A stream-power law for glacial erosion and its implementation in
large-scale landform-evolution models

Hergarten, Stefan

doi:https://doi.org/10.5194/esurf-2021-1

Preprints

https://doi.org/10.5194/esurf-2021-1

Preprints

22 Jan 2021

Review status: a revised version of this preprint was accepted for the journal ESurf and is expected to appear here in due course.

A stream-power law for glacial erosion and its implementation in large-scale landform-evolution models

Stefan Hergarten Stefan Hergarten Stefan Hergarten

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Institut für Geo- und Umweltnaturwissenschaften, Albert-Ludwigs-Universität Freiburg, Albertstr. 23B, 79104 Freiburg, Germany

Received: 14 Jan 2021 – Accepted for review: 20 Jan 2021 – Discussion started: 22 Jan 2021

Abstract. Modeling glacial landform evolution is more challenging than modeling fluvial landform evolution. While several numerical models of large-scale fluvial erosion are available, there are only a few models of glacial erosion, and their application over long time spans requires a high numerical effort. In this paper, a simple formulation of glacial erosion which is similar to the fluvial stream-power model is presented. The model reproduces the occurrence of overdeepenings, hanging valleys, and steps at confluences at least qualitatively. Beyond this, it allows for a seamless coupling to fluvial erosion and sediment transport. The recently published direct numerical scheme for fluvial erosion and sediment transport can be applied to the entire domain, where the numerical effort is only moderately higher than for a purely fluvial system. Simulations over several million years on lattices of several million nodes can be performed on standard PCs. An open-source implementation is freely available as a part of the landform evolution model OpenLEM.

Stefan Hergarten

Status: closed

RC1:
'Comment on esurf-2021-1', Eric Deal, 17 Feb 2021

The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf-2021-1/esurf-2021-1-RC1-supplement.pdf

Citation: https://doi.org/10.5194/esurf-2021-1-RC1
- AC1: 'Reply on RC1', Stefan Hergarten, 22 Feb 2021
  
  Dear Eric Deal,
  thanks a lot for your thorough and even inspiring review! I must apologize that I was indeed not aware of your latest paper in GRL. The first part of the theory was developed in a slightly different way, but is in fact basically the same. And you were definitely earlier, although I spent a lot of time on the numerics before submitting my manuscript.
  There is only one point where I clearly disagree with your opinion. This is the way erosion models of the stream-power type are formulated for non-constant precipitation. Probably owing to the fundamental studies of Hack, erodibilities have been expressed in terms of catchment size instead of discharge until now. If we want to keep this convention, there is no way of avoiding the definition of a reference precipitation p0 and imagining that a given erodibility refers to p0 . Then we can replace the catchment size A by either q/p0 where q is the discharge or even by pA/p0 , but where p is the mean discharge over the upstream catchment. Both versions become
  
  increasingly cumbersome when proceeding to the shared stream-power model and also for the fluvio-glacial version. Beyond this, I do not really like the concept where a mean upstream precipitation occurs in the erosion model and a local precipitation in the climatic component. Therefore, I do not like the version used by the Tübingen/Potsdam groups for some years, which also applies to the version with the product IA in your 2012 paper. My version of defining the ratio q/p0 as the catchment-size equivalent of the actual discharge provides a clear definition that keeps the equations similar to the original stream-power model. I am convinced that reminding
  
  the reader of this definition at some occurrences of A is sufficient to avoid confusion. So there is no realistic chance to convince me here.
  There is another point where I am not sure at all. In your final comment, you suggest to consider the version where the thickness is parameterized in terms of flux and slope instead of flux alone. However, my results already show that such a parameterization leads to a weird scaling of thickness vs. width in a steady state. While width is proportional to q^0.3 then, thickness is proportional to q^0.7. As far as I can see, your recent approach including deformation softens this problem, but thickness still increases more rapidly with flux than width. I guess that the problem already comes in when parameterizing the width by the flux alone without taking into account the slope. If so, it already affects the 2020 EPSL paper by Günther Prasicek where both of us were coauthors, but unfortunately I did not think about this at that time. So it would also be interesting to look at thickness vs. width in your recent results. But for my concept, this mainly tells me that we must consider all approximations as a package and compare it, e.g., to simulations with iSOSIA.
  Best regards,
  
  Stefan
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC1
CC1:
'Comment on esurf-2021-1', Flavien Beaud, 06 Mar 2021
Comment on: A stream-power law for glacial erosion and its implementation in large-scale landform-evolution models, by Stefan Hergarten.
I was not formally asked to review this paper, but I believe my expertise can be useful to improve the manuscript. In that sense, I will focus on very general comments and I am not proposing a detailed review.
The main focus of the study is to develop a landform evolution model that can be solved more readily with less computational power than the current existing models. This would, indeed, be a very useful contribution to glacial geomorphology, especially as the author are making their code open source. To do so, the author derive a stream power law for glacial erosion, including a parametrization of bedrock erosion and sediment transport by subglacial water flow.
My main concern regarding the study is that the author provides little to no explanations for the simplifications they propose, and in many cases, these simplifications are contradictory to the current understanding of these erosion and transport mechanisms. Throughout the paper, there are very few explanations of the current state of knowledge of the physical processes involved in glacial erosion and sediment transport. Most of the papers cited are numerical models that already make significant assumptions. For example, in Section 2, l. 63-64, the derivation of the glacial stream power starts with the statement “If we consider a rectangular cross section […]”. Yet, we know glacial valleys are not rectangular and there is no explanation of that simplification.
Another point of concern is the recurring citation of Prasicek et al. (2020) to justify numerous simplifications. In the Prasicek et al. (2020) study, the goal is to assess timescales of respective processes: glacial erosion, climate and tectonic; not to reproduce landforms. The assumptions Prasicek et al. (2020) are making to assess the relative effectiveness of processes over time, become inadequate when applied to a model that aims at reproducing landforms themselves.
To be more specific, I believe the following reasonings are problematic:
While reading the paper, I was left with the feeling that because there is a stream power erosion law for rivers there should be one for glaciers. In river systems the shear stress can be approximated with discharge, which in turn can be approximated with catchment area and precipitation. Futhermore, observations by Hack (1957) support a simple scaling. For glacier erosion, however, not only do erosion and transport mechanisms remain unclear in general, but there is no large-scale field observations that support such scaling. The paper cited to support such scaling (Bahr, 1997) refers to glaciers themselves, not glacial landscapes, and is therefore not applicable to glacier erosion. In term of the specifics of glacial erosion, even assuming a simple relationship between erosion and sliding velocity to a power, this power is likely >1 (Herman et al., 2015; Koppes et al., 2015). Sliding velocity is intricately linked to basal shear stress and basal water pressure (e.g. Beaud et al., 2014; Ugelvig et al., 2018). Therefore, a thorough explanation is necessary to simplify our current state-of-the-art understanding of the physical processes of glacial erosion into a simple stream power-type law.

Using a shallow ice approximation has be shown to not work very well on steep landscapes (Egholm et al., 2011). These simplifications and their implications for landscape evolution results should be explained.

Model of glacio-fluvial incision (l. 229-232): “While Beaud et al. (2016) developed a more elaborate model for the incision 230 by meltwater within narrow channels, the meltwater component should preferably not introduce a level of complexity much beyond the simple models of fluvial and glacial erosion used here. So let us assume that erosion by meltwater can be described by the same formalism as fluvial erosion.” That statement is in opposition to the results presented in the Beaud et al. (2016) study. Since that model is the only to date to describe such mechanism, dismissing the results or making different assumptions should be substantiated.

Finally, the authors are omitting the vast majority of the glacial landform evolution literature together with the literature explaining the physical underpinning of glacial erosion and sediment transport. In the references at the end of my comment, I propose a non-exhaustive list of papers that should be cited and the finding of which should be included in the current study.

