Scaling between volume and runout of rock avalanches explained by a modified Voellmy rheology

Hergarten, Stefan

doi:https://doi.org/10.5194/egusphere-2023-144

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https://doi.org/10.5194/egusphere-2023-144

Preprints

21 Mar 2023

| 21 Mar 2023

Scaling between volume and runout of rock avalanches explained by a modified Voellmy rheology

Stefan Hergarten

Abstract. Rock avalanches reach considerably greater runout lengths than predicted by Coulomb friction. While it has been known for a long time that runout length increases with volume, explaining the increase qualitatively is still a challenge. In this study, the widely used Voellmy rheology is reinterpreted and modified. Instead of adding a Coulomb friction term and a velocity-dependent term, the modified rheology assigns the two terms to different regimes of velocity. While assuming a transition between Coulomb friction and flow at a given velocity is the simplest approach, a reinterpretation of an existing model for the kinetic energy of random particle motion predicts a dependence of the crossover velocity on the thickness of the rock avalanche. Analytical solutions for a lumped mass on a simple 1-D topography reveal the existence of a slope-dominated and a height-dominated regime within the regime of flow. In the slope-dominated regime, the kinetic energy at the foot of the slope depends mainly on the slope angle, while the absolute height relative to the valley floor has little effect, and vice versa. Both regimes can be distinguished by the ratio of a length scale derived from the rheology and the length scale of the topography. Long runout occurs in the height-dominated regime. In combination with empirical relations between volume, thickness, and height, the approach based on the random kinetic energy model reproduces the scaling of runout length with volume observed in nature very well.

Received: 01 Feb 2023 – Discussion started: 21 Mar 2023

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Stefan Hergarten

Status: closed

RC1:
'Comment on egusphere-2023-144', Anonymous Referee #1, 28 Mar 2023

Paper is definitely interesting, but for me (i'm geologist) it was difficult to flollow. the explanation.
It would be really useful if you will add a list of all parameters used in the equations with all these quantities dimensions and their physical meaning (may be a schematic cross-section). It will help a lot to readers from the geological community to follow your explanations.
Some comments are in the attached file.

Citation: https://doi.org/10.5194/egusphere-2023-144-RC1
- AC1: 'Reply on RC1', Stefan Hergarten, 07 Jun 2023
  
  Dear Reviever,
  
  thank you for your comments! Please allow for some short responses before addressing your points later in a detai.
  
  The theory may indeed be quite tough for "average" geologists. This is why I used a somehow unusual structure (not "Methods", "Results", ...) with two sections (2 and 3) developing the new ideas and already illustrating them with examples. Section 4 becomes more difficult since combining process parameters to length scales (lambda) or to nondimensional proporties (energy rate, epsilon) is not widely done in geology. The consequence is that readers who want to understand why the modified rheology reproduces the scaling of H/L with volume quite well have to go through it step by step. I know that not many readers will do this, but I think that it is not a big problem. As you suggested, I already prepared a schematic cross section that illustrates the geometric properties (H,l,beta,h,S,L). This will probably help to keep track of the variables. I can also check where I can add more explanations to the theory, but overall, it will still be a challenging paper for the majority of the readers.
  
  However, there is one aspect where I disagree to your point of view. It looks a bit to me as if you want to promote the idea that long runout is owing to fragmentation. But even if I believe that the degree of fragmentation increases with increasing runout length, this argument cannot be reverted so easily. I know that some researchers are working on this topic, but to my knowledge is has only been shown that fragmentation can in principle increase the fluctuations in kinetic energy and thus in velocity. Overall, however, fragmentation consumes energy. I am not aware of any experimental or modeling studies that were able to predict the decrease in H/L with volume quantitatively based on fragmentation. As soon as anyone has succeeded in this, we can start the discussion whether my approach with the two flow regimes or fragmentation is better. Until then, however, just claiming that long runout cannot be explained without fragmentation (what some researchers are doing) is not a good argument for me. In turn, however, I do not want to criticize the idea of fragmentation in this paper.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2023-144-AC1
CC1:
'Comment on egusphere-2023-144', Matthias Rauter, 20 May 2023

