Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

Geoscientific Model Development

Volumn 4, Issue 4, 2011, Pages 1133-1149

  Furst J J ^a   Rybak, O ^a   Goelzer, H ^a   De Smedt, B ^a   De Groen, P ^a   Huybrechts, P ^a  

^a VRIJE UNIVERSITEIT BRUSSEL   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

ACCURACY ASSESSMENT; FINITE DIFFERENCE METHOD; ICE SHEET; NUMERICAL MODEL; THREE-DIMENSIONAL MODELING; VELOCITY PROFILE;

EID: 84873631847     PISSN: 1991959X     EISSN: 19919603     Source Type: Journal    
DOI: 10.5194/gmd-4-1133-2011     Document Type: Article

References (54)

1. Albrecht, T., Martin, M., Haseloff, M., Winkelmann, R., and Lever-mann, A.: Parameterization for subgrid-scale motion of ice-shelf calving fronts, The Cryosphere, 5, 35-44, doi:10.5194/tc-5-35-2011, 2011.
2. Benn, D. I., Warren, C. R., and Mottram, R. H.: Calving processes and the dynamics of calving glaciers, Earth-Sci. Rev., 82, 143-179, doi:10.1016/j.earscirev.2007.02.002, 2007.
3. Berman, A. and Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics (SIAM), ISBN:978-0898713213, 1994.
4. Blatter, H.: Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients, J. Glaciol., 41, 333-344, 1995.
5. Bueler, E. and Brown, J.: Shallow shelf approximation as a "sliding law" in a thermomechanically coupled ice sheet model, J. Geophys. Res.-Earth, 114(F03008), 21 pp., doi:10.1029/2008JF001179, 2009.
6. Colinge, J. and Blatter, H.: Stress and velocity fields in glaciers: Part I. Finite-difference schemes for higher-order glacier models, J. Glaciol., 44), 448-456, 1998.
7. De Smedt, B., Pattyn F., and De Groen, P.: Using the unstable manifold correction in a Picard iteration to solve the velocity field in higher-order ice-flow models, J. Glaciol., 56, 257-261, doi:10.3189/002214310791968395, 2010.
8. Dukowicz, J. K., Price, S. F., and Lipscomb, W. H.: Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action, J. Glaciol., 56, 480-496, doi:10.3189/002214310792447851, 2010.
9. Ferziger, J. H. and Perić, M.: Computational Methods for Fluid Dynamics, Springer Verlag, 3, ISBN:978-3540594345, 2002.
10. Fowler, A. C. and Larson, D. A.: The uniqueness of steady state flows of glaciers and ice sheets, Geophys. J. Roy. Astr. S., 63, 333-345, doi:10.1111/j.1365-246X.1980.tb02624.x, 1980.
11. Gagliardini, O., Duran, G., Zwinger, T., Hindmarsh, R. C. A., and Le Meur, E.: Coupling of ice-shelf melting and buttressing is a key process in ice-sheet dynamics, Geophys. Res. Lett., 37, 24 pp., doi:10.1029/2010GL043334, 2010.
12. Goldberg, D., Holland, D. M., and Schoof, C.: Grounding line movement and ice shelf buttressing in marine ice sheets, J. Geo-phys. Res.-Earth, 114, 23 pp., doi:10.1029/2008JF001227, 2009.
13. Harlow, F. and Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8, 2182-2189, doi:10.1063/1.1761178, 1965.
14. Hindmarsch, R. C. A.: A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modelling, J. Geophys. Res.-Earth, 109, F01012, doi:10.1029/2003JF000065, 2004.
15. Hindmarsch, R. C. A.: The role of membrane-like stresses in determining the stability and sensitivity of the Antarctic ice sheets: back pressure and grounding line motion, Philos. T. Roy. Soc. A., 364, 1733-1767, doi:10.1098/rsta.2006.1797, 2006.
16. Hindmarsch, R. C. A. and Payne, A. J.: Time-step limits for stable solutions of the ice-sheet equation, Ann. Glaciol., 23, 74-85, 1996.
17. Hutter, K.: Theoretical glaciology. Material science of ice and the mechanics of glaciers and ice sheets, Reidel, Dordrecht, Netherlands, ISBN:978-9027714732, 1983.
