Skip to main content Accessibility help

Dynamically consistent entrainment laws for depth-averaged avalanche models

  • Dieter Issler (a1)


The bed entrainment rate in a gravity mass flow (GMF) is uniquely determined by the properties of the bed and the flow. In depth-averaging, however, critical information on the flow variables near the bed is lost and empirical assumptions usually are made instead. We study the interplay between bed and flow assuming a perfectly brittle bed, characterized by its shear strength ${\it\tau}_{c}$ , and erosion along the bottom surface of the flow; frontal entrainment is neglected here. The brittleness assumption implies that the shear stress at the bed surface cannot exceed ${\it\tau}_{c}$ . For quasi-stationary flows in a simplified setting, analytic solutions are found for Bingham and frictional–collisional (FC) fluids. Extending this theory to non-stationary flows requires some assumptions for the velocity profile. For the Bingham fluid, the profile of a ‘proxy’ quasi-stationary eroding flow is used; the rheological parameters are chosen to match the instantaneous velocity and shear-layer depth of the non-stationary flow. For the FC fluid, a two-parameter family of functions that closely match the profiles obtained in depth-resolved numerical simulations is assumed; the boundary conditions determine the instantaneous parameter values and allow computation of the erosion rate. Preliminary tests with the FC erosion formula incorporated in a simple slab model indicate that the non-stationary erosion formula matches the depth-resolved simulations asymptotically, but differs in the start-up phase. The non-stationary erosion formulae are valid only up to a limit velocity (and to a limit flow depth if there is Coulomb friction). This appears to mark the transition to another erosion regime – to be described by a different model – where chunks of bed material are intermittently ripped out and gradually entrained into the flow.


Corresponding author

Email address for correspondence:


