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Dynamically consistent entrainment laws for depth-averaged avalanche models

Published online by Cambridge University Press:  28 October 2014

Dieter Issler
Dieter Issler*
Affiliation:
Natural Hazards Division, Norwegian Geotechnical Institute, Postboks 3930 Ullevål Stadion, 0806 Oslo, Norway
*
†Email address for correspondence: di@ngi.no
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Abstract

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.

JFM classification

Mathematical Foundations: General fluid mechanics Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 759 , 25 November 2014 , pp. 701 - 738
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© 2014 Cambridge University Press 

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References

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, 187–196.CrossRefGoogle Scholar
Barbolini, M. & Issler, D.(Eds) 2006 Avalanche Test Sites and Research Equipment in Europe – an Updated Overview. SATSIE Project Team. Accessible at http://satsie.ngi.no/docs/satsie_d08.pdf.Google Scholar
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.Google Scholar
Berger, C., McArdell, B. W. & Schlunegger, F. 2011 Direct measurement of channel erosion by debris flows, Illgraben, Switzerland. J. Geophys. Res. 116, F01002.Google Scholar
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), 1383–1410.Google Scholar
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, 30–34.CrossRefGoogle Scholar
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), 4692–4700.Google Scholar
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), 271–280.Google Scholar
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. 1223–1241. Institute of Low Temperature Science, Hokkaido University.Google Scholar
Brugnot, G. & Pochat, R. 1981 Numerical simulation study of avalanches. J. Glaciol. 27 (95), 77–88.CrossRefGoogle Scholar
Cannon, S. H. & Savage, W. Z. 1988 A mass-change model for the estimation of debris-flow runout. J. Geol. 96, 221–227.Google Scholar
Carroll, C. S., Louge, M. Y. & Turnbull, B. 2013 Frontal dynamics of powder snow avalanches. J. Geophys. Res. 118 (2), 913–924.CrossRefGoogle Scholar
Cherepanov, G. P. & Esparragoza, I. E. 2008 A fracture-entrainment model for snow avalanches. J. Glaciol. 54 (184), 182–188.Google Scholar
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.Google Scholar
Crosta, G. B., Imposimato, S. & Roddeman, D. 2009b Numerical modelling of entrainment/deposition in rock and debris-avalanches. Engng Geol. 109 (1–2), 135–145.Google Scholar
Douady, S., Andreotti, B. & Daerr, A. 1999 On granular surface flow equations. Eur. Phys. J. B 11, 131–142.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Eglit, M. E. & Demidov, K. S. 2005 Mathematical modeling of snow entrainment in avalanche motion. Cold Reg. Sci. Technol. 43 (1–2), 10–23.Google Scholar
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.Google Scholar
Finlayson, B. A. & Scriven, L. E. 1966 The method of weighted residuals – a review. Appl. Mech. Rev. 19 (9), 735–748.Google Scholar
Fukushima, Y. & Parker, G. 1990 Numerical simulation of powder-snow avalanches. J. Glaciol. 36 (123), 229–237.CrossRefGoogle Scholar
Gauer, P. & Issler, D. 2004 Possible erosion mechanisms in snow avalanches. Ann. Glaciol. 38, 384–392.Google Scholar
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 1–29.Google Scholar
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.Google Scholar
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. 405–420. IAHS Press.Google Scholar
Gubler, H. & Hiller, M. 1984 The use of microwave FMCW radar in snow and avalanche research. Cold Reg. Sci. Technol. 9, 109–119.Google Scholar
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. 948–955. J. Wiley & Sons.Google Scholar
Hungr, O. 1995 A model for the runout analysis of rapid flow slides, debris flows, and avalanches. Can. Geotech. J. 32, 610–623.Google Scholar
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), 1240–1252.Google Scholar
