Dear Shaun,

I struggled to get two reviews of your revised paper and I am very sorry it took so long time. As you can see, both reviews have constructive comments. The review #2 is quite detailed and highlights not only mathematical problems, but also important methodological problems. Both require substantial clarifications, especially on what is actually being achieved. This requires an in-depth revision.

Best regards, Daniel

Dear Shaun,

First of all, I would like to clarify that there were only 4 referees and that the first 3 were not "uniformly positive". For example, referee 3 was very critical and referee 2 actually suggested a major revision. Referees 3 and 4 both complained bitterly about the lack of key references to the literature and actually helped to improve the state of the art considerably, although reviewer 4 is still not fully satisfied and believes that this remains a source of problems for your paper. I can also mention that I have received abrupt refusals to review.

The referee 4 had strongly suggested to use the Fourier transform to derive the statistical characteristics of the fractional Langevin equation. The new Appendix A tries to respond to this and I am happy to acknowledge that it is the core of the present revision. However, the referee 4 raises several issues around the main question: how to do it in a very simple way? It has taken me some time to assess the extent to which this calls into question current algebraic developments. I am afraid this is the case since the beginning of the new Appendix A with its unfortunate rush to mathematical details of a given approach instead of trying to optimise the approach and to avoid cumbersome developments. Indeed, it should rely more on physics than a series of “convenient” equations.

This may partly explain why Appendix B relies on a different technique than Appendix A. But the main problem with Appendix B is that it actually addresses a distinct problem. This is an example of the inconsistencies briefly mentioned in the report, some of which I fear are generated by repeated use of incomplete notations, which, for example, do not make explicit the fact that only part of the solution is addressed.

The referee’s report raises other important questions, such as the relationship between this fractional Langevin equation and the actual Budyko-Sellers model, in fact, the actual predictive capabilities of this equation.

Due to the above-mentioned inconsistencies, especially in the new materials, the paper still needs thorough revisions. The safest, simplest and most satisfactory way would be to modify the approach followed in Appendix A. But this would require a major rethink and an alternative might be to eliminate the inconsistencies in the current approach and make it more accessible despite its inherent complexity.

To speed up the process, I choose the option that the next review will be done only by the editor despite the expected thorough revisions.

Best regards,
Daniel

Dear Shaun,

To speed up the review process, I had chosen the option that “the next review will be done only by the editor despite the expected thorough revisions”. Unfortunately, I gradually became aware that there was no response to the referee and the track change copy is unusable, providing only a grey rectangular on the right side of the text. I initially thought that this could be due to bugs of the review system, but after some investigation it does not seem so. Therefore, please provide these documents, slightly updated if necessary.

They may help to provide more complete answers to my own report that was primarily trying to highlight the main points of the referee’s report, which is more detailed. For instance:
The reference made to physics for this Appendix A (instead of a rush to mathematical details) was misunderstood as a justification of the fractional Langevin equations, while the referee specifically challenged the Fourier technique chosen. However, I have accepted that the present paper could remain in the same approach despite the resulting complexity.
There is still a loss of symmetry along the early developments of Appendix A that should at least noted.
The strong reference to the Budiko-Sellers model is not justified, as its main physical process is absent.
The referee also strongly questioned several aspects of the Sect.4, including they considers it as limited to comparisons of prediction skill indicators, themselves discussed.

I apologise for miscounting the referees: they were 5, not 4 and my numbering must be increased by one and you can claim a record. Referee 5 can be thanked for his constructive comments, especially on the role of Fourier techniques, but more generally contributed to this paper being modified.

Best regards,
Daniel

Dear Shaun,

Thank you for uploading a new, effective tracked change copy following requests from me and Copernicus. This document was needed more than ever because of the length and complexity of the manuscript, as well as the incomplete responses to the referees' comments and suggestions, although I summarised them twice. This document shows that many changes in detail have been made in various parts (e.g., changes in notation), but not so much in response to the comments and suggestions. The author's response confirms this, although they could have improved the rigour and readability of the manuscript making it more straightforward and increasing its potential impact.

In spite of this dissatisfaction that I cannot hide, I am pleased to accept your manuscript, up to the "technical corrections", as a given step in the elaboration of a particular fractional Langevin equation, which might give rise to new developments

Some remarks
- Acceptance up to “technical corrections” allows the author to proceed to minor corrections, including typos such as I(z) instead of I(t) in Eq.A1 (I_C(t) might be more appropriate).
- Your claim that you thought to Fourier techniques before Referee 5 could have easily taken hold with the help of an author comment on this important improvement. Indeed, this can be done at any time, and was used in a paper you co-authored.
- I could easily complain about various problems that slowed down the review process, the last one being the faulty tracked change copy that was resolved only a few weeks ago while the Word document crash was known in early August. However, I would certainly not forget the pandemic as a multidimensional factor. I also - remember the author admitted that he “appreciate[s] that it is not easy to find reviewers!”, when I mentioned that I have received abrupt refusals to review.
- I have looked for solutions to speed up the process rather than to complain, such as choosing a minor revision, while still key-questions remained open.
- I find no evidence that the referee 5 has any “ [concealed] desire to kill the paper”, and their reports do not contain specifics. Indeed, both reports include general comments outlining specific key questions as well as “a sample of detailed comments”. It is a pity they were overlooked by the author.
- I still do not understand why Referee 5’s emphasis on Fourier transforms and not on Laplace-Fourier transform was not understood, as well as the rather immediate loss of the symmetry (t->-t, C -> -C) of Eq. A.1 in the equations that follow.
- I fear that the number of papers having FEBE in their title will not solve the problem that FEBE, which depends only on time, does not include the fundamental radiative feedbacks of the Budyko-Sellers type models that require space and time. Therefore, the link between both seems much weaker than claimed.

Best regards,

Daniel