The Entire Landslide Velocity
 Technical University of Munich, School of Engineering and Design, Chair of Landslide Research, Arcisstrasse 21, D80333, Munich, Germany
 Technical University of Munich, School of Engineering and Design, Chair of Landslide Research, Arcisstrasse 21, D80333, Munich, Germany
Abstract. The enormous destructive energy carried by a landslide is principally determined by its velocity. Pudasaini and Krautblatter (2022) presented a simple, physicsbased analytical landslide velocity model that simultaneously incorporates the internal deformation and externally applied forces. They also constructed various general exact solutions for the landslide velocity. However, previous solutions are incomplete as they only apply to accelerating motions. Here, I advance further by constructing several new general analytical solutions for decelerating motions and unify these with the existing solutions for the landslide velocity. This provides the complete and honest picture of the landslide in multiple segments with accelerating and decelerating movements covering its release, motion through the track, the runout as well as deposition. My analytical procedure connects several accelerating and decelerating segments by a junction with a kink to construct a multisectoral unified velocity solution down the entire path. Analytical solutions reveal essentially different novel mechanisms and processes of acceleration, deceleration and the mass halting. I show that there are fundamental differences between the landslide release, acceleration, deceleration and deposition in space and time as the dramatic transition takes place while the motion changes from the driving force dominated to resisting force dominated sector. I uniquely determine the landslide position and time as it switches from accelerating to decelerating state. Considering all the accelerating and decelerating motions, I analytically obtain the exact total travel time and the travel distance for the whole motion. Different initial landslide velocities with ascending or descending fronts result in strikingly contrasting travel distances, and elongated or contracted deposition lengths. Time and space evolution of the marching landslide with initial velocity distribution consisting of multiple peaks and troughs of variable strengths and extents lead to a spectacular propagation pattern with different stretchings and contractings resulting in multiple waves, foldings, crests and settlements. The analytical method manifests that, computationally costly numerical solutions may now be replaced by a highly costeffective, unified and complete analytical solution down the entire track. This offers a great technical advantage for the geomorphologists, landslide practitioners and engineers as it provides immediate and very simple solution to the complex landslide motion.
Shiva P. Pudasaini
Status: final response (author comments only)

RC1: 'Comment on esurf202231', Anonymous Referee #1, 08 Aug 2022
In this manuscript, exact solutions to a recent model by Pudasaini & Krautblatter (2022) are presented. The model is a simplification of the widely used shallowwater approach for modeling shallow granular flows such as avalanches and landslides. The advance in this manuscript is to extend the range of exact solutions computed by Pudasaini & Krautblatter (2022) to decelerating flows, allowing a wider range of solutions to be calculated.
The model presented in this paper is considerably simpler than the shallow water models commonly used in both research and operational contexts for avalanche prediction and mitigation. The advantage of this simplicity is that explicit exact solutions can be found for avalanches that are steady or spatiallyuniform, and implicit exact solutions can be found for flows varying in space and time.
While I am generally supportive of the approach of mathematical analysis of simple models, I have three major concerns about this work, which in my view mean that it is not suitable for publication in ESurf:1. My primary concern is that the simplifications that have been made in the underlying model are so great, that the model is simply not relevant to real earth surface dynamics. I believe the model needs validation, including comparison of its predictions against the wellestablished shallowwater models and/or observations.
2. The paper appears to be aimed at practitioners who would benefit from a simple formula for avalanche velocity, but various aspects of the solutions presented appear make them unsuitable for this task.
3. Some of the results presented are not solutions of the governing equation, due to the exact solution method used being used beyond its point of validity.
Details of these points are given below:
1. The model is not a realistic description of natural avalanches and landslides(a) The model used in this paper is independent of avalanche thickness. This means that the model predicts a landslide runout distance that is independent of the volume of material in the avalanche. This contradicts possibly the most fundamental observation of natural avalanches and landslides, namely that the runout distance (and avalanche velocity) increase with increasing avalanche volume. Consequently, I have serious concerns as to whether the work in this manuscript is of relevance to real avalanches or landslides.
(b) Gradients of the thickness of the avalanche enter into the model equations through the expression for alpha on line 95. However (though it is not stated explicitly) alpha is then assumed to be constant (or a prescribed piecewiseconstant function). This is a very significant assumption that is not justified in either this manuscript or in Pudasaini & Krautblatter (2022). It differs significantly the dominant 'shallowwater' modeling approach where conservation of mass and momentum are used to determine how the thickness varies as a function of space and time.