Imran, J., Harff, P. & Parker, G. 2001 A numerical model of submarine debris flows with graphical user interface. Comput. Geosci. 274 (6), 717–729.Google Scholar
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. 109–160. Springer.Google Scholar
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), 187–192.Google Scholar
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. 58–80. Shaker Verlag.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Issler, D. & Pastor Pérez, M. 2011 Interplay of entrainment and rheology in snow avalanches: a numerical study. Ann. Glaciol. 52 (58), 143–147.Google Scholar
Iverson, R. M. 2012 Elementary theory of bed-sediment entrainment by debris flows and avalanches. J. Geophys. Res. F117, F03006.Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727–730.CrossRefGoogle ScholarPubMed
Kowalski, J. & McElwaine, J. N. 2013 Shallow two-component gravity driven flows with vertical variation. J. Fluid Mech. 714, 434–462.Google Scholar
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. 175–181. Norwegian Geotechnical Institute.Google Scholar
Louge, M. Y. 2003 A model for dense granular flows down bumpy inclines. Phys. Rev. E 67, 061303.Google Scholar
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.Google Scholar
Maeno, N. & Nishimura, K. 1987 Numerical computation of snow avalanche motion in a three-dimensional topography. Low Temp. Sci. A 46, 99–110; (in Japanese).Google Scholar
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.Google Scholar
McElwaine, J. N. 2005 Rotational flow in gravity current heads. Phil. Trans. R. Soc. Lond. A 363, 1603–1623.Google Scholar
Mellor, M. 1977 Engineering properties of snow. J. Glaciol. 19 (81), 15–66.Google Scholar
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), 193–204.Google Scholar
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. 395–404. IAHS Press.Google Scholar
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. 363–380. IAHS Press.Google Scholar
Norem, H. & Schieldrop, B.1991 Stress analyses for numerical modelling of submarine flowslides. NGI Rep. 522090-10. Norges Geotekniske Institutt, Oslo, Norway.Google Scholar
Owen, P. R. 1964 Saltation of uniform grains in air. J. Fluid Mech. 20 (2), 225–242.Google Scholar
Parker, G., Fukushima, Y. & Pantin, H. M. 1986 Self-accelerating turbidity currents. J. Fluid Mech. 171, 145–181.CrossRefGoogle Scholar
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), 143–172.Google Scholar
Rajchenbach, J. 2003 Dense, rapid flows of inelastic grains under gravity. Phys. Rev. Lett. 90, 144302.Google Scholar
Rognon, P. G., Roux, J.-N., Naaim, M. & Chevoir, F. 2008 Dense flows of cohesive granular materials. J. Fluid Mech. 596, 21–47.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
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), 170–183.Google Scholar
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), 183–192.Google Scholar
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183–222.Google Scholar
Scheiwiller, T.1986 Dynamics of powder-snow avalanches. PhD thesis, Laboratory of Hydraulics, Hydrology and Glaciology, ETH Zürich, Zürich, Switzerland, available from http://people.ee.ethz.ch/∼vawweb/vaw_mitteilungen/081/081_g.pdf.Google Scholar
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 http://people.ee.ethz.ch/∼vawweb/vaw_mitteilungen/058/058_g.pdf.Google Scholar
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 http://e-collection.library.ethz.ch/eserv/eth:27359/eth-27359-02.pdf.Google Scholar
Sovilla, B., Burlando, P. & Bartelt, P. 2006 Field experiments and numerical modeling of mass entrainment in snow avalanches. J. Geophys. Res. 111, F03007.Google Scholar
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), 1–13.Google Scholar
Sovilla, B., Sommavilla, F. & Tomaselli, A. 2001 Measurements of mass balance in dense snow avalanche events. Ann. Glaciol. 32, 230–236.Google Scholar
Vallet, J., Gruber, U. & Dufour, F. 2001 Photogrammetric avalanche measurements at Vallée de la Sionne, Switzerland. Ann. Glaciol. 32, 141–146.Google Scholar
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: http://retro.seals.ch/digbib/voltoc?pid=sbz-002:1955:73.Google Scholar