The manuscript deals with the unsolved problem of extreme runouts of large landslides and rock avalanches. I not nesesarrily agree with the approach, but it is a pressing issue and intersting viewpoint and deserves more discussion. The figures are great and clean!
I suggest the publication after a major revision.
Major issues:
The scope of the manuscript is a bit unclear to me. Is it to understand and explain the mechanisms in large avalanches? To develop a better friction model? (Then the issue below should be discussed in more depth) Should it be a physically consistent or an empirical approach? The interpretation of lambda as a length should then also be discussed in more depth.
I do not understand the scaling (section 4). The runout scales with the friction term but reinterpreting it as a length scales does not seem to be useful to me. I also cannot follow Eq. (24). I also think that physical relations and parameters are mixed together with emprical relations which does not always make sense. I would check this section carefully.
The biggest problem I have is the sharp transition between the regimes and the jump in the friction. This seems rather unphysical to me and might lead to sever numerical issues.
I would also bring in some real case examples. Even using the simple model it can showcase the behaviour of the rheology.
Minor issues:
I am not sure how strict this journal is with structuring of the manuscript, but the structure is rather unconventional, mixing methods, results and discussion.
We went trough the same exercise a couple of years ago, deriving a modified Voellmy model from kinetic theory (Rauter et al., 2016). We came to a few different conclusions.
Line 19:" H/L < 0.1, while typical values of mu for Coulomb friction are between 0.5 and 1."
It should be described why and how these are connected/correlated.
Line 34: "Water is present in many rock avalanches and may play a part as well as air. Frictional heating may also have a strong effect on the mechanical properties. Alternatively, the increase in runout length with volume may be an inherent property of granular flow without any specific process beyond the interaction of particles."
This needs to be backed up with references. I also suggest to look at Kesseler et al. (2020) in this context.
Line 46: "The most widely used relation for the basal shear stress".
The role of the basal shear stress should be explained. How is it connected with mu and H/L?
Line 150: "More important, however, it defines the length scale of adjustment of the velocity to the slope."
I do not agree with this statement. A length scale would be the typical height H or length L. I also do not understand "adjustment of the velocity to the slope". Is it the slope length at which terminal velocity is obtained?
Line 156: "the slope length l"
I would prefer a large L, since this is usually a rough scale like as in line 19. If it is not, it deserves an explaination. Generally, i would distinguish clearer between scales and real distances.
Line 165: "phase space trajectories".
This term is new for me for this kind of diagram.
Line 188: "The relation to the RKE model"
This model has (as most models) some issues (Issler et al. 2018). I would make sure that they do not change your conclusions.
Line 235: "Figure 5(a) shows the dependence of S/l on lambda/l"
The runout scales with the friction coefficient. That seems obvious. Why the division with l?
Line 241: "The existence of two different scaling regimes"
I am not sure about this. Are there really two regimes? In the following section you look only at very extreme scenarios. This is hard to say without some real world examples.
Line 244: "For lambda << l, Eq. (18) yields"
So basically an infinite slope?
Line 247: "lambda >> l"
Basically frictionless?
Line "the runout length increases with increasing slope length"
Are they not the same or diffent by a factor cos(beta) at most? I wonder if we are turning in a circle by multiplying all kind of relations.
Line 289: "L/H mostly decreases with l/lambda"
If l=L then this would just mean lambda decreases with H?
Line 289: " So the ratio L/H decreases with increasing slope length"
Isn't L the slope length?
Dieter Issler, James T. Jenkins, and Jim N. McElaine (2018). "Comments on avalanche flow models based on the concept of random kinetic energy." https://doi.org/10.1017/jog.2017.62
Matthew Kesseler, Valentin Heller and Barbara Turnbull (2020), Grain Reynolds Number Scale Effects in Dry Granular Slides. https://doi.org/10.1029/2019JF005347
Matthias Rauter, Jan-Thomas Fischer, Wolfgang Fellin, and Andreas Kofler (2016), Snow avalanche friction relation based on extended kinetic theory. https://doi.org/10.5194/nhess-16-2325-2016

Citation: https://doi.org/10.5194/egusphere-2023-144-CC1
- AC2: 'Reply on CC1', Stefan Hergarten, 13 Jun 2023
  
  Dear Matthias Rauter,
  
  thank you for your comments! It looks a bit to me as if it was intended to be a review, but then accidently registered as a community comment. Anyway, please allow for some short responses before addressing your points later in a detail.
  
