The authors investigate carbon cycle feedbacks under CO2 and non-CO2 GHG forcings. Since non-CO2 GHG lead to warming only, the CO2 concentration induced component of the carbon cycle feedbacks is missing for this forcing. This motivates the authors to investigate what has been termed “non-linearity of carbon cycle feedbacks” in previous studies, but with a focus on non-CO2 forcings. The authors use an impressive set of idealized model experiments to separate the different feedback components. The manuscript is generally well written, well structured, and the methods are sound and well described, although some parts of the manuscript could be improved in clarity and I found some of the results difficult to understand (see my comments below). There are only very few studies dealing with the interactions of non-CO2 GHG forcing and the carbon cycle, even though non-CO2 GHG reduction will be an important climate mitigation measure in pathways that limit global warming to below 2 degrees. Although the main results do not seem to be very surprising, I believe this study is a valuable contribution to this field and I would recommend publication in Earth System Dynamics after my comments listed below have been addressed.

General comments:

1) The topic of this study is complicated and not easy to grasp for a reader without specific knowledge of the carbon-cycle feedback literature. I would therefore encourage the authors to critically review their introduction and provide more explanation of the basic concepts and how they are related to the main topic of the study, the differences between CO2 and non-CO2 GHG forcings. More specifically, I think a link between the non-linearity of carbon cycle feedbacks and the feedbacks due to non-CO2 GHG needs to be made, given that this topic is discussed quite extensively later in the manuscript. It would be a good idea to add a paragraph to the introduction that deals with the fact (and the causes for) that temperature mediated feedbacks can be different under rising or constant CO2, and that this is the main difference between CO2 and non-CO2 GHG mediated feedbacks. Here it would be also pertinent to cite the two (to my knowledge) studies that have investigated the topic of non-linearity previously (Zickfeld et al. 2011 and Schwinger et al. 2014, both studies did not deal with non-CO2 forcings). Also, in the Methods and Table 1, there are some sources of confusion, which should be addressed (see my specific comments below).

On a related note, why do the authors not go a step further and introduce a new symbol for the cross term? A clear definition of the “non-linear” or “cross-term” has been hampered by the fact that in the first studies using the beta/gamma framework (Friedlingstein et al. 2003, 2006), gamma was defined by [CO2]-[CO2bgc]. For this reason, also later studies that actually had a [CO2rad] simulation available continued using the term gamma for both climate carbon feedbacks [CO2rad] and [CO2]-[CO2bgc], as the authors mention themselves. This study might be a good opportunity to clean up with this “notational mess”?

2) In the section on the physical climate (section 3.1), the strongest warming is found in [CO2rad], but it is not explained why. [CO2rad] is warmer, particularly in the Arctic, than both [CO2] and [nonCO2], if I am not mistaken. Results show no very strong CO2 physiological warming in [CO2bgc], but nevertheless the CO2 physiological warming is used to explain the differences in simulations several times (e.g. lines 221-222), and it remains completely unclear to me why then [CO2rad] is the warmest simulation? In previous studies, the strongest CO2 physiological warming was found in the Arctic region for CMIP5 ESMs (Park et al. 2020), with significant regional SAT contributions. This study, which includes the predecessor ESM IPSL-CM5A-LR, could be mentioned in the context of the CO2 physiological warming. In the present study, the authors find the CO2 induced total warming smaller than the radiative warming alone in high latitudes (line 224, Fig. S5e), which is opposite from the results of the Park et al. study. This needs at least to be mentioned and if possible some explanation should be provided (the authors mention differences in snow albedo as an explanation, but this is rather a consequence than a cause of the different surface temperatures?).

3) Table 1 is somewhat confusing. Column 4 refers only to beta and gamma such that both experiments [CO2rad] and [CO2]-[CO2bgc] appear to be the same (they include the carbon cycle feedback “CO2 gamma”), but it is not mentioned that the cross-term is present in [CO2]-[CO2bgc]. The same is true for [nonCO2] and [CO2bgc+nonCO2]-[CO2bgc]. Also, in the 5^th column the only term listed for [CO2]-[CO2bgc] is the cross term, while the actual gamma-term is missing. Again, the same is true for [CO2bgc+nonCO2]-[CO2bgc]. In the footnotes, the terms ΔU_{γ,CO2physiological} and ΔU_{βγ,CO2physilogical} are not defined anywhere. I would suggest to just say that the warming from the physiological CO2 forcing is assumed to be negligible.

4) In the abstract (line 21-22), even if Arora et al 2020 and Schwinger et al. 2014, did not use the term “cross-term” but “non-linearity”, the results are consistent with these studies. So I would suggest adding “consistent with previous studies that considered CO2 forcing only”.

Specific comments:

Equation 2: It might be pertinent to cite Schwinger et al. 2014 here, who used the Taylor expansion to define “nonlinearity” of carbon cycle feedbacks. Please double check the factor 1/2 in the cross-term (also in Equation 5), which is wrong I believe (only the quadratic terms have the factor of 1/2).

Equations 3-4: Why are the quadratic terms included here? They cannot be quantified, so they belong to the residual term in the context of this study.

Line 19-20: Please double check the sentence: Shouldn’t this be the other way around – “Non-CO2 forcing primarily affects temperature driven feedbacks…” or did I misunderstand something here?

Line 22: It is a bit unclear what “both components” refers to. Also, non-CO2 forcing are usually considered in Earth system modelling, e.g., in SSP scenarios. Please reword this sentence to make the main conclusion of this paper clearer.