In summary, in its current form, I do not believe that the current study makes a convincing point that a stream power glacial erosion rule can, or should be used. If the goal of the paper is to reproduce landforms, the author should include and discuss extensively the existing literature and justify their simplifications. If the goal is to produce a model that can be used to test interplay between erosion, climate and tectonic, similarly to Prasicek et al. (2020), that should also be clarified. In any case, I believe, the assumptions should be discussed in light of the existing literature.
I hope you find my comments constructive and am available to answer further questions if necessary (flavien.beaud@ubc.ca).
Best,
Flavien Beaud
References
Anderson, R. S. (2014). Evolution of lumpy glacial landscapes. Geology, doi:10.1130/G35537.1.
Beaud, F., Flowers, G. E., & Pimentel, S. (2014). Seasonal-scale abrasion and quarrying patterns from a two-dimensional ice-flow model coupled to distributed and channelized subglacial drainage. Geomorphology, 219, 176–191.
Beaud, F., Flowers, G. E., & Venditti, J. G. (2016). Efficacy of bedrock erosion by subglacial water flow. Earth Surface Dynamics, 4, 125–145. https://doi.org/doi:10.5194/esurf-4-125-2016
Beaud, F., Flowers, G. E., & Venditti, J. G. (2018). Sediment transport in ice-walled subglacial channels and its implications for esker formation and monitoring glacial erosion. Journal of Geophysical Research, Earth Surface.
Creyts, Timothy T., and Ian Hewitt. "Genesis of esker corridors as erosional features beneath ice sheets." In AGU Fall Meeting Abstracts, vol. 2019, pp. C21E-1496. 2019.
Delaney, Ian, and Surendra Adhikari. "Increased subglacial sediment discharge in a warming climate: Consideration of ice dynamics, glacial erosion, and fluvial sediment transport." Geophysical Research Letters 47, no. 7 (2020): e2019GL085672.
Egholm, D. L., Pedersen, V. K., Knudsen, M. F., & Larsen, N. K. (2011). On the importance of higher order ice dynamics for glacial landscape evolution. Geomorphology, 141--142, 67–80.
Herman, F., Beaud, F., Champagnac, J. D., Lemieux, J. M., & Sternai, P. (2011). Glacial hydrology and erosion patterns: A mechanism for carving glacial valleys. Earth and Planetary Science Letters, 310(3), 498–508.
Herman, F., Beyssac, O., Brughelli, M., Lane, S. N., Leprince, S., Adatte, T., … Cox, S. C. (2015). Erosion by an Alpine glacier. Science, 350(6257), 193–195.
Iverson, N. R. (2012). A theory of glacial quarrying for landscape evolution models. Geology, 40(8), 679–682.
Koppes, M., Hallet, B., Rignot, E., Mouginot, J., Wellner, J. S., & Boldt, K. (2015). Observed latitudinal variations in erosion as a function of glacier dynamics. Nature, 526(7571), 100–103.
MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Waddington, E. D. (2009). Numerical modeling of glacial erosion and headwall processes in alpine valleys. Geomorphology, 28(2), 189–204.
MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Waddington, E. D. (2000). Numerical simulations of glacial longitudinal profile evolution. Geology, 28, 1031–1034.
Riihimaki, C. A., MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Loso, M. G. (2005). Sediment evacuation and glacial erosion rates at a small alpine glacier. Journal of Geophysical Research, 110(F3), F03003.
Ugelvig, S. V., D. L. Egholm, R. S. Anderson, and Neal R. Iverson. "Glacial erosion driven by variations in meltwater drainage." Journal of Geophysical Research: Earth Surface 123, no. 11 (2018): 2863-2877.
Citation: https://doi.org/10.5194/esurf-2021-1-CC1
- AC2: 'Reply on CC1', Stefan Hergarten, 06 Mar 2021
  
  Dear Flavien Beaud,
  thanks for your comments! I hoped that some more researchers from the community of modeling glacial erosion would share their opinion, and that they would perhaps even go beyond keywords. I am convinced that your expertise as a reviewer when considering the proposed concept seriously would have been helpful to improve the paper.
  Let me briefly comment on some of your points.
  For example, in Section 2, l. 63-64, the derivation of the glacial stream power starts with the statement "If we consider a rectangular cross section [...]". Yet, we know glacial valleys are not rectangular and there is no explanation of that simplification.
  This is just some kind of entry gate for the readers. Those who think this way should stop reading at this point and not waste their time.
  
  I hope there will be some readers not arguing on this level, but are able to think about the consequences of this simplification, although
  
  I am aware that not all will be able to solve the exercise of finding out for which shapes of cross sections it remains valid.
  Another point of concern is the recurring citation of Prasicek et al. (2020) to justify numerous simplifications. In the Prasicek et al. (2020)
  
  study, the goal is to assess timescales of respective processes: glacial erosion, climate and tectonic; not to reproduce landforms. The assumptions Prasicek et al. (2020) are making to assess the relative effectiveness of processes over time, become inadequate when applied to a model that aims at reproducing landforms themselves.
  Sorry, I am not such an expert on this since I only developed parts of the theoretical framework of the Prasicek et al. (2020) paper. So far I thought it was about longitudinal equilibrium profiles for a dendritic topology and not about thime scales. But maybe I just missed the key point of that paper.
  The paper cited to support such scaling (Bahr, 1997) refers to glaciers themselves, not glacial landscapes, and is therefore not applicable to
  
  glacier erosion.
  I thought it was about length-width scaling of glaciers and used nothing else from this paper. And in contrast to previous studies, I was honest enough to explain that transferring scaling relations from entire glaciers into individual glaciers is nontrivial.
  In term of the specifics of glacial erosion, even assuming a simple relationship between erosion and sliding velocity to a power, this power
  
  is likely > 1.
  I think I mentioned this, and anyone should feel free to run the model with an exponent > 1. Some researchers may even prefer this for the fluvial regime since there were some studies that suggested exponents > 1 (although I think there are many pitfalls and artifacts there).
  Using a shallow ice approximation has be shown to not work very well on steep landscapes (Egholm et al., 2011). These simplifications and their implications for landscape evolution results should be explained.
  This is not just a problem of steep topographies, but applies to all combinations of shallow-water and shallow-ice equations with
  
  incision-type erosion laws if lateral stresses are not taken into account. Iti s explained briefly in the paper, and in principle it
  
  is an argument in favor of approaches where erosion is driven by a linear element.
  Model of glacio-fluvial incision (l. 229-232): "While Beaud et al. (2016) developed a more elaborate model for the incision by meltwater within narrow channels, the meltwater component should preferably not introduce a level of complexity much beyond the simple models of fluvial and glacial erosion used here. So let us assume that erosion by meltwater can be described by the same formalism as fluvial erosion." That statement is in opposition to the results presented in the Beaud et al. (2016) study. Since that model is the only to date to describe such mechanism, dismissing the results or making different assumptions should be substantiated.
  I cited this paper because it is apparently the first modeling approach towards bedrock incision by meltwater channels. However, I was neither able to recognize any fundamentally new theoretical concepts nor any results that could be generalized to larger scales. So it is not clear to me where the contradiction is. Did you find that meltwater incision in total independent of the meltwater discharge or find that the pressure gradient is totally decoupled from the gradient of the ice surface? Maybe I missed the key point again.
  In summary, in its current form, I do not believe that the current study makes a convincing point that a stream power glacial erosion rule can, or should be used.
  But unfortunately, other authors recently proposed a glacial stream-power law that is very similar to my version (Deal & Prasicek, Geophys. Res. Lett, 2020). They were definitely faster than me. Feel free to write a comment to their paper!
  I hope you find my comments constructive and am available to answer further questions if necessary.
  To be honest, not really. But I definitely appreciate your contribution to the discussion.
  Best,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC2
RC2:
'Comment on esurf-2021-1', Marc Jaffrey, 14 Apr 2021
I recommend rejecting the paper.

In short the author appears to forcing glacial erosion into the framework of fluvial erosion with little or outright incorrect reasoning.

Main Reasons:

The author does not appear to be familiar with nor understand sufficiently glacier dynamics and glacier erosion for which an extensive body of literature exists. The theoretical underpinnings of the paper are unfounded and in several key places in direct conflict with known glacier dynamics.

The author has not sufficiently reviewed the literature.

The paper is not written to address the glaciology community which I presume is a key target audience for the work.

Some key examples of problems highlight above:

Section 10: “In contrast to fluvial erosion, however, glacial erosion has not been extensively considered in modeling large-scale landform evolution.” There is an extensive body of literature on modeling large-scale glacial landform development.

35-40: “A comparable representation of glacial erosion where the erosion rate can be directly computed from properties of the topography is not yet available.” This is not true and stems from the authors misunderstanding of the relationship between glacier dynamics and existing glacier erosion laws implemented within the glacial geomorphology community.

45-50: There are many different form of erosion laws considered by the glacier community of which the author does not acknowledge.

50: If the author is talking about the general mathematical concept of mass balance they need to say so. Mass balance in the glaciology community is something different and the connection to diffusion would require further explanation on the authors part.

60-65: This is false. The author makes a critical assumptions here defining the model presented which is a critical misunderstanding of glacier dynamics and the Shallow Ice-sheet approximation. The ratio of the depth averaged velocity and sliding velocity is categoricallly not controlled by ice thickness, not even a second order control. Sliding velocity is a complex issue which from a dynamic’s perspective, the principal first order controls are effective water pressure, bed temperature, bed roughness and a host of other bed parameters and processes. From a mathematical and numerical simulation perspective sliding velocity is treated independently with a sliding law, a Robin type boundary condition, which is typical defined in terms of the aforementioned bed processes, not ice thickness. The author seems to be unaware of glacier dynamics. Since the assumptions in 60-65 are crucial to the theoretical underpinning of the model the rest of the work, regardless of the care taken with the numerical simulation, is unfounded.

100: “The erodibility K is a lumped parameter that already includes precipitation “ This statement requires a detailed explanation on the author’s parts of what they mean by this statement.

100: This whole section is problematic. The author seems to be connecting concepts without justifications.

225: “These findings support the idea that erosion by meltwater must play an important role, at least in the lower part of glaciers where the flux of water is much higher than the ice flux.” One should always be careful when drawing dynamical conclusions from a numerical model especially when parameter tuning can be used to generate a wide range of results. That is to say, the author is making a circular argument and in this case one that is in conflict with the glacier community. Correlation is not causation.

20-25: References to convexity a concept from fluvial erosion is not one familiar to the glacier community and requires explanation.
Citation: https://doi.org/10.5194/esurf-2021-1-RC2
- AC3: 'Reply on RC2', Stefan Hergarten, 14 Apr 2021
  
  Dear Marc Jaffrey,
  thanks for your comments! You speak a lot about "an extensive body of literature", "known glacier dynamics", and "the glaciology community". Unfortunately, it is difficult for me to recognize what you refer to. I tried to find out from you own work in this field, but only found a conference poster about basin-wide glacial erosion rates. Although I like the concept proposed there, I could not find out where the fundamentals of my work basically differ from what you assume, and I also cannot see how the concept of a mass balance differs as you claim.
  Let me just comment on your point highlighted by boldface line numbers :
  "60-65: This is false. The author makes a critical assumptions here defining the model presented which is a critical misunderstanding of glacier dynamics and the Shallow Ice-sheet approximation. The ratio of the depth averaged velocity and sliding velocity is categoricallly not controlled by ice thickness, not even a second order control. Sliding velocity is a complex issue which from a dynamic’s perspective, the principal first order controls are effective water pressure, bed temperature, bed roughness and a host of other bed parameters and processes. From a mathematical and numerical simulation perspective sliding velocity is treated independently with a sliding law, a Robin type boundary condition, which is typical defined in terms of the aforementioned bed processes, not ice thickness. The author seems to be unaware of glacier dynamics. Since the assumptions in 60-65 are crucial to the theoretical underpinning of the model the rest of the work, regardless of the care taken with the numerical simulation, is unfounded."
  This is just what is typically assumed for the sliding and deformation velocities (not "depth averaged velocity" as you state). I agree that the factors of proportionality in Eqs. (4) and (5) depend on what you mention, but nevertheless the (quite old) model ICE-CASCADE and to some extent also the state-of-the-art model iSOSIA use these relations. They are also used in more theoretical considerations, e.g., Prasicek et al. (2018), doi 10.1029/2017JF004559 and Deal & Prasicek (2021) doi 10.1029/2020GL089263. You are free not to like this approach and propose alterantive concepts. In general, however, I would find it more useful to bring your own ideas to the scientific audience than just tearing down other researchers' work on a keyword level.
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC3
RC4:
'Comment on esurf-2021-1', Marc Jaffrey, 16 Apr 2021
First, there is no conflict of interest: My work and the author’s work are categorically different and furthermore address different spatial and temporal domains. My work is theoretical with the development of an analytical model of glacier erosion rates at the basin scale, while the author’s work addresses glacier erosion at smaller spatial scales utilizing a heuristic approach to define a glacial erosion law for numerical implementation, as is standard for numerical approaches.
Next, please let me express my regret for the tone of my earlier comments without elucidating clear points for the author to address. Since the work makes assumptions in direct conflict with current theories of glacier dynamics, I will not address the computational aspects of the work as the premises on which the model is built are problematic and in this reviewer’s opinion unrealistic rending the erosion law in equation 19 open to sever question. Restating the main issues:
The author has not cited nor sufficiently discussed existing literature: This is an issue as the proposed scientific contribution this work cannot be understood except but subject matter experts.

The theoretical underpinnings, the glacier dynamics, in several key place are incorrect in section two and three rendering the hypothesized glacier erosion law, the equation 19, the critical key equation of the numerical model, open to question.

One of the most concerning issue in terms are some of the scientific conclusions of the paper. Most notably in 225: “These findings support the idea that erosion by meltwater must play an important role, at least in the lower part of glaciers where the flux of water is much higher than the ice flux.” This statement cannot be justified by the a numerical model regardless of soundness of the theoretical underpinnings. While the contributions of numerical simulations to the research and progress of understanding glacier erosion cannot be overstated, drawing such definitive conclusions about mechanisms of glacier erosion is out of reach for this type of approach.

The paper is not written to address the glacial community which I presume is a key target audience for the work.

