We thank the rewiever for his time and valuable comments. We feel that we might have been misunderstood on several points, and we will try to better explain what our original intention was.

As a first general comment, we believe it is important to point out that this article explains how to modify and implement the current MFD algorithms to put an end to mesh dependency. This has a cost: (1) the development of a mathematical framework has been required, (2) even after applying the correction on a MFD algorithm (the consistency), numerical errors coming from non-linear couplings must be handled under penalty of reinjecting an effective mesh dependency. This justifies the second part of this work in which we introduce a filtering strategy derived from the CFD turbulent community.

(1) The grid size dependency of the MFD water flow models has been explained and corrected in detail in a previous paper (Coatléven, 2020). We have completed this study to now be able to include almost all of them. We recognize it implies to introduce several notations and that understanding deeply this part could necessitate to also read (Coatléven, 2020) where more details are sometimes given. This part is however crucial for the geomorphic community: forewarned is forearmed. In our opinion this is the most objective way to explain where was hidden the weakness of the MFD that has produced so much debate around the grid dependency in MFD.  If we understand that the mathematical framework may appear too heavy compared to what is currently published inside this community, we also believe that it will deeply serve the community in their modeling approaches.

We agree however that the link between the MFD and the Gauckler-Manning-Strickler model should be better emphasized in the section on mathematical formalism. We can try to better emphasize the key ideas, for instance by giving the right formula from the very start, explain its interest and then detail how to obtain it. The details are nevertheless important in our opinion because they allow to adapt the results to any of the classical MFD algorithms of table 1 and 2.

 

(2) The "chaotic" nature of drainage network formation is indeed the starting point of this study, more precisely the second part of it  (section 4). But it can be correctly solved only once we have a consistent MFD algorithm, as non-consistency introduces a mesh grid  dependency. In this paper we have developed in the first part (section 3) a solution to fix the non-consistency of the MFD approaches, and this has given us a well-designed mathematical framework on which it has been possible to introduce the filtering strategy of section 4. This filtering strategy is intended to handle numerical errors that will appear because of the self-amplification mechanisms.

We thus discuss two problems, first the non-consistency, then how to manage the self-amplification mechanisms. Each of them is equally important. Moreover, they are highly intricated form the perspective of results quality: model quality will be good enough and mesh dependency will be solved only after having correctly handled these two strategic axes. We do agree that the consistency/grid dependency issue is much more well documented in the literature, and we have originally tried to emphasize those two problems separately in the introduction (paragraphs 3 and 4 respectively) which we can rework to further clarify the objectives of the paper.

 

(3) It is well documented in the mathematical literature that the stationary transport equations like the continuous Gauckler-Manning- Strickler model considered here are well posed if one of the sufficient conditions on the topography we introduced is satisfied (or similar equivalent variants) (for reference: Bardos (1970), Veiga (1987), DiPerna and Lions (1989), Fernandez-Cara et al. (2002), Girault and Tartar (2010)). Otherwise the problem can either be multi-valuated, have infinite solutions, or no solution at all. As the MFD algorithms are discretization of this equation they suffer from the same deficiencies.

The additional "and >0" is a typo, our apologizes for that (originally, we had written the alternative more complex condition on the same line, and the remaining " and >0" is a copy/paste error that has escaped our proof reading).

By "drainage basin" we wanted to designate basins with no accumulation or flat areas which would then satisfy one of the sufficient conditions (the conditions on the Laplacian being the simplest one), and we are probably wrong on the use of the terminology "drainage". 

We did not suggest that the well-posedness problem is limited to flat areas.  It also exists  for instance for accumulation areas the water height associated with MFD algorithms is infinite which is in general circumvented by using "lake" algorithms using a local water balance to correctly set the water height and thus explicitly turn off the MFD algorithms in those accumulation areas, as we briefly mention in section 5.2. Valleys are not necessarily a problem provided they possess a downstream spill point. In our paper we explain it on the example of flat areas, hence the apparent importance given to them. This is a very different problem from the grid size dependency of MFD algorithms, it is more a modeling problem implying that the MFD approach is an oversimplification for too complex topographies. To our knowledge there is no rigorous workaround except using more complex flow models such as the one we present in section 5.2. This is the reason why we claim that MFD algorithms are not well suited for "non drainage basins", which again might be an erroneous use of the term "drainage basins".

Because of the self-amplification mechanisms and the fact that from a practical point of view on cartesian grid the MFD algorithm nevertheless compute a value for the water flow, this model error if overlooked can also lead to non-physical flow patterns. Satisfying one of the sufficient conditions is by no means a solution to numerical instabilities. It is only a requirement for the mathematical well-posedness of the Gauckler Manning Strickler equation. In the numerical experiments of the paper we have been careful to always use topographies that satisfy the required condition. In particular, this is the reason why we have avoided using random noise in our initial conditions . 

 

We can get rid of the term "drainage" if it is problematic, or we could consider to add more details on this well-posedness issue for clarity.

We are aware that this paper might in some respects be better suited to a community of pure modelers, but we also believe that it must be read by the geomorphic community that regularly published on MFD numerical issues. We hope that the cost of the investment required to benefit from the mathematical aspects of the paper will be largely offset by the solutions that the reader can then deploy on these models. We hope that the above clarifications might lead you to reconsider your analysis of the paper.

We renew our thanks for your time and consideration,

Best regards

The authors