27 Aug 2021
27 Aug 2021
Hydrology without dimensions
 Princeton University, Princeton, USA
 Princeton University, Princeton, USA
Abstract. By rigorously accounting for dimensional homogeneity in physical laws, the Pi theorem and the related selfsimilarity hypotheses allow us to achieve a dimensionless reformulation of scientific hypotheses in a lower dimensional context. This paper presents applications of these concepts to the partitioning of water and soil on terrestrial landscapes, for which the process complexity and lack of first principle formulation make dimensional analysis an excellent tool to formulate theories that are amenable to empirical testing and analytical developments. The resulting scaling laws help reveal the dominant environmental controls for these partitionings. In particular, we discuss how the dryness index and the storage index affect the long term rainfall partitioning, the key nonlinear control of the dryness index in global datasets of weathering rates, and the existence of new macroscopic relations among average variables in landscape evolution statistics. The scaling laws for the partitioning of sediments, the elevation profile, and the spectral scaling of selfsimilar topographies also unveil tantalizing analogies with turbulent fluctuations.
Amilcare Porporato
Status: closed

RC1: 'Comment on hess2021442', Anonymous Referee #1, 14 Sep 2021
I am probably very biased given that my PhD was in fluid mechanics and I then transitioned to working on hydrologic systems during my postdoc, but from the very beginning of my research experience it was hammered into me that dimensional analysis and the Pi theorem are two of the most valuable tools available when trying to study anything, particularly when there is any degree of complexity involved. Unfortunately few geoscientists get a truly rigorous exposure to such approaches and while there are several studies that use these principles it is not as widspread as it should be in my opinion. Dimensional analysis allows us to do all of the things that Porporato highlights here and more. It is the reason why we can run a benchtop laboratory experiment and infer things that are happening at global, planetary and astrophysical scales. It should be something that is part of the core curriculum for all earth scientists in my view, whether they are pursing experimental, modeling or purely theoretical work. While I, for personal reasons may have picked a few different examples, I think the ones chosen here are excellent and have indeed taught me something new. I found the paper to be sufficiently comprehensive for someone with a rudimentary analysis of dimensional analysis to get a lot from it as well as deep enough in examples for someone who knows a good bit more to still enjoy it. If I had one minor suggestion it might be that this paper espouses all the benefits of dimensional analysis, but only a very limited discussion of possible shortcomings  this is a very minor point and perhaps does not really apply in the context of this paper, but I have seen very intersting discussions in the context of for example sociohydrology, where in some instances there are elements of interest where it is not even clear what variables and what dimensions are involved (i.e. outside of standard M,L,T,theta approaches). This is a very minor point though and I'm not sure that it really requires anything additional in this paper as it has been touched upon briefly in other review like papers cited in this work.
For all of the reasons above I think this is an excellent contribution and I for one will most definitely use it when educating graduate students in our program. Thank you very much for taking the time to write this paper!

AC1: 'Reply on RC1', Amilcare Porporato, 15 Nov 2021
I would like to thank the anonymous reviewer for his/her encouraging report and thoughtful comments. I am glad the paper was found of interest, also for providing a different angle to this timehonored topic.
The point raised by the reviewer, regarding the application of dimensional analysis to system that do not have a clear and typical physical definition, is very interesting and deserves attention. Unfortunately, however, it is also a difficult problem, one which is not typically considered in the traditional dimensional analysis literature. I plan to revise the paper paying particular attention to this comment emphasizing it, when suitable. There are some references in the economics and social science context, which are already cited in the introduction, where this point is relevant. I plan to expand around this issue, when commenting on these references. Also, the augmented dimensional analysis, and the idea of adding new variables, such as temperature or heat flux, as in the Rayleigh problem, can be seen as a precursor of this interesting problem. Again, I plan to add some comments in the paper to this regard. Thank you.

