# Comments on “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov

## Abstract

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## 1. On the History of the Paper by Grigorian and Ostroumov

## 2. Notes on Editor’s Changes and on Technical Points in the Paper by Grigorian and Ostroumov

**Note****0**- The title was changed following the suggestion of an anonymous reviewer.
**Note****1****Note****2**- Equation (3) can be derived from the modified dry-friction law stated below in Equation (4) by assuming that the normal pressure ${p}_{n}$ decreases linearly with the height. Then the two channel sidewalls combined contribute a force per unit length of $2\mu \rho a{h}^{2}/2$ in the case ${\tau}_{1}<{\tau}_{*}$. Dividing by the density $\rho $, the flow depth h and the channel width L to obtain an acceleration, one verifies the upper part of Equation (3). In order to obtain the formula for the case ${\tau}_{1}>{\tau}_{*}$, one first needs to find the height ${z}_{*}$ where the internal shear stress equals ${\tau}_{*}$ from the equation$$\mu \rho a(h-{z}_{*})={\tau}_{*},$$
**Note****3**- See Section 4 for the editor’s comments on Grigorian’s modified friction law. An obvious misprint in the lower line ($\mu {p}_{n}\le {\tau}_{*}$) was corrected to $\mu {p}_{n}\ge {\tau}_{*}$. The optical appearance of the conditions for the two alternatives was made consistent between Equations (3) and (4) without changing their mathematical content.
**Note****4**- The jump conditions (5) and (6) can be derived by considering the mass and momentum balances in infinitesimally thin boxes enclosing a short piece of the erosion front (see Figure 1). These boxes are assumed to move with the erosion front. The mass and momentum changes inside each of these boxes during a time interval $\Delta t$ must vanish—if this were not the case, the density and/or the speed of the boxes would diverge because the box volume and thus the mass and momentum content are infinitesimally small.Let $\mathrm{d}l$ be the length of the box measured along the interface and $\omega $ and v the interface-normal velocities of the interface and the moving snow, respectively, right above the interface (measured relative to the fixed Earth coordinate system). Then the snow mass per unit width flowing from the snow cover into the box in a time interval $\mathrm{d}t$ is$$\mathrm{d}{m}_{\mathrm{in}}={\rho}_{0}\omega \phantom{\rule{0.166667em}{0ex}}\mathrm{d}l\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t.$$On the avalanche side, the mass$$\mathrm{d}{m}_{\mathrm{out}}={\rho}_{1}(\omega -v)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}l\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t$$In deriving the momentum jump condition, care is required with regard to the signs of momentum fluxes and the choice of coordinate system. Here we choose to describe the process in a coordinate system that moves with the interface and whose x-axis is perpendicular to the interface, pointing from the snowcover into the avalanche. Then the momentum flux (i.e., momentum per unit area and unit time) into the box is given by$${J}_{\mathrm{in}}={\rho}_{0}{\omega}^{2}+{p}_{*}.$$${p}_{*}$ is the compressive strength of the snowcover that is mobilized at the interface and is a non-negative quantity. It leads to momentum entering the box in the $+x$-direction and is therefore counted positive. The corresponding momentum flux on the avalanche side is$${J}_{\mathrm{out}}={\rho}_{1}{(\omega -v)}^{2}+p(h,u).$$The pressure from the avalanche acts in the $-x$-direction, reducing the momentum inside the box. This corresponds to a positive momentum flux out of the box; the pressure $p(h,u)$ being a positive quantity, it is added to the mass-related momentum flux ${\rho}_{1}{(\omega -v)}^{2}$. Equating ${J}_{\mathrm{in}}$ to ${J}_{\mathrm{out}}$ and using ${\rho}_{1}(\omega -v)={\rho}_{0}\omega $ from the jump condition for the mass, one arrives at (6). Equation (7) then follows by simple algebra. For further comments on the assumptions behind this erosion model, see (Reference [9], [A.1.2.1]) and Section 5 of this paper.
**Note****5**- The correct formulation of the momentum balance equation in the presence of entrainment has been a source of confusion and heated debates for decades. The authors formulate the problem such that the eroded snow enters the avalanche with speed 0. Moreover, they write an equation of motion rather than a momentum balance equation and therefore—correctly—include the pseudo-force $-qu/F$, which arises because the eroded snow must be accelerated to the mean speed of the avalanche, u. See the beginning of Section 5.4 of this paper for details.
