General Comments:

Sanderson et al. use a simple climate model composed of several exponential decay terms to model the output of pre-industrial control and abrupt-4×CO₂ simulations from CMIP5, CMIP6, and LongRunMIP. The authors use this simple climate model to estimate potential biases in effective climate sensitivity (EffCS) estimates. This approach is novel and provides an interesting framework to analyze EffCS; however, there are several points that the authors should address in order for me to recommend this manuscript for publication.

Specific Comments:

1) The new framework the authors developed is interesting; however, I am having a hard time deciphering why this paper is important. I am not sure what the main point of the paper is. Is the main point to answer the question the authors stated at the end of the introduction: “How plausible are the higher sensitivity [CMIP6] models”? Is the main point to say that ECS is actually higher than suggested by EffCS given by CMIP6 or IPCC AR6? The authors state “Our results highlight the potential for error in estimates of effective climate sensitivity through the assumptions on the asymptotic radiative balance of climate models (page 9 line 9)”. The authors need to go a step further and provide an indication of what their suggestion for the value of EffCS would be based on their new framework. The authors should discuss their results in the context of recent literature that examines estimates of EffCS. Recent studies have provided estimates of EffCS, such as Zelinka et al. (2020), Tokarska et al. (2020), McBride et al. (2021), Sherwood et al. (2020), and the new comprehensive evaluation conducted by IPCC AR6. Do the authors have a new range of EffCS using their approach compared to these other analyses? Could the authors suggest a way to constrain the estimate of EffCS based on the model’s radiative imbalance between the PICTRL and ABRUPT4X simulation? The authors should add comparisons to recent literature in their results section. In the conclusions section, the authors should expand upon the importance of their results to indicate a revision or addition to current estimates of EffCS, or suggestions on how to revise the current estimate of EffCS using their approach.

2) How did the authors determine the minimum and maximum values of τ for the short timescale, intermediate time scale, and long-time scale given in Table 1? Are these values supported by literature?

3) The authors should explain how assessing the radiative imbalance in the control simulation , R₀^CTRL, impacts the parameters in Eq. 1a or Eq. 1b. As currently written, it is unclear how this assessment is incorporated into Eq. 1a and 1b.

4) Equilibrium climate sensitivity and effective climate sensitivity are the response of the climate system to a doubling of CO₂ relative to preindustrial. The authors use the ABRUPT4X scenario, which is for a quadrupling of CO₂. In other methods, such as Gregory et al. (2004), the temperature response to the quadrupling of CO₂ needs to be divided by 2 to achieve an estimate of the temperature response to the doubling of CO₂. The authors do not discuss how their method accounts for the fact they are using an ABRUPT4X scenario to assess the temperature response to a doubling of CO₂. The authors should elaborate in the methods section how they account for this discrepancy.

5) There is no mention of IPCC AR6 in this paper. How does this analysis compare to the best estimate (3°C) and range of (2 - 5°C) of ECS given by AR6? Does the new framework in this paper support a lower or higher value of EffCS than provided by IPCC?

6) Figures 1 and 2 are barely discussed. The authors should add more discussion of these figures to the results section, especially highlighting any important interpretations of the figures, or move these two figures to the Appendix.

7) In the results section, the authors jump back and forth between discussing Figure 3 or Figure 4 (Page 6 lines 1 – 19), making it difficult to follow the points the authors are trying to make. The authors should consider editing this section by first discussing and interpreting Figure 3, then discussing and interpreting Figure 4.

8) The authors need to verify that the figure captions match the figures. Colors and types of lines described in the figure captions do not match what was plotted in the figure, making it difficult to interpret the figures (see the Technical Corrections related to each figure below).

9) Table A2 is an important table, displaying the difference between EffCS computed using various methods for the LongRunMIP simulations. The authors should consider moving Table A2 into the main part of the text. They can add a discussion of the table to the results section, highlighting why the estimates for ΔT_best-est and ΔT_extrap are similar for some models yet different for others.

Technical Corrections:

Equation 1b: Constant is written as R^4x, but referred to as R₀^4x in the text (page 3 line 1)

Table 1: R_n scaling factors are not listed in Table 1, but S_n scaling factors are listed. Is there a reason why the R_n scaling factors are omitted?

Table 1: R₀ is included in the table, but this variable does not appear in either Eq. 1a or Eq. 1b. How does this variable relate to these two equations?

Why are the lines in figures 1 and 2 labeled as S_LR, S_eff, and S_extrap. In Eq. 1a, 1b, 2, and 3, S refers to a scaling factor. Why are the authors using this variable (S) to label the different lines?