AC1: 'Reply on RC1', Amilcare Porporato, 15 Nov 2021

RC2: 'Comment on hess2021442', Anonymous Referee #2, 27 Sep 2021
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess2021442/hess2021442RC2supplement.pdf

AC2: 'Reply on RC2', Amilcare Porporato, 16 Nov 2021
Thank for the friendly comments and useful suggestions. I am glad the reviewer appreciated the attempt to provide a different approach to dimensional analysis, in particular regarding how the Buckingham Pi theorem be used in an era where data generation is overwhelming the ability of data interpretation.
With regard to the suggestions and comments, I agree with the reviewer that there are too many instances where geophysical problems, when tackled in their most general form, lead to a large number of Pi groups. In these cases, the Pi theorem becomes useless when nk>>1. This is why it is extremely important to emphasize the crucial step of building suitable physical laws. We have further emphasized this point – also in relation to the fermi reasoning, starting from earlier on in the manuscript. Thank you.
I also proceeded to rearrange in part Sec 3.3, as recommended by the reviewer. In particular, I improved and extended the derivation of the the erosion term and the specific contributing area, ‘a’. This is also inline with the recommendations of Ref. 4. Indeed the erosion term is too complicated to derive from first principle, but in the revised version we have better linked it to the water balance equation, which also leads to the equations for ‘a’. We have also provided in an appendix a more direct explanation of the mathematical a analogy between the turbulent velocity profile and the landscape elevation profile.
Unfortunately, while a very interesting suggestion, we are not aware of a derivation of these equations using scaling and dimensional analysis concepts. This valuable suggestion will be kept in mind for future contributions and is actually mentioned in the revised paper as an open direction for research.

AC2: 'Reply on RC2', Amilcare Porporato, 16 Nov 2021

RC3: 'Review on “Hydrology without dimensions” by Demetris Koutsoyiannis', Demetris Koutsoyiannis, 28 Sep 2021
The review report is provided as a PDF attachment.

AC3: 'Reply on RC3', Amilcare Porporato, 17 Nov 2021
Thank you, Demetris, for the very nice words and for the insightful review!
Indeed, the paper tries to be didactic and in the revised version I will pay particular attention to make it more accessible and readable since it examines cases from diverse fields.
Regarding the section “2.3 Augmented and directional dimensional analysis” and its Appendix, I have tried to make this part more precise (besides correcting a few errors). I’ve thus discussed in more detail why, in normal conditions, the last Pi group can be neglected in (A3). This is related to a selfsimilar behavior of the first type, which is supported by experience. As correctly stated in the review, this is not a mathematical necessity, but merely a fact of experience. I've improved the presentation of these concepts in the revised version.
I also added the example of the wave and diffusion, as suggested. This is a simple, but illustrative example, which I also used in the medal lecture, which is worth reproducing in the revised version of this paper. Thank you for the suggestion.
I also agree with the need to improve the introduction of Eq. (7), where the power law (selfsimilarity) may seem to be introduced ‘out of the blue’. It is also important to better explain that, in agreement with the review here and the references by the reviewer (Koutsoyiannis, 2014; Koutsoyiannis et al., 2018), expressions like equation (7) only hold asymptotically. This is at the heart of the very concept of (asymptotic) selfsimilarity.
Regarding the comment that ‘scaling, selfsimilarity and fractal behaviour look to be overemphasized or overpraised in the paper’, as in Figure 4, I should mention that this is not necessarily related to fractal behavior. Moreover fractal behavior can be, but does not need to be linked to stochastic behavior. Like in the Moody diagram in the fully rough regime, the scaling is simply a power law relation among variables (of course it results from some asymptotic behavior of turbulence which in this regime has specific statistical properties).
Of course, stochastic behavior is to be expected along with measurements and model errors. This is emphasized in the revised version of the manuscript. Indeed, the rainfall partitioning model is a stochastic model with random rainfall inputs. This is now emphasized and better explained.
Finally, regarding citations: Kolmogorov is now cited; Theodoratos et al. (2018) is also interesting and now cited – note however that here the authors use the wrong variable, the contributing area, A, instead of specific area, and this is explained in the revised text; and finally, the quote by Strahler is made more precise.
Thank you again for the insightful and useful recommendations.