  First, about the biggest problem from your point of view -- the sharp transition between the two regimes and the jump in friction. You consider sharp transitions "rather unphysical". In turn, I learned parsimony as a fundamental principle when I studied physics in the 1980s. Admittedly, this is long ago, but I still prefer to avoid unnecessary complications of models. The jump in friction also does not cause numerical issues since only the acceleration becomes discontinous, while the velocity remains continuous. The main effect of the jump in friction (beyond the runout length) is that the deposits become hummocky, which is not necessarily unrealistic. However, I cannot address numerical aspects in this paper since it is subject of another manuscript in GMD, which unfortunately has got stuck in the editor search for more than 2 months now.
  
  You say that the scope of the manuscript is a bit unclear to you and mention your 2016 paper (doi 10.5194/nhess-16-2325-2016). While you say "We went trough the same exercise", the scope seems to be completely different to me. You developed a rheology that is somehow similar to Voellmy's rheology and presented a theory that goes much deeper into kinetic theory than the ideas of Salm (1993) (doi 10.3189/S0260305500011551). However, your result was that it fits for snow avalanches as well as the original Voellmy rheology. I think this is ok for snow avalanches, but it will probably not solve the problem of the long runout of rock avalanches or maybe even fail completely at large thickness. It should be clear already from the title that my approach attempts to modifiy Voellmy's rheology seriously in order to predict the long runout of rock avalanches.
  
  Finally, your statement "I do not understand the scaling (section 4)" confuses me a bit. I realize that I lost you quite soon in this section. In turn, I know at least some of your papers and reviewed at least one of these. Since the theory presented there is much more complicated than my approach (only ordinary differential equations and exponential functions), I cannot imagine that the mathematical background is any challenge for you. I guess the extensive usage of nondimensional ratios is the main problem for you, but I am not sure. I can add a schematic diagram illustrating the variables and explain those aspects addressed in your specific comments in more detail, but I am still a bit uncertain where the fundamental problem is.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2023-144-AC2
RC2: 'Comment on egusphere-2023-144', Anonymous Referee #2, 07 Aug 2023

General comments

Manuscript presents a complementary approach to Voellmy rheology based on the separation of behaviour with respect to flow velocities. The approach proposed is demonstrated with a lumped mass model, of which limitations are clearly stated by the author. As demonstration is limited to a very simple application, discussion is also not presented at depth, which obscures the potential of the approach developed. In order to substantiate the claims made by the author especially in the Conclusions section, further demonstrations using other forms of simplified topographies can be presented.

Even though the manuscript has an unorthodox structure for a scientific paper, it is clearly written and conveys its message to its readers. However, it reads more like a technical note than a research paper at its current state. I suggest reconsideration of the manuscript after major revisions.

Specific comments

1. Page 2 - LL 33 – 35: The author makes several claims about the mechanism behind the fluid-like behaviour of rock avalanches. This paragraph sounds rather speculative, unless it is supported by references to published works in the literature. I suggest to include citations to convince the readers that there is no consensus on the aforementioned mechanisms.

2. Page 3 - LL 71 - 73: The author explains the selection of Voellmy rheology only based on its wide use. If the approach by Jop et al. (2005) is not used elsewhere in the manuscript, what is the motivation of introducing it? Please highlight this.

3. Page 3 - Figure 1: This figure is not clear enough in its current form. Choice of colours hinders differentiating the groups of lines. It might be useful to change line type between Voellmy and Jop et al., and use different shades of a colour within the same group to highlight the thickness

used. Readers could also benefit to know about the exact thickness used in the calculations that yielded lines in Figure 1, which can be given with a legend.