Figure 1 Caption:

• Authors state solid yellow lines are linear regressions used to estimate effective climate sensitivity for the first 150 years of data. This should be the dotted yellow lines.
• Authors state solid pink lines are linear regressions used to estimate effective climate sensitivity for the last 15% of warming. This should be the dotted pink lines.
• Authors state vertical dotted pink and yellow lines show corresponding values of effective climate sensitivity. Should be vertical solid pink and yellow lines.
• Authors state solid yellow horizonal line shows the PICTRL net energy imbalance averaged over the final 100 years of the simulation. There are no solid yellow horizontal lines. There are green horizontal lines, which are not included in the caption or legend. Are the green lines supposed to be the PICTRL net energy imbalance? If not, make sure to label what the green lines are showing.
• Solid blue line is not described in the caption
• I am not sure that the dashed blue line is described correctly in the figure caption. Authors say the dashed blue line shows an exponential model fit, but the lines in all of the subplots in Figure 1 are horizontal. Is the solid blue line actually showing the exponential model fit? If so, what do the dashed blue lines represent?
• Green dots are not described in the caption

Figure 1 General Comments:

• Green and blue dots in the legend representing PICTRL and ABRUPT4X are very faint, almost impossible to see. Make them more legible in legend.
• I cannot distinguish the difference between the blue dots representing ABRUPT4X and the light blue ellipse showing the 5-95 CI for ΔT_extrap. It looks like only the light blue ellipse is plotted.
• What does n_yr show? I assume it is the number of years in the LongRunMIP simulation, but the authors should include a description of the parameter in the figure caption for clarity.
• Make sure the lines plotted on the figure do not go through the text (i.e., CNRMCM61 panel has solid blue and dotted yellow lines going through n_yr = 1850)

Figure 2 Caption:

• There is a description of black points, but there are no black points in the figure or legend
• Which dashed horizonal line illustrates ΔT_extrap? Blue? Green?
• A description of the green dashed line does not appear in the figure caption, and the green dashed line is not included in the legend.
• A description of the green dots does not appear in the figure caption
• Authors state the dashed purple line is ΔT_best-est. I do not see a purple line. There is solid pink line. Is this pink line supposed to be ΔT_best-est?

Figure 2 General Comments:

• Missing “of” in the sentence: “Shaded regions and thin dotted lines show the 10^th and 90^th percentiles of the fitted ensemble projections”
• The 4xCO₂ and pictrl is written differently from PICTRL and ABRUPT4X in the first figure caption and the main text. These scenarios should be referred to in a consistent manner
• There are no lines or symbols next to 4xCO₂ and pictrl in the legend

Figure 3 General Comments:

• It is difficult to distinguish the blue dots and the blue shaded region, specially towards the right side of each panel. Making the shaded region a different color, or different shade of blue could help distinguish the points from the shaded region.
• Why do some of the models have visible 10^th and 90^th percentiles at the beginning and ending of the blue line, but others do not? What is different in the models with very small ranges of uncertainty from those with larger ranges?
• 4xCO₂ in the legend does not match ABRUPT4X labeling in figure caption and the main text
• Missing “of” in the sentence: “Shaded regions and thin lines show the 10^th and 90^th percentiles of the fitted ensemble projections”

Figure 4 Caption:

• Left hand column:
  • Caption says there are whiskers in the left-hand column on the light blue diamond symbols. There are no whiskers plotted showing the 10^th & 90^th percentiles of ΔT_extrap
• Central Column:
  • Caption says there are cyan error bars plotted, but they are not on the figure. Only show blue diamonds
  • Solid and dashed yellow lines are not described in the figure caption

Figure 4 General Comments:

• Is there any range of uncertainty for the values of ΔT_best-est shown by the red diamonds? If so, then this uncertainty should be indicated on the figure
• There is no legend included with this figure, whereas the other 3 figures included legends. Consider adding a legend to this figure.

Page 3 Line 31: What does “this estimate” refer to? ΔT_extrap, R_extrap^4x, or both?

Page 3 Line 35: Some other models should be included as described as behaving as expected. GISSE2R and GFDLESM2M show near zero equilibrium TOA balance in both PICTRL and ABRUPT4X simulation in Figure 3. Why were these models excluded from this sentence?

Why are the values in the brackets for ΔT_extrap and ζ_extratp the same in Tables A2, A3, and A4? The table caption explains that the numbers in the brackets represent the 5^th and 95^th percentiles. I find it highly unlikely that the 5^th and 95^th percentiles are the same, especially since the median value is larger than the values in the brackets.