**Note****6**- The original computer code used in Reference [1] is lost, but A. V. Ostroumov [18] described its key features, which are summarized below. The approach anticipates many features of the numerical technique now known as the Material Point Method (MPM) in that two separate grids, one moving and one fixed, are used for numerically solving Equations (1)–(10) in Reference [1]. The equations for $F(S,t)$ (cross-sectional area of the flow), $u(S,t)$ (depth-averaged velocity), $w\left(t\right)$ (front velocity and ${S}_{f}\left(t\right)$ (front position) are solved on a moving grid G${}_{1}$ with a constant, user-selected number of nodes, ${N}_{1}$. The uppermost node $i=1$ is held fixed at the fracture line of the avalanche while the last node $i={N}_{1}$ moves with the avalanche front. To avoid loss of precision due to large gradients of the flow depth $h\left(S\right)$ near the front of the avalanche, the cell length $\Delta {S}_{1}$ is kept constant at a user-selected value between the nodes $i={N}_{1}-{n}_{1}$ and $i={N}_{1}$ near the front. In contrast, the cell lengths between the rear nodes $i=1,\dots ,{N}_{1}-{n}_{1}$ grow uniformly in the process of avalanche propagation.The variables $\delta (S,t)$ (erodible snow depth), $\alpha (S,t)$ (inclination of the snow surface relative to the terrain) and $q(S,t)$ (volumetric entrainment rate integrated over the avalanche width) are calculated on a spatially fixed grid G${}_{2}$ with uniform, user-selected spacing $\Delta {S}_{2}$. At each time step, the currently active portion of G${}_{2}$ is determined as follows: The rearmost active node of G${}_{2}$, ${j}_{e}$, is chosen such that $\delta ({S}_{j},t)=0$ for $j<{j}_{e}$ and $\delta ({S}_{j},t)>0$ for $j\ge {j}_{e}$, that is, behind node ${j}_{e}$ the snow cover is already completely eroded. The frontmost node, ${j}_{f}$, is set at ${j}_{f}=\lceil {S}_{f}\left(t\right)/\Delta {S}_{2}\rceil +5$; thus the number of active G${}_{2}$ nodes is ${N}_{2}\left(t\right)={j}_{f}\left(t\right)-{j}_{e}\left(t\right)+1$.Simulations are advanced by one time step from t to ${t}_{1}=t+\Delta t$ by first calculating the new values $\delta ({S}_{j},{t}_{1})$, $\alpha ({S}_{j},{t}_{1})$ and $q({S}_{j},{t}_{1})$ on G${}_{2}$ by applying the formulas (5)–(9) to the known values $F({S}_{j},t)$ and $u({S}_{j},t)$, which are obtained by interpolating from G${}_{1}$ to G${}_{2}$. After interpolating the so obtained values from G${}_{2}$ to G${}_{1}$, $\delta ({S}_{i},{t}_{1})$, $\alpha ({S}_{i},{t}_{1})$ and $q({S}_{i},{t}_{1})$ can be used to compute the new values $F({S}_{i},{t}_{1})$, $u({S}_{i},{t}_{1})$, $w(t-1)$, ${S}_{f}\left({t}_{1}\right)$ on G${}_{1}$.For calculating each of the different cases listed in (Reference [1], Table 1), suitable values of ${N}_{1}$, ${n}_{1}$, $\Delta {S}_{1}$ and $\Delta {S}_{2}$ are selected to obtain sufficiently precise results. The variable time step $\Delta t$ is determined from the stability condition of the numerical scheme.
**Note****7**- In the original manuscript, the authors erroneously state that they omitted the last two terms of Equation (1) in Reference [1], that is, $-{f}_{1}-{f}_{2}$. In reality, they drop the longitudinal pressure gradient $-(L/2F){\partial}_{S}(a{F}^{2}/{L}^{2})$ and the entrainment deceleration $-{f}_{2}$, but retain the resistance term $-{f}_{1}$.Similar assumptions have been used in applications to various types of gravity mass flows. If the momentum balance equations are integrated over the entire (instantaneous) length of the flow, the pressure-gradient term integrates to zero. Grigorian and Ostroumov do not explicitly carry out such an integration and therefore have to neglect the pressure-gradient term explicitly. If one assumes a constant shape (but variable length and height) of the longitudinal section of the flow, the local velocities and flow depths can be expressed in terms of the center-of-mass velocity (or, alternatively, the front velocity), the maximum instantaneous height over the entire body of the flow, and the assumed shape functions. With this, important local variables like the bed shear stress or the entrainment rate can be computed and integrated. In an early model of powder-snow avalanches, a half-ellipse was assumed as the shape function [19], whereas uniform height leads to so-called “box models” (e.g., Reference [20]).Grigorian and Ostroumov integrate the mass balance equation over the avalanche length, but do not carry out this step for the momentum balance (or equation of motion). Instead, they argue that only the frontal region is of importance and, effectively, only write down the equation of motion of the front. The combined assumptions of no entrainment and constant longitudinal profile shape allow expressing the frontal flow depth simply as $H={H}_{0}{l}_{0}/l$.
**Note****8**- In the original manuscript, the gravitational metric system (GMS) is used. This means that densities and stresses have units of kg s${}^{2}$ m${}^{-4}$ and kg m${}^{-2}$, respectively, that is, they are densities and stresses divided by the standard gravitational acceleration, $g\approx 9.806\phantom{\rule{0.166667em}{0ex}}65\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$. In the edited text, SI units have been used consistently together with the approximation $g\approx 10\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$, but GMS units are used in some of the original figures.