18. Huybrechts, P. and de Wolde, J.: The dynamic response of the Greenland and Antarctic ice sheets to multiple-century climatic warming, J. Climate, 12, 2169-2188, doi:10.1175/1520-0442(1999)012<2169:TDROTG>2.0.CO;2, 1999.
19. Johnson, J. and Fastook, J. L.: Northern Hemisphere glaciation and its sensitivity to basal melt water, Quatern. Int., 95-96, 65-74, doi:10.1016/S1040-6182(02)00028-9, 2002.
20. Jouvet, G., Huss, M., Blatter, H., Picasso, M., and Rappaz, J.: Numerical simulation of Rhonegletscher from 1824 to 2100, J. Comput. Phys., 228, 6426-6439, doi:10.1016/j.jcp.2009.05.033, 2009.
21. LeVeque, R. J.: Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, ISBN: 978-0898716290, 2007.
22. MacAyeal, D. R.: Large-Scale Ice Flow Over a Viscous Basal Sediment: Theory and Application to Ice Stream B, Antarctica, J. Geophys. Res.-Solid, 94, 4071-4087, doi:10.1029/JB094iB04p04071, 1989.
23. Mattheij, R. M. M., Rienstra, S. W., and ten Thije Boonkamp, J. H. M.: Partial differential equations: modeling, analysis, computation, SIAM, Philadelphia, ISBN:978-0898715941, 2005.
24. Morland, L. W.: Unconfined ice-shelf flow, in: Dynamics of the West Antarctic ice sheet, dited by: van der Veen, C. J. and Oer-lemans, J., Kluwer Academic Publishers, 99-116, ISBN:978-9027723703, 1986.
25. Morland, L. W. and Johnson, I. R.: Steady Motion of Ice Sheets, J. Glaciol., 25, 229-246, 1980.
26. Manguoglu, M., Sameh, A. H., Saied, F., Tezduyar, T. E., and Sathe, S.: Preconditioning Techniques for Nonsymmetric Linear Systems in the Computation of Incompressible Flows, J. Appl. Mech., 76, 7 pp., doi:10.1115/1.3059576, 2009.
27. Otero, J., Navarro, F. J., Martin, C., Cuadrado, M. L., and Cor-cuera, M. I.: A three-dimensional calving model: numerical experiments on Johnsons Glacier, Livingston Island, Antarctica, J. Glaciol., 56, 200-214, doi:10.3189/002214310791968539, 2010.
28. Pattyn, F.: A new three-dimensional higher-order thermomechani-cal ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes, J. Geophys. Res.-Sol. Earth, 108, 2382, doi:10.1029/ 2002JB002329, 2003.
29. Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., de Smedt, B., Gagliardini, O., Gudmundsson, G. H., Hindmarsh, R. C. A., Hubbard, A., Johnson, J. V., Kleiner, T., Konovalov, Y., Martin, C., Payne, A. J., Pollard, D., Price, S., Rückamp, M., Saito, F., Soucček, O., Sugiyama, S., and Zwinger, T.: Benchmark experiments for higher-order and full-Stokes ice sheet models (IS-MIPU HOM), The Cryosphere, 2, 95-108, doi:10.5194/tc-2-95-2008, 2008.
30. Pimentel, S., Flowers, G. E., and Schoof, C. G.: A hydrologi-cally coupled higher-order flow-band model of ice dynamics with a Coulomb friction sliding law, J. Geophys. Res.-Earth, 115, F04023, doi:10.1029/2009JF001621, 2010.
31. Pollard, D. and DeConto, R. M.: Modelling West Antarctic ice sheet growth and collapse through the past five million years, Nature, 458, 7236, 329-332, doi:10.1038/nature07809, 2009.
32. Press, W. H., Teukoslky, S. A., Vetterling, W. T., and Flannery, B. P.: Numerical Recipes in Fortran, Second Edition: The Art of Scientific Computing, Volume 1 of Fortran Numerical Recipes, Cambridge University Press, New York, ISBN:978-0521430647, 2003.
33. Pritchard, H. D., Arthern, R. J., Vaughan, D. G., and Edwards, L. A.: Extensive dynamic thinning on the margins of the Greenland and Antarctic ice sheets, Nature, 461, 7266, doi:10.1038/nature08471, 2009.