Page 9 Line 21: Missing closing parentheses after Table A1

References:

Gregory, J. M., Ingram, W. J., Palmer, M. A., Jones, G. S., Stott, P. A., Thorpe, R. B., Lowe, J. A., Johns, T. C., and Williams, K. D.: A new method for diagnosing radiative forcing and climate sensitivity, Geophys. Res. Lett., 31, 2–5, https://doi.org/10.1029/2003GL018747, 2004.

McBride, L. A., Hope, A. P., Canty, T. P., Bennett, B. F., Tribett, W. R., and Salawitch, R. J.: Comparison of CMIP6 historical climate simulations and future projected warming to an empirical model of global climate, Earth Syst. Dynam., 12, 545–579, https://doi.org/10.5194/esd-12-545-2021, 2021.

Sherwood, S. C., Webb, M. J., Annan,J. D., Armour, K. C., Forster, P. M.,Hargreaves, J. C., et al.: An assessment of Earth's climate sensitivity using multiple lines of evidence. Reviews of Geophysics,58,e2019RG000678. https://doi.org/10.1029/, (2020).

Tokarska, K. B., Hegerl, G. C., Schurer, A. P., Forster, P. M., and Marvel, K.: Observational constraints on the effective climate sensitivity from the historical period, Environ. Res. Lett., 15, 1–12, https://doi.org/10.1088/1748-9326/ab738f, 2020.

Zelinka, M. D., Myers, T. A.,McCoy, D. T., Po-Chedley, S.,Caldwell, P. M., Ceppi, P., et al.: Causes of higher climate sensitivity in CMIP6 models. Geophysical ResearchLetters, 47, e2019GL085782. https://doi.org/10.1029/2019GL085782, 2020.

General Comments:

In the manuscript under discussion, the assumption that global climate models evolve to some top-of-atmosphere radiative balance is put to the test. For this, millenia-long runs from LongrunMIP are used, alongside a linear response model with responses on three time scales. Based on the found energetic imbalances in some models and equilibrium temperature estimates, the biases in the latter are related to the former, concluding that energy leaks might influence common equilibrium climate sensitivity estimates much.

I find this an interesting and important exercise, with conclusions that could have big consequences for long-term projections with some global climate models. However, I am not fully convinced by the used methodology. Further, I think the text could be clearer at certain points. Finally, the presentation of the figures feels a bit sloppy with especially colors and line styles not matching with the captions. These issues should be resolved before I would recommend publication.

Specific Comments:

(1) Central in the manuscript is the linear response type model in equations (1a)-(1b). I do not think that these equations are explained well enough nor that made choices are acknowledged and defended well enough. I also have some problems with their use for non-constant forcings.

(1a) First of all, the form of (1a)-(1b) is now defended as consistent with some simple (linear) climate models. However, it also fits with linear response theory as the response of a non-linear model "in the linear response regime". In [Proistosescu and Huybers (2017)], they already frame it in this way, and e.g. in my recent paper [Bastiaansen et al (2021)] this link is made even more explicitly. I think it would be good to clarify these things, which also would further communicate the validity of (1a)-(1b). Further, nowhere is it mentioned that equations (1a)-(1b) only hold for constant forcings, and that the parameters would be different for other forcing levels. These important 'terms and conditions' for the use of (1a)-(1b) should be added.

(1b) It is now assumed that all climate models have a response on three distinct time scales. This choice for the number of time scales should be stated explicitly and a better justification needs to be given. Why should all models have the same number of response time scales? Why should there be precisely three time scales? For me, this now seemingly arbitrarily made choice is one of the weakest points of the paper and could render all your conclusions moot: what if a system actually has more than three time scales and all the remaining observed radiative imbalance disappears if you were to take all these time scales into account? So, did you check if results remain similar when a different number of time scales are used?

(1c) For a few models in LongrunMIP, the abrupt4xCO2 experiment was not long enough, and the results for a different forcing scenario were added at the end of the abrupt4xCO2 simulation in an attempt to construct a long enough simulation. However, the used linear response model in equations (1a)-(1b) is only valid for constant forcings, but the used runs have non-constant forcings (1pct2x, 1pct4x and RCP8.5). To me, that means the equations simply cannot be used. In particular, the timing of forcing in these experiments is of uttermost importance to properly assess the response over time, and splicing runs together like this therefore makes no sense to me. An alternative would be to derive a linear response model for the used forcing scenarios, and use that to fit the parameters from which the abrupt response could be inferred (taking some liberty with the 'ensemble-average' assumptions underlying linear response theory). 

