the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Large structures simulation for landscape evolution models
Abstract. Because of the chaotic behavior of the coupling between water flow and sediment erosion and transport, without any special treatment the practical results of landscape evolution models (LEM) are likely to be dominated by numerical errors. This paper describes two areas of improvement that we believe are necessary for the successful simulation of landscape evolution models. The first one concerns the expression of the water flux that was initially rebuilt in a recent paper in a mathematically consistent way for the celltocell multiple flow direction algorithms, thanks to a reinterpretation as a well chosen discretization of the GaucklerManningStrickler continuous equation. Building on those results, we introduce here a general framework allowing to derive consistent expressions of the water flux for the most commonly used multiple/single flow direction (MFD/SFD) water flow routines, including nodetonode versions. If having a consistent water flux is crucial to avoid any mesh size dependence in a LEM and controlling the consistency error, the expected nonlinear self amplification mechanisms of the water and sediment coupling can still lead to simulations blurred by numerical errors. Those numerical instabilities being highly reminiscent of turbulence induced instabilities in computational fluid dynamics (CFD), in the second part of our paper we present a ``large structure simulation'' (LSS) approach for LEM, mimicking the largeeddy simulations (LES) used for turbulent CFD. The LSS allows to control numerical errors while preserving the major physical based geomorphic patterns.
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RC1: 'Comment on egusphere2023687', Stefan Hergarten, 21 Jul 2023
This manuscript is about flow patterns on discrete grids and about their importance for modeling landform evolution at large scales. Focus is on an improved multipleflowdirection scheme, which is more based on physical/empirical relations(turbulent flow) than existing schemes.
The paper is quite mathematical at least in parts. Personally, I think that there should be more work on mathematical descriptions in geomorphology than there was in the past. On the other hand, I am not convinced that the recent manuscript would fit well into Earth Surface Dynamics for the following reasons.
(1) Style of presentation
I think that large parts of the theory  in particular Section 3  are presumably too heavy for the landform evolution modeling community concerning the mathematical jargon. I know that a certain level of accuracy is necessary in mathematics, but it took me much time to find the main ideas behind the formalism. My own papers are not cited very often, and I think that the majority of the community considers them them too mathematical. If I extrapolate to this manuscript, I am quite sure that it will be lost completely, regardless of its quality. Bringing mathematical concepts into geomorphology requires much more explanation of the ideas and less formalism. My low rating of the presentation quality on the score sheet is not absolute, but refers to the background of the potential readership.(2) Premises
The "chaotic" nature of drainage network formation seems to be the starting point of this study. If I got it correctly, small errors in distributing the flux of water to the lower neighbors on the grid grow through time by erosion and thus finally dominate the topography. This is in principle true, but not the problem as far as I know. At least in those papers cited in the introduction that I know, the problem is the coupling of hillslope processes with fluvial processes, which tend to concentrate the flow pattern. There the results become even dependent on the spatial resolution, which is much worse than the flow pattern finally being dominated by small disturbances. So I feel that the introduction does not address the problem thoroughly and even starts from wrong premises.(3) The assumption about the topography
In Eq. 5, it is stated that the differential equation for the water flux including its boundary conditions is only wellposed if the Laplacian of the surface (bed + sediment) is negative, although I have no idea what the additional "or > 0" means. If I am not completely wrong, almost each topography with valleys does not meet this condition. Why should this mean that the flow model is well justified for drainage basins as stated in line 197? Or why should the problem be limited to flat areas as suggested by the heading of Sect. 5.2? This does not make sense to me. For the approach used by Bonetti et al. (2018), (Eq. 3), valleys indeed result in irregular solutions. If there are tributaries of multiple orders, these will occur in the form of fibers with different values of the water flux density in the trunk stream. If we proceed downstream, these fibers become closer and closer to each other, which makes an irregular solution. So is the constraint in Eq. 5 what we need for avoiding this problem? If indeed only topographies with negative Laplacian can be considered, the approach is very limited. Then it could perhaps be shown that the flow pattern on a circular volcano is more regular than with one of the conventional multiflowdirection schemes, but not much more.
In Sect. 5.2, an extension that overcomes the limitation is proposed. This extension uses the slope of the water surface instead of the slope of the topography. In terms of hydrology, this is the transition from the kinematic wave equation to the diffusive wave equation. This makes sense, but as already stated in Sect. 3 (after Eq. 5), it costs much of the simplicity and computational efficiency.Overall, I am afraid that this study still needs more work to be of added value in landform evolution modeling. I think this is not impossible, but it is not just revising the manuscript.
Anyway, I may be wrong. In case I misunderstood any of the points fundamentally, I would ask the authors to let me know and to give me the chance to revise my opinion.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023687RC1 
AC1: 'Reply on RC1', Julien Coatléven, 24 Jul 2023
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 nonlinear 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 GaucklerManningStrickler 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 nonconsistency introduces a mesh grid dependency. In this paper we have developed in the first part (section 3) a solution to fix the nonconsistency of the MFD approaches, and this has given us a welldesigned 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 selfamplification mechanisms.We thus discuss two problems, first the nonconsistency, then how to manage the selfamplification 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 GaucklerManning 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), FernandezCara et al. (2002), Girault and Tartar (2010)). Otherwise the problem can either be multivaluated, 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 wellposedness 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 selfamplification 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 nonphysical 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 wellposedness 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 wellposedness 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
Citation: https://doi.org/10.5194/egusphere2023687AC1