Based on these issues I cannot recommend publication of the work as in my opinion the prosed erosion law is unjustified calling into question the numerical model and its results.
Focused Comments for Section 1:
1 -5: “In contrast to fluvial erosion, however, glacial erosion has not been extensively considered in modeling large-scale landform evolution”
Here is a list of reference the author might consult:
Egholm, D. L., et al. "On the importance of higher order ice dynamics for glacial landscape evolution." Geomorphology 141 (2012): 67-80.
Harbor, J., 1989. Early Discoverers XXXVI: W J McGee on glacial erosion laws and the development of glacial valleys. Journal of Glaciology 35, 419–425.
Harbor, J., Hallet, B., Raymond, C. A numerical model of landform development by glacial erosion. Nature 333, 347–349 (1988).
Harbor, J. Numerical modeling of the development of U-shaped valleys by glacial erosion. GSA Bulletin 104, 1364–1375 (1992).
Herman, F., Beaud, F., Champagnac, J.D., Lemieux, J-M., Sternai, P. Glacial hydrology and erosion patterns: A mechanism for carving glacial valleys. Earth and Planetary Science Letters, 310, 498–508 (2011).
MacGregor, K.R., Anderson, R.S., Anderson, S.P., Waddington, E.D. Numerical simulations of glacial–valley longitudinal profile evolution. Geology 28, 1031–1034 (2000). Oerlemans, J. Numerical experiments of large-scale glacial erosion. Zeitschrift fuer Gletscherkunde und Glazialgeologie 20, 107–126 (1984)
Oerlemans, J. Numerical experiments of large-scale glacial erosion. Zeitschrift fuer Gletscherkunde und Glazialgeologie 20, 107–126 (1984).
Tomkin, J.H. Numerically simulating alpine landscapes: The geomorphologic consequences of incorporating glacial erosion in surface process models. Geomorphology 103, 180-188 (2009).
Ugelvig, S.V., Egholm, D.L., Iverson, N.R. Glacial landscape evolution by subglacial quarrying: A multiscale computational approach. J. Geo. Res. E. Sur. 121, 2042-2068 (2016).
Ugelvig, S. V., et al. "Overdeepening development in a glacial landscape evolution model with quarrying." AGU Fall Meeting Abstracts. Vol. 2013. 2013.
35-40: “A comparable representation of glacial erosion where the erosion rate can be directly computed from properties of the topography is not yet available.”
It is essential discuss the scales, both temporal and spatial, on which erosion rates are being considered. Though not explicitly discuss in the introduction, it can be implied from context that the spatial scales considered are smaller than the scale of the glacial landforms which is sub basin scale.
Alley, R. B., K. M. Cuffey, and L. K. Zoet. "Glacial erosion: status and outlook." Annals of Glaciology 60.80 (2019): 1-13.
Alley, R. B., et al. "Stabilizing feedbacks in glacier-bed erosion." Nature 424.6950 (2003): 758-760.
Andrews J.T. Glacier power, mass balances, velocities and erosion potential. Zeitschrift fur Geomorphologie 13, 1-17 (1972).
oulton, Geoffrey S. "Processes and patterns of glacial erosion." Glacial geomorphology. Springer, Dordrecht, 1982. 41-87.
Boulton, G. S. "Theory of glacial erosion, transport and deposition as a consequence of subglacial sediment deformation." Journal of Glaciology 42.140 (1996): 43-62.
Cook, Simon J., et al. "The empirical basis for modelling glacial erosion rates." Nature communications 11.1 (2020):
Delmas, M., Calvet, M., Gunnell, Y. Variability of quaternary glacial erosion rates— a global perspective with special reference to the Eastern Pyrenees. Quat. Sci. Rev. 28, 484–498 (2009).
Hall, Adrian M., et al. "Glacial ripping: geomorphological evidence from Sweden for a new process of glacial erosion." Geografiska Annaler: Series A, Physical Geography (2020): 1-21.
Hallet, B. Glacial abrasion and sliding: their dependence on the debris concentration in basal ice. Annals of Glaciology 2, 23-28 (1981).
Hallet, B. Glacial quarrying: A simple theoretical model. Annals of Glaciology 22, 1–8 (1996).
Iverson, N.R. A theory of glacial quarrying for landscape evolution models. Geology 40, 679–682 (2012).
Menzies, J., Jaap JM van der Meer, and W. W. Shilts. "Subglacial processes and sediments." Past glacial environments. Elsevier, 2018. 105-158.
Steinemann, Olivia, et al. "Quantifying glacial erosion on a limestone bed and the relevance for landscape development in the Alps." Earth Surface Processes and Landforms 45.6 (2020): 1401-1417
Ugelvig, S. V., et al. "Glacial erosion driven by variations in meltwater drainage." Journal of Geophysical Research: Earth Surface 123.11 (2018): 2863-2877.
Equation 4 is incorrect: See chapter 7, eq's 7.6, 7.10, 7.17, 8.35, 8.36, 8.65 and sections 8.1, 8.4, 8.5 and 8.6 in Cuffey, Kurt M., and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010. Sliding velocity cannot be reduced to ice thickness and slope under any approximation. As Cuffey and Patterson in discussion the Shallow Ice Approximation say in chapter 8, "The rate of basal slip must be specified directly or through a relation to bed stress such as Eq. 8.25"
Equation 6: Yes erosion laws of this form are typically implemented, however there are many other forms that have been used within numerical simulation.
50: This section requires substantially further discussion and justification. Ice thickness is not a diffusion process. See section 8.5.5 and equation 8.65 , 8.70, 8.77, 8.78 and 8.79 in Cuffey and Patterson. In the Shallow Ice Approximation ice flux q~h not the partial derivative of h wrt to x. There is a divergence relationship, first order partial derivatives, but diffusion is second order in the spatial partial derivatives. Without substantial justification, ice thickness cannot be treated as diffusion with strong diffusivity.
Section 2:
Again equation 4 is unfounded so that section 2 begins with a false premise. This problem then follows through into equations 8, 9, and the key equation 14 rendering it unfounded.
70-80: This section requires further explanation.
100: The authors treatment of catchment size, precipitation, and discharge may have a clear rationale for fluvial systems, but it is not clear why this would apply to glacial system except perhaps at the terminus. The author needs a detail justification.
Equation 19: Taken as a whole, the validity of the proposed erosion law is questionable.
Citation: https://doi.org/10.5194/esurf-2021-1-RC4
- AC4:
  'Reply on RC4', Stefan Hergarten, 18 Apr 2021
  Dear Marc Jaffrey,
  thanks for the more specific version of the review and in particular for the personal email you sent to me. This makes your point about some of the numerical models used on this field more clear to me. I am fully with you that it is a serious problem of science that things become "true" just by publishing and citing repeatedly.
  Nevertheless, it seems indispensable for me to state that your arguments about the equations are completely wrong in my opinion.
  
  Concerning Eq. (4), you state: "Equation 4 is incorrect: See chapter 7, eq's 7.6, 7.10, 7.17, 8.35, 8.36, 8.65 and sections 8.1, 8.4, 8.5 and 8.6 in Cuffey, Kurt M., and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010. Sliding velocity cannot be reduced to ice thickness and slope under any approximation. As Cuffey and Paterson in discussion the Shallow Ice Approximation say in chapter 8, `The rate of basal slip must be specified directly or through a relation to bed stress such as Eq. 8.25'"
  
  Equation 4 just requires v_s ~ τ^ψ/σ (Budd et al., J. Glacial.,23, 157-170, 1979) where τ = shear stress and σ = effective normal stress. According to the shallow ice approximation, τ ~ hS (h = ice thickness, S = slope of the ice surface). In this form, it yields v_s ~ h^ψ/(h-p) S^ψ where p = water pressure. Among the references you provided, at least Harbor, Hallet & Raymond, Nature 333, 347-349 (1988) and Tomkin, Geomorphology 103, 180-188 (2009) assumed p ~ h, which exactly yields Eq. 4, v_s ~ h^ψ-1S^ψ. You may question the assumption that the fluid pressure at the bed is proportional to the ice thickness and claim that coupling with a distinct model for melting and for the flow of melt water may yield better results, and that such models are available. However, your statement "Sliding velocity cannot be reduced to ice thickness and slope under any approximation" is wrong.
  
  About the type of the differential equation: "Ice thickness is not a diffusion process. See section 8.5.5 and equation 8.65 , 8.70, 8.77, 8.78 and 8.79 in Cuffey and Paterson. In the Shallow Ice Approximation ice flux q~h not the partial derivative of h wrt to x. There is a divergence relationship, first order partial derivatives, but diffusion is second order in the spatial partial derivatives. Without substantial justification, ice thickness cannot be treated as diffusion with strong diffusivity."
  
