Skip to main content Accessibility help
×
Home

A Mathematical Approach to the Theory of Glacier Sliding

  • A. C. Fowler (a1)

Abstract

Previous theories of glacier sliding are subject to the criticism that they are not properly formulated. Here we describe how the basal ice flow may be related to the bulk ice flow by means of the formal mathematical method of matched asymptotic expansions. A complete model of the basal sliding (involving coupled problems in ice, water film, and bedrock) may be rationally reduced by a dimensional analysis to a consideration of the ice flow only, and regelation may be neglected provided roughness is absent on the finest scales (< c. 1 mm). If the viscosity is supposed to be independent of the moisture content, then complementary variational principles exist which allow bounds on the drag to be obtained. In particular, these determine the magnitude of the basal velocity in terms of two crucial dimensionless parameters. Arguments are presented as to why realistic sliding laws should be taken as continuous functions of the temperature, and a (major) consequence of this assumption is mentioned. Finally the effect of cavitation is discussed, via an (exact) leading-order solution of the ice flow in the particular case of a Newtonian fluid and a “small” bedrock slope.

Résumé

Les théories précédentes du glissement des glaciers sont sujettes à la critique de n’être pas convenablement formulées. Nous décrivons ici comment l’écoulement de la glace au fond peut être relié à l’écoulement d’ensemble de la masse de glace par le moyen de la méthode mathématique formelle des développements asymptotiques équivalents. Un modèle complet du glissement au fond (comprenant les problèmes connexes sur la glace, le film liquide et le lit roeheux) peut être rationnellement réduit par une analyse dimensionnelle à une considération ď écoulcment de glace uniquement, et le regel peut être négligé pourvu que la rugosité soit absente aux échelles les plus fines (<1 mm). Si on suppose que la viscosité est indépendante de la teneur en eau, il existe des principes complémentaires de variations qui permettent de fixer des limites au frottement. En particulier, ces principes déterminent l’ordre de grandeur de la vitesse au fond par le biais de deux paramètres adimensionnels fondamentaux. On présente des arguments expliquant pourquoi des lois réalistes de glissement doivent être considérées comme des fonctions de la température et une conséquence (essentielle) de cette hypothèse est mentionnée. L’effet de la teneur en eau liquide est briévement évoqué et finalement celui de la cavitation est discuté à partir ďune solution théorique (exacte) de l’écoulement de la glace dans le cas particulier d’un fluide Newtonien et ďune pente “faible” du fond du lit.

Zusammenfassung

Gegen frühere Theorien des Gletschergleitens wird der Einwand erhoben, sie seien nicht sachgemäss formuliert. In diesem Beitrag wird gezeigt, wie der Eisfluss am Untergrund mit Hilfe der formalen mathematischen Methode der angepassten asymptotischen Expansion zum Fluss der gesamten Eismasse in Beziehung gesetzt werden kann. Ein vollständiges Modell des Gleitens am Untergrund (einschliesslich der gekoppelten Probleme von Eis, Wasserfilm und Felsbett) lässt sich durch eine Dimensionsanalyse auf die Betrachtung des Eisflusses allein reduzieren; dabei ist die Regelation vernachlässigbar, sofern keine Rauhigkeit im kleinsten Ausmass (< 1 mm) vorhanden ist. Nimmt man die Viskosität als unabhängig vom Feuchtigkeitsgehalt an, so bestehen zusätliche Variationsprinzipien, die es gestatten, Schranken für den Zugwiderstand festzustellen. Im einzelnen bestimmen diese die Grösse der Geschwindigkeit am Untergrund in Funktion zweier wesentlicher, dimensionsloser Parameter. Es werden Gründe dafür aufgeführt, warum realistische Gleitgesetze als kontinuierliche Funktionen der Temperatur formuliert werden sollten; eine (bedeutsamere) Folge dieser Annahme wird erwähnt. Die wirkung des Feuchtigkeitsgehalts wird kurz betrachtet; schliesslich wird die Wirkung der Hohlraumbildung diskutiert, wobei eine exakte Lösung für das Fliessen des Eises im speziellen Fall einer Newton’schen Flüssigkeit und einer “geringen” Felsbettneigung herangezogen wird.

