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Waves on glaciers

Published online by Cambridge University Press:  20 April 2006

A. C. Fowler
A. C. Fowler
Affiliation:
School of Mathematics, 39 Trinity College, Dublin, Ireland Present address: Department of Mathematics, 2–336, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
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Abstract

This paper is an attempt at a mathematical synopsis of the theory of wave motions on glaciers. These comprise surface waves (analogous to water waves) and seasonal waves (more like compression waves). Surface waves have been often treated and are well understood, but seasonal waves, while observed, do not seem to have attracted any theoretical explanation. Additionally, the spectacular phenomenon of glacier surges, while apparently a dynamic phenomenon, has not been satisfactorily explained.

The present thesis is that the two wave motions (and probably also surging, though a discussion of this is not developed here) can both be derived from a rational theory based on conservation laws of mass and momentum, provided that the basal kinematic boundary condition involving boundary slip is taken to have a certain reasonable form. It is the opinion of this author that the form of this ‘sliding law’ is the crux of the difference between seasonal and surface waves, and that a further understanding of these motions must be based on a more satisfactory analysis of basal sliding.

Since ice is here treated in the context of a slow, shallow, non-Newtonian fluid flow, the theory that emerges is that of non-Newtonian viscous shallow-water theory; rather than balance inertia terms with gravity in the momentum equation, we balance the shear-stress gradient. The resulting set of equationsis, in essence, a first-order nonlinear hyperbolic (kinematic) wave equation, and susceptible to various kinds of analysis. We show how both surface and seasonal waves are naturally described by such a model when the basal boundary condition is appropriately specified. Shocks can naturally occur, and we identify the (small) diffusive parameters that are present, and give the shock structure: in so doing, we gain a useful understanding of the effects of surface slope and longitudinal stress in these waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 120 , July 1982 , pp. 283 - 321
Copyright
© 1982 Cambridge University Press

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References

Aharon, P., Chappel, J. & Compston, W. 1980 Stable isotope and sea level data from New Guinea supports Antarctic ice surge theory of ice ages. Nature 283, 649–651.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.
Bindschadler, R. 1979 Basal sliding and bed separation: is there a connection? J. Glaciol. 23, 407–408.Google Scholar
Bird, R. B. 1976 Useful non-Newtonian models. Ann. Rev. Fluid Mech. 8, 13–34.Google Scholar
Blümcke, A. & Finsterwalder, S. 1905 Zeitliche Änderungen in der Geschwindigkeit der Gletscherbewegung. Sitz. k. Bayerischen Akad. Wiss., Math.-phys. Klasse 35, 109–131.Google Scholar
Bowen, D. Q. 1980 Antarctic ice surges and theories of glaciation. Nature 283, 619–620.Google Scholar
Budd, W. F. 1975 A first simple model for periodically self surging glaciers. J. Glaciol. 14, 3–21.Google Scholar
Budd, W. F. & Radok, U. 1971 Glaciers and other large ice masses. Rep. Prog. Phys. 34, 1–70.Google Scholar
Campbell, W. J. & Rasmussen, L. A. 1969 Three-dimensional surges and recoveries in a numerical glacier model. Can. J. Earth Sci. 6, 979–986.Google Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Cary, P. W., Clarke, G. K. C. & Peltier, W. R. 1979 A creep instability analysis of the Antarctic and Greenland ice sheets. Can. J. Earth Sci. 16, 182–188.Google Scholar
Clarke, G. K. C., Nitsan, U. & Paterson, W. S. B. 1977 Strain heating and creep instability in glaciers and ice sheets. Rev. Geophys. Space Phys. 15, 235–247.Google Scholar
Colbeck, S. C. & Evans, R. J. 1973 A flow law for temperate glaciers. J. Glaciol. 12, 71–86.Google Scholar
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Collins, I. F. 1968 On the use of equilibrium equations and flow law in relating surface and bed topography of glaciers and ice sheets. J. Glaciol. 7, 199–204.Google Scholar
Copson, E. T. 1975 Partial Differential Equations. Cambridge University Press.
Deeley, R. M. & Parr, P. H. 1914 On the Hintereis glacier. Phil. Mag. 27 (6), 153–176.Google Scholar
Dorsey, N. E. 1940 Properties of Ordinary Water-Substance. Reinhold.
Drake, L. D. & Shreve, R. L. 1973 Pressure melting and regelation of ice by round wires. Proc. R. Soc. Lond. A 332, 51–83.Google Scholar
Finsterwalder, S. 1907 Die Theorie der Gletscherschwankungen. Z. Gletscherkunde 2, 81–103.Google Scholar
Fowler, A. C. 1979a The use of a rational model in the mathematical analysis of a polythermal glacier. J. Glaciol. 24, 443–456.Google Scholar
Fowler, A. C. 1979b A mathematical approach to the theory of glacier sliding. J. Glaciol. 23, 131–141.Google Scholar
Fowler, A. C. 1980 The existence of multiple steady states in the flow of large ice masses. J. Glaciol. 25, 183–184.Google Scholar
Fowler, A. C. 1981 A theoretical treatment of the sliding of glaciers in the absence of cavitation. Phil. Trans. R. Soc. Lond. A 298, 637–684.Google Scholar
Fowler, A. C. & Larson, D. A. 1978 On the flow of polythermal glaciers I. Model and preliminary analysis. Proc. R. Soc. Lond. A 363, 217–242.Google Scholar
Fowler, A. C. & Larson, D. A. 1980a The uniqueness of steady state flows of glaciers and ice sheets. Geophys. J. R. Astr. Soc. 63, 333–345.Google Scholar
Fowler, A. C. & Larson, D. A. 1980b On the flow of polythermal glaciers II. Surface wave analysis. Proc. R. Soc. Lond. A 370, 155–171.Google Scholar
Fowler, A. C. & Larson, D. A. 1980c Thermal stability properties of a model of glacier flow. Geophys. J. R. Astr. Soc. 63, 347–359.Google Scholar
Gavalas, G. R. 1968 Nonlinear Differential Equations of Chemically Reacting Systems. Springer.
Glen, J. W. 1955 The creep of polycrystalline ice. Proc. R. Soc. Lond. A 228, 519–538.Google Scholar
Gruntfest, I. J. 1963 Thermal feedback in liquid flow: plane shear at constant stress. Trans. Soc. Rheol. 7, 195–207.Google Scholar
Hodge, S. M. 1974 Variations in the sliding of a temperate glacier. J. Glaciol. 13, 349–369.Google Scholar
Hollin, J. T. 1980 Climate and sea-level in isotope stage 5: an East Antarctic ice surge at 95000 BP? Nature 283, 629–633.Google Scholar
Hutter, K. 1980 Time dependent surface elevation of an ice slope. J. Glaciol. 25, 247–266.Google Scholar
Hutter, K. 1981 The effect of longitudinal strain on the shear stress of an ice sheet: in defense of using stretched coordinates. J. Glaciol. 27, 39–56.Google Scholar
Kamb, W. B. 1970 Sliding motion of glaciers: theory and observation. Rev. Geophys. Space Phys. 8, 673–728.Google Scholar
Kamb, W. B. & LaChapelle, E. 1964 Direct observations of the mechanism of glacier sliding over bedrock. J. Glaciol. 5, 159–172.Google Scholar
Lick, W. 1970 The propagation of disturbances on glaciers. J. Geophys. Res. 75, 2189–2197.Google Scholar
Lliboutry, L. A. 1965 Traité de Glaciologie, t. 2. Masson.
Lliboutry, L. A. 1968 General theory of subglacial cavitation and sliding of temperate glaciers. J. Glaciol. 7, 21–58.Google Scholar
Lliboutry, L. A. 1971 The glacier theory. In Advances in Hydroscience, vol. 7 (ed. Ven te Chow), pp. 81–167.
Lliboutry, L. A. 1976 Physical processes in temperate glaciers. J. Glaciol. 16, 151–158.Google Scholar
Lliboutry, L. A. 1978 Glissement d'un glacier sur un plan parsemé d'obstacles hémisphériques. Ann. Géophys. 34, 147–162.Google Scholar
Lliboutry, L. A. 1979 Local friction laws for glaciers: a critical review and new openings. J. Glaciol. 23, 67–95.Google Scholar
Lliboutry, L. A. & Rayneaud, L. 1981 Global dynamics of a temperate valley glacier, Mer de Glace, and past velocities deduced from Forbes bands. J. Glaciol. 27, 207–226.Google Scholar
Meier, M. F. 1979 Variations in time and space of the velocity of lower Columbia Glacier, Alaska. J. Glaciol. 23, 408.Google Scholar
Meier, M. F. & Post, A. S. 1969 What are glacier surges? Can. J. Earth Sci. 6, 807–817.Google Scholar
Morland, L. W. 1976 Glacier sliding down an inclined wavy bed. J. Glaciol. 17, 447–462.Google Scholar
Morris, E. M. 1976 An experimental study of the motion of ice past obstacles by the process of regelation. J. Glaciol. 17, 79–98.Google Scholar
Morris, E. M. 1979 The flow of ice, treated as a newtonian viscous liquid, around a cylindrical obstacle near the bed of a glacier. J. Glaciol. 23, 117–129.Google Scholar
Murray, J. D. 1970 Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations. SIAM J. Appl. Math. 19, 273–298.Google Scholar
Nye, J. F. 1952 A method of calculating the thickness of the ice sheets. Nature 169, 529–530.Google Scholar
Nye, J. F. 1960 The response of glaciers and ice-sheets to seasonal and climatic changes. Proc. R. Soc. Lond. A 256, 559–584.Google Scholar
Nye, J. F. 1963 On the theory of the advance and retreat of glaciers. Geophys. J.R. Astr. Soc. 7, 431–456.Google Scholar
Nye, J. F. 1967 Theory of regelation. Phil. Mag. 16 (8), 1249–1266.Google Scholar
Nye, J. F. 1969 A calculation of the sliding of ice over a wavy surface using a newtonian viscous approximation. Proc. R. Soc. Lond. A 311, 445–467.Google Scholar
Nye, J. F. 1970 Glacier sliding without cavitation in a linear viscous approximation. Proc. R. Soc. Lond. A 315, 381–403.Google Scholar
Nye, J. F. 1973 The motion of ice past obstacles. In Physics and Chemistry of Ice (ed. E. Whalley, S. J. Jones & L. W. Gold), pp. 387–394. Ottawa: Royal Society of Canada.
Paterson, W. S. B. 1969 The Physics of Glaciers. Pergamon.
Pearson, J. R. A. 1977 Variable-viscosity flows in channels with high heat generation. J. Fluid Mech. 83, 191–206.Google Scholar
Pearson, J. R. A. 1978 Polymer flows dominated by high heat generation and low heat transfer. Polymer Engng Sci. 18, 222–229.Google Scholar
Richardson, S. 1973 On the no-slip boundary condition. J. Fluid Mech. 59, 707–719.Google Scholar
Robin, G. De Q. 1955 Ice movement and temperature distribution in glaciers and ice sheets. J. Glaciol. 2, 523–532.Google Scholar
Robin, G. De Q. 1967 Surface topography of ice sheets. Nature 215, 1029–1032.Google Scholar
Robin, G. De Q. 1969 Initiation of glacier surges. Can. J. Earth Sci. 6, 919–928.Google Scholar
Robin, G. De Q. & Weertman, J. 1973 Cyclic surging of glaciers. J. Glaciol. 12, 3–18.Google Scholar
Smirnova, G. N. 1963 Linear parabolic equations which degenerate on the boundary of the region (in Russian). Sibirsk Mat. ž. 4, 343–358.Google Scholar
Stocker, R. L. & Ashby, M. F. 1973 On the rheology of the upper mantle. Rev. Geophys. Space Phys. 11, 391–426.Google Scholar
Theakstone, W. H. 1979 Observations within cavities at the bed of the glacier østerdalsisen, Norway. J. Glaciol. 23, 273–281.Google Scholar
Turcotte, D. L. & Oxburgh, E. R. 1972 Mantle convection and the new global tectonics. Ann. Rev. Fluid Mech. 4, 33–68.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Vivian, R. A. & Bocquet, G. 1973 Subglacial cavitation phenomena under the Glacier d'Argentière, Mont Blanc, France. J. Glaciol. 12, 439–451.Google Scholar
Weertman, J. 1957 On the sliding of glaciers. J. Glaciol. 3, 33–38.Google Scholar
Weertman, J. 1958 Traveling waves on glaciers. In Proc. IUGG Symp. Chamonix, France, pp. 162–168. Publication no. 47 de l'Association Internationale d'Hydrologie Scientifique.
Weertman, J. 1964 The theory of glacier sliding. J. Glaciol. 5, 287–303.Google Scholar
Weertman, J. 1977 Penetration depth of closely spaced water-free crevasses. J. Glaciol. 18, 37–46.Google Scholar
Weertman, J. 1979 The unsolved general glacier sliding problem. J. Glaciol. 23, 97–115.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Yuen, D. A. & Schubert, G. 1977 Asthenospheric shear flow: thermally stable or unstable? Geophys. Res. Lett. 4, 503–506.Google Scholar
Yuen, D. A. & Schubert, G. 1979 The role of shear heating in the dynamics of large ice masses. J. Glaciol. 24, 195–212.Google Scholar