AC3: 'Reply on RC3', Amilcare Porporato, 17 Nov 2021

RC4: 'Comment on hess2021442', Stefan Hergarten, 29 Sep 2021
The article "Hydrology without Dimensions" by Amilcare Porporato addresses scaling laws and nondimensional properties in hydrology with emphasis on applying the Pitheorem. As stated in the acknowledgments, the article is related to the Dalton medal lecture by the author. So the paper is somewhere between an original research paper and a review paper. Although somewhat unusual, such a focus makes sense, and I am sure that the community (including advanced students) will appreciate this paper.
Similar to the other reviewers (who were faster than me), my assessment of the manuscript is overall positive. I would suggest a publication with very few moderate adjustments, including one more check for typos.
(1) Right in the beginning of the theory, at Eqs. (1) and (2), I stumbled over the definition and the meaning of the scaling factor lambda. Formally, the argument of the function f should always have the same physical dimension, so that lambda should be nondimensional. And I would typically assume that lambda must be the same for all values of x, which makes lambda = 1/x problematic. Your point is clear, and I think it will not be a serious problem for the readers. But maybe you find another elegant formulation of the equations that avoids the issue.
(2) From my own background, Section 3.3 about landform evolution modeling is particularly interesting, and it nicely shows some recent work of your group. However, I am a bit wary about the concept of the specific drainage area (Eq. 28) and its application in the landform evolution model (Eq. 27). Let us assume a smooth (so with continuous derivatives) topography with a dendritic network of valleys. Then the flow pattern in a large valley consists of many fibers with different upstream lengths and thus with different specific discharges. Due to the dendritic structure, the acrossvalley pattern of the specific discharges is quite irregular, and the fibers even come closer to each other downstream. For me, this concept differs from the "classical" idea of a river with a given width, so that the model with the specific discharge differs in its spirit from what was previously assumed in this context.
A fully agree that the widely used version with the drainage area instead of the specific drainage area is inconsistent when proceeding from a discrete network to a continuous topography and leads to results depending on the spatial resolution of the grid. Unfortunately, it even seems that scaling relations were recently developed without taking care of this problem (Theodoratos et al. 2018). The alternative approach of considering a river as a line with a finite width (Howard 1994, Perron 2008) or rescaling the diffusion term (Pelletier 2010) are also not free of problem, which also applies to my own approach (Hergarten 2020). However, if you apply your model to a discrete grid, you practically integrate a^m (where a = specific drainage area). The result will differ from A^m (where A = drainage area). which means that the river consisting of fibers erode at a different rate than a river with a given discharge. So I am not completely convinced that your approach avoids the problem of the gridspacing dependence unless the grid is fine enough to resolve all the small fibers (which would practically not be feasible).
Please do not get me wrong  I do not want to criticize published work of your group too much. I just think that your reasoning about replacing A by a might be somewhat oversimplified and not free of caveats. I would be happy if you could add some discussion about these aspects. And in case I am wrong, please accept my apologies.
Best regards,
Stefan HergartenReferences
Pelletier, J. D.: Minimizing the gridresolution dependence of flowrouting algorithms for geomorphic applications, Geomorphology, 122, 9198, https://doi.org/10.1016/j.geomorph.2010.06.001, 2010
Perron, J. T., Dietrich, W. E., and Kirchner, J. W.: Controls on the spacing of firstorder valleys, J. Geophys. Res. Earth Surf., 113, F04 016, https://doi.org/10.1029/2007JF000977, 2008
Hergarten, S.: Rivers as linear elements in landform evolution models, Earth Surf. Dynam., 8, 367377, https://doi.org/10.5194/esurf83672020, 2020
Theodoratos, N., Seybold, H., and Kirchner, J. W.: Scaling and similarity of a streampower incision and linear diffusion landscape evolution model, Earth Surf. Dynam., 6, 779–808, https://doi.org/10.5194/esurf67792018, 2018