34. Ritz, C., Rommelaere, V., and Dumas, C.: Modeling the evolution of Antarctic ice sheet over the last 420, 000 years: Implications for altitude changes in the Vostok region, J. Geophys. Res.-Atmos., 106, 31943-31964, doi:10.1029/2001JD900232, 2001.
35. Saad, Y. and Schultz, M. H.: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM, Philadelphia, 7, 856-869, doi:10.1137/0907058, 1986.
36. Schoof, C.: A variational approach to ice streams flow, J. Fluid. Mech., 556, 227-251, doi:10.1017/S0022112006009591, 2006.
37. Schoof, C.: Ice sheet grounding line dynamics: Steady states, stability, and hysteresis, J. Geophys. Res., 112, F03S28, doi:10.1029/2006JF000664, 2007.
38. Schoof, C. and Hindmarsh, R. C. A.: Thin-Film Flows with Wall Slip: An Asymptotic Analysis of Higher Order Glacier Flow Models, Q. J. Mech. Appl. Math., 63, 73-114, doi:10.1093/qjmam/hbp025, 2010.
39. Shepherd, A. and Wingham, D.: Recent sea-level contributions of the Antarctic and Greenland ice sheets, Science, 315, 1529-1532, doi:10.1126/science.1136776, 2007.
40. Sleijpen, G. L. G. and van Gijzen, M. B.: Exploiting BiCGstab(l) Strategies to Induce Dimension Reduction, SIAM, Philadelphia, 32, 2687-2709, doi:10.1137/090752341, 2010.
41. Sonneveld, P. and Van Gijzen, M. B.: IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM, Philadelphia, 31, 1053-1062, doi:10.1137/070685804, 2008.
42. Trottenberg, U., Oosterlee, C. W. and Schüller, A.: Multigrid, Academic Press, ISBN: 978-0127010700, 2001.
43. Truffer, M. and Fahnestock, M.: Rethinking ice sheet time scales, Science, 315, 1508-1510, doi:10.1126/science.1140469, 2007.
44. Tulaczyk, S., Kamb, W. B., and Engelhardt, H. F.: Basal mechanics of Ice Stream B, West Antarctica 1. Till mechanics, J. Geophys. Res.-Sol. Earth, 105, 463-481, 2000.
45. van Gijzen, M. B. and Sonneveld, P.: An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties, Technical Report 10-16, Department of Applied Mathematical Analysis, Delft University of Technology, Delft, 2010.
46. Van Der Veen, C. J.: A Numerical Scheme for Calculating Stresses and Strain Rates in Glacier, Math. Geol., 21, 363-377, doi:10.1007/BF00893696, 1989.
47. Van Der Veen, C. J.: Fundamentals of Glacier Dynamics, edited by: Balkema, A. A., Rotterdam, Netherlands, ISBN: 978-9054104704, 1999.
48. Van der Vorst, H. A.: Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems, SIAM, Philadelphia, doi:10.1137/0913035, 1992.
49. Vaughan, D. G. and Arthern, R.: Why is it hard to predict the future of ice sheets?, Science, 315, 1503-1504, doi:10.1126/science.1141111, 2007.
50. Veldman, A. E. P. and Rinzema, K.: Playing with nonuniform grids, J. Eng. Math., 26, 119-130, doi:10.1007/BF00043231, 1992.
51. Weis, M., Greve, R., and Hutter, K.: Theory of shallow ice shelves, Continuum Mech. Therm., 11, 15-50, doi:10.1007/s001610050102, 1999.
52. Wesseling, P. and Sonneveld, P.: Numerical Experiments with a Multiple Grid and a Preconditioned Lanczos Type Method, Lecture Notes in Mathematics 771, Springer Verlag, 543-562, 1980.
53. Yeung, M.-C. and Chan, T. F.: ML(K)BiCGSTAB: A BiCGSTAB Variant based on Multiple Lanczos Starting Vectors, SIAM, Philadelphia, 21, 1263-1290, doi:10.1137/S1064827597321581, 1999.
54. Zwinger, T., Greve, R., Gagliardini, O., Takayuki, S., and Lyly, M.: A full Stokes-flow thermo-mechanical model for firn and ice applied to the Gorshkov crater glacier, Kamchatka, Ann. Glaciol., 45, 29-37, doi:10.3189/172756407782282543, 2007.