TY - JOUR

T1 - The uniqueness of steady state flows of glaciers and ice sheets

AU - Fowler, A. C.

AU - Larson, D. A.

PY - 1980/11

Y1 - 1980/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0019173080&partnerID=8YFLogxK

U2 - 10.1111/j.1365-246X.1980.tb02624.x

DO - 10.1111/j.1365-246X.1980.tb02624.x

M3 - Article

AN - SCOPUS:0019173080

SN - 0016-8009

VL - 63

SP - 333

EP - 345

JO - Geophysical Journal of the Royal Astronomical Society

JF - Geophysical Journal of the Royal Astronomical Society

IS - 2

ER -