  • 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.

      A Mathematical Approach to the Theory of Glacier Sliding
      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.

      A Mathematical Approach to the Theory of Glacier Sliding
      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.

      A Mathematical Approach to the Theory of Glacier Sliding
      Available formats
      ×

Copyright

References

Hide All
Batchelor, G. K. 1967. An introduction to fluid mechanics. Cambridge, Cambridge University Press.
Carol, H. 1947. The formation of roches moutonnees. Journal of Glaciology, Vol. 1, No. 2, p. 5759.10.1017/S0022143000007589
Carrier, G. F., and others. 1966. Functions of a complex variable, by Carrier, G. F., Krook, M. and Pearson, C. E. New York, McGraw-Hill Book Co., Inc.
Fowler, A. C. 1979. The use of a rational model in the mathematical analysis of a polythermal glacier. Journal of Glaciology, Vol. 24, No. 90.10.1017/S002214300001491X
Fowler, A. C. Unpublished. Glacier dynamics. [D.Phil. thesis, University of Oxford, 1977.]
Fowler, A. C., and Larson, D. A. 1978. On the flow of polythermal glaciers. I: model and preliminary analysis. Proceedings of the Royal Society of London, Ser. A, Vol. 363, No. 1713, p. 21742.10.1098/rspa.1978.0165
Hodge, S. M. 1974. Variations in the sliding of a temperate glacier. Journal of Glaciology, Vol. 13, No. 69, P. 34969 10.1017/S0022143000023157
Johnson, M. W. jr., 1960. Some variational theorems for non-Newtonian flow. Physics of Fluids, Vol. 3, No. 6, p. 87178.10.1063/1.1706150
Johnson, M. W. jr., 1961. On variational principles for non-Newtonian fluids. Transactions of the Society of Rheology, Vol. 5, p. 921.10.1122/1.548882
Kamb, W. B. 1970. Sliding motion of glaciers: theory and observation. Reviews of Geophysics and Space Physics, Vol. 8, No. 4, p. 673728.10.1029/RG008i004p00673
Lliboutry, L. A. 1968. General theory of subglacial cavitation and sliding of temperate glaciers. Journal of Glaciology, Vol. 7, No. 49, p. 2158.10.1017/S0022143000020396
Lliboutry, L. A. 1976. Physical processes in temperate glaciers. Journal of Glaciology, Vol. 16, No. 74, p. 15158.10.1017/S002214300003149X
Morland, L. W. 1976[a]. Glacier sliding down an inclined wavy bed with friction. Journal of Glaciology, Vol. 17, No. 77 p. 44762.10.1017/S0022143000013733
Morland, L. W. 1976[b]. Glacier sliding down an inclined wavy bed with friction. Journal of Glaciology, Vol. 17, No. 77 p. 46377.10.1017/S0022143000013745
Nye, J. F. 1960. The response of glaciers and ice-sheets to seasonal and climatic changes. Proceedings of the Royal Society of London, Ser. A, Vol. 256, No. 1287, p. 55984.
Nye, J. F. 1969. A calculation on the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proceedings of the Royal Society of London, Ser. A, Vol. 311, No. 1506, p. 44567.10.1098/rspa.1969.0127
Nye, J. F. 1970. Glacier sliding without cavitation in a linear viscous approximation. Proceedings of the Royal Society of London, Ser. A, Vol. 315, No. 1522, p. 381403.10.1098/rspa.1970.0050
Robin, G. de Q. 1976. Is the basal ice of a temperate glacier at the pressure melting point? Journal of Glaciology, Vol. 16, No. 74, p. 18396.10.1017/S002214300003152X
Van Dyke, M. D. 1975. Perturbation methods in fluid mechanics. Second edition. Stanford, California, Parabolic Press.
Weertman, J. 1957. On the sliding of glaciers. Journal of Glaciology, Vol. 3, No. 21, p. 3338.10.1017/S0022143000024709

A Mathematical Approach to the Theory of Glacier Sliding

  • A. C. Fowler (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed