The Flow of Ice Sheets and Ice Shelves

  • L. W. Morland
Part of the International Centre for Mechanical Sciences book series (CISM, volume 337)


A brief survey of the mathematical and numerical modelling of ice sheet and ice shelf flows is presented, emphasising the developments over the last decade. For the very slow gravity driven flow of natural ice masses over time scales of thousands of years, the ice is modelled by an incompressible non-linearly viscous fluid with a temperature dependent rate factor. The conservation laws of mass, linear momentum and energy are formulated, together with the free surface conditions and basal conditions for a grounded sheet and a floating shelf. A systematic reduction of the equations for both sheet and shelf is developed by the introduction of dimensionless variables and stretched co-ordinates, which are motivated by the forms of the sheet profiles sought, and the necessary assumptions are drawn out. Analytic progress is made for restricted flow configurations. The failure of depth-integrated equations to provide a valid approximation in more general configurations is noted. A binary mixture theory is developed for co-existing ice and water in extensive melt zones.


Deviatoric Stress Order Unity Plane Flow Rear Edge Tangential Traction 
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  1. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972).MATHGoogle Scholar
  2. Bentley, C. R.: Antarctic ice streams: a review. J. Geophys. Res. (1987) 92, 8843–8851.CrossRefADSGoogle Scholar
  3. Blatter, H. and Hutter, K.: Polythermal conditions in Arctic glaciers. J. Glaciology (1991) 37, 261–269.ADSGoogle Scholar
  4. Bodvardsson, G.: On the flow of ice sheets and glaciers. Jökull (1955) 5, 1–8.Google Scholar
  5. Boulton, G. S., Smith, G. D. and Morland, L. W.: The reconstruction of former ice sheets and their mass balance characteristics using a non-linearly viscous model. J. Glaciology (1984) 30, 140–152.ADSGoogle Scholar
  6. Budd, W. F.: The Dynamics of Ice Masses. Australian Nat. Antarctic Res. Expeds Rep. A (IV) No 108 (1969).Google Scholar
  7. Budd, W. F.: Ice flow over bedrock perturbations. J. Glaciology (1970) 9, 29–48.ADSGoogle Scholar
  8. Budd, W. F.: Stress variations with ice flow over undulations. J. Glaciology (1971) 10, 177–195.ADSGoogle Scholar
  9. Collins, I. F.: On the use of the equilibrium equation and flow law in relating the surface and bed topography of glaciers and ice sheets. J. Glaciology (1968) 7, 199–204.ADSGoogle Scholar
  10. Fowler, A. C.: Glacier Dynamics. D. Phil. Thesis (1977) Oxford University.Google Scholar
  11. Fowler, A. C.: A mathematical approach to the theory of glacier sliding. J. Glaciology (1979) 23, 131–141.ADSGoogle Scholar
  12. Fowler, A. C.: The use of a rational model in the mathematical analysis of a polythermal glacier. J. Glaciology (1979) 24, 443–456.ADSGoogle Scholar
  13. Fowler, A. C.: A theoretical treatment of the sliding of glaciers in the absence of cavitation. Phil. Trans. Roy. Soc. (1981) 298, 637–685.MathSciNetCrossRefMATHADSGoogle Scholar
  14. Fowler, A. C.: On the transport of moisture in polythermal glaciers. Geophys. Astrophys. Fluid Dynamics (1984) 28, 99–140.CrossRefMATHADSGoogle Scholar
  15. Fowler, A. C.: Modelling ice sheet dynamics. Geophys. Astrophys. Fluid Dynamics (1991) to appear.Google Scholar
  16. Fowler, A. C. and Larson, D. A.: On the flow ofpolythermal glaciers. I. Model and preliminary analysis. Proc. Roy. Soc. A. (1978) 363, 217–242.MathSciNetCrossRefADSGoogle Scholar
  17. Glen, J. W.: The creep of polycrystalline ice. Proc. Roy. Soc. A (1955) 228, 515–538.ADSGoogle Scholar
  18. Glen, J. W.: The flow law of ice. A discussion of the assumptions made in glacier theory, their experimental foundations and consequences. Proc. Iash Symp. 47, Chamonix (1958), 171–183.Google Scholar
  19. Herterich, K.: On the flow within the transition zone between ice sheet and ice shelf. Proc. Workshop Dynamics of The West Antarctic Ice Sheet, Utrecht 1985. Ed. Van der Veen, C. J. and Oerlemans, J. Reidel, Dordrecht (1987), 185–202.Google Scholar
  20. Herterich, K.: A three-dimensional model of the Antarctic Ice Sheet. Annals of Glaciology (1988) 11, 32–35.ADSGoogle Scholar
  21. Hindmarsh, R. C. A.: Modelling the dynamics of ice sheets. Prog. Phys. Geog. ( 1992 submitted).Google Scholar
  22. Hindmarsh, R. C. A., Morland, L. W., Boulton, G. S. and Butter, K.: The unsteady plane flow of ice sheets: a parabolic problem with two moving boundaries. Geophys. Astrophys. Fluid Dynamics (1987), 39, 183–225.CrossRefMATHADSGoogle Scholar
  23. Hindmarsh, R. C. A., Boulton, G. S. and Hutter, K.: Modes of operation of thermomechanically coupled ice sheets. Annals of Glaciology (1989) 12, 57–69.ADSGoogle Scholar
  24. Hindmarsh, R. C. A. and Hutter, K.: Numerical fixed domain mapping solution of free-surface flows coupled with an evolving interior field. Int. J. Num. Anal. Meth. Geomech. (1988) 12, 437–459.CrossRefMATHGoogle Scholar
  25. Hughes, T. J.: Is the West Antarctic Ice Sheet distintegrating? J. Geophys. Res. (1973) 78, 7884–7910.CrossRefADSGoogle Scholar
  26. Hughes, T. J.: The West Antarctic Ice Sheet: instability, disintegration and initiation of ice ages. Rev. Geophys. Space Phys. (1975) 13, 502–526.CrossRefADSGoogle Scholar
  27. Hutter, K.: The effect of longitudinal strain on the shear stress of an ice sheet: in defence of using stretched co-ordinates. J. Glaciology (1981) 27, 39–56.ADSGoogle Scholar
  28. Hutter, K.: Dynamics of glaciers and large ice masses. Ann. Rev. Fluid Mech. (1982) 14, 87–130.MathSciNetCrossRefADSGoogle Scholar
  29. Hutter, K.: A mathematical model of polythermal glaciers and ice sheets. Geophys. Astrophys. Fluid Dynamics (1982b) 21, 201–224.CrossRefMATHADSGoogle Scholar
  30. Hutter, K.: Theoretical Glaciology, Reidel, Dordrecht (1983).CrossRefGoogle Scholar
  31. Hutter, K.: Thermo-mechanically coupled ice sheet response. Cold, polythermal, temperate. J. Glaciology in press (1993).Google Scholar
  32. Hutter, K., Blatter, H. and Funk, M.: A model computation of moisture content in polythermal glaciers. J. Geophys. Res. (1988) 93, 12205–12214.CrossRefADSGoogle Scholar
  33. Nutter, K., Legerer, F. and Spring, U.: First order stresses and deformation in glaciers and ice sheets. J. Glaciology (1981) 27, 227–270.ADSGoogle Scholar
  34. Nutter, K., YaicowItz, S. and SzIdarovsiy, F.: A numerical study of plane ice sheet flow. J. Glaciology (1986) 32, 139–160.ADSGoogle Scholar
  35. Nutter, K., Yakowitz, S. and Szidarovsky, F.: Coupled thermomechanical response of an axisymmetric cold ice sheet. Water Resources Res. (1987) 23, 1327–1339.CrossRefADSGoogle Scholar
  36. Huybrecits, P.: The Antarctic Ice Sheet and environmental change: a three-dimensional modelling study. Reports on Polar Research 99, Alfred Wegener Institute for Polar and Marine Research (1992).Google Scholar
  37. Johnson, R. E. and Mcmeeking, R. M.: Near-surface flow in glaciers obeying Glen’s law. Quart. J. Mech. Appl. Math. (1984) 37, 273–291.MathSciNetCrossRefMATHGoogle Scholar
  38. Kamb, W. B.: Sliding motion of glaciers: theory and observation. Rev. Geophys. Space Phys. (1970) 8, 673–728.CrossRefADSGoogle Scholar
  39. Lliboutry, L. A.: General theory of subglacial cavitation and sliding of temperate glaciers. J. Glaciology (1968) 7, 21–58.ADSGoogle Scholar
  40. Macayeal, D. R.: Ice shelf back pressure: form drag versus dynamics drag. Proc. Workshop Dynamics of The West Antarctic Ice Sheet, Utrecht 1985. Ed. Van der Veen, C. J. and Oerlemans, J. Reidel, Dordrecht (1987), 141–160.Google Scholar