**Note****9**- The Tables 3.2 and 4.1 in the original manuscript were combined into the single Table 1 to make the text more concise and to facilitate direct comparison between the results of the full model and different approximations.
**Note****10**- In the argument of the first exponential function of the second term under the root sign, a minus sign was omitted in the original manuscript. Note also that this formula assumes tacitly that the value of D is constant within each segment. If D changes at some point because $\tau $ crosses the value ${\tau}_{*}$, the segment needs to be subdivided at that point.
**Note****11**- The original manuscript reads $\gamma S=A/B$, which clearly is a misprint.
**Note****12**- The point-mass model can be derived from the full model by integrating over the length of the avalanche body. The gradient of the hydrostatic pressure thereby integrates to the difference of its values at the front and tail of the avalanche, which individually are 0. this reflects the fact that this is an internal force and does not influence the motion of the center of mass. Note, however, that in this derivation $sin\psi \left(S\right)$ and $cos\psi \left(s\right)$ should be understood as their mean values along the avalanche body at this instant.
**Note****13**- Subsection titles added by the editor.
**Note****14**- In the original manuscript, $\psi \left(\xi \right)$ is used instead of ${\psi}_{1}$. Since $\psi $ is declared constant, including an explicit $\xi $-dependence may be confusing. To emphasize that the value of $\psi $ to use is the one from the first (inclined) segment, the subscript ${}_{1}$ was added. Moreover, the condition ${u}_{0}\ge 0$ was added because the formula is not valid in the (admittedly unrealistic) case where the avalanche has an uphill initial velocity.
**Note****15**- The original manuscript reads u instead of ${u}^{2}$.
**Note****16**- The full run-out distance is measured along the path and not along its projection onto the horizontal plane, as would be more customary in hazard-mapping applications.
**Note****17**- The formula given in the original manuscript is$$u={u}_{0}\sqrt{1-\frac{gH(sin{\psi}_{1}-\mu cos{\psi}_{1})}{k{u}_{0}^{2}}\left(\right)open="["\; close="]">exp\left(\right)open="("\; close=")">-\frac{2k}{H}(S-{S}_{0})-1}.$$This is correct only in the case ${u}_{0}=0$. If ${u}_{0}\ne 0$ and $S\to \infty $, it gives$$u\u27f6\sqrt{{u}_{0}^{2}+{\displaystyle \frac{gH}{k}}(sin{\psi}_{1}-\mu cos{\psi}_{1})},$$$${u}_{\infty}=\sqrt{{\displaystyle \frac{gH}{k}}(sin{\psi}_{1}-\mu cos{\psi}_{1})}.$$Solving the differential equation, one finds that a factor $exp[-(2k/h)(S-{S}_{0})]$ was omitted from the first term under the root. Moreover, if ${u}_{0}<0$, the original equation gives $u\left(S\right)<0$ all the while the avalanche is moving in the $+S$-direction.After Equation (5.16) in the original manuscript (corresponding to (57) in the edited text), the authors stated that the run-out distance does not depend on the size of the avalanche, even though both the original and corrected equation show an approximately linear dependence on the flow depth H. This statement has therefore been deleted.
**Note****18**- A small misprint (${L}_{2}$ instead of L on the left-hand side) has been corrected.
**Note****19**- The original manuscript erroneously subtracts ${l}_{1}$ from the right-hand side of Equation (64). That would give the same result as for ${l}_{2}$ in (63).
**Note****20**- A spurious factor of H in the denominator of the expression for ${u}_{**}$ has been removed.
**Note****21**- The original manuscript states ${l}_{2}/{l}_{1}\approx 0.125/8=0.0156$ in the first case with Coulomb dry friction and ${l}_{2}/{l}_{1}\approx 0.25/8=0.0313$ for the case of stress-limited dry friction.
**Note****22**- Compared to friction parameters typically used in the simulation of medium-size to large avalanches with Voellmy-type models (see, e.g., Reference [21]), $k=0.02$ (corresponding to $\xi =500\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}}^{-2}$) appears rather large. With a more typical value $k=0.005$ (or $\xi \approx 2000\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}}^{-2}$), ${l}_{2}/{l}_{1}\approx ...$ results with Coulomb-type dry friction and ${l}_{2}/{l}_{1}\approx ...$ for the stress-limited friction law.
**Note****23**- A. V. Ostroumov confirmed that the authors observed this behavior only in simulations with both entrainment and the stress-limited friction law [18].
**Note****24**- This table was compiled by the editor on the basis of the information contained in Figures 8 and 9 and S2.10–S2.14.