Hide All
Aradian, A., Raphaël, E. & de Gennes, P.-G. 2002 Surface flows of granular materials: a short introduction to some recent models. C. R. Phys. 3, 187196.
Barbolini, M. & Issler, D.(Eds) 2006 Avalanche Test Sites and Research Equipment in Europe – an Updated Overview. SATSIE Project Team. Accessible at
Berger, C., McArdell, B. W., Fritschi, B. & Schlunegger, F. 2010 A novel method for measuring the timing of bed erosion during debris flows and floods. Water Resour. Res. 46, W02502.
Berger, C., McArdell, B. W. & Schlunegger, F. 2011 Direct measurement of channel erosion by debris flows, Illgraben, Switzerland. J. Geophys. Res. 116, F01002.
Bouchaud, J.-P., Cates, M. E., Ravi Prakash, J. & Edwards, S. F. 1994 A model for the dynamics of sandpile surfaces. J. Phys. I Paris 4 (10), 13831410.
Bouchet, A., Naaim, M., Bellot, H. & Ousset, F. 2004 Experimental study of dense flow avalanches: velocity profiles in steady and fully developed flows. Ann. Glaciol. 38, 3034.
Boutreux, T., Raphaël, E. & de Gennes, P.-G. 1998 Surface flows of granular materials: a modified picture for thick avalanches. Phys. Rev. E 58 (4), 46924700.
Breien, H., De Blasio, F. V., Elverhøi, A. & Høeg, K. 2008 Erosion and morphology of a debris flow caused by a glacial lake outburst flood, Western Norway. Landslides 5 (3), 271280.
Briukhanov, A. V., Grigorian, S. S., Miagkov, S. M., Plam, M. Ya., Shurova, I. Ya., Eglit, M. E. & Yakimov, Yu. L. 1967 On some new approaches to the dynamics of snow avalanches. In Physics of Snow and Ice, Proceedings International Conference Low Temperature Science, Sapporo, Japan, 1966, vol. I, Part 2 (ed. Ôura, H), pp. 12231241. Institute of Low Temperature Science, Hokkaido University.
Brugnot, G. & Pochat, R. 1981 Numerical simulation study of avalanches. J. Glaciol. 27 (95), 7788.
Cannon, S. H. & Savage, W. Z. 1988 A mass-change model for the estimation of debris-flow runout. J. Geol. 96, 221227.
Carroll, C. S., Louge, M. Y. & Turnbull, B. 2013 Frontal dynamics of powder snow avalanches. J. Geophys. Res. 118 (2), 913924.
Cherepanov, G. P. & Esparragoza, I. E. 2008 A fracture-entrainment model for snow avalanches. J. Glaciol. 54 (184), 182188.
Crosta, G. B., Imposimato, S. & Roddeman, D. 2009a Numerical modeling of 2-d granular step collapse on erodible and nonerodible surface. J. Geophys. Res. F114, F03020.
Crosta, G. B., Imposimato, S. & Roddeman, D. 2009b Numerical modelling of entrainment/deposition in rock and debris-avalanches. Engng Geol. 109 (1–2), 135145.
Douady, S., Andreotti, B. & Daerr, A. 1999 On granular surface flow equations. Eur. Phys. J. B 11, 131142.
Dufour, F., Gruber, U., Issler, D., Schaer, M., Dawes, N. & Hiller, M.1999 Grobauswertung der Lawinenereignisse 1998/1999 im Grosslawinenversuchsgelände Vallée de la Sionne. Interner Bericht 732. Eidg. Institut für Schnee- und Lawinenforschung, CH-7260 Davos Dorf, Switzerland.
Eglit, M. E.1968 Teoreticheskie podkhody k raschetu dvizheniia snezhnyk lavin. (Theoretical approaches to avalanche dynamics). Itogi Nauki. Gidrologiia Sushi. Gliatsiologiia pp. 69–97 (in Russian). English transl. Soviet Avalanche Research – Avalanche Bibliography Update: 1977–1983. Glaciological Data Report GD-16, pp. 63–116. World Data Center A for Glaciology (Snow and Ice), 1984.
Eglit, M. E. & Demidov, K. S. 2005 Mathematical modeling of snow entrainment in avalanche motion. Cold Reg. Sci. Technol. 43 (1–2), 1023.
Eglit, M. E. & Yakubenko, A. E. 2014 Numerical modeling of slope flows entraining bottom material. Cold Reg. Sci. Technol., doi:10.1016/j.coldregions.2014.07.002.
Finlayson, B. A. & Scriven, L. E. 1966 The method of weighted residuals – a review. Appl. Mech. Rev. 19 (9), 735748.
Fukushima, Y. & Parker, G. 1990 Numerical simulation of powder-snow avalanches. J. Glaciol. 36 (123), 229237.
Gauer, P. & Issler, D. 2004 Possible erosion mechanisms in snow avalanches. Ann. Glaciol. 38, 384392.