Surprisingly, neither this manuscript nor Pudasaini & Krautblatter (2022) attempt to validate the new model. To validate the assumption of (piecewise) constant alpha, I would like to see comparison of numerical solutions of a shallowwater type model (e.g. equations 1 and 2 of Pudasaini & Krautblatter 2022) with the corresponding velocity equation (equation 5 of Pudasaini & Krautblatter 2022). The commonlystudied initial conditions of a 'dambreak' release of a finite mass of material could be used as one text case. Good agreement between the velocity fields and runout lengths of the two models would provide some reassurance that the assumption of constant alpha is reasonable.
(c) The origin of the term beta u^2 in equation (1) is unclear. It is described as a viscous drag coefficient, but is not of the correct form for either a Newtonian viscosity, nor a Chezy or Voellmy drag (in the latter cases I would expect a term scaling with u^2/h, introducing the volume dependence mentioned in point 1(a)). What is the physical derivation of this term? Is it validated by any field or laboratory observations?
2. Setting aside the realism of the model, the exact solutions presented in this paper provide very limited value to the practitioners at which this paper appears to be aimed
(a) The model used in this paper is not predictive of avalanche thickness. This is an absolutely fundamental problem for using this model to assess landslide hazard (e.g. design protective structures, line 33), because the momentum and kinetic energy of a flow are proportional to the flow thickness.
(b) In the simple solutions presented in section 5.1, the parameters alpha and beta are given various constant values. How are these values chosen? (e.g. in line 263, what is the process to 'properly choose' the parameters? In particular, how is the free surface gradient chosen, given that it is spatially varying and can take any value?) The solutions of the model are clearly sensitive to the values of alpha and beta. If the model were to be applied to a real avalanche, how could the value of alpha and beta be found (as a function of distance downslope)?
(c) The model solutions in section 4 are not explicit: they rely on numerical solution of implicit algebraic equations (19) and (22). As such, the model equations presented here require numerical solution (and potentially, identification of multiple solutions), and therefore do not have the advantage of simplicity associated with explicit exact solutions.
(d) The introduction discusses the value of computing exact solutions to equations, and I am in agreement that exact solutions are have significant value for assessing models and numerical methods. However, the problems associated with solving full shallowwater models numerically (line 50) are overstated in this paper. Numerical methods for hyperbolic systems, such as that of Kurganov and Petrova https://doi.org/cms/1175797625, are robust and well validated, and have become very widely used. Importantly, they are not computationally expensive, and can find accurate numerical approximations to onedimensional problems, such as those studied in this manuscript, within a few seconds. Numerical shallowwater calculations of this sort have been the primary tool used in operational avalanche hazard mapping for some years. The numerical shallowwater approach avoids many of the shortcomings of the present manuscript, in that it can predict both avalanche thickness and velocity, using a realistic rheology, and can be applied directly to real digital elevation model topography.
3. As noted in Pudasaini & Krautblatter 2022, the implicit equations 18, 19, 21, 22, have multiple solutions, and these are interpreted in this manuscript as a `folding' wave. This is incorrect mathematics: the solutions to equations (18,19,21,22) cease to become solutions to the governing PDE (equation 1) at the point that multiplicity of solutions starts (ie. when the gradient du/dx diverges). The `folding' process described in section 5.2.2 and 5.2.3 is therefore simply a mathematical artifact that occurs when the particular implicit solution method used in this paper is pushed beyond its point of validity. Therefore, the analogies made in section 5.2.3 between the shape of plots in figure 8, and folding depositional behavior in avalanche deposits are not valid. As the author is no doubt aware, solutions to shockforming PDEs (such as equation (1) of this paper) only exist up to the formation of a shock, and require an additional equation (a jump condition) to be integrated beyond this point. AC2: 'Reply on RC1', Shiva P. Pudasaini, 20 Nov 2022

RC2: 'Comment on esurf202231', Anonymous Referee #2, 01 Sep 2022
This article presents a synthetic physical model describing the propagation of a landslide over a slope and details analytical solutions of the equations in various cases.
My main concern regarding this work it that, though many results are said to be useful to practitioners, overall they are presented in a very abstract way, which makes it particularly difficult for the reader to see their relevance and potential applications. The solutions that are exhibited all derive from initial configurations that appear as very arbitrary, and the results are not related to any concrete examples. Nor is any comparison made to simple benchmarks (either numerical, experimental or fieldbased) that might be available in the literature. I understand that the given examples are useful to demonstrate the possibilities of the model, but they do not tell about its relevance or validity.