4. Page 5 - LL 113 – 115: Could the author suggest even approximate ranges for ”high velocities” and ”low velocities”?

5. Page 5 - Figure 2: Similar to Comment #2, a legend indicating the thicknesses used in the calculations would be beneficial.

6. Page 15 - Line 341: "The new approach should be able to improve numerical continuum simulations". I think this is a strong statement without testing it. I suggest reformulating this sentence.

Technical corrections
I did not detect any obvious typos in the manuscript at its current state.

Citation: https://doi.org/10.5194/egusphere-2023-144-RC2

Status: closed

RC1:
'Comment on egusphere-2023-144', Anonymous Referee #1, 28 Mar 2023

Paper is definitely interesting, but for me (i'm geologist) it was difficult to flollow. the explanation.
It would be really useful if you will add a list of all parameters used in the equations with all these quantities dimensions and their physical meaning (may be a schematic cross-section). It will help a lot to readers from the geological community to follow your explanations.
Some comments are in the attached file.

Citation: https://doi.org/10.5194/egusphere-2023-144-RC1
- AC1: 'Reply on RC1', Stefan Hergarten, 07 Jun 2023
  
  Dear Reviever,
  
  thank you for your comments! Please allow for some short responses before addressing your points later in a detai.
  
  The theory may indeed be quite tough for "average" geologists. This is why I used a somehow unusual structure (not "Methods", "Results", ...) with two sections (2 and 3) developing the new ideas and already illustrating them with examples. Section 4 becomes more difficult since combining process parameters to length scales (lambda) or to nondimensional proporties (energy rate, epsilon) is not widely done in geology. The consequence is that readers who want to understand why the modified rheology reproduces the scaling of H/L with volume quite well have to go through it step by step. I know that not many readers will do this, but I think that it is not a big problem. As you suggested, I already prepared a schematic cross section that illustrates the geometric properties (H,l,beta,h,S,L). This will probably help to keep track of the variables. I can also check where I can add more explanations to the theory, but overall, it will still be a challenging paper for the majority of the readers.
  
  However, there is one aspect where I disagree to your point of view. It looks a bit to me as if you want to promote the idea that long runout is owing to fragmentation. But even if I believe that the degree of fragmentation increases with increasing runout length, this argument cannot be reverted so easily. I know that some researchers are working on this topic, but to my knowledge is has only been shown that fragmentation can in principle increase the fluctuations in kinetic energy and thus in velocity. Overall, however, fragmentation consumes energy. I am not aware of any experimental or modeling studies that were able to predict the decrease in H/L with volume quantitatively based on fragmentation. As soon as anyone has succeeded in this, we can start the discussion whether my approach with the two flow regimes or fragmentation is better. Until then, however, just claiming that long runout cannot be explained without fragmentation (what some researchers are doing) is not a good argument for me. In turn, however, I do not want to criticize the idea of fragmentation in this paper.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2023-144-AC1
CC1:
'Comment on egusphere-2023-144', Matthias Rauter, 20 May 2023