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 129.
Grigorian, S. S. & Ostroumov, A. V.1977 The mathematical model for slope processes of avalanche type (in Russian). Scientific Report 1955. Institute for Mechanics, Moscow State University, Moscow, Russia.
Gubler, H. 1987 Measurements and modelling of snow avalanche speeds. In Avalanche Formation, Movement and Effects (Proceedings of the Davos Symposium, September 1986) (ed. Salm, B. & Gubler, H.), IAHS Publication, vol. 162, pp. 405420. IAHS Press.
Gubler, H. & Hiller, M. 1984 The use of microwave FMCW radar in snow and avalanche research. Cold Reg. Sci. Technol. 9, 109119.
Hermann, F., Issler, D. & Keller, S. 1994 Towards a numerical model of powder snow avalanches. In Proceedings of the Second European Computational Fluid Dynamics Conference, Stuttgart (Germany), September 5–8, 1994 (ed. Wagner, S., Hirschel, E. H., Périaux, J. & Piva, R.), pp. 948955. J. Wiley & Sons.
Hungr, O. 1995 A model for the runout analysis of rapid flow slides, debris flows, and avalanches. Can. Geotech. J. 32, 610623.
Hungr, O. & Evans, S. G. 2004 Entrainment of debris in rock avalanches: an analysis of a long run-out mechanism. Bull. Geol. Soc. Am. 116 (9–10), 12401252.
Imran, J., Harff, P. & Parker, G. 2001 A numerical model of submarine debris flows with graphical user interface. Comput. Geosci. 274 (6), 717729.
Issler, D. 2003 Experimental information on the dynamics of dry-snow avalanches. In Dynamic Response of Granular and Porous Materials Under Large and Catastrophic Deformations (ed. Hutter, K. & Kirchner, N.), Lecture Notes in Applied and Computational Mechanics, vol. 11, pp. 109160. Springer.
Issler, D., Errera, A., Priano, S., Gubler, H., Teufen, B. & Krummenacher, B. 2008 Inferences on flow mechanisms from snow avalanche deposits. Ann. Glaciol. 49 (1), 187192.
Issler, D., Gauer, P. & Barbolini, M. 2000 Continuum models of particle entrainment and deposition in snow drift and avalanche dynamics. In Models of Continuum Mechanics in Analysis and Engineering. Proceedings of a Conference Held at the Technische Universität Darmstadt, September 30 to October 2, 1998 (ed. Balean, R.), pp. 5880. Shaker Verlag.
Issler, D., Gauer, P., Schaer, M. & Keller, S.1996 Staublawinenereignisse im Winter 1995: Seewis (GR), Adelboden (BE) und Col du Pillon (VD). Interner Bericht 694. Eidg. Institut für Schnee- und Lawinenforschung, Davos, Switzerland.
Issler, D. & Jóhannesson, T.2006 On the formulation of entrainment in gravity mass flow models. NGI Rep. 20021048-12. Norwegian Geotechnical Institute, N-0806 Oslo, Norway.
Issler, D. & Jóhannesson, T.2011 Dynamically consistent entrainment and deposition rates in depth-averaged gravity mass flow models. NGI Tech. Note 20110112-01-TN. Norwegian Geotechnical Institute, Oslo, Norway.
Issler, D. & Pastor Pérez, M. 2011 Interplay of entrainment and rheology in snow avalanches: a numerical study. Ann. Glaciol. 52 (58), 143147.
Iverson, R. M. 2012 Elementary theory of bed-sediment entrainment by debris flows and avalanches. J. Geophys. Res. F117, F03006.
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727730.
Kowalski, J. & McElwaine, J. N. 2013 Shallow two-component gravity driven flows with vertical variation. J. Fluid Mech. 714, 434462.
Lied, K., Instanes, B., Domaas, U. & Harbitz, C. 1998 Avalanche at Bleie, Ullensvang, January 1994. In 25 Years of Snow Avalanche Research, Voss 1998 (ed. Hestnes, E.), NGI Publication, vol. 203, pp. 175181. Norwegian Geotechnical Institute.
Louge, M. Y. 2003 A model for dense granular flows down bumpy inclines. Phys. Rev. E 67, 061303.
Louge, M. Y., Carroll, C. S. & Turnbull, B. 2011 Role of pore pressure gradients in sustaining frontal particle entrainment in eruption currents: the case of powder snow avalanches. J. Geophys. Res. 116 (F4), 002065.
Maeno, N. & Nishimura, K. 1987 Numerical computation of snow avalanche motion in a three-dimensional topography. Low Temp. Sci. A 46, 99110; (in Japanese).
