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# Comparison of Three Contemporary Flow Laws in a Three-Dimensional, Time-Dependent Glacier Model

## Abstract

A numerical model for three-dimensional, time-dependent glacier flow (Campbell and Rasmussen, 1970) treated the ice as a Newtonian viscous fluid and related its dynamics to two large-scale bulk parameters: the viscosity v determining the ice-to-ice friction, and a basal friction parameter A determining the ice-to-rock friction. The equations were solved using the relatively simple flow law of Bodvarsson (1955) in which the basal shear stress is proportional to volume transport. Recent research suggests that a more realistic basal flow law is one in which the basal shear stress to some lower power (1–3) is either proportional to the vertically averaged velocity (Glen, 1958; Nye, 1960, 1963[a], [b], [c], 1965[a], [b], [c]) or to the ratio of the vertically averaged velocity to glacier thickness (Budd and Jenssen, in press).

In the present study a generalized flow law incorporating all of the above bulk basal flow laws is applied to the Campbell–Rasmussen momentum equation to form a generalized two-dimensional transport equation, which, when combined with the continuity equation, yields a numerically tractable set of equations for three-dimensional, time-dependent glacier flow. Solutions of the model are shown for steady-state flow and surge advance and recovery for a typical valley glacier bed for powers 1, 2, and 3 for each of the basal flow laws for a steady-state climate input and a given ice-to-ice viscosity parameter.

Un modèle numérique à trois dimensions pour un écoulement de glacier fonction du temps (Campbell et Rasmussen, 1970) traitait la glace comme un fluide visqueux Newtonien et rapportait sa dynamique à deux paramètres d’ensemble à grande échelle: la viscosité v représentant le frottement glace–glace et un paramètre de frottement basal A représentant le frottement glace–rocher. On résolvait les équations en utilisant la loi d’écoulement relativement simple de Bodvarsson (1955) dans laquelle l’effort de cisaillement basal est proportional au volume transporté. Des recherches récentes suggèrent qu’une loi d’écoulement basal plus réaliste serait celle selon laquelle l’effort de cisaillement à la base à une puissance inférieure (1–3) est proportionnelle soit à la vitesse moyenne le long d’une verticale (Glen, 1958; Nye, 1960, 1963[a], [b], [c], 1965[a], [b], [c]) soit au rapport entre la vitesse moyenne le long d’une verticalle à l’épaisseur du glacier (Budd et Jenssen, sous presse).

Dans la présente étude, une loi d’écoulement générale englobant toutes les lois d’écoulement d’ensemble ci-dessus est appliquée à l’équation des moments de Campbell–Rasmussen pour former une équation générale du transport à deux dimensions qui, combinée avec l’équation de continuité fournit un système d’équation traitable numériquement pour l’écoulement tridimensionnel du glacier en fonction du temps. On montre les solutions de ce modèle pour un écoulement permanent et pour un épisode d’avance et de retrait après une crue dans une vallée glaciaire typique avec des puissances de 1, 2 et 3 pour chacune des lois d’écoulement à la base sous un climat constant et une valeur donnée du paramètre viscosité glace–glace.

Zusammenfassung

Ein numerisches Modell für dreidimensionales, zeitabhängiges Gletscherfliessen (Campbell und Rasmussen, 1970) behandelte das Eis als eine Newton’sehe viskose Flüssigkeit und beschrieb seine Bewegung mit Hilfe zweier makroskopischer Massenparameter: der Viskosität v, welche die Eis-zu-Eis-Reibung bestimmt, und einem Grundreibungsparameter A, der die Eis-zu-Fels-Reibung bestimmt. Die Gleichungen wurden mit dem relativ einfachen Fliessgesetz von Bodvarsson (1955) gelöst, in dem die Scherspannung am Untergrund proportional zum Massentransport ist. Neuere Untersuchungen deuten daraufhin, dass in einem realistischeren Gesetz für das Fliessen am Untergrund eine gewisse niedrige Potenz (1–3) der Scherspannung proportional entweder zur vertikalen Durchschnittsgoschwindigkcit (Glen, 1958; Nye, 1960, 1963[a], [b], [c], 1965[a], [b], [c]) oder zum Verhältnis von vertikaler Durchschnittsgeschwindigkeit und Gletscherdicke (Budd und Jenssen, im Druck) angenommen werden sollte.

In der vorliegenden Untersuchung wird ein allgemeines Fliessgesetz, das alle oben genannten Fliessgesetze für Massen am Untergrund einschliesst, auf die Campbell–Rasmussen’sche Momentengleichung angewandt; daraus ergibt sich eine allgemeine zweidimensionale Transportgleichung, die in Verbindung mit der Kontinuitäisgleichung ein numerisch lösbares Gleichungssystem für ein dreidimensionales zeitabhängiges Gletscherfliessen liefert, Lösungen des Modells werden für stationäres Fliessen sowie für einen ausbruchartigen Vorstoss und dessen Regeneration in einem typischen Talgletscherbett mit den Potenzen 1, 2 und 3 in jedem der drei Fliessgesetze unter der Annahme stationärer Klimaverhältnisse und mit einem gegebenen Eis-zu-Eis-Viskositätsparameter aufgezeigt.

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## References

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