the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Apr 2024
Modeling the formation of toma hills based on fluid dynamics with a modified Voellmy rheology
Abstract. Toma hills are the perhaps most enigmatic morphological feature found in rock avalanche deposits. While it was already proposed that toma hills might emerge from the fluid-like behavior of rock avalanches, there still seems to be no consistent explanation for their occurrence. This paper presents numerical results based on a modified version of Voellmy's rheology, which was recently developed for explaining the long runout of rock avalanches. In contrast to the widely used original version, the modified Voellmy rheology defines distinct regimes of Coulomb friction at low velocities and velocity-dependent friction at high velocities. When movement slows down, falling back to Coulomb friction may cause a sudden increase in friction. Material accumulates in the region upstream of a point where this happens. In turn, high velocities may persist for some time in the downstream and lateral range, resulting in a thin deposit layer finally. In combination, both processes generate more or less isolated hills with shapes and sizes similar to toma hills found in real rock avalanche deposits. So the modified Voellmy rheology suggests a simple mechanism for the formation of toma hills.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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RC1: 'Comment on egusphere-2024-1070', Anonymous Referee #1, 04 Jun 2024
This paper suggests that toma hills are formed when movement slows down increasing friction (Coulomb friction) causing an accumulation of material on the material source direction (upstream side) of a point at which this happens. High velocity may continue on the downstream and lateral sides and this results in thin deposits and more or less “isolated” toma hills.
I am reviewing as an expert on how hummocks are formed in debris avalanche deposits based on analogue granular experiments.
My full review is as follows:
- Does the paper address relevant scientific questions within the scope of ESurf?
The paper suggests a mechanism of forming toma hills (hummocks in volcanic debris avalanches), on the discussion of physical processes shaping the Earth’s surface using numerical modelling; thus, within the scope of Esurf journal.
- Does the paper present novel concepts, ideas, tools, or data?
This work uses a modified Voellmy’s rheology. This idea has two distinct regimes of granular flow; (1) the original Voellmy rheology adopted for high velocities that result in an effective (velocity-dependent) friction proportional to the square of the velocity and (2) Coulomb friction for low velocities where friction may be lower in fast regime than slow regime and velocity at transition was an interpretation of the concept of random kinetic energy. In this regard, this work presents a novel concept.
- Are substantial conclusions reached?
Yes, toma hills are reproduced from numerical simulations when there is a significant decrease in friction from original Voellmy rheology to Coulomb friction causing an accumulation of material while adjacent sides (downstream and lateral sides) continue moving at high velocities.
- Are the scientific methods and assumptions valid and clearly outlined?
Yes, assumptions are outlined. Although, this paper assumes that Voellmy rheology and Coulomb Friction are known to the reader. It would be beneficial to the reader if a brief in-depth description of these would be presented in this work.
A clear distinction between, if any, between toma hills and hummocks would also widen the audience of this work with applicability not only for rock avalanches but also for volcanic and non-volcanic debris avalanches.
Line 48-49: It is not clear what this sentence means. A brief description of Hergarten (2012)’s rockslide disposition will be helpful. In this paragraph, it says the model does not consider faults and rock properties but in the same paragraph, it also says, the most recent version of the model was used to account for local orientation of failure surface. A confirmation of how much consideration was accounted for on the original source area geomorphology would make this clear.
In line 55: As the study area has been impacted by rock avalanches in the past, how will previous avalanches’ depositional surface affect the current surface simulation (roughness of bed)? I assume that if a toma already exists in the current avalanche pathway, it will affect the emplacement and runout of current avalanche? How would volume affect the result of this work’s simulations?
The simulation has been applied assuming a valley-filling rockslide debris avalanche. This work would be of interest to a wider audience, for example, researchers looking at volcanic and non-volcanic debris avalanches if a simulation could also be done on a topography without topographical barriers such as an adjacent elevated area. This would chase out how the impact on the the transition and changes in velocity as avalanche material is spreading freely.
- Are the results sufficient to support the interpretations and conclusions?
