the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
MinVoellmy v1: a lightweight model for simulating rapid mass movements based on a modified Voellmy rheology
Abstract. The Voellmy rheology has been widely used for simulating snow avalanches and also for rock avalanches. Recently, a modified version of this rheology was proposed. While the conventional version of Voellmy's rheology uses the sum of Coulomb friction and a velocitydependent friction term, the modified version assigns the two terms to different regimes of velocity. The software MinVoellmy presented here provides the first numerical implementation of the modified rheology. It consists of MATLAB and Python classes, where simplicity and parsimony were the design goals. In contrast to the majority of the models in this field, MinVoellmy uses a Cartesian coordinate system and a simple upstream scheme, which turns out to be sufficient for rheologies of the Voellmy type. Numerical tests reveal that the modified Voellmy rheology reproduces the empirical relation between runout length, height drop, and volume of large rock avalanches quite well. Furthermore, there seems to be a large potential for further research on hummocky deposit morphologies and longitudinal striations. However, the MinVoellmy software is only designed for research and teaching, but not for operational use in realworld hazard assessment.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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RC1: 'Comment on egusphere2023802', Anonymous Referee #1, 09 Jul 2023
The paper introduces a numerical model based on a modified Voellmy friction model that is currently under review in a different manuscript. The mechanical concept follows Hergarten and Robl (2015).
From a scientific point of view, this work does not provide many novelties and innovations. The modified friction model is mentioned as the motivation for this work and seems to be its focal point. Therefore I would merge this manuscript with the other manuscript under review.
The model seems to follow the same concept as SHALTOP (Brunet et al. 2017) but comes to different conclusions and governing equations. SHALTOP also provides an exact solution of the SavageHutter model projected to a flat surface, while this work provides an approximation only. This should be discussed. What is the upside form the approximations done in this work? How do they compare?
Considering this is a new piece of code, I miss basic tests, e.g. comparisons with experiments, analytical solutions or existing software.
Citation: https://doi.org/10.5194/egusphere2023802RC1 
AC1: 'Reply on RC1', Stefan Hergarten, 12 Jul 2023
Dear Reviewer,
thanks for your comments! I guess that merging two manuscripts in journals with different foci is not thought to be a serious suggestion. Model description papers in GMD should not present scientific results (the runout scaling as a function of volume in the ESurf manuscript), while ESurf is not a good place for technical descriptions of software.
A 100 line code (for the timestep) probably cannot compete with the model SHALTOP, and I do not mind mentioning this. On the other hand, readers have to pick information from multiple papers and to rely on the authors providing them with the source code before being able to work with SHALTOP. For me, this is an argument in favor of having all relevant information in one paper and to be able to download the code.
Concerning the approach, the main difference towards the SavageHutter model (and presumably also SHALTOP and the rather crude Cartesian approach by Hergarten & Robl 2015) is that averaging is not performed normal to the bed, but vertically. As long as the layer
is sufficiently thin in relation to the radius of curvature of the bed, this seems to be an unnecessary approximation to the SavageHutter model.
In many applications, however, this condition is not satisfied, e.g., if the bed is rough. Then measuring and averaging vertically instead
of normal to the rough bed is presumably better, although I cannot quantify it mathematically and do not want to put too much weight on this aspect. So your statement that SHALTOP provides and exact solution of the SavageHutter model is formally correct, but is not the whole truth.
As a second aspect, it is shown that shockpreserving numerical schemes are not as important as usually assumed for a certain type of rheologies (if friction decreases with thickness). This is illustrated quite in detail, although probably unimportant for you because SHALTOP uses a shockpreserving scheme. Anyway, the manuscript is not about arguing against any established model, but rather for illustrating how simple the numerical treatment can be.
Finally, your comment about the missing basic tests confuses me. I thought that the comparison with the analytical solution for a front (with and without the approximation) was such a test.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023802AC1