Furthermore, all calculations account for the velocity field but at no point do the results include the landslide's volume, thickness or shape, which are obviously quantities of interest for practical applications: is it implicit that the thickness is constant and uniform? or what does the model predict for its variations with x and t?In consequence, I think that the manuscript would be easier to follow and more suitable for publication in ESurfD provided that more effort is made to relate its conclusions to (even simple) physical/geological configurations.
Below are some more precise remarks and questions about the manuscript:
# Section 2
Presentation of the model in section 2 is rather confusing. In the following 'Results' sections, eqs (1) and (2) are referred to as radically different (though they only differ by a sign convention), and it looks like eq (1) stands for 'accelerated' and eq (2) for 'decelerated'. However from section 2, one gets the impression that equation (1) covers all cases (l.119 'we have the following two situations' and l.129 'alpha^a<0') and eq (2) is a subcase.
It would be much clearer to start from the beginning with either two distinct equations (say (a) and (d)) that include only positive coefficients, or (perhaps even simpler) a unique equation with two cases (alpha>0 and alpha<0). Additionally, a sketch presenting the physical system modelled by these equations would be most useful.
l.89: Is the solid fraction supposed to be constant (and independent on the local velocity or other varying parameters)? if it is indeed the case it should be precised. Similarly, h_g is included among the other 'external' parameters, but it has to be intrinsically linked to the landslide dynamics: can the author detail the assumptions made here?
l.95: please associate more explicitely each term in alpha to its physical meaning. What does the term 'liquefaction' cover here?
l.121: 'the initial velocity u_0': doesn't is depend on the position x? Where is the condition verified?
l.131: what is the 'decelerating velocity', and why is it obviously always larger in the case II.1 than II.2?
# Section 3
l.166: the similarity between equations (5) and (6) would be more obvious if expressed in a more uniform way (e.g. not switch from 1/exp(A) to exp(A) and keep the same first factor)
l.186: I do not understand here what the travel time is (from where to where? what is a sector?)
# Section 5
l.265: if I am not mistaken, equations (1) and (2) are not dimensionless. Coefficients alpha and beta should therefore be given units.
What justifies the ranges adopted here? (and should the range for beta read 0.0010.0025 or rather 0.00250.01?)
And how realistic are these values? Perhaps the author can give an example of common values for each physical control parameter (slope, gamma, mu...) and the resulting value of alpha.
Same comment for beta: what values for the viscous drag coefficient are commonly used, typically in the abundant literature about shallowlayer ('SaintVenant') models for landslides?l.282286: please introduce earlier (maybe within a sketch) what the 'lower portion of the track', 'transition zone', 'fan region' are regarding to the model.
The whole paragraph is written in such a way that it is very hard to make out the concrete situation that is modelled here. Maybe this can be reformulated starting from the example that is actually computed in figure 1, for which I do not understand the initial configuration (what is the length of the sliding mass? is the velocity u0 uniform?)figure 1: if I am not mistaken, at this point of the analysis, u is a function of both space and time. If so, I do not understand what is plotted here: in figure (a) is it the velocity at a given position (and which), and in plot (b) at a given time?
In this figure as in the others, units are missing for u_0, alpha and beta.
l.307: I guess that 'ascending' and 'descending' refer here to the velocity, but 'ascending sector' sounds like it refers to an upward slope.
'Accelerating' and 'decelerating' might be more appropriate.l.312: please justify the transition from alpha=3.5 to alpha=1.2: what would physically cause such a transition (kink in the slope for instance?)
 figure 2: same question as for figure 1 (and as for figs 3,4,5): to what position (a) and time (b) do the plots correspond?
In the caption, the coordinates are given without units in two different coordinate spaces.l.332: what is a 'variable track'? Please give a physical example that would produce the results presented in the following figures (for instance, all other parameters being constant, what shape of the slope would lead to such successive values of alpha).
l.344: repetitive explanations of all ascending/descending connections do not seem necessary, terms being selfexplanatory.
l.357: I do not understand the sentences 'alpha values are relative to each other' and 'perceived as relatively negative to alpha^a'.
l.363370: the velocity is observed to change dramatically at the major kink, but this sounds intuitive if we impose a dramatic change in the value of alpha. Is this a realistic case?