Several simulations were conducted, it is not clear why a bed roughness of 500 ms-2 and factor of proportionality of 4 ms-1 was chosen for an in-depth analysis and how changing these parameters affect the model.
Is the longer runout in Figure 2 using the model due to the volume that is also x2 that of Ostermann et al (2012)?
The biggest hills in the simulation is bigger than the real toma hills. How does the total number of toma hills and distribution compare between both simulation and reality?
- Is the description of experiments and calculations sufficiently complete and precise to allow their reproduction by fellow scientists (traceability of results)?
I appreciate the availability of matlab codes, to recreate figures and simulations. The results can be traced if you use the same data input as described in the text, which is good. It is however, imperative, for the codes to have an accompanying better documentation (as comments on the code or an accompanying read me file) to define each element in the code. An as example, fs, lw, cm in figure codes, and uhs, vhs, and others in simulation code are not defined. The article, if published, would immensely benefit if other researchers will understand the code and use their own DEM with the code.
- Do the authors give proper credit to related work and clearly indicate their own new/original contribution?
Yes, proper credit was given on related work and new and original contributions of this work are clearly indicated.
- Does the title clearly reflect the contents of the paper?
Yes, the title clearly reflects the contents of the paper, which is on modelling the formation of toma hills based on a fluid dynamics using a modified Voellmy rheology.
- Does the abstract provide a concise and complete summary? Yes.
- Is the overall presentation well structured and clear? Yes.
- Is the language fluent and precise? Yes.
- Are mathematical formulae, symbols, abbreviations, and units correctly defined and used?
smin and smax are not defined in this work, although it points to Agentin et al. (2021), would be useful to say this here.
Would be better to put vc on page 3, where the others are also described.
- Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated?
Figures 3 and 4 if they are put either side by side (or top and bottom) to make them easier to compare between simulation and real toma hills.
- Are the number and quality of references appropriate?
Yes, but I would suggest looking at a few more as these might help support, or expand discussions in this work:
Kelfoun, K., & Druitt, T. H. (2005). Numerical modeling of the emplacement of Socompa rock avalanche, Chile. Journal of Geophysical Research: Solid Earth, 110(12), 1–13. https://doi.org/10.1029/2005JB003758
Thompson, N., Bennett, M. R., & Petford, N. (2010). Development of characteristic volcanic debris avalanche deposit structures: New insight from distinct element simulations. Journal of Volcanology and Geothermal Research, 192(3–4), 191–200. https://doi.org/10.1016/j.jvolgeores.2010.02.021
- Is the amount and quality of supplementary material appropriate?
Codes need to have better documentation to as explained in item 6 above to make it more useful for those who want to apply this using another DEM. In Figure6.m code, it is hard to see what to change to be able to reproduce the graph for toma hills, 3,5,10, 11 (center, Fig 6) and 4,6,7,12 (right, Fig 6).
Citation: https://doi.org/10.5194/egusphere-2024-1070-RC1 -
AC2: 'Reply on RC1', Stefan Hergarten, 24 Jul 2024
Dear Reviewer,
thank you for your constructive comments! In particular, I appreciate your suggestions that could widen the audience.
As one point, you mention debris avalanches, which could make the model interesting in the field of volcanic flows. I have been thinking for some time about an extension towards rock-water mixtures, but the assumed sudden increase in friction below a certain threshold velocity is not so trivial for mixtures. So I think it is better to keep the limitation to dry rock avalanches in this paper.Another question, which also goes into the direction of volcanic debris avalanches, is about the distinction between toma hills and hummocks. To my understanding, toma hills are particular hummocks with quite low (or even zero) deposit thickness between the hummocks. In contrast, the majority of the observed hummocks seem to sit on rather thick deposits, where the thickness may be much larger than the height of the hummocks. In this study, I retreated to toma hills because all previous numerical and laboratory models I am aware of run into problems here. The
modified Voellmy rheology also produces hummocks on thicker deposits, but I cannot be sure that it is better than existing approaches here.You also detected the main point I plan to address in a subsequent study -- constraining the conditions under which toma hills form. I already performed several simulations, but just inspected
to deposit morphologies visually. It indeed looks as if flowing along a valley with a quite flat floor promotes the formation of toma hills compared to scenarios with free radial spreading. However, it is not so trivial and I need to understand better why this is the case before writing anything about it.In a nutshell, I have to think a bit more whether I want to extend the paper into any of these directions or whether it would be better to keep the focus on proposing a simple mechanism
for the formation of toma hills.
Finally, about the MATLAB codes in the repository. I do not mind extending the codes by explanations, although I am not sure how helpful this will be. At least some of the analyses behind the figures are quite specific and I do not expect that anyone will repeat exactly the same procedure with different data. It will require some programming experience, and then it will become clear quite soon which variables are just font size, line width, color order, etc. Anyway, it will not do any harm, so I will extend the codes.Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere-2024-1070-AC2
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RC2: 'Comment on egusphere-2024-1070', Martin Mergili, 14 Jun 2024
The author presents an approach to numerically reproduce toma hills (more or less isolated hills in the distal area of rock avalanche deposits). For this purpose, he uses the relatively simple and straightforward Voellmy approach, which builds on bed friction and turbulent friction. Reinterpreting the idea of the random kinetic energy (e.g., Buser and Bartelt, 2009), he defines a flow thickness-dependent threshold velocity above which the bed friction does not act. Using this approach, he simulates a generic rock avalanche in the Obernberg Valley, Tyrol, Austria, and compares the resulting toma hills with those produced by a prehistoric event in the same area. The results are plausible, and a strong dependency of the formation of toma hills on the local topography is revealed.
This topic is of high scientific interest and significantly contributes to the ongoing scientific debate on the formation of toma hills. The discussion paper is clearly within the scope of the Earth Surface Dynamics journal. It is very well written, structured, and illustrated. Appropriate references are given to previous work, and the method and results are described and discussed in a clear and comprehensive way. I would definitely like to see this work published in Earth Surface Dynamics.
One aspect I thought about when reading the results and discussion section is the influence of the spatial resolution on the model results, and whether there would be some maximum cell size (in relation to the toma hill size) beyond which the formation of toma hills is blurred in the simulation. From my point of view, it is not mandatory to do some additional simulations with varying cell size within this publication. Such an exercise could also be a possible direction for follow-up research.
Therefore, I recommend acceptance of the paper for publication in Earth Surface Dynamics.
Citation: https://doi.org/10.5194/egusphere-2024-1070-RC2 -
AC1: 'Reply on RC2', Stefan Hergarten, 22 Jul 2024
Dear Martin,
thanks a lot for your review, which makes me quite happy!
The spatial resolution is indeed an interesting point. It is finally more (worse) than just hills becoming blurred if the grid is too coarse.
I made a few additional tests with different resolutions. The attached figure shows basically the same properties as in Fig. 2 (final deposit thickness and maximum extent of the flowing material). The versions with dx = 10 m and 20 m use the 5 m DEM averaged over 2x2 and 4x4 pixel patches. The version with dx = 2.5 m was obtained by bilinear interpolation.
The hills indeed become a bit blurred with increasing dx. Similar to the variation of the Voellmy parameters (Sect. 3.4), however, the effect of the resolution on runout length seems to be stronger than on the formation of hills. A finer resolution leads to a shorter runout. In turn, the position of some hills is even similar for dx = 2.5 m and dx = 5 m.
As far as I know, the spatial resolution is a problem in all depth-averaged models. I would guess that the main problem is that even short-wavelength oscillations of the fluid surface affect the
pressure of the entire column down to the bed, which finally causes artificial variations in velocity. In combination with friction proportional to v^2, these variations lead to an increase in effective friction with increasing spatial resolution. I wanted to solve this problem when developing MinVoellmy version 2, but I was not satisfied and postponed it. I also did not yet test whether including horizontal stresses, resulting in a diffusion of momentum (as offered as an option, e.g., in AvaFrame) solves the problem.
So I am still thinking about whether I should start this discussion in this paper or whether it would be better to address it seriously later, including a discussion of potential solutions.
Best regards,
Stefan
-
AC1: 'Reply on RC2', Stefan Hergarten, 22 Jul 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-1070', Anonymous Referee #1, 04 Jun 2024
This paper suggests that toma hills are formed when movement slows down increasing friction (Coulomb friction) causing an accumulation of material on the material source direction (upstream side) of a point at which this happens. High velocity may continue on the downstream and lateral sides and this results in thin deposits and more or less “isolated” toma hills.
I am reviewing as an expert on how hummocks are formed in debris avalanche deposits based on analogue granular experiments.
My full review is as follows:
- Does the paper address relevant scientific questions within the scope of ESurf?
The paper suggests a mechanism of forming toma hills (hummocks in volcanic debris avalanches), on the discussion of physical processes shaping the Earth’s surface using numerical modelling; thus, within the scope of Esurf journal.
- Does the paper present novel concepts, ideas, tools, or data?
This work uses a modified Voellmy’s rheology. This idea has two distinct regimes of granular flow; (1) the original Voellmy rheology adopted for high velocities that result in an effective (velocity-dependent) friction proportional to the square of the velocity and (2) Coulomb friction for low velocities where friction may be lower in fast regime than slow regime and velocity at transition was an interpretation of the concept of random kinetic energy. In this regard, this work presents a novel concept.
- Are substantial conclusions reached?
Yes, toma hills are reproduced from numerical simulations when there is a significant decrease in friction from original Voellmy rheology to Coulomb friction causing an accumulation of material while adjacent sides (downstream and lateral sides) continue moving at high velocities.
- Are the scientific methods and assumptions valid and clearly outlined?
Yes, assumptions are outlined. Although, this paper assumes that Voellmy rheology and Coulomb Friction are known to the reader. It would be beneficial to the reader if a brief in-depth description of these would be presented in this work.
A clear distinction between, if any, between toma hills and hummocks would also widen the audience of this work with applicability not only for rock avalanches but also for volcanic and non-volcanic debris avalanches.
Line 48-49: It is not clear what this sentence means. A brief description of Hergarten (2012)’s rockslide disposition will be helpful. In this paragraph, it says the model does not consider faults and rock properties but in the same paragraph, it also says, the most recent version of the model was used to account for local orientation of failure surface. A confirmation of how much consideration was accounted for on the original source area geomorphology would make this clear.
In line 55: As the study area has been impacted by rock avalanches in the past, how will previous avalanches’ depositional surface affect the current surface simulation (roughness of bed)? I assume that if a toma already exists in the current avalanche pathway, it will affect the emplacement and runout of current avalanche? How would volume affect the result of this work’s simulations?
The simulation has been applied assuming a valley-filling rockslide debris avalanche. This work would be of interest to a wider audience, for example, researchers looking at volcanic and non-volcanic debris avalanches if a simulation could also be done on a topography without topographical barriers such as an adjacent elevated area. This would chase out how the impact on the the transition and changes in velocity as avalanche material is spreading freely.
- Are the results sufficient to support the interpretations and conclusions?
Several simulations were conducted, it is not clear why a bed roughness of 500 ms-2 and factor of proportionality of 4 ms-1 was chosen for an in-depth analysis and how changing these parameters affect the model.
Is the longer runout in Figure 2 using the model due to the volume that is also x2 that of Ostermann et al (2012)?
The biggest hills in the simulation is bigger than the real toma hills. How does the total number of toma hills and distribution compare between both simulation and reality?
- Is the description of experiments and calculations sufficiently complete and precise to allow their reproduction by fellow scientists (traceability of results)?