AC1: 'Reply on RC1', Stefan Hergarten, 12 Jul 2023

RC2: 'Comment on egusphere2023802', Fabian Walter, 25 Sep 2023
This submission by Hergarten presents a new implementation of the shallow water approach to granular flow modeling under the assumption of friction that switches between a Coulomb and velocitydependent parameterization. The novelty is the adaption of a Cartesian coordinate system, which makes the implementation more tractable but may cause numerical problems. With a redefined expression for the hydrostatic pressure, the author is able to produce simple granular flows that agree with conventional solutions in a bedparallel coordinate system.
The manuscript is written well, and most parts are straightforward to follow. The findings are supported with test model runs and analytical arguments. My main criticism concerns a lacking discussion in the context of the SavageHutter approach to the shallow water approximation. In the detailed comments below, I elaborate on this point and in various other parts of the text I ask the author to add clarification. Overall, I enjoyed reading the manuscript and believe that with the additional clarifications it can be brought to publication quality.
I would like to stress that I am not a model developer and hence I had to go through the literature background to provide this review. This explains why I took longer to submit my feedback for which I apologize. In addition, some of my criticism may not be justified or my questions may be trivial to answer. Since a simplified model framework as presented here is of particular interest to the modeling novice like myself, my review should nevertheless be of use.
Fabian Walter.
MAJOR COMMENTS
=================
The reformulated pressure expression is aimed to circumvent the numerical singularity that arises for situations when bed and surface gradients are perpendicular. The author acknowledges that this situation is not representative for the SavageHutter approximation but of interest in view of numerical considerations. At this point I wonder if the defiance of the SavageHutter approximation can be dismissed in such an easy way: After all, their approach is not only a geometric argument on the bedsurface configuration. Instead, it is used to simplify the NavierStokes equations by neglecting terms with the help of scaling arguments. Can these scaling arguments be brought in agreement with the perpendicular bed/surface geometry? I suggest clarifying this point in the context of a discussion on the SavageHutter approximation.
Most equations were discussed and presented in a way that allows the reader to verify and understand them. However, I strongly suggest adding a sketch in which bed and surface are shown and angles are defined. This sketch should also define the signs of different quantities (e.g., angles and acceleration), which are crucial for the presented material.
Finally, the reader needs more information where certain equations come from or how assertions are justified. First, Lines 5063 and Lines 8084 include important statements without references. Second, the balance equations (6) and (7) are given without a reference. I was able to derive the former one from the form given in Savage and Hutter (1989), but the second one is not so straightforward. The reader should be presented with a clear source in the literature. Moreover, all assumptions that go into these equations should be stated (e.g., incompressibility?). As another remark on these equations: please state if the del operators act on the product of v_x and h or on the first factor, only.
SPECIFIC COMMENTS
=================
Equation 5: Here and elsewhere, are the bed and free surface b(x,y) and s(x,y) defined as level sets? I.e., b(x,y)z(x,y)=0? This should be stated.
Line 112: “where f is the absolute value” of what?
Equation (8): Is “p” defined? Provide a reference for this equation. Perhaps trivial, but why is a dot n = 0 a requirement?
Equation (12): Is hydrostatic pressure assumed here? Or what is the motivation for this equation?
Line 134: See main comment above. Also, it may help referring to the trigonometric identity tan(theta)=cot(thetapi/2) to concisely pinpoint the singularity.
Figure 1: Here it would be extremely helpful to see a sketch for the geometry of the depicted situations (see main comment above). Similar, the asymmetry discussion (Lines 145ff) would benefit from a sketch depicting uphillfacing and downhillfacing fronts. Is “a” the acceleration in the bedparallel direction? The dashed lines should be defined in the captions and not only in the main text. The description of the singularity makes sense, but I do not understand why only the dashed parts of the curve are shifted right under the coordinate transformation. Perhaps this can be rephrased.
Line 159: Which “curves of the two models”?
Line 164: “the good properties of the equation”: Which properties of which equations?
Line 169: This may sound like splitting hairs, but I would stick to acceleration and call it either uphill or downhill, ideally associating the two with a sign using a sketch (see major comments).
Lines 174175: I suggest including a reference for “earth pressures” since this seems widely used in the soil mechanics literature.
Equation 21: Define c.
Equation 22: Should min be max?
Equation 23: The Euler scheme references I found include a + rather than a – sign between the two RHS terms.
Equation 24 and the fowling equations: The primes are not derivatives, right? If so, I suggest specifying this.
Line 241: The referenced equation is a proportionality. How is the constant of proportionality determined?
Line 255: Give a reference for the CourantFriedrichsLewy criterion.
Lines 259260: Avoid 1sentence paragraphs. Define the «rectangle».
Lines 267: Which «versions»?
Line 273: Rewrite “some waves”.
Line 280: melts down decreases
Line 288: “straight slope” is inappropriate since slope is a scalar value.
Equation 37: Should the cosines be squared?
Lines 310ff: How is this supported? Is there a corresponding figure?
Figure 3: There seem to be several curves with the same color, please allow for better distinction. I could not distinguish lines with and without markers. For some of the lines in (b) there seems to be an overhang near x=0. If intended, I suggest commenting.
Line 331: Robustness with respect to what?
Line 332333: “works well technically” should be qualified better or even quantified.
Figure 4: I suggest labeling the colors directly in the plot (e.g., with a legend or text boxes). The reader will appreciate this.
Line 339: Delete one “necessary”.
343: Why does the velocity remain constant at the kink? Because it is a single point in space?
Figure 5: I suggest outlining the release areas rather by encircling it. Referring to them as “red lines” in the caption is confusing. Also, I strongly recommend pointing out the striations and hummocks discussed in the text directly in the figure.
Lines 368ff: It is not obvious to me why transverse diffusion is not allowed in Equation 7. After all, both horizontal components and gradients are present. Perhaps this is a wellknown fact, but it would be interesting to have more information on this phenomenon.
Line 371: obviously > apparently
Line 372: A reason or ideally a reference why the longitudinal striations are realistic should be given.
Line 373: into > in
Line 376: Rather than using “strongly” I suggest quantifying the grid orientation effect.
Line 381: I am surprised that this was the motivation for model development. After all, other models have reproduced long runouts. Perhaps a clarifying sentence would help.
Figure 7: The different alphas should be labeled directly in the plot.
Line 394: Which proportionality factor was used?
Lines 396397: Rewrite “without getting into conflict …”.
Line 419: What makes the other models more comprehensive?
Lines 420421: Why would the time it takes to develop a model affect its use in hazard assessment?
Lines 423424: From my experience even when applying the same model to two different sites, parameter transferability is limited.
MINOR COMMENTS
================
Line 137: A singularity also exists if grad h grows without bounds. I suggest using other terminology.
Fewer adjectives and adverbs reflecting subjectivity in argumentation should be used. Line 8: Delete “quite well”. Line 60: Delete “quite”. Line 269: Delete “very”. Line 284: Rewrite “works well”. Line 347: delete “quite”. Line 425: Delete “very”.
Line 37: Rewrite/correct “the lowest friction a low velocities”.
I may have missed it, but I could not find a plugandplay matlab or python script on the code repository. It would be helpful if there was a setup that can simply be executed to reproduce the shown figures.
Citation: https://doi.org/10.5194/egusphere2023802RC2 
AC2: 'Reply on RC2', Stefan Hergarten, 18 Oct 2023
Dear Fabian Walter,
thanks for your constructive comments! Your arguments about the starting level make sense to me. The approach and the implementation are indeed simpler than in other models, so that there could be a chance to make the stuff accessible to researchers who are not so familiar with numerical modeling. On the other hand, however, a model description paper in GMD has to be concise. So I am still not sure how much I can improve the accessibility to novices in model development, but I will give my best.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023802AC2