(2) In (1a) and (1b) the parameters T0 and R4x are playing similar roles. However, they are not determined in the same way, as T0 is derived from the control experiment instead of fitted with the abrupt4xCO2 experiment. The reason for doing this should be explained.

(3) To obtain the model parameters from the data, one way or another a nonlinear fitting procedure needs to be used. Those can be sensitive to the choices for metaparameters -- in this case, the choices for the priors (i.e. the mentioned distributions in Table 1). Did the authors check to make sure the presented results do not depend too much on these priors? Additionally, the choices for the prios should also be explained better; now, it just seems to be some made up numbers, but there certainly is some sort of rationale behind them?

(4) Part of the goal of the paper seems to be to estimate the 'equilibrium' imbalance for abrupt4xCO2 experiments. Why would we want to use equations (1a) and (1b) for that? If one is only interested in that long-term imbalance, why would you not fit a decaying exponential to the last part of the transient of the imbalance instead? In any way, such kinds of choices should be addressed more explicitly in the text, including the rationale of making these choices.

(5) Figures and captions are not in line with each other. For instance, in Figure 1, the caption talks about a yellow horizontal line but in the figures it seems to be a green horizontal line, regression lines are said to be solid lines but they appear to be dotted lines and vertical lines are said to be dotted but they appear to be solid. There are also blue lines, not all of which seem to be explained in the caption. The authors should verify that the captions match with the figures and explain all lines.

(6) For me, one of the questions remaining after having read the text is what we should consider an equilibrium of the climate system. Would that just be the long-term response of the system, or do we actually want the system to have achieved radiative balance in some way? Most of the equilibrium climate sensitivity methods, including EffCS in the text, are basing their estimation technique on the requirement that there is radiative balance in equilibrium. However, the equations (1a)-(1b) explicitly do not require this. So for instance, the text on page 6, lines 43-44 stating that "if we do not know what the radiative imbalance will be when temperatures stabilise in an ABRUPT4X simulation, we in turn cannot predict the climate sensitivity with precision", hinges on what we interpret as equilibrium; in fact, you could argue that the method used in this paper is an example of a climate sensitivity prediction that does not require prior knowledge on the radiative imbalance in equilibrium, making this statement in the discussion incorrect with regard to the rest of the text. But above all, I think all these points relate to what we define as equilibrium: Originally we would say that it refers to a state in which there is radiative balance. Then when we found consistent imbalance even in the control simulation, we redefined equilibrium to mean having an imbalance similar to the control simulation. And now this paper seems to shake up even that definition in some models. In short, I think the paper would benefit from a discussion of the definition of an equilibrium climate state, relating it to the radiative imbalance and incorporating the implications of the current paper.

(7) For me, the discussion is missing some sort of recommendation or directive to follow-up on the found energy balance issue. I know that on page 6, lines 45-51 there is some text on this, but I feel like it could be a bit more concrete. Should we conclude from this paper that estimates based on inferred equilibrium radiative (im)balance are inapt for some (or all) global climate models? And should we therefore not use such estimation methods anymore? Should we conclude from the paper that global climate models have energy leaks? And should we therefore not trust these on long time scales? Should we make sure that our global climate models have no energy leaks, and e.g. move towards climate models that are discretized in a way that prevents energy leaks or prevent energy leaks by going to finer resolutions? I don't expect the authors to answer all these questions, but at least posing some of these might give the paper some more direction and might make the implications of their findings more clear.

Technical Corrections:

(1) The top-of-atmosphere radiative balance in Figure 1 and Figure 3 have different signs. That should be made consistent.

(2) In section 2.1, it is said that Table 1 contains parameter ranges and constraints. The caption does, however, say these are the prior ranges. Could the authors confirm that the values in Table 1 indeed correspond to the prior values and that the parameter ranges are not constrained in the fitting procedure? If so, also the text in section 2.1 should be changed to reflect that.

(3) Table 1 misses the parameters R_n.

(4) Figure 4: the whiskers in the sensitivity and zeta plots are missing.

(5) In the tables at the end of the paper, I was not able to find the fitted values for Sn, Rn and especially tau_n.

(6) The tables at the end of the paper, the 95% values seem to just state the 5% values again for some of the parameters.

Literature

Bastiaansen, R., Dijkstra, H. A., & Heydt, A. S. V. D. (2021). Projections of the Transient State-Dependency of Climate Feedbacks. Geophysical Research Letters, 48(20), e2021GL094670.

Proistosescu, C., & Huybers, P. J. (2017). Slow climate mode reconciles historical and model-based estimates of climate sensitivity. Science advances, 3(7), e1602821.