Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T22:38:27.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Mechanical error estimators for shallow ice flow models

Published online by Cambridge University Press:  18 October 2016

G. Jouvet
G. Jouvet*
Affiliation:
ETH Zürich, VAW, Hönggerbergring 26, 8093 Zürich, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a posteriori ‘mechanical’ error estimators that are able to evaluate the solution discrepancy between two ice flow models. We first reformulate the classical shallow ice flow models by applying simplifications to the weak formulation of the Glen–Stokes model. This approach leads to a unified hierarchical formulation which relates the Glen–Stokes model, the Blatter model, the shallow ice approximation and the shallow shelf approximation. Based on this formulation and on residual techniques commonly used to estimate numerical errors, we derive three a posteriori estimators, each of which compares a pair of models using measures of the velocity field from the simpler (shallower) model. Numerical experiments confirm that these estimators can be used to assess the validity of the shallow ice models that are commonly used in glacier and ice sheet modelling.

JFM classification

Geophysical and Geological Flows: Ice sheets Non-Newtonian Flows: Non-Newtonian Flows Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 807 , 25 November 2016 , pp. 40 - 61
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, R. A. & Fournier, J. J. F. 2003 Sobolev spaces. Pure and Applied Mathematics. Elsevier Science.Google Scholar
Ahlkrona, J., Lotstedt, P., Kirchner, N. & Zwinger, T. 2016 Dynamically coupling the non-linear Stokes equations with the shallow ice approximation in glaciology: description and first applications of the ISCAL method. J. Comput. Phys. 308, 1–19.CrossRefGoogle Scholar
Barrett, J. W. & Liu, W. B. 1993 Finite element approximation of the p-Laplacian. Math. Comput. 61, 523–537.Google Scholar
Blatter, H. 1995 Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients. J. Glaciol. 41 (138), 333–344.CrossRefGoogle Scholar
Brezis, H. 1999 Analyse Fonctionnelle (Théorie et Applications). Dunod.Google Scholar
Brown, J., Smith, B. F. & Ahmadia, A. 2013 Achieving textbook multigrid efficiency for hydrostatic ice sheet flow. SIAM J. Sci. Comput. 35, 8359–8375.CrossRefGoogle Scholar
Bueler, E. & Brown, J. 2009 Shallow shelf approximation as a ‘sliding law’ in a thermomechanically coupled ice sheet model. J. Geophys. Res. 114, F03008.Google Scholar
Colinge, J. & Rappaz, J. 1999 A strongly nonlinear problem arising in glaciology. M2AN Math. Model. Numer. Anal. 33 (2), 395–406.Google Scholar
Dukowicz, J. K. 2012 Reformulating the full-Stokes ice sheet model for a more efficient computational solution. Cryosphere 6 (1), 21–34.CrossRefGoogle Scholar
Ern, A. & Guermond, J. L. 2004 Theory and Practice of Finite Elements. Springer.Google Scholar
Gagliardini, O. & Zwinger, T. 2008 The ISMIP-HOM benchmark experiments performed using the finite-element code Elmer. Cryosphere Discuss. 2, 75–109.Google Scholar
Girault, V. & Raviart, P. A. 1986 Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms (Springer Series in Computational Mathematics). Springer.CrossRefGoogle Scholar
Glen, J. W. 1953 Rate of flow of polycrystalline ice. Nature 172, 721–722.CrossRefGoogle Scholar
Glowinski, R. & Le Tallec, P. 1989 Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (SIAM Studies in Applied Mathematics), Society for Industrial and Applied Mathematics.Google Scholar
Greve, R. & Blatter, H. 2009 Dynamics of Ice Sheets and Glaciers. Springer.Google Scholar
Hindmarsh, R. C. A. 2004 A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling. J. Geophys. Res.: Earth Surf. 109, F01012.Google Scholar
Hutter, K. 1983 Theoretical Glaciology. Reidel.Google Scholar
Jouvet, G.2010 Modélisation, analyse mathématique et simulation numérique de la dynamique des glaciers. PhD thesis, EPF Lausanne.Google Scholar
Jouvet, G. & Gräser, C. 2013 An adaptive Newton multigrid method for a model of marine ice sheets. J. Comput. Phys. 252, 419–437.Google Scholar
Jouvet, G. & Picasso, M. A posteriori estimator for the accuracy of the shallow shelf approximation. Special issue of Mathematical Geosciences (submitted).Google Scholar
Jouvet, G. & Rappaz, J. 2011 Analysis and finite element approximation of a nonlinear stationary Stokes problem arising in glaciology. Adv. Numer. Anal. 2011, 164581.Google Scholar
Lingle, C. S. & Troshina, E. N. 1998 Relative magnitudes of shear and longitudinal strain rates in the inland Antarctic ice sheet, and response to increasing accumulation. Ann. Glaciol. 27 (1), 187–193.CrossRefGoogle Scholar
MacAyeal, D. R. 1989 Large-scale ice flow over a viscous basal sediment: theory and application to ice stream b, Antarctica. J. Geophys. Res.: Solid Earth 94 (B4), 4071–4087.CrossRefGoogle Scholar
Morland, L. W. 1987 Unconfined ice-shelf flow. In Dynamics of the West Antarctic Ice Sheet (Glaciology and Quaternary Geology) (ed. Veen, C. J. & Oerlemans, J.), vol. 4, pp. 99–116. Springer.Google Scholar
Paterson, W. S. B. 1994 The Physics of Glaciers, 3rd edn. Pergamon.Google Scholar
Pattyn, F. et al. 2008 Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP-HOM). Cryosphere 2 (2), 95–108.Google Scholar
Quarteroni, A. & Valli, A. 1994 Numerical Approximation of Partial Differential Equations. Springer.CrossRefGoogle Scholar
Rappaz, J. & Reist, A. 2005 Mathematical and numerical analysis of a three-dimensional fluid flow model in glaciology. Math. Models Meth. Appl. Sci. 15 (1), 37–52.CrossRefGoogle Scholar
Ritz, C., Rommelaere, V. & Dumas, C. 2001 Modeling the evolution of Antarctic ice sheet over the last 420 000 years: implications for altitude changes in the Vostok region. J. Geophys. Res.: Atmospheres 106 (D23), 31943–31964.CrossRefGoogle Scholar
Schoof, C. 2006 A variational approach to ice stream flow. J. Fluid Mech. 556, 227–251.Google Scholar
Schoof, C. 2010 Coulomb friction and other sliding laws in a higher order glacier flow model. Math. Models Meth. Appl. Sci. 20 (01), 157–189.Google Scholar
Schoof, C. & Hewitt, I. 2013 Ice-sheet dynamics. Annu. Rev. Fluid Mech. 45 (1), 217–239.Google Scholar
Schoof, C. & Hindmarsh, R. C. A. 2010 Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models. Q. J. Mech. Appl. Math. 63 (1), 73–114.Google Scholar
Schoof, C., Hindmarsch, R. C. A. & Pattyn, F. 2000 MISMIP: Marine ice sheet model intercomparison project. In Tech. Rep., Laboratoire de Glaciologie, Départment des Sciences de la Terre et de l’Environnement, Université Libre de Bruxelles.Google Scholar
Seroussi, H.2011 Modeling ice flow dynamics with advanced multi-model formulations. PhD thesis, Ecole centrale, Paris.Google Scholar
Weis, M., Greve, R. & Hutter, K. 1999 Theory of shallow ice shelves. Contin. Mech. Thermodyn. 11 (1), 15–50.CrossRefGoogle Scholar