AC4: 'Reply on RC4', Amilcare Porporato, 18 Nov 2021
Dear Professor Hergarten,
Thank you for your kind review and useful suggestions. I’ve appreciated the positive criticism and made sure that the revised version clarifies some of these points (reported here in italic when necessary) as much as possible.
Regarding Eqs. (1) and (2) and the definition of the scaling factor lambda, the reviewer raises an interesting point that is worth better explaining in the revised version of the manuscript. Formally, the argument of the function f, especially if the function is transcendental, should be dimensionless (see e.g. Barenblatt 1996). On the other hand, lambda as a scale factor, should have the dimension of the inverse of x. The fact is that, as a matter of common use, we often omit writing explicitly unit factors which fix the dimensions in this regard. Thus, one either considers the scaling in Eq 1 with regard to dimensionless quantities (then lambda, x, and f are all dimensionless and things work well – I believe this what mathematicians have in mind when they speak of homogeneous functions like here), or uses dimensions and then either explicitly (but this becomes very cumbersome) or implicitly assumes the presence of these unit factors that convert the units. E.g.
f(lambda*x)=(1*lambda)^n f(1*x).
where the different ‘1’ have different dimensions…
I guess, this is somewhat similar to the fact that log(x) only makes sense if x is dimensionless, so in practice it is log(x/1), but in common practice this is not done, also because if one then writes this as log(x)log(1) then we start over again… or it is enough to think of power series expansions, where obviously there are omitted factors that make the dimensions consistent …
The revised version of the paper clarifies these points. Thank you.
(2) From my own background, Section 3.3 about landform evolution modeling is particularly interesting, and it nicely shows some recent work of your group. However, I am a bit wary about the concept of the specific drainage area (Eq. 28) and its application in the landform evolution model (Eq. 27). Let us assume a smooth (so with continuous derivatives) topography with a dendritic network of valleys. Then the flow pattern in a large valley consists of many fibers with different upstream lengths and thus with different specific discharges. Due to the dendritic structure, the acrossvalley pattern of the specific discharges is quite irregular, and the fibers even come closer to each other downstream.
I fully agree with you here. This is a perfect description of the differential geometry of a (smooth) landscape and is exactly what happens in terms of streamlines and specific contributing area (see Bonetti et al. PRSA 2018).
For me, this concept differs from the "classical" idea of a river with a given width, so that the model with the specific discharge differs in its spirit from what was previously assumed in this context.
Defining a river is in my view a different story from what we do here and in general should not be confused with numerical (or theoretical) issues related to the solution of a given equation, which needs to be well posed. One should keep in mind that these minimalist models of landscape evolution have a very limited and rudimentary representation of the physical processes. For example, if one also considers the surface water flowfield, then one sees that there is water everywhere all the time, which obviously makes no physical sense. In our case, defining rivers is not the goal, but rather the understanding of the mechanisms of hierarchical branching and formation of valleys and ridges (of course one could define thresholds of certain water height to define rivers, but this would be quite unsatisfactory, I think).
I fully agree that the widely used version with the drainage area instead of the specific drainage area is inconsistent when proceeding from a discrete network to a continuous topography and leads to results depending on the spatial resolution of the grid. Unfortunately, it even seems that scaling relations were recently developed without taking care of this problem (Theodoratos et al. 2018). The alternative approach of considering a river as a line with a finite width (Howard 1994, Perron 2008) or rescaling the diffusion term (Pelletier 2010) are also not free of problem, which also applies to my own approach (Hergarten 2020).
I fully agree.
However, if you apply your model to a discrete grid, you practically integrate a^m (where a = specific drainage area). The result will differ from A^m (where A = drainage area). which means that the river consisting of fibers erode at a different rate than a river with a given discharge. So I am not completely convinced that your approach avoids the problem of the gridspacing dependence unless the grid is fine enough to resolve all the small fibers (which would practically not be feasible).
The fact is that ‘a’, the specific contributing area is a variable defined in the continuum (i.e. pde) representation. Once the domain is discretized, then suitable operators should be found and for each grid point one has to decide what the contributing area is. This is an important numerical problem, but not a theoretical one.
Please do not get me wrong  I do not want to criticize published work of your group too much. I just think that your reasoning about replacing A by a might be somewhat oversimplified and not free of caveats. I would be happy if you could add some discussion about these aspects. And in case I am wrong, please accept my apologies.
Thank you again for the interesting points of discussion and for spurring me to clarify and better explain these issues. In the revised version I’ve tried to clarify better these points and the derivation of the landscape evolution equation to show that ‘a’ is the right variables. No PDE in continuum mechanics has directly an extensive variable like A in their terms (it would be like having volume V of diameter D of the pipe directly in the Navier Stokes equations rather than using them as boundary conditions.
Barenblatt, G. I.: Scaling, selfsimilarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, 14, Cambridge University Press, 1996.
Bonetti, S., Bragg, A., and Porporato, A.: On the theory of drainage area for regular and nonregular points, Proceedings of the Royal Society 670 A: Mathematical, Physical and Engineering Sciences, 474, 20170 693, 2018.