I appreciate the availability of matlab codes, to recreate figures and simulations. The results can be traced if you use the same data input as described in the text, which is good. It is however, imperative, for the codes to have an accompanying better documentation (as comments on the code or an accompanying read me file) to define each element in the code. An as example, fs, lw, cm in figure codes, and uhs, vhs, and others in simulation code are not defined. The article, if published, would immensely benefit if other researchers will understand the code and use their own DEM with the code.
- Do the authors give proper credit to related work and clearly indicate their own new/original contribution?
Yes, proper credit was given on related work and new and original contributions of this work are clearly indicated.
- Does the title clearly reflect the contents of the paper?
Yes, the title clearly reflects the contents of the paper, which is on modelling the formation of toma hills based on a fluid dynamics using a modified Voellmy rheology.
- Does the abstract provide a concise and complete summary? Yes.
- Is the overall presentation well structured and clear? Yes.
- Is the language fluent and precise? Yes.
- Are mathematical formulae, symbols, abbreviations, and units correctly defined and used?
smin and smax are not defined in this work, although it points to Agentin et al. (2021), would be useful to say this here.
Would be better to put vc on page 3, where the others are also described.
- Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated?
Figures 3 and 4 if they are put either side by side (or top and bottom) to make them easier to compare between simulation and real toma hills.
- Are the number and quality of references appropriate?
Yes, but I would suggest looking at a few more as these might help support, or expand discussions in this work:
Kelfoun, K., & Druitt, T. H. (2005). Numerical modeling of the emplacement of Socompa rock avalanche, Chile. Journal of Geophysical Research: Solid Earth, 110(12), 1–13. https://doi.org/10.1029/2005JB003758
Thompson, N., Bennett, M. R., & Petford, N. (2010). Development of characteristic volcanic debris avalanche deposit structures: New insight from distinct element simulations. Journal of Volcanology and Geothermal Research, 192(3–4), 191–200. https://doi.org/10.1016/j.jvolgeores.2010.02.021
- Is the amount and quality of supplementary material appropriate?
Codes need to have better documentation to as explained in item 6 above to make it more useful for those who want to apply this using another DEM. In Figure6.m code, it is hard to see what to change to be able to reproduce the graph for toma hills, 3,5,10, 11 (center, Fig 6) and 4,6,7,12 (right, Fig 6).
Citation: https://doi.org/10.5194/egusphere-2024-1070-RC1 -
AC2: 'Reply on RC1', Stefan Hergarten, 24 Jul 2024
Dear Reviewer,
thank you for your constructive comments! In particular, I appreciate your suggestions that could widen the audience.
As one point, you mention debris avalanches, which could make the model interesting in the field of volcanic flows. I have been thinking for some time about an extension towards rock-water mixtures, but the assumed sudden increase in friction below a certain threshold velocity is not so trivial for mixtures. So I think it is better to keep the limitation to dry rock avalanches in this paper.Another question, which also goes into the direction of volcanic debris avalanches, is about the distinction between toma hills and hummocks. To my understanding, toma hills are particular hummocks with quite low (or even zero) deposit thickness between the hummocks. In contrast, the majority of the observed hummocks seem to sit on rather thick deposits, where the thickness may be much larger than the height of the hummocks. In this study, I retreated to toma hills because all previous numerical and laboratory models I am aware of run into problems here. The
modified Voellmy rheology also produces hummocks on thicker deposits, but I cannot be sure that it is better than existing approaches here.You also detected the main point I plan to address in a subsequent study -- constraining the conditions under which toma hills form. I already performed several simulations, but just inspected
to deposit morphologies visually. It indeed looks as if flowing along a valley with a quite flat floor promotes the formation of toma hills compared to scenarios with free radial spreading. However, it is not so trivial and I need to understand better why this is the case before writing anything about it.In a nutshell, I have to think a bit more whether I want to extend the paper into any of these directions or whether it would be better to keep the focus on proposing a simple mechanism
for the formation of toma hills.