  41. Macayeal, D. R. and Barcilon, V.: Ice shelf response to ice stream discharge fluctuations, I Unconfined ice tongues. J. Glaciology (1988) 34, 121–127.ADSGoogle Scholar
  42. Macayeal, D. R. and Lange, M. A.: Ice shelf response to ice stream discharge fluctuations, II Ideal rectangular basin. J. Glaciology (1988) 34, 128–135.ADSGoogle Scholar
  43. Mellor, M. and Testa, R.: Effect of temperature on the creep of ice. J. Glaciology (1969) 8, 131–145.ADSGoogle Scholar
  44. Mellor, M.: Mechanical properties of polycrystalline ice. Proc. Iutam Symp. Physics and Mechanics of Ice, Copenhagen 1979. Ed. Tryde, P. Springer-Verlag, Berlin (1980), 217–245.Google Scholar
  45. Mercer, J. H.: West Antarctic Ice Sheet and CO2 greenhouse effect: a threat of disaster. Nature (1978) 271, 321–325.CrossRefADSGoogle Scholar
  46. Morland, L. W.: Glacier sliding down an inclined wavy bed. J. Glaciology (1976) 17, 447–462.ADSGoogle Scholar
  47. Morland, L. W.: Glacier sliding down an inclined wavy bed with friction. J. Glaciology (1976) 17, 463–477.Google Scholar
  48. Morland, L. W.: Constitutive laws for ice. Cold Reg. Sci. Tech. (1979) 1, 101–108.Google Scholar
  49. Morland, L. W.: Thermomechanical balances of ice sheet flows. Geophys. Astrophys. Fluid Dynamics (1984) 29, 237–266.CrossRefMATHADSGoogle Scholar
  50. Morland, L. W.: Unconfined ice sheet flow. Proc. Workshop Dynamics of The West Angarctic Ice Sheet, Utrecht 1985. Ed. Van der Veen, C. J. and Oerlemans, J. Reidel, Dordrecht (1987), 99–116.Google Scholar
  51. Morland, L. W.: Non-linear viscoelastic response of ice. Applied Oceans Research (1992) 13, 31–38.Google Scholar
  52. Morland, L. W.: Flow of viscous fluids through a porous deformable matrix. Surveys in Geophys. (1992) 13, 209–268.CrossRefADSGoogle Scholar
  53. Morland, L. W. and Johnson, I. R.: Steady motion of ice sheets. J. Glaciology (1980) 25, 229–246.ADSGoogle Scholar
  54. Morland, L. W. and Johnson, I. R.: Effects of bed inclination and topography on steady isothermal ice sheets. J. Glaciology (1982) 28, 71–90.ADSGoogle Scholar
  55. Morland, L. W. and Shoemaker, E. M.: Ice shelf balances. Cold Reg. Sci. Tech. (1982) 5, 235–251.CrossRefGoogle Scholar
  56. Morland, L. W. and Smith, G. D.: Influence of non-uniform temperature distribution on the steady motion of ice sheets. J. Fluid Mech. (1984) 140, 113–133.CrossRefMATHADSGoogle Scholar
  57. Morland, L. W., Smith, G. D. and Boulton, G. S.: Basal sliding relations deduced from ice sheet data. J. Glaciology (1984) 30, 131–139.ADSGoogle Scholar
  58. Morland, L. W. and Spring, U.: Viscoelastic fluid relation for the deformation of ice. Cold Reg. Sci. Tech. (1981) 4, 225–268.Google Scholar
  59. Morland, L. W. and Spring, U.: Single integral representations for non-linear viscoelastic solids. Mech. Mat. (1982) 1, 161–170.CrossRefGoogle Scholar
  60. Morland, L. W. and Zainuddin, R.: Plane and radial ice-shelf flow. Proc. Workshop Dynamics of The West Antarctic Ice Sheet, Utrecht 1985. Ed. Van der Veen, C. J. and Oerlemans, J. Reidel, Dordrecht (1987), 117–140.Google Scholar
  61. Nye, J. F.: The flow law of ice from measurements in glacier tunnels, laboratory experiments and the Jungfrau firn borehole experiments. Proc. Roy. Soc. A (1953) 219, 477–489.CrossRefADSGoogle Scholar
  62. Nye, J. F.: The motion of ice sheets and glaciers. J. Glaciology (1959) 3, 493–507.ADSGoogle Scholar
  63. Nye, J. F.: A calculation on the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc. Roy. Soc. A (1969) 311, 445–467.CrossRefADSGoogle Scholar
  64. Nye, J. F.: The effect of longitudinal stress on the shear stress at the base of an ice sheet. J. Glaciology (1969) 8, 207–213.ADSGoogle Scholar
  65. Nye, J. F.: Glacier sliding without cavitation in a linear viscous approximation. Proc. Roy. Soc. A (1970) 315, 381–403.CrossRefADSGoogle Scholar
