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Ice Flow Over Bedrock Perturbations

  • W.F. Budd (a1)

Abstract

The use of well known simple periodic solutions of the two-dimensional biharmonic stress equation for studying the flow over undulations of an ice mass of small surface slope is examined. The model considered is one in which most of the shear (deformation or. sliding) takes place near the base and the upper part moves largely as a block, with longitudinal strain-rates varying linearly with the longitudinal stress deviations. For bedrock perturbations of a given wavelength the steady-state surface shape consists of similar waves but out of phase by ½π, such that the steepest slope occurs over the highest bedrock; and the amplitude is reduced by a “damping factor”, depending on the speed, viscosity, ice thickness and wavelength.

Minimum damping occurs for λ m ≈ 3.3 times the ice thickness, while waves much longer or much shorter than this are almost completely damped out. The energy dissipation and the resistance to the ice flow is also a maximum for an undulation scale of several times the ice thickness, whereas the effects of small basal irregularities die out exponentially with distance into the ice, and only have an effect in so far as the average basal stress is related to the average surface slope. As a result of this a revision of present glacier sliding theories becomes possible.

Various predictions of the theory have been confirmed from spectral analysis of surface and bedrock profiles of ice caps.

Résumé

L'emploi de solutions périodiques simples et bien connues de l'équation de contrainte biharmonique à deux dimensions pour l'étude de l'écoulement sur des ondulations d'une masse de glace de faible pente de surface est examinée. Le modèle considéré est celui dans lequel la plus grande partie de l'effort de cisaillement (déformation ou glissement) prend place près de la base, la partie supérieure bougeant largement comme un bloc avec des vitesses de déformation longitudinales variant linéairement avec les déviations de la contrainte longitudinale. Pour les perturbations du socle rocheux d'une longueur d'onde donnée, la forme superficielle permanente consiste en vagues similaires mais hors de phase de ½π de sorte que la pente la plus forte arrive sur le plus haut sommet du socle rocheux; et l'amplitude est réduite par un facteur d'amortissement dépendant de la vitesse, viscosité, épaisseur de glace et longueur d'onde.

L'amortissement minimum a lieu pour λm≈ 3,3 fois l'épaisseur de glace, tandis que des ondes plus longues ou plus courtes sont presque complètement amorties. La dissipation d'énergie et la résistance à l'écoulement glaciaire est aussi maximum pour une échelle d'ondulation de plusieurs fois l'épaisseur de glace, tandis que les effets des petites irrégularitiés basale meurent exponentiellement avec la distance dans la glace et ont seulement un effet aussi loin que la contrainte basale moyenne est liée à la pente moyenne de surface. En conséquence, une révision des théories de glissement actuel de glacier devient possible.

Des prédictions variées de la théorie ont été confirmées par l'analyse spectrale des profils de la surface et du socle rocheux des calottes glaciaires.

Zusammenfassung

Die Anwendbarkeit bekannter periodischer Lösungen der zweidimensionalen biharmonischen Spannungsgleichung auf das Studium der Bewegung einer Eismasse mit geringer Oberflächenneigung über Unebenheiten wird untersucht. In dem betrachteten Modell findet der Grossteil der Scherung (Verformung oder Gleiten) in der Nähe der Grundfläche statt, während der obere Teil sich weitgehend als Block bewegt. Die Deformationsgeschwindigkeit in Längsrichtung ändert sich linear mit den Abweichungen der Längsspannung. Bei Unebenheiten im Untergrund mit bekannter Wellenlänge zeigen sich auf der stationären Oberfläche ähnliche Wellen, jedoch mit einer Phasenverschiebung von ½π, so dass die grösseren Neigungen über den höchsten Stellen des Untergrundes auftreten. Die Amplitude wird durch einen “Dämpfungsfaktor” verringert, der von der Geschwindigkeit, der Viskosität, der Eisdicke und der Wellenlänge abhängt.

Die geringste Dämpfung tritt für λ m ≈ 3.3 mal der Eisdicke ein, während Wellen mit sehr viel grösseren oder sehr viel kleineren Längen fast ganz verschwinden. Der Energieverlust und der Widerstand gegen die Eisbewegung ist ebenfalls bei einer Wellengrösse von mehrfacher Eisdicke maximal, während die Wirkung kleiner Unregelmässigkeiten am Untergrund exponentiell zum Abstand im Eis ausläuft und nur insoweit von Bedeutung ist, als die mittlere Spannung am Untergrund von der mittleren Oberflächenneigung abhängt. Als Folge hieraus scheint eine Überprüfung gegenwärtiger Theorien des Gletschergleitens möglich.

Verschiedene Voraussagen der Theorie wurden durch Geschwindigkeitsanalysen von Oberflächen- und Untergrundprofilen in Eisschilden bestätigt.

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References

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Ice Flow Over Bedrock Perturbations

  • W.F. Budd (a1)

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