Finally, about the MATLAB codes in the repository. I do not mind extending the codes by explanations, although I am not sure how helpful this will be. At least some of the analyses behind the figures are quite specific and I do not expect that anyone will repeat exactly the same procedure with different data. It will require some programming experience, and then it will become clear quite soon which variables are just font size, line width, color order, etc. Anyway, it will not do any harm, so I will extend the codes.Best regards,
Stefan HergartenCitation: https://doi.org/10.5194/egusphere-2024-1070-AC2
-
RC2: 'Comment on egusphere-2024-1070', Martin Mergili, 14 Jun 2024
The author presents an approach to numerically reproduce toma hills (more or less isolated hills in the distal area of rock avalanche deposits). For this purpose, he uses the relatively simple and straightforward Voellmy approach, which builds on bed friction and turbulent friction. Reinterpreting the idea of the random kinetic energy (e.g., Buser and Bartelt, 2009), he defines a flow thickness-dependent threshold velocity above which the bed friction does not act. Using this approach, he simulates a generic rock avalanche in the Obernberg Valley, Tyrol, Austria, and compares the resulting toma hills with those produced by a prehistoric event in the same area. The results are plausible, and a strong dependency of the formation of toma hills on the local topography is revealed.
This topic is of high scientific interest and significantly contributes to the ongoing scientific debate on the formation of toma hills. The discussion paper is clearly within the scope of the Earth Surface Dynamics journal. It is very well written, structured, and illustrated. Appropriate references are given to previous work, and the method and results are described and discussed in a clear and comprehensive way. I would definitely like to see this work published in Earth Surface Dynamics.
One aspect I thought about when reading the results and discussion section is the influence of the spatial resolution on the model results, and whether there would be some maximum cell size (in relation to the toma hill size) beyond which the formation of toma hills is blurred in the simulation. From my point of view, it is not mandatory to do some additional simulations with varying cell size within this publication. Such an exercise could also be a possible direction for follow-up research.
Therefore, I recommend acceptance of the paper for publication in Earth Surface Dynamics.
Citation: https://doi.org/10.5194/egusphere-2024-1070-RC2 -
AC1: 'Reply on RC2', Stefan Hergarten, 22 Jul 2024
Dear Martin,
thanks a lot for your review, which makes me quite happy!
The spatial resolution is indeed an interesting point. It is finally more (worse) than just hills becoming blurred if the grid is too coarse.
I made a few additional tests with different resolutions. The attached figure shows basically the same properties as in Fig. 2 (final deposit thickness and maximum extent of the flowing material). The versions with dx = 10 m and 20 m use the 5 m DEM averaged over 2x2 and 4x4 pixel patches. The version with dx = 2.5 m was obtained by bilinear interpolation.
The hills indeed become a bit blurred with increasing dx. Similar to the variation of the Voellmy parameters (Sect. 3.4), however, the effect of the resolution on runout length seems to be stronger than on the formation of hills. A finer resolution leads to a shorter runout. In turn, the position of some hills is even similar for dx = 2.5 m and dx = 5 m.
As far as I know, the spatial resolution is a problem in all depth-averaged models. I would guess that the main problem is that even short-wavelength oscillations of the fluid surface affect the
pressure of the entire column down to the bed, which finally causes artificial variations in velocity. In combination with friction proportional to v^2, these variations lead to an increase in effective friction with increasing spatial resolution. I wanted to solve this problem when developing MinVoellmy version 2, but I was not satisfied and postponed it. I also did not yet test whether including horizontal stresses, resulting in a diffusion of momentum (as offered as an option, e.g., in AvaFrame) solves the problem.
So I am still thinking about whether I should start this discussion in this paper or whether it would be better to address it seriously later, including a discussion of potential solutions.
Best regards,
Stefan
-
AC1: 'Reply on RC2', Stefan Hergarten, 22 Jul 2024
Peer review completion
Journal article(s) based on this preprint
Model code and software
Formation of toma hills Stefan Hergarten https://doi.org/10.5281/zenodo.10932346
Video supplement
Formation of toma hills Stefan Hergarten http://hergarten.at/minvoellmy/tomahills
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Stefan Hergarten
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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