  Sorry, but the shallow water equations and its derivates (shallow ice equations and Savage-Hutter equations for granular flow) assume a free surface and hydrostatic pressure in vertical direction. Then the horizontal force is proportional to the gradient of the free surface and not to the gradient of the bed (as you presumably assume in your reasoning). As a consequence, the velocity in the shallow ice equation also depends on h and on the gradient of the ice surface. Then the divergence term from the mass balance yields second-order spatial derivatives of h in total. The resulting spatial differential operator is elliptic, so that the entire time-dependent equation is a parabolic equation (= diffusion type for the broader readership).
  
  Best regards,
  
  Stefan
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC4

Status: closed

RC1:
'Comment on esurf-2021-1', Eric Deal, 17 Feb 2021

The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf-2021-1/esurf-2021-1-RC1-supplement.pdf

Citation: https://doi.org/10.5194/esurf-2021-1-RC1
- AC1: 'Reply on RC1', Stefan Hergarten, 22 Feb 2021
  
  Dear Eric Deal,
  thanks a lot for your thorough and even inspiring review! I must apologize that I was indeed not aware of your latest paper in GRL. The first part of the theory was developed in a slightly different way, but is in fact basically the same. And you were definitely earlier, although I spent a lot of time on the numerics before submitting my manuscript.
  There is only one point where I clearly disagree with your opinion. This is the way erosion models of the stream-power type are formulated for non-constant precipitation. Probably owing to the fundamental studies of Hack, erodibilities have been expressed in terms of catchment size instead of discharge until now. If we want to keep this convention, there is no way of avoiding the definition of a reference precipitation p0 and imagining that a given erodibility refers to p0 . Then we can replace the catchment size A by either q/p0 where q is the discharge or even by pA/p0 , but where p is the mean discharge over the upstream catchment. Both versions become
  
  increasingly cumbersome when proceeding to the shared stream-power model and also for the fluvio-glacial version. Beyond this, I do not really like the concept where a mean upstream precipitation occurs in the erosion model and a local precipitation in the climatic component. Therefore, I do not like the version used by the Tübingen/Potsdam groups for some years, which also applies to the version with the product IA in your 2012 paper. My version of defining the ratio q/p0 as the catchment-size equivalent of the actual discharge provides a clear definition that keeps the equations similar to the original stream-power model. I am convinced that reminding
  
  the reader of this definition at some occurrences of A is sufficient to avoid confusion. So there is no realistic chance to convince me here.
  There is another point where I am not sure at all. In your final comment, you suggest to consider the version where the thickness is parameterized in terms of flux and slope instead of flux alone. However, my results already show that such a parameterization leads to a weird scaling of thickness vs. width in a steady state. While width is proportional to q^0.3 then, thickness is proportional to q^0.7. As far as I can see, your recent approach including deformation softens this problem, but thickness still increases more rapidly with flux than width. I guess that the problem already comes in when parameterizing the width by the flux alone without taking into account the slope. If so, it already affects the 2020 EPSL paper by Günther Prasicek where both of us were coauthors, but unfortunately I did not think about this at that time. So it would also be interesting to look at thickness vs. width in your recent results. But for my concept, this mainly tells me that we must consider all approximations as a package and compare it, e.g., to simulations with iSOSIA.
  Best regards,
  
  Stefan
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC1
CC1:
'Comment on esurf-2021-1', Flavien Beaud, 06 Mar 2021
Comment on: A stream-power law for glacial erosion and its implementation in large-scale landform-evolution models, by Stefan Hergarten.
I was not formally asked to review this paper, but I believe my expertise can be useful to improve the manuscript. In that sense, I will focus on very general comments and I am not proposing a detailed review.
The main focus of the study is to develop a landform evolution model that can be solved more readily with less computational power than the current existing models. This would, indeed, be a very useful contribution to glacial geomorphology, especially as the author are making their code open source. To do so, the author derive a stream power law for glacial erosion, including a parametrization of bedrock erosion and sediment transport by subglacial water flow.
My main concern regarding the study is that the author provides little to no explanations for the simplifications they propose, and in many cases, these simplifications are contradictory to the current understanding of these erosion and transport mechanisms. Throughout the paper, there are very few explanations of the current state of knowledge of the physical processes involved in glacial erosion and sediment transport. Most of the papers cited are numerical models that already make significant assumptions. For example, in Section 2, l. 63-64, the derivation of the glacial stream power starts with the statement “If we consider a rectangular cross section […]”. Yet, we know glacial valleys are not rectangular and there is no explanation of that simplification.
Another point of concern is the recurring citation of Prasicek et al. (2020) to justify numerous simplifications. In the Prasicek et al. (2020) study, the goal is to assess timescales of respective processes: glacial erosion, climate and tectonic; not to reproduce landforms. The assumptions Prasicek et al. (2020) are making to assess the relative effectiveness of processes over time, become inadequate when applied to a model that aims at reproducing landforms themselves.
To be more specific, I believe the following reasonings are problematic:
While reading the paper, I was left with the feeling that because there is a stream power erosion law for rivers there should be one for glaciers. In river systems the shear stress can be approximated with discharge, which in turn can be approximated with catchment area and precipitation. Futhermore, observations by Hack (1957) support a simple scaling. For glacier erosion, however, not only do erosion and transport mechanisms remain unclear in general, but there is no large-scale field observations that support such scaling. The paper cited to support such scaling (Bahr, 1997) refers to glaciers themselves, not glacial landscapes, and is therefore not applicable to glacier erosion. In term of the specifics of glacial erosion, even assuming a simple relationship between erosion and sliding velocity to a power, this power is likely >1 (Herman et al., 2015; Koppes et al., 2015). Sliding velocity is intricately linked to basal shear stress and basal water pressure (e.g. Beaud et al., 2014; Ugelvig et al., 2018). Therefore, a thorough explanation is necessary to simplify our current state-of-the-art understanding of the physical processes of glacial erosion into a simple stream power-type law.

Using a shallow ice approximation has be shown to not work very well on steep landscapes (Egholm et al., 2011). These simplifications and their implications for landscape evolution results should be explained.

Model of glacio-fluvial incision (l. 229-232): “While Beaud et al. (2016) developed a more elaborate model for the incision 230 by meltwater within narrow channels, the meltwater component should preferably not introduce a level of complexity much beyond the simple models of fluvial and glacial erosion used here. So let us assume that erosion by meltwater can be described by the same formalism as fluvial erosion.” That statement is in opposition to the results presented in the Beaud et al. (2016) study. Since that model is the only to date to describe such mechanism, dismissing the results or making different assumptions should be substantiated.

Finally, the authors are omitting the vast majority of the glacial landform evolution literature together with the literature explaining the physical underpinning of glacial erosion and sediment transport. In the references at the end of my comment, I propose a non-exhaustive list of papers that should be cited and the finding of which should be included in the current study.