## 3. Grigorian and Ostroumov’s Work in the Context of Present-Day Avalanche Dynamics Research—Is There Still a Use for Simple Avalanche Models in the 21st Century?

## 4. A Brief Discussion of Grigorian’s Stress-Limited Friction Law

## 5. Some Remarks on the Grigorian–Ostroumov Erosion Model

#### 5.1. How Does the Grigorian–Ostroumov Erosion Formula Relate to Other Erosion and Entrainment Models?

#### 5.2. The Eglit–Grigorian–Yakimov Model for Frontal Entrainment

#### 5.3. Main Features of the Tangential-Jump Entrainment Model

#### 5.4. Particular Features of the Grigorian–Ostroumov Erosion Model

#### 5.5. Can the Grigorian–Ostroumov Erosion Formula Be Used in Practice?

## 6. Do Avalanches Behave Non-Monotonically in the Friction Coefficients?

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Grigorian, S.S.; Ostroumov, A.V. On a continuum model for avalanche flow and its simplified variants. Geosciences
**2020**, 10, 35. [Google Scholar] [CrossRef] [Green Version] - Voellmy, A. Über die Zerstörungskraft von Lawinen. Schweiz. Bauztg.
**1955**, 73, 159–165, 212–217, 246–249, 280–285. [Google Scholar] - Briukhanov, A.V.; Grigorian, S.S.; Miagkov, S.M.; Plam, M.Y.; Shurova, I.Y.; Eglit, M.E.; Yakimov, Y.L. On some new approaches to the dynamics of snow avalanches. In Physics of Snow and Ice, Proc. Intl. Conf. Low Temperature Science, Sapporo, Japan, 1966; Ôura, H., Ed.; Institute of Low Temperature Science, Hokkaido University: Sapporo, Japan, 1967; Volume I, Part 2; pp. 1223–1241. [Google Scholar]
- Savage, S.B.; Hutter, K. The motion of a finite mass of granular material down a rough incline. J. Fluid Mech.
**1989**, 199, 177–215. [Google Scholar] [CrossRef] - Eglit, M.E. Teoreticheskie podkhody k raschetu dvizheniia snezhnyk lavin. (Theoretical approaches to avalanche dynamics). Itogi Nauki. Gidrologiia Sushi. Gliatsiologiia
**1967**, 69–97. (In Russian) English translation in: Soviet Avalanche Research—Avalanche Bibliography Update: 1977–1983. Glaciological Data Report GD–16, pages 63–116. World Data Center A for Glaciology [Snow and Ice], 1984 [Google Scholar] - Bakhvalov, N.S.; Eglit, M.E. Issledovanie reshenii upravlenii dvizheniia snezhnykh lavin (Investigations of the solutions to snow avalanche movement equations). Akad. Nauk SSSR Inst. Geogr. Mat. Gliatsol. Issledov. Khr. Obsuzhdeniia
**1969**, 15, 31–38. (In Russian) Translated in: Glaciological Data, Soviet Avalanche Research—Avalanche Bibliography Update: 1977–1983, World Data Center A for Glaciology, Boulder, CO, USA, Report GD-16, 1984 [Google Scholar] - Eglit, E.M. Some mathematical models of snow avalanches. In Advances in the Mechanics and the Flow of Granular Materials, 1st ed.; Shahinpoor, M., Ed.; Trans Tech Publications: Clausthal-Zellerfeld, Germany, 1983; Volume II, pp. 577–588. [Google Scholar]
- Eglit, M.E. Mathematical and physical modelling of powder-snow avalanches in Russia. Ann. Glaciol.