The manuscript deals with the unsolved problem of extreme runouts of large landslides and rock avalanches. I not nesesarrily agree with the approach, but it is a pressing issue and intersting viewpoint and deserves more discussion. The figures are great and clean!
I suggest the publication after a major revision.
Major issues:
The scope of the manuscript is a bit unclear to me. Is it to understand and explain the mechanisms in large avalanches? To develop a better friction model? (Then the issue below should be discussed in more depth) Should it be a physically consistent or an empirical approach? The interpretation of lambda as a length should then also be discussed in more depth.
I do not understand the scaling (section 4). The runout scales with the friction term but reinterpreting it as a length scales does not seem to be useful to me. I also cannot follow Eq. (24). I also think that physical relations and parameters are mixed together with emprical relations which does not always make sense. I would check this section carefully.
The biggest problem I have is the sharp transition between the regimes and the jump in the friction. This seems rather unphysical to me and might lead to sever numerical issues.
I would also bring in some real case examples. Even using the simple model it can showcase the behaviour of the rheology.
Minor issues:
I am not sure how strict this journal is with structuring of the manuscript, but the structure is rather unconventional, mixing methods, results and discussion.
We went trough the same exercise a couple of years ago, deriving a modified Voellmy model from kinetic theory (Rauter et al., 2016). We came to a few different conclusions.
Line 19:" H/L < 0.1, while typical values of mu for Coulomb friction are between 0.5 and 1."
It should be described why and how these are connected/correlated.
Line 34: "Water is present in many rock avalanches and may play a part as well as air. Frictional heating may also have a strong effect on the mechanical properties. Alternatively, the increase in runout length with volume may be an inherent property of granular flow without any specific process beyond the interaction of particles."
This needs to be backed up with references. I also suggest to look at Kesseler et al. (2020) in this context.
Line 46: "The most widely used relation for the basal shear stress".
The role of the basal shear stress should be explained. How is it connected with mu and H/L?
Line 150: "More important, however, it defines the length scale of adjustment of the velocity to the slope."
I do not agree with this statement. A length scale would be the typical height H or length L. I also do not understand "adjustment of the velocity to the slope". Is it the slope length at which terminal velocity is obtained?
Line 156: "the slope length l"
I would prefer a large L, since this is usually a rough scale like as in line 19. If it is not, it deserves an explaination. Generally, i would distinguish clearer between scales and real distances.
Line 165: "phase space trajectories".
This term is new for me for this kind of diagram.
Line 188: "The relation to the RKE model"
This model has (as most models) some issues (Issler et al. 2018). I would make sure that they do not change your conclusions.
Line 235: "Figure 5(a) shows the dependence of S/l on lambda/l"
The runout scales with the friction coefficient. That seems obvious. Why the division with l?
Line 241: "The existence of two different scaling regimes"
I am not sure about this. Are there really two regimes? In the following section you look only at very extreme scenarios. This is hard to say without some real world examples.
Line 244: "For lambda << l, Eq. (18) yields"
So basically an infinite slope?
Line 247: "lambda >> l"
Basically frictionless?
Line "the runout length increases with increasing slope length"
Are they not the same or diffent by a factor cos(beta) at most? I wonder if we are turning in a circle by multiplying all kind of relations.
Line 289: "L/H mostly decreases with l/lambda"
If l=L then this would just mean lambda decreases with H?
Line 289: " So the ratio L/H decreases with increasing slope length"
Isn't L the slope length?
Dieter Issler, James T. Jenkins, and Jim N. McElaine (2018). "Comments on avalanche flow models based on the concept of random kinetic energy." https://doi.org/10.1017/jog.2017.62
Matthew Kesseler, Valentin Heller and Barbara Turnbull (2020), Grain Reynolds Number Scale Effects in Dry Granular Slides. https://doi.org/10.1029/2019JF005347
Matthias Rauter, Jan-Thomas Fischer, Wolfgang Fellin, and Andreas Kofler (2016), Snow avalanche friction relation based on extended kinetic theory. https://doi.org/10.5194/nhess-16-2325-2016

Citation: https://doi.org/10.5194/egusphere-2023-144-CC1
- AC2: 'Reply on CC1', Stefan Hergarten, 13 Jun 2023
  
  Dear Matthias Rauter,
  
  thank you for your comments! It looks a bit to me as if it was intended to be a review, but then accidently registered as a community comment. Anyway, please allow for some short responses before addressing your points later in a detail.
  
  First, about the biggest problem from your point of view -- the sharp transition between the two regimes and the jump in friction. You consider sharp transitions "rather unphysical". In turn, I learned parsimony as a fundamental principle when I studied physics in the 1980s. Admittedly, this is long ago, but I still prefer to avoid unnecessary complications of models. The jump in friction also does not cause numerical issues since only the acceleration becomes discontinous, while the velocity remains continuous. The main effect of the jump in friction (beyond the runout length) is that the deposits become hummocky, which is not necessarily unrealistic. However, I cannot address numerical aspects in this paper since it is subject of another manuscript in GMD, which unfortunately has got stuck in the editor search for more than 2 months now.
  