In summary, in its current form, I do not believe that the current study makes a convincing point that a stream power glacial erosion rule can, or should be used. If the goal of the paper is to reproduce landforms, the author should include and discuss extensively the existing literature and justify their simplifications. If the goal is to produce a model that can be used to test interplay between erosion, climate and tectonic, similarly to Prasicek et al. (2020), that should also be clarified. In any case, I believe, the assumptions should be discussed in light of the existing literature.
I hope you find my comments constructive and am available to answer further questions if necessary (flavien.beaud@ubc.ca).
Best,
Flavien Beaud
References
Anderson, R. S. (2014). Evolution of lumpy glacial landscapes. Geology, doi:10.1130/G35537.1.
Beaud, F., Flowers, G. E., & Pimentel, S. (2014). Seasonal-scale abrasion and quarrying patterns from a two-dimensional ice-flow model coupled to distributed and channelized subglacial drainage. Geomorphology, 219, 176–191.
Beaud, F., Flowers, G. E., & Venditti, J. G. (2016). Efficacy of bedrock erosion by subglacial water flow. Earth Surface Dynamics, 4, 125–145. https://doi.org/doi:10.5194/esurf-4-125-2016
Beaud, F., Flowers, G. E., & Venditti, J. G. (2018). Sediment transport in ice-walled subglacial channels and its implications for esker formation and monitoring glacial erosion. Journal of Geophysical Research, Earth Surface.
Creyts, Timothy T., and Ian Hewitt. "Genesis of esker corridors as erosional features beneath ice sheets." In AGU Fall Meeting Abstracts, vol. 2019, pp. C21E-1496. 2019.
Delaney, Ian, and Surendra Adhikari. "Increased subglacial sediment discharge in a warming climate: Consideration of ice dynamics, glacial erosion, and fluvial sediment transport." Geophysical Research Letters 47, no. 7 (2020): e2019GL085672.
Egholm, D. L., Pedersen, V. K., Knudsen, M. F., & Larsen, N. K. (2011). On the importance of higher order ice dynamics for glacial landscape evolution. Geomorphology, 141--142, 67–80.
Herman, F., Beaud, F., Champagnac, J. D., Lemieux, J. M., & Sternai, P. (2011). Glacial hydrology and erosion patterns: A mechanism for carving glacial valleys. Earth and Planetary Science Letters, 310(3), 498–508.
Herman, F., Beyssac, O., Brughelli, M., Lane, S. N., Leprince, S., Adatte, T., … Cox, S. C. (2015). Erosion by an Alpine glacier. Science, 350(6257), 193–195.
Iverson, N. R. (2012). A theory of glacial quarrying for landscape evolution models. Geology, 40(8), 679–682.
Koppes, M., Hallet, B., Rignot, E., Mouginot, J., Wellner, J. S., & Boldt, K. (2015). Observed latitudinal variations in erosion as a function of glacier dynamics. Nature, 526(7571), 100–103.
MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Waddington, E. D. (2009). Numerical modeling of glacial erosion and headwall processes in alpine valleys. Geomorphology, 28(2), 189–204.
MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Waddington, E. D. (2000). Numerical simulations of glacial longitudinal profile evolution. Geology, 28, 1031–1034.
Riihimaki, C. A., MacGregor, K. R., Anderson, R. S., Anderson, S. P., & Loso, M. G. (2005). Sediment evacuation and glacial erosion rates at a small alpine glacier. Journal of Geophysical Research, 110(F3), F03003.
Ugelvig, S. V., D. L. Egholm, R. S. Anderson, and Neal R. Iverson. "Glacial erosion driven by variations in meltwater drainage." Journal of Geophysical Research: Earth Surface 123, no. 11 (2018): 2863-2877.
Citation: https://doi.org/10.5194/esurf-2021-1-CC1
- AC2: 'Reply on CC1', Stefan Hergarten, 06 Mar 2021
  
  Dear Flavien Beaud,
  thanks for your comments! I hoped that some more researchers from the community of modeling glacial erosion would share their opinion, and that they would perhaps even go beyond keywords. I am convinced that your expertise as a reviewer when considering the proposed concept seriously would have been helpful to improve the paper.
  Let me briefly comment on some of your points.
  For example, in Section 2, l. 63-64, the derivation of the glacial stream power starts with the statement "If we consider a rectangular cross section [...]". Yet, we know glacial valleys are not rectangular and there is no explanation of that simplification.
  This is just some kind of entry gate for the readers. Those who think this way should stop reading at this point and not waste their time.
  
  I hope there will be some readers not arguing on this level, but are able to think about the consequences of this simplification, although
  
  I am aware that not all will be able to solve the exercise of finding out for which shapes of cross sections it remains valid.
  Another point of concern is the recurring citation of Prasicek et al. (2020) to justify numerous simplifications. In the Prasicek et al. (2020)
  
  study, the goal is to assess timescales of respective processes: glacial erosion, climate and tectonic; not to reproduce landforms. The assumptions Prasicek et al. (2020) are making to assess the relative effectiveness of processes over time, become inadequate when applied to a model that aims at reproducing landforms themselves.
  Sorry, I am not such an expert on this since I only developed parts of the theoretical framework of the Prasicek et al. (2020) paper. So far I thought it was about longitudinal equilibrium profiles for a dendritic topology and not about thime scales. But maybe I just missed the key point of that paper.
  The paper cited to support such scaling (Bahr, 1997) refers to glaciers themselves, not glacial landscapes, and is therefore not applicable to
  
  glacier erosion.
  I thought it was about length-width scaling of glaciers and used nothing else from this paper. And in contrast to previous studies, I was honest enough to explain that transferring scaling relations from entire glaciers into individual glaciers is nontrivial.
  In term of the specifics of glacial erosion, even assuming a simple relationship between erosion and sliding velocity to a power, this power
  
  is likely > 1.
  I think I mentioned this, and anyone should feel free to run the model with an exponent > 1. Some researchers may even prefer this for the fluvial regime since there were some studies that suggested exponents > 1 (although I think there are many pitfalls and artifacts there).
  Using a shallow ice approximation has be shown to not work very well on steep landscapes (Egholm et al., 2011). These simplifications and their implications for landscape evolution results should be explained.
  This is not just a problem of steep topographies, but applies to all combinations of shallow-water and shallow-ice equations with
  
  incision-type erosion laws if lateral stresses are not taken into account. Iti s explained briefly in the paper, and in principle it
  
  is an argument in favor of approaches where erosion is driven by a linear element.
  Model of glacio-fluvial incision (l. 229-232): "While Beaud et al. (2016) developed a more elaborate model for the incision by meltwater within narrow channels, the meltwater component should preferably not introduce a level of complexity much beyond the simple models of fluvial and glacial erosion used here. So let us assume that erosion by meltwater can be described by the same formalism as fluvial erosion." That statement is in opposition to the results presented in the Beaud et al. (2016) study. Since that model is the only to date to describe such mechanism, dismissing the results or making different assumptions should be substantiated.
  I cited this paper because it is apparently the first modeling approach towards bedrock incision by meltwater channels. However, I was neither able to recognize any fundamentally new theoretical concepts nor any results that could be generalized to larger scales. So it is not clear to me where the contradiction is. Did you find that meltwater incision in total independent of the meltwater discharge or find that the pressure gradient is totally decoupled from the gradient of the ice surface? Maybe I missed the key point again.
  In summary, in its current form, I do not believe that the current study makes a convincing point that a stream power glacial erosion rule can, or should be used.
  But unfortunately, other authors recently proposed a glacial stream-power law that is very similar to my version (Deal & Prasicek, Geophys. Res. Lett, 2020). They were definitely faster than me. Feel free to write a comment to their paper!
  I hope you find my comments constructive and am available to answer further questions if necessary.
  To be honest, not really. But I definitely appreciate your contribution to the discussion.
  Best,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC2
RC2:
'Comment on esurf-2021-1', Marc Jaffrey, 14 Apr 2021
I recommend rejecting the paper.

In short the author appears to forcing glacial erosion into the framework of fluvial erosion with little or outright incorrect reasoning.

Main Reasons:

The author does not appear to be familiar with nor understand sufficiently glacier dynamics and glacier erosion for which an extensive body of literature exists. The theoretical underpinnings of the paper are unfounded and in several key places in direct conflict with known glacier dynamics.

The author has not sufficiently reviewed the literature.

The paper is not written to address the glaciology community which I presume is a key target audience for the work.

Some key examples of problems highlight above:

Section 10: “In contrast to fluvial erosion, however, glacial erosion has not been extensively considered in modeling large-scale landform evolution.” There is an extensive body of literature on modeling large-scale glacial landform development.

