Viscous Flow: Experiments and Theory

The great naturalists of the nineteenth century discovered glacier flow and described its main features. They showed that the flow properties of glaciers resemble those of highly viscous fluids, and attributed the ductility of the apparently brittle substance ice to the effects of regelation or to the relative motion of individual ice crystals.

Accurate measurements of glacier flow were begun in the Alps in the 1890’s and continued for several decades.Reference Blümcke and Hess ¹ Surface velocities were measured at many points on glacier tongues, bore-holes reaching down to the rock bed were drilled and their progression was carefully recorded. Finally, in the 1930’s, reliable and less tiresome methods of measuring glacier depths became possible with the introduction of seismic soundings.

Many of the experiments and observations were accompanied by attempts at theoretical interpretation. One of the earliest of these was Finsterwalder’s geometric streamline theory,Reference Finsterwalder and Vernagtferner ² which correlated accumulation and ablation with glacier flow and explained the periodic advance and recession of glaciers. Finsterwalder’s theory is mainly descriptive and free from all arbitrary assumptions; on the other hand, this makes the theory so general that it is difficult to derive from it any quantitative predictions. Hess actually employed it to obtain equations for the total mechanical energy of a glacier, involving integration over six dependent variables which cannot be evaluated without a host of dubious assumptions. The solutions of these equations led him to some exceedingly improbable conclusions concerning the stress distribution in glaciers. More important are Weinberg’s and Somigliana’s attempts to formulate a quantitative theory of glacier flow on a hydrodynamic basis.Reference Somigliana and Finsterwalder ^3, Reference Weinberg ⁴ Somigliana treats the stationary flow of a viscous liquid in an evenly inclined cylindrical channel under the influence of gravity. His work is outstanding for the clear exposition of the underlying assumptions and important because it leads to a relation between the depth and surface velocity of a glacier. Lagally extended his theory and used it to derive the average viscosity of glacier ice from the change in flow velocity between the edges and the centre of a glacier surfaceReference Lagally ⁵ ; he obtained a value of μ = 1 × 10¹⁴ g. cm.⁻¹ sec.⁻¹. Taking the cross-section of the glacier bed to be a semi-ellipse whose major axis is very large compared to the minor one, and the velocity of flow on the rock bed to be negligibly small, he obtains the equation

where z is the depth of the glacier, μ the viscosity of ice, V₀ the surface velocity at the centre of the glacier, ρ the density of ice, g the acceleration of gravity and α the inclination of the rock bed. On the Pasterze (Austria), where the ideal conditions of Somigliana’s theory seem to be very nearly fulfilled, Lagally was able to predict a value for the depth of the glacier which was verified by the subsequent seismic soundings of Mothes and Brockamp.Reference Brockamp and Mothes ⁶ This remarkable success established beyond any reasonable doubt that the theory of continuous, stationary, viscous flow is substantially correct provided that all external conditions liable to lead to discontinuities are absent. The experiments of Blümcke and Hess had shown that the temperatures in Alpine glacier tongues remain at the pressure melting-point throughout the yearReference Blümcke and Hess, ⁷ except for an insignificant surface layer affected by the winter cold wave. Crevassing is confined to the top 50–100 ft., and the distribution of crevasses can easily be interpreted on the basis of the stresses introduced by differential flow and the changes in direction and inclination of the rock-bed.

Limitations of Simple Theory.

The Plastic Properties of Ice

Clearly, explanations of these diverse phenomena must be sought from the point of view of the general mechanism of deformation of plastic bodies. Such attempts are not likely to be successful, unless account is taken of the peculiar plastic properties of ice itself. In this connection Höppler’s recent measurements of the apparent viscosity of ice under various conditions are of the greatest interest.Reference Höppler ¹³ As expected, Höppler found the apparent viscosity to be a function of the applied shear stress, the temperature and the crystal orientation in his specimens. The actual figures, however, are surprising and allow a revealing insight into some of the more puzzling features of glacier flow. At a constant shear stress of 63 Kg./cm.² the apparent viscosity at —20° C. was 1000 times greater than at—1⁰ C.; at a constant temperature of —1° C. an increase in the shear stress from 10 to 63 Kg./cm.² reduced the apparent viscosity by a factor of 200. In a sample of frozen snow, possessing no preferred crystal orientation, the apparent viscosity was 70 times greater than in a multicrystalline block where all the crystal glide planes were parallel to the direction of shear. (This last result shows the enormous gain in plasticity of local regions, e.g. blue bands, brought about by preferred orientation.) Unfortunately, none of Höppler’s actual viscosity measurements can be applied to glacier problems directly, since he did not determine the viscosity at the pressure melting-point and all but one of his test specimens consisted of crystals in preferred orientation. His observations do point the way, however, to the kind of experimental data that will have to be collected before some of the peculiarities of glacier flow can be fully explained.

One important factor not considered by Höppler concerns the probable changes in viscosity with increasing depth below the glacier surface, due to hydrostatic pressure. This causes the melting of

where p is the pressure in Kg./cm.². Thus at a depth of 500 m., say, over 3 per cent. of the glacier mass at pressure melting-point will consist of water, whose presence should greatly reduce the viscosity.

Micro-mechanism of Flow

Very little is known as yet about the micro-mechanism of flow in glacier tongues, since the work of the Jungfraujoch Research Party was concerned chiefly with the firn region. There are indications that locally, at any rate, crystals tend to assume a preferred orientation with their glide plane parallel to the direction of shear stress, and that large scale thrust planes do not consist of actual cracks but rather of thin strata of large crystals in preferred orientation.Reference Perutz and Seligman ⁸ Plastic flow is probably the main factor responsible for the astonishing growth of the glacier crystals, though, as Seligman has pointed out, regelation or thaw-freeze processes on the surface may also play a part. Winterhalter’s recent studies of ice sections under tension are interesting in this connection.Reference Winterhalter ¹⁴ The internal stress, orientation and growth of each crystal in a test strip can be observed between crossed nicols while tension is being applied. The method thus offers an almost unique opportunity for the study of the micromechanism of plastic deformation in a transparent polycrystalline material. Again observations have been too few to permit any definite deductions, but the general potentialities of the method are obvious.

Conclusions

For a more complete understanding of the mechanism of glacier flow, advances are needed along three main lines; first, there is a call for precise data on the actual flow mechanism in glaciers, so far as it can be made accessible to observation; second, studies of the plastic properties and the micro-mechanism of flow of polycrystalline ice at the pressure melting-point are required; and third, theoretical and experimental work on the behaviour of large masses of ice in gravitational fields will have to be done. The first is a matter for field work on alpine and arctic glaciers, while the second and third are subjects for laboratory work. I shall not discuss the possible forms of attack upon these problems in detail, but rather give an outline of the separate points which, in my view, merit investigation. For the sake of completeness I shall also include lines of work which have only an indirect but nevertheless important bearing on the subjects discussed above.

Proposed Research