AC1: 'Reply on RC1', Julien Coatléven, 24 Jul 2023

RC2: 'Comment on egusphere2023687', Anonymous Referee #2, 07 Aug 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere2023687/egusphere2023687RC2supplement.pdf
Status: closed

RC1: 'Comment on egusphere2023687', Stefan Hergarten, 21 Jul 2023
This manuscript is about flow patterns on discrete grids and about their importance for modeling landform evolution at large scales. Focus is on an improved multipleflowdirection scheme, which is more based on physical/empirical relations(turbulent flow) than existing schemes.
The paper is quite mathematical at least in parts. Personally, I think that there should be more work on mathematical descriptions in geomorphology than there was in the past. On the other hand, I am not convinced that the recent manuscript would fit well into Earth Surface Dynamics for the following reasons.
(1) Style of presentation
I think that large parts of the theory  in particular Section 3  are presumably too heavy for the landform evolution modeling community concerning the mathematical jargon. I know that a certain level of accuracy is necessary in mathematics, but it took me much time to find the main ideas behind the formalism. My own papers are not cited very often, and I think that the majority of the community considers them them too mathematical. If I extrapolate to this manuscript, I am quite sure that it will be lost completely, regardless of its quality. Bringing mathematical concepts into geomorphology requires much more explanation of the ideas and less formalism. My low rating of the presentation quality on the score sheet is not absolute, but refers to the background of the potential readership.(2) Premises
The "chaotic" nature of drainage network formation seems to be the starting point of this study. If I got it correctly, small errors in distributing the flux of water to the lower neighbors on the grid grow through time by erosion and thus finally dominate the topography. This is in principle true, but not the problem as far as I know. At least in those papers cited in the introduction that I know, the problem is the coupling of hillslope processes with fluvial processes, which tend to concentrate the flow pattern. There the results become even dependent on the spatial resolution, which is much worse than the flow pattern finally being dominated by small disturbances. So I feel that the introduction does not address the problem thoroughly and even starts from wrong premises.(3) The assumption about the topography
In Eq. 5, it is stated that the differential equation for the water flux including its boundary conditions is only wellposed if the Laplacian of the surface (bed + sediment) is negative, although I have no idea what the additional "or > 0" means. If I am not completely wrong, almost each topography with valleys does not meet this condition. Why should this mean that the flow model is well justified for drainage basins as stated in line 197? Or why should the problem be limited to flat areas as suggested by the heading of Sect. 5.2? This does not make sense to me. For the approach used by Bonetti et al. (2018), (Eq. 3), valleys indeed result in irregular solutions. If there are tributaries of multiple orders, these will occur in the form of fibers with different values of the water flux density in the trunk stream. If we proceed downstream, these fibers become closer and closer to each other, which makes an irregular solution. So is the constraint in Eq. 5 what we need for avoiding this problem? If indeed only topographies with negative Laplacian can be considered, the approach is very limited. Then it could perhaps be shown that the flow pattern on a circular volcano is more regular than with one of the conventional multiflowdirection schemes, but not much more.
In Sect. 5.2, an extension that overcomes the limitation is proposed. This extension uses the slope of the water surface instead of the slope of the topography. In terms of hydrology, this is the transition from the kinematic wave equation to the diffusive wave equation. This makes sense, but as already stated in Sect. 3 (after Eq. 5), it costs much of the simplicity and computational efficiency.Overall, I am afraid that this study still needs more work to be of added value in landform evolution modeling. I think this is not impossible, but it is not just revising the manuscript.
Anyway, I may be wrong. In case I misunderstood any of the points fundamentally, I would ask the authors to let me know and to give me the chance to revise my opinion.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023687RC1 
AC1: 'Reply on RC1', Julien Coatléven, 24 Jul 2023
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 nonlinear 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 GaucklerManningStrickler 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 nonconsistency introduces a mesh grid dependency. In this paper we have developed in the first part (section 3) a solution to fix the nonconsistency of the MFD approaches, and this has given us a welldesigned 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 selfamplification mechanisms.We thus discuss two problems, first the nonconsistency, then how to manage the selfamplification 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 GaucklerManning 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), FernandezCara et al. (2002), Girault and Tartar (2010)). Otherwise the problem can either be multivaluated, 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 wellposedness 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 selfamplification 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 nonphysical 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 wellposedness 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 wellposedness 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
Citation: https://doi.org/10.5194/egusphere2023687AC1

AC1: 'Reply on RC1', Julien Coatléven, 24 Jul 2023

RC2: 'Comment on egusphere2023687', Anonymous Referee #2, 07 Aug 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere2023687/egusphere2023687RC2supplement.pdf
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