**1998**, 26, 281–284. [Google Scholar] [CrossRef] - Eglit, M.E.; Demidov, K.S. Mathematical modeling of snow entrainment in avalanche motion. Cold Reg. Sci. Technol.
**2005**, 43, 10–23. [Google Scholar] [CrossRef] - Kulibaba, V.S.; Eglit, M.E. Numerical modeling of an avalanche impact against an obstacle with account of snow compressibility. Ann. Glaciol.
**2008**, 49, 27–32. [Google Scholar] [CrossRef] [Green Version] - Eglit, M.E.; Yakubenko, A.E. Numerical modeling of slope flows entraining bottom material. Cold Reg. Sci. Technol.
**2014**, 108, 139–148. [Google Scholar] [CrossRef] - Grigorian, S.S. Mechanics of snow avalanches. In Snow Mechanics Symposium Mécanique de la Neige— Proceedings of the Grindelwald Symposium, April 1974; LaChapelle, E.R., Kuroiwa, D., Salm, B., Eds.; International Association of Hydrological Sciences: Wallingford, UK, 1975; pp. 355–368. [Google Scholar]
- Grigoryan, S.S.; Urumbayev, N.A. On the nature of an avalanche air wave. Nauchnye trudy Instituta Mekhaniki Moskovskogo Gosudarstvennogo Universiteta [Scientific Works of the Institute of Mechanics of Moscow State University]
**1975**, 42, 74–82. (In Russian) English translation in: 34 Selected Papers on Main Ideas of the Soviet Glaciology, 1940s–1980s. Kotlyakov, V.M., Ed.; Glaciological Association of Russia: Moscow, Russia, 1997; pp. 289–296. [Google Scholar] - Grigoryan, S.S. A new law of friction and mechanism for large-scale avalanches and landslides. Sov. Phys. Dokl.
**1979**, 24, 110–111. [Google Scholar] - Sadovnichii, V.A.; Nigmatulin, R.I. Samvel Samvelovich Grigoryan (On his Eightieth Birthday). J. Appl. Math. Mech.
**2010**, 74, 255–266. [Google Scholar] [CrossRef] - Available online: https://cordis.europa.eu/event/rcn/1575/en (accessed on 31 May 2019).
- Eglit, M.E.; Yakubenko, A.; Zayko, J. A review of Russian snow avalanche models—From analytical solutions to novel 3D models. Geosciences
**2020**, 10, 77. [Google Scholar] [CrossRef] [Green Version] - Ostroumov, A.V.; (Institute of Mechanics, Lomonossov State University, Moscow, Russia). Personal communication, to D. Issler. September 2019.
- Kulikovskiy, A.G.; Sveshnikova, Y.I. Model’dlya rasheta dvizheniya pylevoy snezhnoy laviny (A model for calculating the motion of a powder-snow avalanche). Mater. Glyatsiol. Issled.
**1977**, 31, 74–80. (In Russian) [Google Scholar] - Dade, W.B.; Huppert, H.E. A box model for non-entraining, suspension-driven gravity surges on horizontal surfaces. Sedimentology
**1995**, 42, 453–471. [Google Scholar] [CrossRef] - Bartelt, P.; Bühler, Y.; Christen, M.; Deubelbeiss, Y.; Salz, M.; Schneider, M.; Schumacher, L. RAMMS::AVALANCHE User Manual; WSL Institute for Snow and Avalanche Research SLF: Davos Dorf, Switzerland, 2017. [Google Scholar]
- Salm, B. A short and personal history of snow avalanche dynamics. Cold Reg. Sci. Technol.