Mangeney, A., Roche, O., Hungr, O., Mangold, N., Faccanoni, G. & Lucas, A. 2010 Erosion and mobility in granular collapse over sloping beds. J. Geophys. Res. 115, F03040.
McElwaine, J. N. 2005 Rotational flow in gravity current heads. Phil. Trans. R. Soc. Lond. A 363, 16031623.
Mellor, M. 1977 Engineering properties of snow. J. Glaciol. 19 (81), 1566.
Naaim, M., Naaim-Bouvet, F., Faug, T. & Bouchet, A. 2004 Dense snow avalanche modeling: flow, erosion, deposition and obstacle effects. Cold Reg. Sci. Technol. 39 (2–3), 193204.
Nishimura, K. & Maeno, N. 1987 Experiments on snow-avalanche dynamics. In Avalanche Formation, Movements and Effects. Proceedings of the Davos Symposium, September 1986 (ed. Salm, B. & Gubler, H.), IAHS Publication, vol. 162, pp. 395404. IAHS Press.
Norem, H., Irgens, F. & Schieldrop, B. 1987 A continuum model for calculating snow avalanche velocities. In Avalanche Formation, Movement and Effects. Proceedings of the Davos Symposium, September 1986 (ed. Salm, B. & Gubler, H.), IAHS Publication, vol. 162, pp. 363380. IAHS Press.
Norem, H. & Schieldrop, B.1991 Stress analyses for numerical modelling of submarine flowslides. NGI Rep. 522090-10. Norges Geotekniske Institutt, Oslo, Norway.
Owen, P. R. 1964 Saltation of uniform grains in air. J. Fluid Mech. 20 (2), 225242.
Parker, G., Fukushima, Y. & Pantin, H. M. 1986 Self-accelerating turbidity currents. J. Fluid Mech. 171, 145181.
Pastor, M., Haddad, B., Sorbino, G., Cuomo, S. & Drempetic, V. 2009 A depth-integrated, coupled SPH model for flow-like landslides and related phenomena. Intl J. Numer. Anal. Meth. Geomech. 33 (2), 143172.
Rajchenbach, J. 2003 Dense, rapid flows of inelastic grains under gravity. Phys. Rev. Lett. 90, 144302.
Rognon, P. G., Roux, J.-N., Naaim, M. & Chevoir, F. 2008 Dense flows of cohesive granular materials. J. Fluid Mech. 596, 2147.
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357369.
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14 (1), 170183.
Sailer, R., Fellin, W., Fromm, R., Jörg, Ph., Rammer, L., Sampl, P. & Schaffhauser, A. 2008 Snow avalanche mass-balance calculation and simulation-model verification. Ann. Glaciol. 48 (1), 183192.
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183222.
Scheiwiller, T.1986 Dynamics of powder-snow avalanches. PhD thesis, Laboratory of Hydraulics, Hydrology and Glaciology, ETH Zürich, Zürich, Switzerland, available from∼vawweb/vaw_mitteilungen/081/081_g.pdf.
Scheiwiller, T. & Hutter, K. 1982 Lawinendynamik – Übersicht über Experimente und theoretische Modelle von Fliess- und Staublawinen, Mittlg. VAW 58. Laboratory of Hydraulics, Hydrology and Glaciology. ETH Zürich, (in German), available from∼vawweb/vaw_mitteilungen/058/058_g.pdf.
Sovilla, B.2004 Field experiments and numerical modelling of mass entrainment and deposition processes in snow avalanches. PhD thesis, ETH Zürich, Zürich, Switzerland, available from
Sovilla, B., Burlando, P. & Bartelt, P. 2006 Field experiments and numerical modeling of mass entrainment in snow avalanches. J. Geophys. Res. 111, F03007.
Sovilla, B., McElwaine, J. N., Schaer, M. & Vallet, J. 2010 Variation of deposition depth with slope angle in snow avalanches: measurements from Vallée de la Sionne. J. Geophys. Res. 115 (F2), 113.
Sovilla, B., Sommavilla, F. & Tomaselli, A. 2001 Measurements of mass balance in dense snow avalanche events. Ann. Glaciol. 32, 230236.
Vallet, J., Gruber, U. & Dufour, F. 2001 Photogrammetric avalanche measurements at Vallée de la Sionne, Switzerland. Ann. Glaciol. 32, 141146.
Voellmy, A. 1955 Über die Zerstörungskraft von Lawinen. Schweizerische Bauzeitung 73, (12, 15, 17, 19), 159–165, 212–217, 246–249, 280–285. Available online at:
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Dynamically consistent entrainment laws for depth-averaged avalanche models

  • Dieter Issler (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.