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Stress Variations with Ice Flow Over Undulations

  • W.F. Budd (a1)

Abstract

The analysis of steady-state flow over bedrock perturbations of Budd(1970[a]) is extended to evaluate the stress variations associated with the flow. In particular the four separate terms of the fundamental one-dimensional (longitudinal) stress equation (Budd,1968) are examined to show explicitly how the longitudinal stress gradient G, the base stress τ, the surface slope stress s, and the integrated second derivative of the shear stress T all vary over-a given wave.

The result shows an error of a factor of 2 in some earlier studies relating longitudinal stress variations to the slope. This error resulted from the erroneous assumption that remains constant at its regional value. In fact the basal stress varies over the undulations in a similar way to the longitudinal stress gradient, and this is the same as the surface slope stress for long waves, but for short waves or less) requires the addition of the fourth term T, which then becomes dominant.

Résumé

L'analyse de lécoulement permanent sur les irrégularités du lit rocheux de Budd (1970 [a]) est étendue pour estimer les variations de l'effort associées à l'écoulement. En particulier, les quatre termes séparés de l'équation fondamentale (Budd, 1968) des efforts selon une seule, dimension (longitudinale) sont examinés pour montrer explicitement comment le gradient G de l'effort longitudinal, l'effort à la base τ, l'effort à la surface s et la dérivée seconde intégrée de l'effort de cisaillement T varient tous suivant un rythme ondulatoire donné.

Le résultat montre une erreur d'un facteur 2 dans quelques travaux antérieurs relatant la variation longitudinale des efforts selon la pente. Cette erreur résultait de l'hypothèse erronée que l'effort à la base restait constant à sa valeur moyenne pour le secteur considéré. En réalité, l'effort à la base varie avec les ondulations de la même manière que le gradient de l'effort longitudinal et il en est de même pour l'effort à la surface dans le case des longues ondulations, alors que dans le case des courtes ondulations (à peu près ou moins) il faut ajouter le quatrième terme T qui devient prédominant.

Zusammenfassung

Die Analyse des stetigen Fliessens über Störungen im Felsbctt von Budd (1970 [a]) wird zur Bestimmung der mit dem Flicssen verbundenen Spannungsschwankungen erweitert. Im einzelnen werden die vier unabhängigen Grossen der eindimensionalen (longitudinalen) Spannungsgleichung (Budd, 1968) untersucht, um explizit zu zeigen, wie der longitudinale Spannungsgradient G, die Spannung am Grund τ, die Spannung infolge Oberflächenneigung s und das Integral der zweiten Ableitung der Seherspannung T über einer gegebenen Welle variieren.

Das Ergebnis weist einen Fehler mit dem Faktor 2 in einigen früheren Untersuchungen nach, die longitudinale Spauiiungsuntcisrhiede auf die Neigung zurückführen. Dieser Fehler gehl auf die irrige Annahme zurück, dass die Spannung am Grunde ihren örtlichen Wert konstant beibehält. Tatsächlich ändert sie sich über Undulationen ähnlich wie der longitudinale Spannungsgradient; dieser ist derselbe wie die Spannung infolge Oberflächenneigung für lange Wellen, für kurze Wellen ( oder weniger), hingegen ist die Addition der vierten Grosse T notwendig, die dann überwiegt.

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References

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Nye, J.F. 1969[b] The effect of longitudinal stress on the shear stress at the base of an ice sheet. Journal of Glaciology, Vol. 8, No. 53, p. 20713.
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Weertman, J. 1962 Catastrophic glacier advances. Union Géodésique et Géophysique Internationale. Association Internationale d'Hydrologie Scientifique. Commission des Neiges et Glaces. Colloque d'Obergurgl, 10–9–18–9 1962, p. 3139.
Weertman, J. 1964 The theory of glacier sliding. Journal of Glaciology, Vol. 5, No. 39, p. 287303.

Stress Variations with Ice Flow Over Undulations

  • W.F. Budd (a1)

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