The paragraph is concluded with the sentence 'this can be a scenario for a track': the section should start with the example of such a scenario, that is investigated here: what physical configuration (e.g. with alpha being controlled by the slope profile only) would lead to a brutal transition from alpha=6 to alpha=0.15?Overall, all situations studied here (figures 3,4,5) appear rather arbitrary and abstract. Though it is useful to demonstrate the capacities of the analytical model, it would be more convincing to apply them to concrete configurations: a first step would be to plot the slope profile that would lead to each calculated dynamics. Even better would be to compare the outcomes (e.g. runout distance) to other models in known, simple configurations (such as a constant slope followed by a horizontal plane). Numerical works of Mangeney et al. with SaintVenant equations and Staron et al. with DEM simulations, experimental works on inclines or even simplified versions of field cases should be used as benchmarks to validate the results obtained here.
 figures 3 and 4: I am not convinced that the list of all kink coordinates brings much to the results (especially since their positions are imposed). Focus should be brought upon travel time or runout distance.
 figure 5: though keeping the same colors is useful, the two different solutions have to be distinguishable on the plot (e.g. dotted vs plain lines).
Why is the second case totally unrealistic? Some landslides are known to travel more than 3.5 km and alpha could keep getting beyond that point.l.480: here again 'ascending' and 'descending' are equivocal and one might think that they refer to the shape of the front (i.e. h(x) and not u(x)), whose evolution it would be most interesting to plot here.
l.485: for the reader unfamiliar with the previous work, on what basis are these initial velocity profiles chosen? Once again concluding that the runout distance differ is most useful, but it is hard to relate the arbitrary 'initial' configurations to a practical situation (or, for that matter, to previous examples such as figure 5). Starting with the release of a given mass at zero velocity, how does the landslide end up in the 6a rather than 6b configuration?
section 5.2.3: the predictions of the model regarding the geometry of the deposit would indeed be of much interest, but the link between the results (velocity profile only) and the geomorphology (that is, the thickness profile of the deposit) remains only implicit here.

AC3: 'Reply on RC2', Shiva P. Pudasaini, 20 Nov 2022
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202231/esurf202231AC3supplement.pdf

AC3: 'Reply on RC2', Shiva P. Pudasaini, 20 Nov 2022

EC1: 'Associate editor comment', Eric Lajeunesse, 04 Nov 2022
Dear Shiva Pudasaini,
I have now received two anonymous reviews of your manuscript «The Entire Landslide Velocity ». Both reviewers appreciate the simplicity of your model with respect to the shallow water models, commonly used in the community. Yet both of them identify issues, which need to be addressed to make the manuscript accessible for the wider readership of ESurf. I would therefore advise you to revise your manuscript in line with the points raised by the reviewers. I would particularly insist on the following recommendations.
 Although the manuscript is presented as an effort to develop a model useful for practitioners, it focuses on the maths, sometimes to the detriment of Physics. There are many places where the manuscript — and the reader — would benefit from additional discussions about the relevance of the model, its potential applications, and the physical meaning of the parameters it involves.
 In the same vein, information about the assumptions that support the model and their range of validity are often implicit.The model has been presented in a previous publication, and the reader does not need a comprehensive mathematical derivation. Yet some basic information would help to make the manuscript accessible for a wider readership. What is the physics at work in the model? How are the lubrication, liquefaction and viscous forces parameterized? Does your model assume that the solid fraction is constant — and thus independent on the local velocity or other varying parameters ? But what does the model predict or assume regarding the landslide's volume, thickness and shape? How do you set the values of the parameters alpha and beta? etc…
 Given that your model is a simplification of the wellestablished shallowwater model, I agree with the reviewers that a comparison of the outcomes of the two models is essential for the reader to assess the validity and the potential benefits of your approach. A comparison of the predictions of your model (velocity, runout distance, …) to DEM simulations and/or experimental works in simple configurations would also help to convince the reader of what he might gained by adopting your approach.
 Like reviewer #1, I am concerned by the fact that your model seems independent of the landslide thickness. This point needs clarification. This is also one more reason to compare your model’s predictions against the shallowwater equations and, if possible, against experimental data available in the literature. Good agreement between the two would indeed provide reassurance about the validity of your simplified model.
 Over the last 10 years, the physics community has done considerable work on the rheology of granular media. I believe that your manuscript would strongly benefit from a discussion of your result in the light of recent results in the field of granular rheology. How, for example, does your lubrication, liquefaction and viscous forces connect to the wellestablished « mu of I » rheological framework? See, for example, Jop et al. (2006) or Pouliquen, O., & Forterre, Y. (2009).
Kind regards,
Eric Lajeunesse
Jop, P., Forterre, Y., & Pouliquen, O. (2006). A constitutive law for dense granular flows. Nature, 441(7094), 727730.Pouliquen, O., & Forterre, Y. (2009). A nonlocal rheology for dense granular flows. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367(1909), 50915107.
 AC1: 'Reply on EC1', Shiva P. Pudasaini, 20 Nov 2022
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