AC4: 'Reply on RC4', Amilcare Porporato, 18 Nov 2021

EC1: 'Editor's comment', Nunzio Romano, 05 Oct 2021
Dear Amilcare,
To feed the open discussion and start possible interactions with the reviewers, I invite you to start uploading some preliminary responses or comments from your side (apart from your final reply as the author at the closure of this first step).
 AC5: 'Reply on EC1', Amilcare Porporato, 18 Nov 2021
Status: closed

RC1: 'Comment on hess2021442', Anonymous Referee #1, 14 Sep 2021
I am probably very biased given that my PhD was in fluid mechanics and I then transitioned to working on hydrologic systems during my postdoc, but from the very beginning of my research experience it was hammered into me that dimensional analysis and the Pi theorem are two of the most valuable tools available when trying to study anything, particularly when there is any degree of complexity involved. Unfortunately few geoscientists get a truly rigorous exposure to such approaches and while there are several studies that use these principles it is not as widspread as it should be in my opinion. Dimensional analysis allows us to do all of the things that Porporato highlights here and more. It is the reason why we can run a benchtop laboratory experiment and infer things that are happening at global, planetary and astrophysical scales. It should be something that is part of the core curriculum for all earth scientists in my view, whether they are pursing experimental, modeling or purely theoretical work. While I, for personal reasons may have picked a few different examples, I think the ones chosen here are excellent and have indeed taught me something new. I found the paper to be sufficiently comprehensive for someone with a rudimentary analysis of dimensional analysis to get a lot from it as well as deep enough in examples for someone who knows a good bit more to still enjoy it. If I had one minor suggestion it might be that this paper espouses all the benefits of dimensional analysis, but only a very limited discussion of possible shortcomings  this is a very minor point and perhaps does not really apply in the context of this paper, but I have seen very intersting discussions in the context of for example sociohydrology, where in some instances there are elements of interest where it is not even clear what variables and what dimensions are involved (i.e. outside of standard M,L,T,theta approaches). This is a very minor point though and I'm not sure that it really requires anything additional in this paper as it has been touched upon briefly in other review like papers cited in this work.
For all of the reasons above I think this is an excellent contribution and I for one will most definitely use it when educating graduate students in our program. Thank you very much for taking the time to write this paper!

AC1: 'Reply on RC1', Amilcare Porporato, 15 Nov 2021
I would like to thank the anonymous reviewer for his/her encouraging report and thoughtful comments. I am glad the paper was found of interest, also for providing a different angle to this timehonored topic.
The point raised by the reviewer, regarding the application of dimensional analysis to system that do not have a clear and typical physical definition, is very interesting and deserves attention. Unfortunately, however, it is also a difficult problem, one which is not typically considered in the traditional dimensional analysis literature. I plan to revise the paper paying particular attention to this comment emphasizing it, when suitable. There are some references in the economics and social science context, which are already cited in the introduction, where this point is relevant. I plan to expand around this issue, when commenting on these references. Also, the augmented dimensional analysis, and the idea of adding new variables, such as temperature or heat flux, as in the Rayleigh problem, can be seen as a precursor of this interesting problem. Again, I plan to add some comments in the paper to this regard. Thank you.

AC1: 'Reply on RC1', Amilcare Porporato, 15 Nov 2021

RC2: 'Comment on hess2021442', Anonymous Referee #2, 27 Sep 2021
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess2021442/hess2021442RC2supplement.pdf

AC2: 'Reply on RC2', Amilcare Porporato, 16 Nov 2021
Thank for the friendly comments and useful suggestions. I am glad the reviewer appreciated the attempt to provide a different approach to dimensional analysis, in particular regarding how the Buckingham Pi theorem be used in an era where data generation is overwhelming the ability of data interpretation.
With regard to the suggestions and comments, I agree with the reviewer that there are too many instances where geophysical problems, when tackled in their most general form, lead to a large number of Pi groups. In these cases, the Pi theorem becomes useless when nk>>1. This is why it is extremely important to emphasize the crucial step of building suitable physical laws. We have further emphasized this point – also in relation to the fermi reasoning, starting from earlier on in the manuscript. Thank you.
I also proceeded to rearrange in part Sec 3.3, as recommended by the reviewer. In particular, I improved and extended the derivation of the the erosion term and the specific contributing area, ‘a’. This is also inline with the recommendations of Ref. 4. Indeed the erosion term is too complicated to derive from first principle, but in the revised version we have better linked it to the water balance equation, which also leads to the equations for ‘a’. We have also provided in an appendix a more direct explanation of the mathematical a analogy between the turbulent velocity profile and the landscape elevation profile.
Unfortunately, while a very interesting suggestion, we are not aware of a derivation of these equations using scaling and dimensional analysis concepts. This valuable suggestion will be kept in mind for future contributions and is actually mentioned in the revised paper as an open direction for research.