**2004**, 39, 83–92. [Google Scholar] [CrossRef] - Gauer, P. Comparison of avalanche front velocity measurements and implications for avalanche models. Cold Reg. Sci. Technol.
**2014**, 97, 132–150. [Google Scholar] [CrossRef] - McClung, D.M.; Gauer, P. Maximum frontal speeds, alpha angles and deposit volumes of flowing snow avalanches. Cold Reg. Sci. Technol.
**2018**, 153, 78–85. [Google Scholar] [CrossRef] - Bozhinskiy, A.N.; Losev, K.S. The Fundamentals of Avalanche Science; Institut für Schnee- und Lawinenforschung: Davos, Switzerland, 1998. [Google Scholar]
- De Blasio, F.V.; Medici, L. Microscopic model of rock melting beneath landslides calibrated on the mineralogical analysis of the Köfels frictionite. Landslides
**2017**, 14, 337–350. [Google Scholar] [CrossRef] - Sovilla, B. Field Experiments and Numerical Modelling of Mass Entrainment and Deposition Processes in Snow Avalanches. Ph.D. Thesis, ETH Zürich, Zürich, Switzerland, 2004. [CrossRef]
- Sovilla, B.; Burlando, P.; Bartelt, P. Field experiments and numerical modeling of mass entrainment in snow avalanches. J. Geophys. Res.
**2006**, 111, F03007. [Google Scholar] [CrossRef] - Issler, D.; Jóhannesson, T. Dynamically Consistent Entrainment and Deposition Rates in Depth-Averaged Gravity Mass Flow Models; NGI Technical Note 20110112-01-TN; Norwegian Geotechnical Institute: Oslo, Norway, 2011. [Google Scholar] [CrossRef]
- Issler, D. Dynamically consistent entrainment laws for depth-averaged avalanche models. J. Fluid Mech.
**2014**, 759, 701–738. [Google Scholar] [CrossRef] [Green Version] - Shapiro, L.H.; Johnson, J.B.; Sturm, M.; Blaisdell, G.H. Snow Mechanics—Review of the State of Knowledge and Applications; CRREL Report 97-3; US Army Corps of Engineers, Cold Regions Research and Engineering Laboratory: Hanover, NH, USA, 1997.
- Sovilla, B.; Sommavilla, F.; Tomaselli, A. Measurements of mass balance in dense snow avalanche events. Ann. Glaciol.
**2001**, 32, 230–236. [Google Scholar] [CrossRef] [Green Version] - Issler, D. Experimental information on the dynamics of dry-snow avalanches. In Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformations; Lecture Notes in Applied and Computational Mechanics; Hutter, K., Kirchner, N., Eds.; Springer: Berlin, Germany, 2003; Volume 11, pp. 109–160. [Google Scholar] [CrossRef]
- Gauer, P.; Issler, D. Possible erosion mechanisms in snow avalanches. Ann. Glaciol.
**2004**, 38, 384–392. [Google Scholar] [CrossRef] [Green Version] - Sovilla, B.; Margreth, S.; Bartelt, P. On snow entrainment in avalanche dynamics calculations. Cold Reg. Sci. Technol.
**2007**, 47, 69–79. [Google Scholar] [CrossRef] - Rauter, M.; Köhler, A. Constraints on Entrainment and Deposition Models in Avalanche Simulations from High-Resolution Radar Data. Geosciences
**2020**, 10, 9. [Google Scholar] [CrossRef] [Green Version] - Issler, D.; Pastor Pérez, M. Interplay of entrainment and rheology in snow avalanches: A numerical study. Ann. Glaciol.
**2011**, 52, 143–147. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic representation of the integration volume (blue box) for the mass and momentum jump conditions across the interface between snowcover (density ${\rho}_{0}$) and avalanche (density ${\rho}_{1}$). The normal velocity components on both sides of the interface (red arrows) refer to the coordinate system moving with the front.

