Skip to main content
×
×
Home

An exact solution for a steady, flowline marine ice sheet

  • Ed Bueler (a1)
Abstract
Abstract

G. Böðvarsson’s 1955 plug-flow solution for an Icelandic glacier problem is shown to be an exact solution to the steady form of the simultaneous stress-balance and mass-continuity equations widely used in numerical models of marine ice sheets. The solution, which has parabolic ice thickness and constant vertically integrated longitudinal stress, solves the steady shallow-shelf approximation with linear sliding on a flat bed. It has an elevation-dependent surface mass-balance rate and, in the interpretation given here, a contrived location-dependent ice hardness distribution. By connecting Böðvarsson’s solution to the Van der Veen (1983) solution for floating ice, we construct an exact solution to the ‘rapid-sliding’ marine ice-sheet problem, continuously across the grounding line. We exploit this exact solution to examine the accuracy of two numerical methods, one grid-free and the other based on a fixed, equally spaced grid.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      An exact solution for a steady, flowline marine ice sheet
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      An exact solution for a steady, flowline marine ice sheet
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      An exact solution for a steady, flowline marine ice sheet
      Available formats
      ×
Copyright
References
Hide All
Balay S and 13 others (2014) PETSc users manual. (Tech. Rep. ANL-95/11, Rev. 3.4) Argonne National Laboratory, Argonne, IL
Böðvarsson G (1955) On the flow of ice-sheets and glaciers. Jökull, 5, 18
British Glaciological Society (BGS) (1949) Joint meeting of the British Glaciological Society, the British Rheologists’ Club and the Institute of Metals. J. Glaciol., 1(5), 231240 (doi: 10.3189/002214349793702827)
Bueler E and Brown J (2009) Shallow shelf approximation as a ‘sliding law’ in a thermomechanically coupled ice sheet model. J. Geophys. Res., 114(F3), F03008 (doi: 10.1029/2008JF001179)
Bueler E, Lingle CS, Kallen-Brown JA, Covey DN and Bowman LN (2005) Exact solutions and verification of numerical models for isothermal ice sheets. J. Glaciol., 51(173), 291306 (doi: 10.3189/172756505781829449)
Cogley JG and 10 others (2011) Glossary of glacier mass balance and related terms. (IHP-VII Technical Documents in Hydrology 86) UNESCO–International Hydrological Programme, Paris
Feldmann J, Albrecht T, Khroulev C, Pattyn F and Levermann A (2014) Resolution-dependent performance of grounding line motion in a shallow model compared with a full-Stokes model according to the MISMIP3d intercomparison. J. Glaciol., 60(220), 353360 (doi: 10.3189/2014JoG13J093)
Fowler AC (1992) Modelling ice sheet dynamics. Geophys. Astrophys. Fluid Dyn., 63(1–4), 2965 (doi: 10.1080/03091929208228277)
Gladstone RM, Payne AJ and Cornford SL (2010) Parameterising the grounding line in flow-line ice sheet models. Cryosphere, 4(4), 605619 (doi: 10.5194/tc-4–605–2010)
Higham DJ and Trefethen LN (1993) Stiffness of ODEs. BIT Num. Math., 33(2), 285303 (doi: 10.1007/BF01989751)
Hindmarsh AC (1983) ODEPACK, a systematized collection of ODE solvers. In Stepleman RS, Carver M, Peskin R, Ames WF and Vichnevetsky R eds. Scientific computing, applications of mathematics and computing to the physical sciences. 10th IMACS World Congress on Systems Simulation and Scientific Computation, 8–13 August 1982, Montreal, Canada. (IMACS Transactions on Scientific Computation 1) North-Holland, Amsterdam, 5564
Kelley CT (1987) Solving nonlinear equations with Newton’s Method. Society for Industrial and Applied Mathematics, Philadelphia, PA
Leguy GR, Asay-Davis XS and Lipscomb WH (2014) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet model. Cryosphere, 8(4), 12391259 (doi: 10.5194/tc-8–1239–2014)
MacAyeal DR (1989) Large-scale ice flow over a viscous basal sediment: theory and application to Ice Stream B, Antarctica. J. Geophys. Res., 94(B4), 40714087 (doi: 10.1029/JB094iB04p04071)
Morton KW and Mayers DF (2005) Numerical solution of partial differential equations: an introduction, 2nd edn. Cambridge University Press, Cambridge
Nye JF (1952) A method of calculating the thickness of ice sheets. Nature, 169(4300), 529530
Pattyn F and 18 others (2012) Results of the Marine Ice Sheet Model Intercomparison Project, MISMIP. Cryosphere, 6(3), 573588 (doi: 10.5194/tc-6–573–2012)
Pollard D and DeConto RM (2009) Modelling West Antarctic ice sheet growth and collapse through the past five million years. Nature, 458(7236), 329332 (doi: 10.1038/nature07809)
Press WH, Teukolsky SA, Vetterling WT and Flannery BP (1992) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge
Schoof C (2005) The effect of cavitation on glacier sliding. Proc. R. Soc. London, Ser. A, 461(2055), 609627 (doi: 10.1098/rspa. 2004.1350)
Schoof C (2006) A variational approach to ice stream flow. J. Fluid Mech., 556, 227251 (doi: 10.1017/S0022112006009591)
Schoof C (2007) Marine ice-sheet dynamics. Part 1. The case of rapid sliding. J. Fluid Mech., 573, 2755 (doi: 10.1017/S0022112006003570)
Van der Veen CJ (1983) A note on the equilibrium profile of a free floating ice shelf. (IMOU Report V83–15) Rijksuniversiteit Instituut voor Meteorologie en Oceanografie, Utrecht
Van der Veen CJ (2013) Fundamentals of glacier dynamics, 2nd edn. CRC Press, Boca Raton, FL
Vialov SS (1958) Regularities of glacial shields movement and the theory of plastic viscous flow. IASH Publ. 47 (Symposium at Chamonix, 1958 – Physics of the Movement of the Ice), 266275
Weertman J (1961) Stability of ice-age ice sheets. J. Geophys. Res., 66(11), 37833792 (doi: 10.1029/JZ066i011p03783)
Weis M, Greve R and Hutter K (1999) Theory of shallow ice shelves. Contin. Mech. Thermodyn., 11(1), 1550 (doi: 10.1007/s001610050102)
Wesseling P (2001) Principles of computational fluid dynamics. Springer Verlag, Berlin
Winkelmann R and 6 others (2011) The Potsdam Parallel Ice Sheet Model (PISM-PIK) – Part 1: Model description. Cryosphere, 5(3), 715726 (doi: 10.5194/tc-5–715–2011)
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Glaciology
  • ISSN: 0022-1430
  • EISSN: 1727-5652
  • URL: /core/journals/journal-of-glaciology
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 6
Total number of PDF views: 28 *
Loading metrics...

Abstract views

Total abstract views: 26 *
Loading metrics...

* Views captured on Cambridge Core between 10th July 2017 - 24th February 2018. This data will be updated every 24 hours.