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Improvements to shear-deformational models of glacier dynamics through a longitudinal stress factor

  • Surendra Adhikari (a1) and Shawn J. Marshall (a1)


In a two-dimensional (plane strain) glacier domain, gravity-driven ice flow is balanced by basal drag and the resistance associated with longitudinal stress gradients. The plane strain Stokes model accommodates both these resistances, whereas several simpler models only account for basal drag. Solving the Stokes equations is numerically challenging and computationally expensive, but simpler models may lead to unrealistic dynamical behaviour. Here, we propose a factor which can be introduced in shear-deformational flow models to yield results comparable to those from the plane strain Stokes model. As this factor adapts simpler models to capture the effects of missing dynamics, i.e. longitudinal stress gradients, we refer to it as the longitudinal stress (L-)factor. We assess the usefulness of this factor for idealized domains with complex basal topography and evolving geometry. We apply the model to Haig Glacier, Canadian Rockies, in order to present an illustration of how simulations of glacier response to climate forcing can be improved through the introduction of the L-factor in a shear-deformational flow model.

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Adhikari, S. and Huybrechts, P.. 2009. Numerical modelling of historical front variations and the 21st-century evolution of glacier AX010, Nepal Himalaya. Ann. Glaciol, 50(52), 2740.
Baral, D., Hutter, K. and Greve, R.. 2001. Asymptotic theories of large-scale motion, temperature and moisture distribution in land-based polythermal ice sheets: a critical review and new developments. Appl. Mech. Rev, 54(3), 215256.
Blatter, H. 1995. Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients. J. Glaciol, 41(138), 333344.
Blatter, H., Greve, R. and Abe-Ouchi, A.. 2010. A short history of the thermomechanical theory and modelling of glaciers and ice sheets. J. Glaciol, 56(200), 10871094.
Budd, W.F., Keage, P.L. and Blundy, N.A.. 1979. Empirical studies of ice sliding. J. Glaciol, 23(89), 157170.
Donea, J. and Huerta, A.. 2003. Finite element methods for flow problems. Chichester, Wiley.
Flowers, G.E., Marshall, S.J., Björnsson, H. and Clarke, G.K.C.. 2005. Sensitivity of Vatnajökull ice cap hydrology and dynamics to climate warming over the next 2 centuries. J. Geophys. Res, 110(F2), F02011. (10.1029/2004JF000200.)
Franca, L.P. and Frey, S.L.. 1992. Stabilized finite element methods: II. The incompressible Navier–Stokes equations. Comput. Meth. Appl. Mech. Eng, 99(2–3), 209233.
Gagliardini, O. and Zwinger, T.. 2008. The ISMIP-HOM benchmark experiments performed using the finite-element code Elmer. Cryosphere, 2(1), 6776.
Glen, J.W. 1955. The creep of polycrystalline ice. Proc. R. Soc. London, Ser. A, 228(1175), 519538.
Gudmundsson, G.H. 2003. Transmission of basal variability to a glacier surface. J. Geophys. Res, 108(B5), 2253. (10.1029/2002JB0022107.)
Hindmarsh, R.C.A. 2004. A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling. J. Geophys. Res, 109(F1), F01012. (10.1029/2003JF000065.)
Hood, P. and Taylor, C.. 1974. Navier–Stokes equations using mixed interpolation. In Oden, J.T., Zienkiewicz, O.C., Gallagher, R.H. and Taylor, C., eds. Finite element methods for flow problems. Huntsville, AL, UAH Press, 121132.
Hubbard, A. 2000. The verification and significance of three approaches to longitudinal stresses in high-resolution models of glacier flow. Geogr. Ann., Ser. A, 82(4), 471487.
Hutter, K. 1983. Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets. Dordrecht, etc., D. Reidel Publishing Co./Tokyo, Terra Scientific Publishing Co.
Huybrechts, P. 1990. A 3-D model for the Antarctic ice sheet: a sensitivity study on the glacial–interglacial contrast. Climate Dyn, 5(2), 7992.
Jansson, P., Hock, R. and Schneider, T.. 2003. The concept of glacier storage: a review. J. Hydrol, 282(1–4), 116129.
Jouvet, G., Huss, M., Blatter, H., Picasso, M. and Rappaz, J.. 2009. Numerical simulation of Rhonegletscher from 1874 to 2100. J. Comput. Phys, 228(17), 64266439.