  You say that the scope of the manuscript is a bit unclear to you and mention your 2016 paper (doi 10.5194/nhess-16-2325-2016). While you say "We went trough the same exercise", the scope seems to be completely different to me. You developed a rheology that is somehow similar to Voellmy's rheology and presented a theory that goes much deeper into kinetic theory than the ideas of Salm (1993) (doi 10.3189/S0260305500011551). However, your result was that it fits for snow avalanches as well as the original Voellmy rheology. I think this is ok for snow avalanches, but it will probably not solve the problem of the long runout of rock avalanches or maybe even fail completely at large thickness. It should be clear already from the title that my approach attempts to modifiy Voellmy's rheology seriously in order to predict the long runout of rock avalanches.
  
  Finally, your statement "I do not understand the scaling (section 4)" confuses me a bit. I realize that I lost you quite soon in this section. In turn, I know at least some of your papers and reviewed at least one of these. Since the theory presented there is much more complicated than my approach (only ordinary differential equations and exponential functions), I cannot imagine that the mathematical background is any challenge for you. I guess the extensive usage of nondimensional ratios is the main problem for you, but I am not sure. I can add a schematic diagram illustrating the variables and explain those aspects addressed in your specific comments in more detail, but I am still a bit uncertain where the fundamental problem is.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2023-144-AC2
RC2: 'Comment on egusphere-2023-144', Anonymous Referee #2, 07 Aug 2023

General comments

Manuscript presents a complementary approach to Voellmy rheology based on the separation of behaviour with respect to flow velocities. The approach proposed is demonstrated with a lumped mass model, of which limitations are clearly stated by the author. As demonstration is limited to a very simple application, discussion is also not presented at depth, which obscures the potential of the approach developed. In order to substantiate the claims made by the author especially in the Conclusions section, further demonstrations using other forms of simplified topographies can be presented.

Even though the manuscript has an unorthodox structure for a scientific paper, it is clearly written and conveys its message to its readers. However, it reads more like a technical note than a research paper at its current state. I suggest reconsideration of the manuscript after major revisions.

Specific comments

1. Page 2 - LL 33 – 35: The author makes several claims about the mechanism behind the fluid-like behaviour of rock avalanches. This paragraph sounds rather speculative, unless it is supported by references to published works in the literature. I suggest to include citations to convince the readers that there is no consensus on the aforementioned mechanisms.

2. Page 3 - LL 71 - 73: The author explains the selection of Voellmy rheology only based on its wide use. If the approach by Jop et al. (2005) is not used elsewhere in the manuscript, what is the motivation of introducing it? Please highlight this.

3. Page 3 - Figure 1: This figure is not clear enough in its current form. Choice of colours hinders differentiating the groups of lines. It might be useful to change line type between Voellmy and Jop et al., and use different shades of a colour within the same group to highlight the thickness

used. Readers could also benefit to know about the exact thickness used in the calculations that yielded lines in Figure 1, which can be given with a legend.

4. Page 5 - LL 113 – 115: Could the author suggest even approximate ranges for ”high velocities” and ”low velocities”?

5. Page 5 - Figure 2: Similar to Comment #2, a legend indicating the thicknesses used in the calculations would be beneficial.

6. Page 15 - Line 341: "The new approach should be able to improve numerical continuum simulations". I think this is a strong statement without testing it. I suggest reformulating this sentence.

Technical corrections
I did not detect any obvious typos in the manuscript at its current state.

Citation: https://doi.org/10.5194/egusphere-2023-144-RC2

Stefan Hergarten

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Short summary

Large landslides turn into an avalanche-like mode of flow at high velocities, which allows for a much longer runout than predicted for a sliding solid body. In this study, the Voellmy rheology widely used in models for hazard assessment is reinterpreted and extended. The new approach predicts the increase in runout length with volume observed in nature quite well and may thus be a major step towards a more consistent modeling of rock avalanches and improve hazard assessment.


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