AC2: 'Reply on RC2', Stefan Hergarten, 18 Oct 2023

EC1: 'Comment on egusphere2023802', Thomas Poulet, 27 Sep 2023
Dear author,
I am looking forward to reading your official response to the reviewers' comments and the corresponding revision of the manuscript.
Best regards,
Thomas Poulet.
Citation: https://doi.org/10.5194/egusphere2023802EC1
Interactive discussion
Status: closed

RC1: 'Comment on egusphere2023802', Anonymous Referee #1, 09 Jul 2023
The paper introduces a numerical model based on a modified Voellmy friction model that is currently under review in a different manuscript. The mechanical concept follows Hergarten and Robl (2015).
From a scientific point of view, this work does not provide many novelties and innovations. The modified friction model is mentioned as the motivation for this work and seems to be its focal point. Therefore I would merge this manuscript with the other manuscript under review.
The model seems to follow the same concept as SHALTOP (Brunet et al. 2017) but comes to different conclusions and governing equations. SHALTOP also provides an exact solution of the SavageHutter model projected to a flat surface, while this work provides an approximation only. This should be discussed. What is the upside form the approximations done in this work? How do they compare?
Considering this is a new piece of code, I miss basic tests, e.g. comparisons with experiments, analytical solutions or existing software.
Citation: https://doi.org/10.5194/egusphere2023802RC1 
AC1: 'Reply on RC1', Stefan Hergarten, 12 Jul 2023
Dear Reviewer,
thanks for your comments! I guess that merging two manuscripts in journals with different foci is not thought to be a serious suggestion. Model description papers in GMD should not present scientific results (the runout scaling as a function of volume in the ESurf manuscript), while ESurf is not a good place for technical descriptions of software.
A 100 line code (for the timestep) probably cannot compete with the model SHALTOP, and I do not mind mentioning this. On the other hand, readers have to pick information from multiple papers and to rely on the authors providing them with the source code before being able to work with SHALTOP. For me, this is an argument in favor of having all relevant information in one paper and to be able to download the code.
Concerning the approach, the main difference towards the SavageHutter model (and presumably also SHALTOP and the rather crude Cartesian approach by Hergarten & Robl 2015) is that averaging is not performed normal to the bed, but vertically. As long as the layer
is sufficiently thin in relation to the radius of curvature of the bed, this seems to be an unnecessary approximation to the SavageHutter model.
In many applications, however, this condition is not satisfied, e.g., if the bed is rough. Then measuring and averaging vertically instead
of normal to the rough bed is presumably better, although I cannot quantify it mathematically and do not want to put too much weight on this aspect. So your statement that SHALTOP provides and exact solution of the SavageHutter model is formally correct, but is not the whole truth.
As a second aspect, it is shown that shockpreserving numerical schemes are not as important as usually assumed for a certain type of rheologies (if friction decreases with thickness). This is illustrated quite in detail, although probably unimportant for you because SHALTOP uses a shockpreserving scheme. Anyway, the manuscript is not about arguing against any established model, but rather for illustrating how simple the numerical treatment can be.
Finally, your comment about the missing basic tests confuses me. I thought that the comparison with the analytical solution for a front (with and without the approximation) was such a test.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023802AC1