Kamb, B. and Echelmeyer, K.A.. 1986. Stress-gradient coupling in glacier flow: I. Longitudinal averaging of the influence of ice thickness and surface slope. J. Glaciol, 32(111), 267284.
Kaser, G., Cogley, J.G., Dyurgerov, M.B., Meier, M.F. and Ohmura, A.. 2006. Mass balance of glaciers and ice caps: consensus estimates for 1961–2004. Geophys. Res. Lett, 33(19), L19501. (10.1029/2006GL027511.)
Le Meur, E., Gagliardini, O., Zwinger, T. and Ruokolainen, J.. 2004. Glacier flow modelling: a comparison of the Shallow Ice Approximation and the full-Stokes equation. C. R. Phys, 5(7), 709722.
Lemke, P. and 10 others. 2007. Observations: changes in snow, ice and frozen ground. In Solomon, S. and 7 others, eds. Climate change 2007: the physical science basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, etc., Cambridge University Press, 339383.
Leysinger Vieli, G.J.M.C. and Gudmundsson, G.H.. 2004. On estimating length fluctuations of glaciers caused by changes in climatic forcing. J. Geophys. Res, 109(F1), F01007. (10.1029/2003JF000027.)
MacAyeal, D.R. 1993. A tutorial on the use of control methods in ice-sheet modeling. J. Glaciol, 39(131), 9198.
Marshall, S.J., Björnsson, H., Flowers, G.E. and Clarke, G.K.C.. 2005. Simulation of Vatnajökull ice cap dynamics. J. Geophys. Res, 110(F3), F03009. (10.1029/2004JF000262.)
Marshall, S.J. and 7 others. 2011. Glacier water resources on the eastern slopes of the Canadian Rocky Mountains. Can. Water Res. J, 36(2), 109134.
Meier, M.F. and 7others. 2007. Glaciers dominate eustatic sea-level rise in the 21st century. Science, 317(5841), 10641067.
Nye, J.F. 1952. A method of calculating the thickness of ice sheets. Nature, 169(4300), 529530.
Nye, J.F. 1965. The flow of a glacier in a channel of rectangular, elliptic or parabolic cross-section. J. Glaciol, 5(41), 661690.
Nye, J.F. 1969. The effect of longitudinal stress on the shear stress at the base of an ice sheet. J. Glaciol, 8(53), 207213.
Oerlemans, J. 2001. Glaciers and climate change. Lisse, etc., A.A. Balkema.
Oerlemans, J. 2005. Extracting a climate signal from 169 glacier records. Science, 308(5722), 675677.
Oerlemans, J. and 10 others. 1998. Modelling the response of glaciers to climate warming. Climate Dyn, 14(4), 267274.
Pattyn, F. 2003. A new three-dimensional higher-order thermomechanical ice-sheet model: basic sensitivity, ice stream development, and ice flow across subglacial lakes. J. Geophys. Res, 108(B8), 2382. (10.1029/2002JB002329.)
Pattyn, F. 2008. Investigating the stability of subglacial lakes with a full Stokes ice-sheet model. J. Glaciol, 54(185), 353361.
Pattyn, F. and 20 others. 2008. Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP-HOM). Cryosphere, 2(2), 95108.
Price, S.F., Waddington, E.D. and Conway, H.. 2007. A full-stress, thermomechanical flow band model using the finite volume method. J. Geophys. Res, 112(F3), F03020. (10.1029/2006JF000724.)
Rigsby, G.P. 1958. Effect of hydrostatic pressure on velocity of shear deformation on single ice crystals. J. Glaciol, 3(24), 273278.
Schneeberger, C., Albrecht, O., Blatter, H., Wild, M. and Hock, R.. 2001. Modelling the response of glaciers to a doubling in atmospheric CO2: a case study of Storglaciären. Climate Dyn, 17(11), 825834.
Shea, J.M., Anslow, F.S. and Marshall, S.J.. 2005. Hydrometeorological relationships on Haig Glacier, Alberta, Canada. Ann. Glaciol, 40, 5260.
Shoemaker, E.M. and Morland, L.W.. 1984. A glacier flow model incorporating longitudinal deviatoric stresses. J. Glaciol, 30(106), 334340.
Souček, O. and Martinec, Z.. 2008. Iterative improvement of the shallow-ice approximation. J. Glaciol, 54(188), 812822.
Van der Veen, C.J. 1999. Fundamentals of glacier dynamics.Rotterdam, A.A. Balkema.
Whillans, I.M. 1987. Force budget of ice sheets. In Van der Veen, C.J. and Oerlemans, J., eds. Dynamics of the West Antarctic ice sheet. Dordrecht, etc., D. Reidel Publishing Co., 1736.
Zwinger, T. and Moore, J.C.. 2009. Diagnostic and prognostic simulations with a full Stokes model accounting for superimposed ice of Midtre Lovénbreen, Svalbard. Cryosphere, 3(2), 217229.
Zwinger, T., Greve, R., Gagliardini, O., Shiraiwa, T. and Lyly, M.. 2007. A full Stokes-flow thermo-mechanical model for firn and ice applied to the Gorshkov crater glacier, Kamchatka. Ann. Glaciol, 45, 2937.


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