35-40: “A comparable representation of glacial erosion where the erosion rate can be directly computed from properties of the topography is not yet available.” This is not true and stems from the authors misunderstanding of the relationship between glacier dynamics and existing glacier erosion laws implemented within the glacial geomorphology community.

45-50: There are many different form of erosion laws considered by the glacier community of which the author does not acknowledge.

50: If the author is talking about the general mathematical concept of mass balance they need to say so. Mass balance in the glaciology community is something different and the connection to diffusion would require further explanation on the authors part.

60-65: This is false. The author makes a critical assumptions here defining the model presented which is a critical misunderstanding of glacier dynamics and the Shallow Ice-sheet approximation. The ratio of the depth averaged velocity and sliding velocity is categoricallly not controlled by ice thickness, not even a second order control. Sliding velocity is a complex issue which from a dynamic’s perspective, the principal first order controls are effective water pressure, bed temperature, bed roughness and a host of other bed parameters and processes. From a mathematical and numerical simulation perspective sliding velocity is treated independently with a sliding law, a Robin type boundary condition, which is typical defined in terms of the aforementioned bed processes, not ice thickness. The author seems to be unaware of glacier dynamics. Since the assumptions in 60-65 are crucial to the theoretical underpinning of the model the rest of the work, regardless of the care taken with the numerical simulation, is unfounded.

100: “The erodibility K is a lumped parameter that already includes precipitation “ This statement requires a detailed explanation on the author’s parts of what they mean by this statement.

100: This whole section is problematic. The author seems to be connecting concepts without justifications.

225: “These findings support the idea that erosion by meltwater must play an important role, at least in the lower part of glaciers where the flux of water is much higher than the ice flux.” One should always be careful when drawing dynamical conclusions from a numerical model especially when parameter tuning can be used to generate a wide range of results. That is to say, the author is making a circular argument and in this case one that is in conflict with the glacier community. Correlation is not causation.

20-25: References to convexity a concept from fluvial erosion is not one familiar to the glacier community and requires explanation.
Citation: https://doi.org/10.5194/esurf-2021-1-RC2
- AC3: 'Reply on RC2', Stefan Hergarten, 14 Apr 2021
  
  Dear Marc Jaffrey,
  thanks for your comments! You speak a lot about "an extensive body of literature", "known glacier dynamics", and "the glaciology community". Unfortunately, it is difficult for me to recognize what you refer to. I tried to find out from you own work in this field, but only found a conference poster about basin-wide glacial erosion rates. Although I like the concept proposed there, I could not find out where the fundamentals of my work basically differ from what you assume, and I also cannot see how the concept of a mass balance differs as you claim.
  Let me just comment on your point highlighted by boldface line numbers :
  "60-65: This is false. The author makes a critical assumptions here defining the model presented which is a critical misunderstanding of glacier dynamics and the Shallow Ice-sheet approximation. The ratio of the depth averaged velocity and sliding velocity is categoricallly not controlled by ice thickness, not even a second order control. Sliding velocity is a complex issue which from a dynamic’s perspective, the principal first order controls are effective water pressure, bed temperature, bed roughness and a host of other bed parameters and processes. From a mathematical and numerical simulation perspective sliding velocity is treated independently with a sliding law, a Robin type boundary condition, which is typical defined in terms of the aforementioned bed processes, not ice thickness. The author seems to be unaware of glacier dynamics. Since the assumptions in 60-65 are crucial to the theoretical underpinning of the model the rest of the work, regardless of the care taken with the numerical simulation, is unfounded."
  This is just what is typically assumed for the sliding and deformation velocities (not "depth averaged velocity" as you state). I agree that the factors of proportionality in Eqs. (4) and (5) depend on what you mention, but nevertheless the (quite old) model ICE-CASCADE and to some extent also the state-of-the-art model iSOSIA use these relations. They are also used in more theoretical considerations, e.g., Prasicek et al. (2018), doi 10.1029/2017JF004559 and Deal & Prasicek (2021) doi 10.1029/2020GL089263. You are free not to like this approach and propose alterantive concepts. In general, however, I would find it more useful to bring your own ideas to the scientific audience than just tearing down other researchers' work on a keyword level.
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC3
RC4:
'Comment on esurf-2021-1', Marc Jaffrey, 16 Apr 2021
First, there is no conflict of interest: My work and the author’s work are categorically different and furthermore address different spatial and temporal domains. My work is theoretical with the development of an analytical model of glacier erosion rates at the basin scale, while the author’s work addresses glacier erosion at smaller spatial scales utilizing a heuristic approach to define a glacial erosion law for numerical implementation, as is standard for numerical approaches.
Next, please let me express my regret for the tone of my earlier comments without elucidating clear points for the author to address. Since the work makes assumptions in direct conflict with current theories of glacier dynamics, I will not address the computational aspects of the work as the premises on which the model is built are problematic and in this reviewer’s opinion unrealistic rending the erosion law in equation 19 open to sever question. Restating the main issues:
The author has not cited nor sufficiently discussed existing literature: This is an issue as the proposed scientific contribution this work cannot be understood except but subject matter experts.

The theoretical underpinnings, the glacier dynamics, in several key place are incorrect in section two and three rendering the hypothesized glacier erosion law, the equation 19, the critical key equation of the numerical model, open to question.

One of the most concerning issue in terms are some of the scientific conclusions of the paper. Most notably in 225: “These findings support the idea that erosion by meltwater must play an important role, at least in the lower part of glaciers where the flux of water is much higher than the ice flux.” This statement cannot be justified by the a numerical model regardless of soundness of the theoretical underpinnings. While the contributions of numerical simulations to the research and progress of understanding glacier erosion cannot be overstated, drawing such definitive conclusions about mechanisms of glacier erosion is out of reach for this type of approach.

The paper is not written to address the glacial community which I presume is a key target audience for the work.