Line 75-81: “like many contemporary models” could be made more specific by saying “like all other ESMs participating in CMIP6” or similar. Generally, I think this paragraph is not necessary here. These are idealized concentration-driven experiments, so why discuss the lack of CH4 and N2O emission-driven capability in the Introduction? Particularly, since a section on “limitations” exists at the end of the manuscript. I would suggest deleting this paragraph and move parts of the text to Section 4.

Line 210-214: Although the physiological warming might be “significant” it is still quite small. Also, I would suggest being more careful here (and elsewhere in the manuscript), since the ensemble size is small and decadal scale variability can still be present in the ensemble mean. For example, the “significant CO2 physiological warming” in [CO2bgc] over “the high latitudes of land and ocean during stabilization period” could very well be an effect of AMOC, which happens to be significantly stronger over much of the stabilization period of [CO2bgc] compared to [piControl] in two of three ensemble members (Fig. S1a).

Line 219: “… the higher sensitivity to non-CO2 forcing compared to CO2 forcing”. This should be the other way round (SAT is higher under CO2 forcing)?

Line 220-221: “The combined effect of CO2 physiological and radiative forcing leads to more warming in the coupled [CO2] experiment compared to both the [CO2rad] experiment.” I guess the “both” should be deleted? Also, I cannot see this in Fig 2a, here [CO2rad] shows a stronger warming than [CO2].  This is consistent with the figures in the supplementary, which also show that [CO2rad] seems to be warmer than both [CO2] and [nonCO2], particularly in the high latitudes (Fig S3a). What is the reason for this? Also, as mentioned above, this is different from the CMIP5 study of Park et al. 2020.

Line 223-224: “…the CO2-induced total surface warming is larger than CO2-induced radiative warming almost everywhere, except for the high northern latitudes over the land and ocean (Fig. S3).” I can’t see this from Fig S3, because [CO2]-[CO2rad] is not shown there. Again, the most striking difference is that [CO2rad] is warmer than [nonCO2], particularly in high latitudes (and by comparison with the next column also warmer than [CO2] in the high latitudes. What is the reason for this difference?

Line 238-243: This paragraph is very confusing. It seems to repeat things that have been explained in the Methods section, but in a way that I doubt is helpful for the reader. I would suggest either rewording and expanding this paragraph or deleting it. Again, the terms ΔU_{γ,CO2physiological} and ΔU_{βγ,CO2physilogical} have never been defined in the manuscript.

Table 2: While CO2 (and non-CO2 GHG) concentrations are all the same in the different concentration driven experiments, this is not the case for the temperature increase. For example, SAT is 10-15% lower for [nonCO2] compared to [CO2] and [CO2rad] (estimated from Fig.2). Therefore, I am wondering if it would not make more sense to give values for gammas in this table? I would expect ΔU_γ,nonCO2 be somewhat lower than ΔU_γ,CO2rad just because of the lower temperature increase, while it is actually gamma which makes the most useful comparison between the simulations. More importantly, how are the cross-term carbon uptakes (first line in the lower part of the table) calculated? Shouldn’t this be the difference between the second and fourth line of the upper part of the table? I cannot see this is the case.

Line 282-284: As mentioned above, it is a choice to “attribute” the cross-term to the carbon-climate feedback, which makes sense in the context of previous studies. But I don’t see why this would be necessary, and I would encourage the authors to drop this attribution and just go ahead with beta, gamma, and the cross-term (as mentioned above, maybe introduce a new symbol for the cross term?).

Line 304: “larger climate change driven carbon source” is not precise. It is rather a larger climate change driven reduction of the ocean sink. The ocean remains a sink throughout. Same comment applies for line 312.

Line 324: Why would reducing non-CO2 GHG only change ΔU_γ? By changing temperature, the cross-term would be affected, too.

Line 369-370: Again, the highest GSAT is found in [CO2rad] which is inconsistent with this conclusion.

Technical:

Line 37: delete “over”

Line 69: consider changing to “to clarify whether the climate responses to declining CO2 and non-CO2 GHGs differ globally and regionally.”

Line 86: Place reference to Boucher et al. 2020 after the model name, not after CMIP. Replace CMIP by CMIP6

Line 96: Confusing sentence, please consider rewording. Maybe “… between a model experiment with perturbed GHG concentration but fixed sea surface and ice temperatures and a control simulation with pre-industrial GHG concentrations.” or similar.

Line 108: “referred to” could be understood as if the effective concentrations are used in the text and figures. I would suggest rewording this sentence.

Line 156: Delete “atmospheric CO2 induced”.

Line 199: thermostatic -> thermosteric

Line 201: Consider replacing “under considered timescale” by “within the time-horizon considered here” or similar.

Line 250: “… which induces carbon sink…” -> “which represents the CO2 induced carbon sink…”

Line 254: Complicated sentence. Why not say “Over the ocean beta is positive (carbon sink) in all regions …”

Line 278: What do you mean by “prolonged duration of beta”? Please clarify.

Line 286: Please spell out what “equivalent” means (within one standard deviation?).

Line 287: Remove subscript betas before “in Table 2”.

Line 295: the gamma -> gamma

References

Park, SW., Kim, JS. & Kug, JS. The intensification of Arctic warming as a result of CO₂ physiological forcing. Nat Commun 11, 2098 (2020). https://doi.org/10.1038/s41467-020-15924-3

Schwinger, J., and Coauthors, 2014: Nonlinearity of Ocean Carbon Cycle Feedbacks in CMIP5 Earth System Models. J. Climate, 27, 3869–3888, https://doi.org/10.1175/JCLI-D-13-00452.1.

Zickfeld, K., M. Eby, H. D. Matthews, A. Schmittner, and A. J. Weaver, 2011: Nonlinearity of Carbon Cycle Feedbacks. J. Climate, 24, 4255–4275, https://doi.org/10.1175/2011JCLI3898.1.