The uniqueness of steady state flows of glaciers and ice sheets

A. C. Fowler, D. A. Larson

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the uniqueness of solutions to a model set of equations describing the temperature and flow of cold (or sub‐temperate) glaciers. These equations are based on the model developed by Fowler & Larson (1978), and incorporate the essential features of a free surface boundary, free melting boundary, and a fully non‐Newtonian flow law. The only real simplification which is made is that of taking small Peclet number (or Graetz number), which implies that advective heat transport is negligible. Parametric estimates indicate that this is unrealistic, but nevertheless it represents a formally self‐consistent way of effectively adopting a ‘slab’ model, while avoiding the inherent disadvantages of such models. Since the possibility of non‐uniqueness is associated with the non‐linear viscous heating term, we expect that non‐zero advection of heat will only affect the results quantitatively, but not qualitatively. We find that for the various kinds of basal boundary condition which can occur, the solutions for the temperature (and hence the flow field) are unique, with the possible exception of regions where the basal ice is temperate and sliding, but the rest of the glacier is cold. In such regions one can have up to three solutions (even with the exponential approximation to the Arrhenius term), and such solutions can exhibit hysteretic instability: however, we then show that such multiple solutions must transgress the condition that the ice be below its freezing point, and hence they are not relevant in the present study. We therefore conclude that the solutions are unique, and so the non‐linear heating term is unlikely to cause surge‐like instabilities in real glaciers (or ice sheets). This does not preclude the possibility that the unique solution is linearly unstable to infinitesimal perturbations: such an instability might also lead to surging states, but not in the explosive manner that non‐uniqueness would suggest.

Original languageEnglish
Pages (from-to)333-345
Number of pages13
JournalGeophysical Journal of the Royal Astronomical Society
Volume63
Issue number2
DOIs
Publication statusPublished - Nov 1980
Externally publishedYes

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