AC2: 'Reply on RC2', Amilcare Porporato, 16 Nov 2021

RC3: 'Review on “Hydrology without dimensions” by Demetris Koutsoyiannis', Demetris Koutsoyiannis, 28 Sep 2021
The review report is provided as a PDF attachment.

AC3: 'Reply on RC3', Amilcare Porporato, 17 Nov 2021
Thank you, Demetris, for the very nice words and for the insightful review!
Indeed, the paper tries to be didactic and in the revised version I will pay particular attention to make it more accessible and readable since it examines cases from diverse fields.
Regarding the section “2.3 Augmented and directional dimensional analysis” and its Appendix, I have tried to make this part more precise (besides correcting a few errors). I’ve thus discussed in more detail why, in normal conditions, the last Pi group can be neglected in (A3). This is related to a selfsimilar behavior of the first type, which is supported by experience. As correctly stated in the review, this is not a mathematical necessity, but merely a fact of experience. I've improved the presentation of these concepts in the revised version.
I also added the example of the wave and diffusion, as suggested. This is a simple, but illustrative example, which I also used in the medal lecture, which is worth reproducing in the revised version of this paper. Thank you for the suggestion.
I also agree with the need to improve the introduction of Eq. (7), where the power law (selfsimilarity) may seem to be introduced ‘out of the blue’. It is also important to better explain that, in agreement with the review here and the references by the reviewer (Koutsoyiannis, 2014; Koutsoyiannis et al., 2018), expressions like equation (7) only hold asymptotically. This is at the heart of the very concept of (asymptotic) selfsimilarity.
Regarding the comment that ‘scaling, selfsimilarity and fractal behaviour look to be overemphasized or overpraised in the paper’, as in Figure 4, I should mention that this is not necessarily related to fractal behavior. Moreover fractal behavior can be, but does not need to be linked to stochastic behavior. Like in the Moody diagram in the fully rough regime, the scaling is simply a power law relation among variables (of course it results from some asymptotic behavior of turbulence which in this regime has specific statistical properties).
Of course, stochastic behavior is to be expected along with measurements and model errors. This is emphasized in the revised version of the manuscript. Indeed, the rainfall partitioning model is a stochastic model with random rainfall inputs. This is now emphasized and better explained.
Finally, regarding citations: Kolmogorov is now cited; Theodoratos et al. (2018) is also interesting and now cited – note however that here the authors use the wrong variable, the contributing area, A, instead of specific area, and this is explained in the revised text; and finally, the quote by Strahler is made more precise.
Thank you again for the insightful and useful recommendations.

