Proceedings of the Royal Society of London A Mathematical and Physical Sciences · 1957 · 594 citations · 0 references
Natural SciencesPhysical GeographyFluid DynamicsGlaciologyGlaciophysicsGlacier DynamicsGlacier MechanicsIce DynamicsIce RheologyIce Sheet DynamicsGlacier Flow DynamicsGlacier Flow MechanicsIce Flow DynamicsIce Flow MechanicsLaminar Flow RegimesLongitudinal Stress GradientsPerfect PlasticitySteady Flow
A theoretical glacier model treats a block of ice on a rough slope, ignoring valley sides, and distinguishes laminar flow (zero longitudinal velocity gradient) from extending/compressive flow (non‑zero gradient), the former derived under a general ice flow law and the latter only under perfect plasticity. This study derives steady‑flow equations for a glacier using a general flow law that permits a non‑zero longitudinal velocity gradient. The equations allow depth‑dependent density, temperature, or flow law, and given known depth profiles, surface slope, velocity, and gradient, they compute stresses and velocity; the model is tested against borehole data from the Jungfraufirn, where observed gradients and Glen’s law yield a theoretical curve compared to experiment. At 50 m depth the omission of the gradient underestimates shear rate by a factor of 50, whereas the new theory overestimates it by 1.33, with the remaining discrepancy likely due to glacier sides.
A block of ice resting upon a rough slope forms a theoretical model of a glacier or an ice-sheet, the sides of the glacier valley being ignored. Previous papers have described two types of steady flow in this model: ( a ) laminar flow, in which the longitudinal velocity gradient r is zero, and ( b ) extending or compressive flow, in which r is non-zero, ( a ) was derived under the assumption of a general flow law for ice, but ( b ) was only derived under the assumption of perfect plasticity. In the present paper a general flow law is used throughout, and the equations for steady flow, with r allowed to be non-zero, are found. The previous results ( a ) and ( b ) appear as special cases. Possible variations of density, temperature or flow law with depth are allowed for. If the density and the flow law are known as functions of depth in any region, and if the surface slope, the surface velocity, and the value of r are known, the equations give the stresses and velocity as functions of depth. The borehole experiment on the Jungfraufirn (1948-50) allows an experimental test. From the observed value of r , and Glen’s laboratory flow law for ice, a theoretical curve for the result of the experiment is calculated which is compared with the experimental curve. At a depth of 50 m the effect of ignoring r , as has been done hitherto, is to underestimate the shear rate by a factor of 50; on the present theory it is overestimated by a factor of 1∙33. The remaining discrepancy is probably mainly due to the effect of the glacier sides.