**Figure 2.**Volume dependence of dry-friction coefficient $\mu $ in the recommended calibration for the flow model RAMMS::AVALANCHE for different altitude zones and two types of terrain characteristics. With an assumed scaling relation $h\propto {V}^{-0.25}$, the Grigorian’s friction law (GFL) captures that behavior well up to $V=60,000\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}$ (brown curve). To avoid large discrepancies for large volumes, the scaling exponent of h needs to be reduced to about 0.1 (yellow curve).

**Figure 3.**Control volumes used in the derivation of the Eglit–Grigorian (EGYEM, left), Grigorian–Ostroumov (GOEM, middle), and tangential-jump (TJEM, right) entrainment models. Note that the EGYEM and GOEM evaluate the jump equations for the velocity and momentum components normal to the “long” side of the control volume while the TJEM considers tangential velocity and shear stresses.

**Figure 4.**Schematic representation of the entrainment process for entire slabs. At the avalanche front, the new-snow layer (yellow, density ${\rho}_{0}$) is compressed to density ${\rho}_{1}$, but only partially accelerated, as shown by the rightmost velocity profile. Friction from the overriding avalanche body (turquoise) reduces the velocity deficit of the slab (represented by color changing from yellow to turquoise) until it becomes a part of the main body. During the process, scouring entrainment may also occur (small arrows).

**Figure 5.**Strongly simplified representation of a hypothetical structure of the $(H,U)$ phase space of the “enhanced simplified” Grigorian–Ostroumov model. Three copies of the phase-space plane, representing different values of the friction coefficient $\mu $, are shown; the parameters k, ${\tau}_{*}$ and ${p}_{*}$ are held fixed. The red dots mark an (arbitrarily chosen) initial condition $({H}_{0},{U}_{0})$. The avalanche evolves either to a final rest state indicated by the blue points on the axis $U=0$ (if $\mu <{\mu}_{1}$ or $\mu >{\mu}_{2}$) or towards and along a critical line (in blue) that approaches either a fixed point corresponding to a moving steady state or to a state where the avalanche grows and/or accelerates indefinitely. The colored areas indicate the basin of attraction of the fixed points on the axis $U=0$.

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Issler, D.
Comments on “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov. *Geosciences* **2020**, *10*, 96.
https://doi.org/10.3390/geosciences10030096

**AMA Style**

Issler D.
Comments on “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov. *Geosciences*. 2020; 10(3):96.
https://doi.org/10.3390/geosciences10030096

**Chicago/Turabian Style**

Issler, Dieter.
2020. "Comments on “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov" *Geosciences* 10, no. 3: 96.
https://doi.org/10.3390/geosciences10030096