AC3: 'Reply on RC3', Amilcare Porporato, 17 Nov 2021

RC4: 'Comment on hess2021442', Stefan Hergarten, 29 Sep 2021
The article "Hydrology without Dimensions" by Amilcare Porporato addresses scaling laws and nondimensional properties in hydrology with emphasis on applying the Pitheorem. As stated in the acknowledgments, the article is related to the Dalton medal lecture by the author. So the paper is somewhere between an original research paper and a review paper. Although somewhat unusual, such a focus makes sense, and I am sure that the community (including advanced students) will appreciate this paper.
Similar to the other reviewers (who were faster than me), my assessment of the manuscript is overall positive. I would suggest a publication with very few moderate adjustments, including one more check for typos.
(1) Right in the beginning of the theory, at Eqs. (1) and (2), I stumbled over the definition and the meaning of the scaling factor lambda. Formally, the argument of the function f should always have the same physical dimension, so that lambda should be nondimensional. And I would typically assume that lambda must be the same for all values of x, which makes lambda = 1/x problematic. Your point is clear, and I think it will not be a serious problem for the readers. But maybe you find another elegant formulation of the equations that avoids the issue.
(2) From my own background, Section 3.3 about landform evolution modeling is particularly interesting, and it nicely shows some recent work of your group. However, I am a bit wary about the concept of the specific drainage area (Eq. 28) and its application in the landform evolution model (Eq. 27). Let us assume a smooth (so with continuous derivatives) topography with a dendritic network of valleys. Then the flow pattern in a large valley consists of many fibers with different upstream lengths and thus with different specific discharges. Due to the dendritic structure, the acrossvalley pattern of the specific discharges is quite irregular, and the fibers even come closer to each other downstream. For me, this concept differs from the "classical" idea of a river with a given width, so that the model with the specific discharge differs in its spirit from what was previously assumed in this context.
A fully agree that the widely used version with the drainage area instead of the specific drainage area is inconsistent when proceeding from a discrete network to a continuous topography and leads to results depending on the spatial resolution of the grid. Unfortunately, it even seems that scaling relations were recently developed without taking care of this problem (Theodoratos et al. 2018). The alternative approach of considering a river as a line with a finite width (Howard 1994, Perron 2008) or rescaling the diffusion term (Pelletier 2010) are also not free of problem, which also applies to my own approach (Hergarten 2020). However, if you apply your model to a discrete grid, you practically integrate a^m (where a = specific drainage area). The result will differ from A^m (where A = drainage area). which means that the river consisting of fibers erode at a different rate than a river with a given discharge. So I am not completely convinced that your approach avoids the problem of the gridspacing dependence unless the grid is fine enough to resolve all the small fibers (which would practically not be feasible).
Please do not get me wrong  I do not want to criticize published work of your group too much. I just think that your reasoning about replacing A by a might be somewhat oversimplified and not free of caveats. I would be happy if you could add some discussion about these aspects. And in case I am wrong, please accept my apologies.
Best regards,
Stefan HergartenReferences
Pelletier, J. D.: Minimizing the gridresolution dependence of flowrouting algorithms for geomorphic applications, Geomorphology, 122, 9198, https://doi.org/10.1016/j.geomorph.2010.06.001, 2010
Perron, J. T., Dietrich, W. E., and Kirchner, J. W.: Controls on the spacing of firstorder valleys, J. Geophys. Res. Earth Surf., 113, F04 016, https://doi.org/10.1029/2007JF000977, 2008
Hergarten, S.: Rivers as linear elements in landform evolution models, Earth Surf. Dynam., 8, 367377, https://doi.org/10.5194/esurf83672020, 2020
Theodoratos, N., Seybold, H., and Kirchner, J. W.: Scaling and similarity of a streampower incision and linear diffusion landscape evolution model, Earth Surf. Dynam., 6, 779–808, https://doi.org/10.5194/esurf67792018, 2018