Based on these issues I cannot recommend publication of the work as in my opinion the prosed erosion law is unjustified calling into question the numerical model and its results.
Focused Comments for Section 1:
1 -5: “In contrast to fluvial erosion, however, glacial erosion has not been extensively considered in modeling large-scale landform evolution”
Here is a list of reference the author might consult:
Egholm, D. L., et al. "On the importance of higher order ice dynamics for glacial landscape evolution." Geomorphology 141 (2012): 67-80.
Harbor, J., 1989. Early Discoverers XXXVI: W J McGee on glacial erosion laws and the development of glacial valleys. Journal of Glaciology 35, 419–425.
Harbor, J., Hallet, B., Raymond, C. A numerical model of landform development by glacial erosion. Nature 333, 347–349 (1988).
Harbor, J. Numerical modeling of the development of U-shaped valleys by glacial erosion. GSA Bulletin 104, 1364–1375 (1992).
Herman, F., Beaud, F., Champagnac, J.D., Lemieux, J-M., Sternai, P. Glacial hydrology and erosion patterns: A mechanism for carving glacial valleys. Earth and Planetary Science Letters, 310, 498–508 (2011).
MacGregor, K.R., Anderson, R.S., Anderson, S.P., Waddington, E.D. Numerical simulations of glacial–valley longitudinal profile evolution. Geology 28, 1031–1034 (2000). Oerlemans, J. Numerical experiments of large-scale glacial erosion. Zeitschrift fuer Gletscherkunde und Glazialgeologie 20, 107–126 (1984)
Oerlemans, J. Numerical experiments of large-scale glacial erosion. Zeitschrift fuer Gletscherkunde und Glazialgeologie 20, 107–126 (1984).
Tomkin, J.H. Numerically simulating alpine landscapes: The geomorphologic consequences of incorporating glacial erosion in surface process models. Geomorphology 103, 180-188 (2009).
Ugelvig, S.V., Egholm, D.L., Iverson, N.R. Glacial landscape evolution by subglacial quarrying: A multiscale computational approach. J. Geo. Res. E. Sur. 121, 2042-2068 (2016).
Ugelvig, S. V., et al. "Overdeepening development in a glacial landscape evolution model with quarrying." AGU Fall Meeting Abstracts. Vol. 2013. 2013.
35-40: “A comparable representation of glacial erosion where the erosion rate can be directly computed from properties of the topography is not yet available.”
It is essential discuss the scales, both temporal and spatial, on which erosion rates are being considered. Though not explicitly discuss in the introduction, it can be implied from context that the spatial scales considered are smaller than the scale of the glacial landforms which is sub basin scale.
Alley, R. B., K. M. Cuffey, and L. K. Zoet. "Glacial erosion: status and outlook." Annals of Glaciology 60.80 (2019): 1-13.
Alley, R. B., et al. "Stabilizing feedbacks in glacier-bed erosion." Nature 424.6950 (2003): 758-760.
Andrews J.T. Glacier power, mass balances, velocities and erosion potential. Zeitschrift fur Geomorphologie 13, 1-17 (1972).
oulton, Geoffrey S. "Processes and patterns of glacial erosion." Glacial geomorphology. Springer, Dordrecht, 1982. 41-87.
Boulton, G. S. "Theory of glacial erosion, transport and deposition as a consequence of subglacial sediment deformation." Journal of Glaciology 42.140 (1996): 43-62.
Cook, Simon J., et al. "The empirical basis for modelling glacial erosion rates." Nature communications 11.1 (2020):
Delmas, M., Calvet, M., Gunnell, Y. Variability of quaternary glacial erosion rates— a global perspective with special reference to the Eastern Pyrenees. Quat. Sci. Rev. 28, 484–498 (2009).
Hall, Adrian M., et al. "Glacial ripping: geomorphological evidence from Sweden for a new process of glacial erosion." Geografiska Annaler: Series A, Physical Geography (2020): 1-21.
Hallet, B. Glacial abrasion and sliding: their dependence on the debris concentration in basal ice. Annals of Glaciology 2, 23-28 (1981).
Hallet, B. Glacial quarrying: A simple theoretical model. Annals of Glaciology 22, 1–8 (1996).
Iverson, N.R. A theory of glacial quarrying for landscape evolution models. Geology 40, 679–682 (2012).
Menzies, J., Jaap JM van der Meer, and W. W. Shilts. "Subglacial processes and sediments." Past glacial environments. Elsevier, 2018. 105-158.
Steinemann, Olivia, et al. "Quantifying glacial erosion on a limestone bed and the relevance for landscape development in the Alps." Earth Surface Processes and Landforms 45.6 (2020): 1401-1417
Ugelvig, S. V., et al. "Glacial erosion driven by variations in meltwater drainage." Journal of Geophysical Research: Earth Surface 123.11 (2018): 2863-2877.
Equation 4 is incorrect: See chapter 7, eq's 7.6, 7.10, 7.17, 8.35, 8.36, 8.65 and sections 8.1, 8.4, 8.5 and 8.6 in Cuffey, Kurt M., and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010. Sliding velocity cannot be reduced to ice thickness and slope under any approximation. As Cuffey and Patterson in discussion the Shallow Ice Approximation say in chapter 8, "The rate of basal slip must be specified directly or through a relation to bed stress such as Eq. 8.25"
Equation 6: Yes erosion laws of this form are typically implemented, however there are many other forms that have been used within numerical simulation.
50: This section requires substantially further discussion and justification. Ice thickness is not a diffusion process. See section 8.5.5 and equation 8.65 , 8.70, 8.77, 8.78 and 8.79 in Cuffey and Patterson. In the Shallow Ice Approximation ice flux q~h not the partial derivative of h wrt to x. There is a divergence relationship, first order partial derivatives, but diffusion is second order in the spatial partial derivatives. Without substantial justification, ice thickness cannot be treated as diffusion with strong diffusivity.
Section 2:
Again equation 4 is unfounded so that section 2 begins with a false premise. This problem then follows through into equations 8, 9, and the key equation 14 rendering it unfounded.
70-80: This section requires further explanation.
100: The authors treatment of catchment size, precipitation, and discharge may have a clear rationale for fluvial systems, but it is not clear why this would apply to glacial system except perhaps at the terminus. The author needs a detail justification.
Equation 19: Taken as a whole, the validity of the proposed erosion law is questionable.
Citation: https://doi.org/10.5194/esurf-2021-1-RC4
- AC4:
  'Reply on RC4', Stefan Hergarten, 18 Apr 2021
  Dear Marc Jaffrey,
  thanks for the more specific version of the review and in particular for the personal email you sent to me. This makes your point about some of the numerical models used on this field more clear to me. I am fully with you that it is a serious problem of science that things become "true" just by publishing and citing repeatedly.
  Nevertheless, it seems indispensable for me to state that your arguments about the equations are completely wrong in my opinion.
  
  Concerning Eq. (4), you state: "Equation 4 is incorrect: See chapter 7, eq's 7.6, 7.10, 7.17, 8.35, 8.36, 8.65 and sections 8.1, 8.4, 8.5 and 8.6 in Cuffey, Kurt M., and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010. Sliding velocity cannot be reduced to ice thickness and slope under any approximation. As Cuffey and Paterson in discussion the Shallow Ice Approximation say in chapter 8, `The rate of basal slip must be specified directly or through a relation to bed stress such as Eq. 8.25'"
  
  Equation 4 just requires v_s ~ τ^ψ/σ (Budd et al., J. Glacial.,23, 157-170, 1979) where τ = shear stress and σ = effective normal stress. According to the shallow ice approximation, τ ~ hS (h = ice thickness, S = slope of the ice surface). In this form, it yields v_s ~ h^ψ/(h-p) S^ψ where p = water pressure. Among the references you provided, at least Harbor, Hallet & Raymond, Nature 333, 347-349 (1988) and Tomkin, Geomorphology 103, 180-188 (2009) assumed p ~ h, which exactly yields Eq. 4, v_s ~ h^ψ-1S^ψ. You may question the assumption that the fluid pressure at the bed is proportional to the ice thickness and claim that coupling with a distinct model for melting and for the flow of melt water may yield better results, and that such models are available. However, your statement "Sliding velocity cannot be reduced to ice thickness and slope under any approximation" is wrong.
  
  About the type of the differential equation: "Ice thickness is not a diffusion process. See section 8.5.5 and equation 8.65 , 8.70, 8.77, 8.78 and 8.79 in Cuffey and Paterson. In the Shallow Ice Approximation ice flux q~h not the partial derivative of h wrt to x. There is a divergence relationship, first order partial derivatives, but diffusion is second order in the spatial partial derivatives. Without substantial justification, ice thickness cannot be treated as diffusion with strong diffusivity."
  
  Sorry, but the shallow water equations and its derivates (shallow ice equations and Savage-Hutter equations for granular flow) assume a free surface and hydrostatic pressure in vertical direction. Then the horizontal force is proportional to the gradient of the free surface and not to the gradient of the bed (as you presumably assume in your reasoning). As a consequence, the velocity in the shallow ice equation also depends on h and on the gradient of the ice surface. Then the divergence term from the mass balance yields second-order spatial derivatives of h in total. The resulting spatial differential operator is elliptic, so that the entire time-dependent equation is a parabolic equation (= diffusion type for the broader readership).
  
  Best regards,
  
  Stefan
  
  Citation: https://doi.org/10.5194/esurf-2021-1-AC4

Stefan Hergarten

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Short summary

This paper presents a new approach to modeling glacial erosion on large scales. The formalism is similar to large-scale models of fluvial erosion, so glacial and fluvial processes can be easily combined. The model is simpler and numerically less demanding than established models based on a more detailed description of the ice flux. The numerical implementation achieves almost the efficiency of purely fluvial models, so that simulations over millions of years can be performed on standard PCs.


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