AC1: 'Reply on RC1', Stefan Hergarten, 12 Jul 2023

RC2: 'Comment on egusphere2023802', Fabian Walter, 25 Sep 2023
This submission by Hergarten presents a new implementation of the shallow water approach to granular flow modeling under the assumption of friction that switches between a Coulomb and velocitydependent parameterization. The novelty is the adaption of a Cartesian coordinate system, which makes the implementation more tractable but may cause numerical problems. With a redefined expression for the hydrostatic pressure, the author is able to produce simple granular flows that agree with conventional solutions in a bedparallel coordinate system.
The manuscript is written well, and most parts are straightforward to follow. The findings are supported with test model runs and analytical arguments. My main criticism concerns a lacking discussion in the context of the SavageHutter approach to the shallow water approximation. In the detailed comments below, I elaborate on this point and in various other parts of the text I ask the author to add clarification. Overall, I enjoyed reading the manuscript and believe that with the additional clarifications it can be brought to publication quality.
I would like to stress that I am not a model developer and hence I had to go through the literature background to provide this review. This explains why I took longer to submit my feedback for which I apologize. In addition, some of my criticism may not be justified or my questions may be trivial to answer. Since a simplified model framework as presented here is of particular interest to the modeling novice like myself, my review should nevertheless be of use.
Fabian Walter.
MAJOR COMMENTS
=================
The reformulated pressure expression is aimed to circumvent the numerical singularity that arises for situations when bed and surface gradients are perpendicular. The author acknowledges that this situation is not representative for the SavageHutter approximation but of interest in view of numerical considerations. At this point I wonder if the defiance of the SavageHutter approximation can be dismissed in such an easy way: After all, their approach is not only a geometric argument on the bedsurface configuration. Instead, it is used to simplify the NavierStokes equations by neglecting terms with the help of scaling arguments. Can these scaling arguments be brought in agreement with the perpendicular bed/surface geometry? I suggest clarifying this point in the context of a discussion on the SavageHutter approximation.
Most equations were discussed and presented in a way that allows the reader to verify and understand them. However, I strongly suggest adding a sketch in which bed and surface are shown and angles are defined. This sketch should also define the signs of different quantities (e.g., angles and acceleration), which are crucial for the presented material.
Finally, the reader needs more information where certain equations come from or how assertions are justified. First, Lines 5063 and Lines 8084 include important statements without references. Second, the balance equations (6) and (7) are given without a reference. I was able to derive the former one from the form given in Savage and Hutter (1989), but the second one is not so straightforward. The reader should be presented with a clear source in the literature. Moreover, all assumptions that go into these equations should be stated (e.g., incompressibility?). As another remark on these equations: please state if the del operators act on the product of v_x and h or on the first factor, only.
SPECIFIC COMMENTS
=================
Equation 5: Here and elsewhere, are the bed and free surface b(x,y) and s(x,y) defined as level sets? I.e., b(x,y)z(x,y)=0? This should be stated.
Line 112: “where f is the absolute value” of what?
Equation (8): Is “p” defined? Provide a reference for this equation. Perhaps trivial, but why is a dot n = 0 a requirement?
Equation (12): Is hydrostatic pressure assumed here? Or what is the motivation for this equation?
Line 134: See main comment above. Also, it may help referring to the trigonometric identity tan(theta)=cot(thetapi/2) to concisely pinpoint the singularity.
Figure 1: Here it would be extremely helpful to see a sketch for the geometry of the depicted situations (see main comment above). Similar, the asymmetry discussion (Lines 145ff) would benefit from a sketch depicting uphillfacing and downhillfacing fronts. Is “a” the acceleration in the bedparallel direction? The dashed lines should be defined in the captions and not only in the main text. The description of the singularity makes sense, but I do not understand why only the dashed parts of the curve are shifted right under the coordinate transformation. Perhaps this can be rephrased.
Line 159: Which “curves of the two models”?
Line 164: “the good properties of the equation”: Which properties of which equations?