AC4: 'Reply on RC4', Amilcare Porporato, 18 Nov 2021
Dear Professor Hergarten,
Thank you for your kind review and useful suggestions. I’ve appreciated the positive criticism and made sure that the revised version clarifies some of these points (reported here in italic when necessary) as much as possible.
Regarding Eqs. (1) and (2) and the definition of the scaling factor lambda, the reviewer raises an interesting point that is worth better explaining in the revised version of the manuscript. Formally, the argument of the function f, especially if the function is transcendental, should be dimensionless (see e.g. Barenblatt 1996). On the other hand, lambda as a scale factor, should have the dimension of the inverse of x. The fact is that, as a matter of common use, we often omit writing explicitly unit factors which fix the dimensions in this regard. Thus, one either considers the scaling in Eq 1 with regard to dimensionless quantities (then lambda, x, and f are all dimensionless and things work well – I believe this what mathematicians have in mind when they speak of homogeneous functions like here), or uses dimensions and then either explicitly (but this becomes very cumbersome) or implicitly assumes the presence of these unit factors that convert the units. E.g.
f(lambda*x)=(1*lambda)^n f(1*x).
where the different ‘1’ have different dimensions…
I guess, this is somewhat similar to the fact that log(x) only makes sense if x is dimensionless, so in practice it is log(x/1), but in common practice this is not done, also because if one then writes this as log(x)log(1) then we start over again… or it is enough to think of power series expansions, where obviously there are omitted factors that make the dimensions consistent …
The revised version of the paper clarifies these points. Thank you.
(2) From my own background, Section 3.3 about landform evolution modeling is particularly interesting, and it nicely shows some recent work of your group. However, I am a bit wary about the concept of the specific drainage area (Eq. 28) and its application in the landform evolution model (Eq. 27). Let us assume a smooth (so with continuous derivatives) topography with a dendritic network of valleys. Then the flow pattern in a large valley consists of many fibers with different upstream lengths and thus with different specific discharges. Due to the dendritic structure, the acrossvalley pattern of the specific discharges is quite irregular, and the fibers even come closer to each other downstream.
I fully agree with you here. This is a perfect description of the differential geometry of a (smooth) landscape and is exactly what happens in terms of streamlines and specific contributing area (see Bonetti et al. PRSA 2018).
For me, this concept differs from the "classical" idea of a river with a given width, so that the model with the specific discharge differs in its spirit from what was previously assumed in this context.
Defining a river is in my view a different story from what we do here and in general should not be confused with numerical (or theoretical) issues related to the solution of a given equation, which needs to be well posed. One should keep in mind that these minimalist models of landscape evolution have a very limited and rudimentary representation of the physical processes. For example, if one also considers the surface water flowfield, then one sees that there is water everywhere all the time, which obviously makes no physical sense. In our case, defining rivers is not the goal, but rather the understanding of the mechanisms of hierarchical branching and formation of valleys and ridges (of course one could define thresholds of certain water height to define rivers, but this would be quite unsatisfactory, I think).
I fully agree that the widely used version with the drainage area instead of the specific drainage area is inconsistent when proceeding from a discrete network to a continuous topography and leads to results depending on the spatial resolution of the grid. Unfortunately, it even seems that scaling relations were recently developed without taking care of this problem (Theodoratos et al. 2018). The alternative approach of considering a river as a line with a finite width (Howard 1994, Perron 2008) or rescaling the diffusion term (Pelletier 2010) are also not free of problem, which also applies to my own approach (Hergarten 2020).
I fully agree.
However, if you apply your model to a discrete grid, you practically integrate a^m (where a = specific drainage area). The result will differ from A^m (where A = drainage area). which means that the river consisting of fibers erode at a different rate than a river with a given discharge. So I am not completely convinced that your approach avoids the problem of the gridspacing dependence unless the grid is fine enough to resolve all the small fibers (which would practically not be feasible).
The fact is that ‘a’, the specific contributing area is a variable defined in the continuum (i.e. pde) representation. Once the domain is discretized, then suitable operators should be found and for each grid point one has to decide what the contributing area is. This is an important numerical problem, but not a theoretical one.
Please do not get me wrong  I do not want to criticize published work of your group too much. I just think that your reasoning about replacing A by a might be somewhat oversimplified and not free of caveats. I would be happy if you could add some discussion about these aspects. And in case I am wrong, please accept my apologies.
Thank you again for the interesting points of discussion and for spurring me to clarify and better explain these issues. In the revised version I’ve tried to clarify better these points and the derivation of the landscape evolution equation to show that ‘a’ is the right variables. No PDE in continuum mechanics has directly an extensive variable like A in their terms (it would be like having volume V of diameter D of the pipe directly in the Navier Stokes equations rather than using them as boundary conditions.
Barenblatt, G. I.: Scaling, selfsimilarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, 14, Cambridge University Press, 1996.
Bonetti, S., Bragg, A., and Porporato, A.: On the theory of drainage area for regular and nonregular points, Proceedings of the Royal Society 670 A: Mathematical, Physical and Engineering Sciences, 474, 20170 693, 2018.

AC4: 'Reply on RC4', Amilcare Porporato, 18 Nov 2021

EC1: 'Editor's comment', Nunzio Romano, 05 Oct 2021
Dear Amilcare,
To feed the open discussion and start possible interactions with the reviewers, I invite you to start uploading some preliminary responses or comments from your side (apart from your final reply as the author at the closure of this first step).
 AC5: 'Reply on EC1', Amilcare Porporato, 18 Nov 2021
Amilcare Porporato
Amilcare Porporato
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