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 Savage-Hutter 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.

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 velocity-dependent 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 bed-parallel 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 Savage-Hutter 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 Savage-Hutter approximation but of interest in view of numerical considerations. At this point I wonder if the defiance of the Savage-Hutter approximation can be dismissed in such an easy way: After all, their approach is not only a geometric argument on the bed-surface configuration. Instead, it is used to simplify the Navier-Stokes 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 Savage-Hutter 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 50-63 and Lines 80-84 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(theta-pi/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 uphill-facing and downhill-facing fronts. Is “a” the acceleration in the bed-parallel 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 174-175: 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 Courant-Friedrichs-Lewy criterion.

Lines 259-260: Avoid 1-sentence 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 332-333: “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 well-known 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 396-397: Rewrite “without getting into conflict …”.

Line 419: What makes the other models more comprehensive?

Lines 420-421: Why would the time it takes to develop a model affect its use in hazard assessment?

Lines 423-424: 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 plug-and-play 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.