  66. Oerlemans, J.: Some basic experiments with a vertically integrated ice sheet model. Tellus (1981) 33, 1–11.CrossRefADSGoogle Scholar
  67. Paterson, W. S. B.: The Physics of Glaciers ( 2nd ed. ). Pergamon, Oxford (1969).Google Scholar
  68. Reeh, N.: A flow-line model for calculating the surface profile and velocity, strain-rate, and stress fields in an ice sheet. J. Glaciology (1988) 34, 46–54.ADSGoogle Scholar
  69. Robin, G. Q.: Surface topography of ice sheets. Nature (1967) 215, 1029–1032.CrossRefADSGoogle Scholar
  70. Robin, G. Q.: Formation, flow, and disintegrations of ice shelves. J. Glaciology (1979) 24, 259–271.ADSGoogle Scholar
  71. Robin, G. DE Q. and Swithinbank, C.: Fifty years of progress in understanding ice sheets. J. Glaciology (1987) Special Issue, 33–47.Google Scholar
  72. Sanderson, T. J. O.: Equilibrium profiles of ice shelves. J. Glaciology (1979) 22, 435–459.ADSGoogle Scholar
  73. Sanderson, T. J. O. and Doake, C. S. M.: Is vertical shear in an ice shelf negligible ? J. Glaciology (1979) 22, 285–292.ADSGoogle Scholar
  74. Shoemaker, E. M. and Morland, L. W.: A glacier flow model incorporating longitudinal devia.toric stresses. J. Glaciology (1984) 30, 334–340.ADSGoogle Scholar
  75. Smith, G. D. and Morland, L. W.: Viscous relations for the steady creep of polycrystalline ice. Cold Reg. Sci. Tech. (1981) 5, 141–150.Google Scholar
  76. Spring, U. and Morland, L. W.: Viscoelastic solid relations for the deformation of ice. Cold Reg. Sci. Tech. (1982) 5, 221–234.Google Scholar
  77. Spring, U. and Morland, L. W.: Integral representations for the viscoelastic deformation of ice. Cold Reg. Sci. Tech. (1983) 6, 185–193.Google Scholar
  78. Steinemann, S.: Flow and recrystallisation of ice. Proc. Iash Symp. 39, Rome (1958), 449–462.Google Scholar
  79. Thomas, R. H.: Flow law of Antarctic ice shelves. Nature, Physical Sciences (1971) 232, 85–87.CrossRefADSGoogle Scholar
  80. Thomas, R. H.: The creep of ice shelves: theory. J. Glaciology (1973) 12, 45–53.ADSGoogle Scholar
  81. Thomas, R. H.: The creep of ice shelves: interpretation of observed behaviour. J. Glaciology (1973) 12, 55–70.MATHADSGoogle Scholar
  82. Thomas, R. H.: Dynamics of marine ice sheets. J. Glaciology (1979) 24, 166–177.ADSGoogle Scholar
  83. Thomas, R. H.: Ice shelves: a review. J. Glaciology (1979) 24, 273–286.ADSGoogle Scholar
  84. Thomas, R. H. and Bentley, C. R.: A model for the Holocene retreat of the West Antarctic Ice Sheet. Quat. Res. (1978) 10, 150–170.CrossRefGoogle Scholar
  85. Thomas, R. H., Sanderson, T. J. O. and Rose, K. E.: Effect of climate warming on the West Antarctic Ice Sheet. Nature (1979) 277, 355–357.CrossRefADSGoogle Scholar
  86. Truesdell, C.: On the foundations of mechanics and energetics, in The Rational Mechanics of Materials. Gordon and Breach, New York (1965), 293–304.Google Scholar
  87. Van Der Veen, C. J.: Longitudinal stresses and basal sliding. Proc. Workshop Dynamics of the West Antarctic Ice Sheet, Utrecht 1985. Ed. Van der Veen, C. J. and Oerlemans, J. Reidel, Dordrecht (1987), 223–248.Google Scholar
  88. Weertman, J: Deformation of floating ice shelves. J. Glaciology (1957) 3, 38–42.ADSGoogle Scholar
  89. Weertman, J: On the sliding of glaciers. J. Glaciology (1957) 3, 33–38.ADSGoogle Scholar
  90. Weertman, J: Equilibrium profile of ice caps. J. Glaciology (1961) 3, 953–964.ADSGoogle Scholar
  91. Weertman, J: The theory of glacier sliding. J. Glaciology (1964) 5, 287–303.ADSGoogle Scholar
  92. Weertman, J: Stability of the junction of an ice sheet and an ice shelf. J. Glaciology (1974) 13, 3–11.ADSGoogle Scholar
  93. Weertman, J: Glaciology’s grand unsolved problem. Nature (1976) 260, 284–286.ADSGoogle Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • L. W. Morland
    • 1
  1. 1.School of MathematicsUniversity of East AngliaNorwichUK

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