentioned. Very briefly, Feldl and Merlis (2023) relies on radiative kernels analysed from numerical models, and is therefore not calculating radiative climate feedbacks purely from data. Koll et al. 2023 calculates only the clear sky longwave feedback, and does not calculate all sky or albedo feedbacks.
Overall, the climate feedback is defined as the sensitivity of outgoing radiation at the top of the atmosphere to a change in surface temperature following a change to climate state. Our approach uses spatial variation in climate state to isolate this temperature sensitivity of outgoing radiation by accounting for the nontemperature properties that also affect outgoing radiation. In a revised version, we will demonstrate that our method successfully accounts for these nontemperature properties for observed climatology over a 1 degree by 1 degree horizontal resolution. Our extended approach will complement existing methods that consider the change in climate state at a particular location over time, as the assumptions and approximations made are independent.
References not appearing in the initial manuscript:
Budyko M.I. (1969) The effect of solar radiation variations on the climate of the earth, Tellus, 21, pp. 611619
Feldl and Merlis (2023), doi:10.1029/2023GL105796.
Koll et al (2023), doi:10.1175/JASD220178.1.
Koll D.D.B., Cronin T.W. (2018) Earth’s outgoing longwave radiation linear due to H_{2}O greenhouse effect, Proc. Natl. Acad. Sci., 115 (41), pp. 1029310298, 10.1073/pnas.1809868115
North G.R. (1975) Theory of energybalance climate models, J. Atmos. Sci., 32 (11) (1975), pp. 20332043, 10.1175/15200469(1975)032<2033:TOEBCM>2.0.CO;2
Citation: https://doi.org/10.5194/egusphere20232307AC2  We will demonstrate the skill of our approach. This will include, for example:

AC2: 'Reply on RC2', P. Goodwin, 26 Apr 2024
Model code and software
Energy_Balance_Climate_Feedback P. Goodwin, R. G. Williams, P. Ceppi, and B. B. Cael https://doi.org/10.5281/zenodo.8421164
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