Numerical simulation of the motion of a three-dimensional glacier
Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, p. 1-28

The motion of a three-dimensional glacier is considered. Ice is modeled as an incompressible non-Newtonian fluid. At each time step, given the shape of the glacier, a nonlinear elliptic system has to be solved in order to obtain the two components of the horizontal velocity field. Then, the shape of the glacier is updated by solving a transport equation. Finite element techniques are used to compute the velocity field and to solve the transport equation. Numerical results are compared to experiments on Storglaciaren (Sweden) between 1959 and 1990.

DOI : https://doi.org/10.5802/ambp.236
Classification:  65N30,  76M10
Keywords: glacier, ice, non-Newtonian fluid, finite elements
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author = {Picasso, Marco and Rappaz, Jacques and Reist, Adrian},
title = {Numerical simulation of the motion of a three-dimensional glacier},
journal = {Annales math\'ematiques Blaise Pascal},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {15},
number = {1},
year = {2008},
pages = {1-28},
doi = {10.5802/ambp.236},
mrnumber = {2418010},
zbl = {1141.76038},
language = {en},
url = {http://www.numdam.org/item/AMBP_2008__15_1_1_0}
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Picasso, Marco; Rappaz, Jacques; Reist, Adrian. Numerical simulation of the motion of a three-dimensional glacier. Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, pp. 1-28. doi : 10.5802/ambp.236. http://www.numdam.org/item/AMBP_2008__15_1_1_0/

[1] Albrecht, Olaf; Jansson, Peter; Blatter, Heinz Modelling Glacier Response to Measured Mass-Balance Forcing, Annals of Glaciology, Tome 31 (2000), pp. 91-96 | Article

[2] Alley, R.; Clark, P.U.; Huybrechts, P.; Joughin, I. Ice sheets and sea-level change, Science, Tome 310 (2005), pp. 456-460 | Article

[3] Blatter, Heinz Velocity and Stress Fields in Grounded Glaciers: A Simple Algorithm for Including Deviatoric Stress Gradients, Journal of Glaciology, Tome 41 (1995) no. 138, pp. 333-344

[4] Boffi, D.; Gastaldi, L. Stability and geometric conservation laws for ALE formulations, Comput. Methods Appl. Mech. Engrg., Tome 193 (2004) no. 42-44, pp. 4717-4739 | Article | MR 2091121 | Zbl 1112.76382

[5] Brooks, Alexander N.; Hughes, Thomas J. R. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., Tome 32 (1982) no. 1-3, pp. 199-259 (FENOMECH ’81, Part I (Stuttgart, 1981)) | Article | MR 679322 | Zbl 0497.76041

[6] Carey, G.F.; Barth, W.; Woods, J.A.; Kirk, B.S.; Anderson, M.L.; Chow, S.; Bangerth, W. Modelling error and constitutive relations in simulation of flow and transport, Int. J. Numer. Meth. Fluids, Tome 46 (2004), pp. 1211-1236 | Article | MR 2106595 | Zbl 1064.76064

[7] Carey, Graham F. Computational grids, Taylor & Francis, Washington, DC, Series in Computational and Physical Processes in Mechanics and Thermal Sciences (1997) (Generation, adaptation, and solution strategies) | MR 1483891 | Zbl 0955.74001

[8] Chow, S.; Carey, G.F.; Anderson, M.L. Finite element approximations of a glaciology problem, Math. Model. Numer. Anal., Tome 38 (741–756) no. 5, pp. 2004 | Numdam | MR 2104426 | Zbl 1130.86300

[9] Colinge, J.; Blatter, H. Stress and velocity fields in glaciers: Part I. Finite difference schemes for higher-order glacier models, Journal of Glaciology, Tome 44 (1998) no. 149, pp. 457-466

[10] Colinge, J.; Rappaz, J. A strongly nonlinear problem arising in glaciology, Math. Model. Numer. Anal., Tome 33 (1999) no. 2, pp. 395-406 | Article | Numdam | MR 1700041 | Zbl 0946.65115

[11] Erikson, K.; Estep, D.; Hansbo, P.; Johnson, C. Computational Differential Equations, Cambridge University Press (1996) | Zbl 0946.65049

[12] Fowler, A. C. A mathematical analysis of glacier surges, SIAM J. Appl. Math., Tome 49 (1989) no. 1, pp. 246-263 | Article | MR 978837 | Zbl 0695.35009

[13] Fowler, A. C. Glaciers and ice sheets, The mathematics of models for climatology and environment (Puerto de la Cruz, 1995), Springer, Berlin (NATO ASI Ser. Ser. I Glob. Environ. Change) Tome 48 (1997), pp. 301-336 | MR 1635292 | Zbl 0897.73054

[14] Glowinski, R.; Rappaz, J. Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, Math. Model. Numer. Anal., Tome 37 (175–186) no. 1, pp. 2003 | Numdam | MR 1972657 | Zbl 1046.76002

[15] Gudmundsson, G.H. A three-dimensional numerical model of the confluence area of unteraargletscher, bernese alps, Switzerland, J. Glaciol, Tome 45 (1999) no. 150, pp. 219-230 | Article

[16] Herzfeld, U. C.; Eriksson, M. G.; Holmlund, P. On the Influence of Kriging Parameters on the Cartographic Output - A study in Mapping Subglacial Topography, Mathematical Geol., Tome 27 (1993) no. 7, pp. 881-900 | Article

[17] Holmlund, P. Maps of Storglaciären and their use in glacier monitoring studies. (incl. 2 maps of the glaciers in the Tarfala valley in the scale 1:10 000), Geogr. Ann., Tome 78 A (1996) no. 2–3, pp. 193-196 | Article

[18] Huybrechts, P.; Payne, T.; Intercomparison Group, The Eismint The EISMINT benchmark for testing ice-sheet models, Annals of Glaciology, Tome 23 (1996)

[19] Löner, R.; Yang, C. Improved ALE mesh velocities for moving boundaries, Comm. Num. Meth. Eng., Tome 12 (1996), pp. 599-608 | Article | Zbl 0858.76042

[20] Martin, C.; Navarro, F.; Otero, J.; Cuadrado, M.L.; Corcuera, M.L. Three-dimensional modelling of the dynamics of Johnsons Glacier, Livingston Island, Antarctica, Annals of Glaciology, Tome 39 (2004), pp. 1-8 | Article

[21] Picasso, Marco; Rappaz, Jacques; Reist, Adrian; Blatter, Heinz; Funk, Martin Numerical simulation of the motion of a two-dimensional glacier, Int. J. Numer. Meth. Eng., Tome 60 (2004), pp. 995-1009 | Article | MR 2058758 | Zbl 1060.76577

[22] Rappaz, J.; Reist, A. Mathematical and Numerical Analysis of a Three Dimensional Fluid Flow Model in Glaciology, M3AS, Tome 15 (2005) no. 1, pp. 37-52 | MR 2110451 | Zbl pre02170863

[23] Reist, A. Mathematical analysis and numerical simulation of the motion of a glacier, 1015 Lausanne, Ecole Polytechnique Fédérale de Lausanne (2005) (Ph. D. Thesis)

[24] Robertson, I.; Sherwin, S. Free-surface flow simulation using hp/spectral elements, J. Comp. Phys., Tome 155 (1999), pp. 26-53 | Article | MR 1716505 | Zbl 0953.76055