Line 169: This may sound like splitting hairs, but I would stick to acceleration and call it either uphill or downhill, ideally associating the two with a sign using a sketch (see major comments).
Lines 174175: I suggest including a reference for “earth pressures” since this seems widely used in the soil mechanics literature.
Equation 21: Define c.
Equation 22: Should min be max?
Equation 23: The Euler scheme references I found include a + rather than a – sign between the two RHS terms.
Equation 24 and the fowling equations: The primes are not derivatives, right? If so, I suggest specifying this.
Line 241: The referenced equation is a proportionality. How is the constant of proportionality determined?
Line 255: Give a reference for the CourantFriedrichsLewy criterion.
Lines 259260: Avoid 1sentence paragraphs. Define the «rectangle».
Lines 267: Which «versions»?
Line 273: Rewrite “some waves”.
Line 280: melts down decreases
Line 288: “straight slope” is inappropriate since slope is a scalar value.
Equation 37: Should the cosines be squared?
Lines 310ff: How is this supported? Is there a corresponding figure?
Figure 3: There seem to be several curves with the same color, please allow for better distinction. I could not distinguish lines with and without markers. For some of the lines in (b) there seems to be an overhang near x=0. If intended, I suggest commenting.
Line 331: Robustness with respect to what?
Line 332333: “works well technically” should be qualified better or even quantified.
Figure 4: I suggest labeling the colors directly in the plot (e.g., with a legend or text boxes). The reader will appreciate this.
Line 339: Delete one “necessary”.
343: Why does the velocity remain constant at the kink? Because it is a single point in space?
Figure 5: I suggest outlining the release areas rather by encircling it. Referring to them as “red lines” in the caption is confusing. Also, I strongly recommend pointing out the striations and hummocks discussed in the text directly in the figure.
Lines 368ff: It is not obvious to me why transverse diffusion is not allowed in Equation 7. After all, both horizontal components and gradients are present. Perhaps this is a wellknown fact, but it would be interesting to have more information on this phenomenon.
Line 371: obviously > apparently
Line 372: A reason or ideally a reference why the longitudinal striations are realistic should be given.
Line 373: into > in
Line 376: Rather than using “strongly” I suggest quantifying the grid orientation effect.
Line 381: I am surprised that this was the motivation for model development. After all, other models have reproduced long runouts. Perhaps a clarifying sentence would help.
Figure 7: The different alphas should be labeled directly in the plot.
Line 394: Which proportionality factor was used?
Lines 396397: Rewrite “without getting into conflict …”.
Line 419: What makes the other models more comprehensive?
Lines 420421: Why would the time it takes to develop a model affect its use in hazard assessment?
Lines 423424: From my experience even when applying the same model to two different sites, parameter transferability is limited.
MINOR COMMENTS
================
Line 137: A singularity also exists if grad h grows without bounds. I suggest using other terminology.
Fewer adjectives and adverbs reflecting subjectivity in argumentation should be used. Line 8: Delete “quite well”. Line 60: Delete “quite”. Line 269: Delete “very”. Line 284: Rewrite “works well”. Line 347: delete “quite”. Line 425: Delete “very”.
Line 37: Rewrite/correct “the lowest friction a low velocities”.
I may have missed it, but I could not find a plugandplay matlab or python script on the code repository. It would be helpful if there was a setup that can simply be executed to reproduce the shown figures.
Citation: https://doi.org/10.5194/egusphere2023802RC2 
AC2: 'Reply on RC2', Stefan Hergarten, 18 Oct 2023
Dear Fabian Walter,
thanks for your constructive comments! Your arguments about the starting level make sense to me. The approach and the implementation are indeed simpler than in other models, so that there could be a chance to make the stuff accessible to researchers who are not so familiar with numerical modeling. On the other hand, however, a model description paper in GMD has to be concise. So I am still not sure how much I can improve the accessibility to novices in model development, but I will give my best.
Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere2023802AC2

AC2: 'Reply on RC2', Stefan Hergarten, 18 Oct 2023

EC1: 'Comment on egusphere2023802', Thomas Poulet, 27 Sep 2023
Dear author,
I am looking forward to reading your official response to the reviewers' comments and the corresponding revision of the manuscript.
Best regards,
Thomas Poulet.
Citation: https://doi.org/10.5194/egusphere2023802EC1
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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