IV. Relationships of Stress and Temperature

The stress, movement, temperature and viscosity conditions in cold ice affect each other very closely.

V. Viscosity Relationships in Cold Ice

The methed now described and their application to the determination of the rheological properties of cold glacier ice serve to supplement the investigations of the stress-strain rate curve by measurements on artifically groen ice which have recently been carried out in variouse laboratoriesReference Perutz ^{9 ,} Reference Nye ^{10 ,} Reference Steinmann ^{11 ,} Reference Glen ²⁴ .

The creep of ice is not only dependent upon shear stress and temperature, but also upon numerous other factors, such as the average principal stress, crystallography, chemical composition, load at earlier times, etc. The influence of these factors is insufficiently known at present, and so it has been found advantageous for the solving of technical problems to simplify the relations, which in reality are very complicated, by the use of an easily understood approximationReference Haefeli ¹² . This purpose is achieved by the concept of the standard apparent viscosity μ which corresponds to a definite point P in the stress-strain curve, according to the equation (cf. Fig. 12., p. 625):

Fig. 12.

Diagrams to illustrate the theory. Left, Flow curve of ice. Right, Deformation of an ice cube at the division of movement, i.e. in the middle of the ice sheet

By reason of this simplification, the calculation of creep movements within a certain area of stress can be worked out, in a similar way as for a Newtonian fluid: in every concrete case, the art lies in the correct estimation of the standard apparent viscosity μ, which cannot be regarded as a material constant in view of its dependence upon the state of stress, etc.

In our particular case, three different methods were employed to arrive at the apparent viscosity, namely:

• a. Measurement of length alterations of the cross-tunnels under approximately known overburden pressure.

• b. Deformation of a circular tunnel under a general stress. (Cross-tunnel Q₁₀₀.)

• c. Deformation of a circular tunnel under uni-axial compression. (Longitudinal tunnel.)

Except for a short section of method (b), we here limit ourselves to an explanation of methods ( a) and (c), using the measurements of strain made in cross-tunnel Q₁₂₀.

(a) If one examines a prismatic ice lamella (Fig. 11. and 12.) in the region of the division of movement,Footnote * bounded on both sides by longitudinal cracks, then the horizontal principal stress σ ₃ = σ _y in the plane of the picture can be approximately put equal to zero, so that in this plane (y—z), the vertical pressure from above alone acts as principal stress σ ₁ = σ _z If one ignores the insufficiently well known, yet relatively small alterations in length perpendicular to the plane of the picture, this corresponds to the supposition that the so-called rest pressure is effective in the x-direction (neutral zone).

With these assumptions, the apparent viscosity of the ice can be calculated as follows (Fig. 12.):

The angular rate of change ω is thus identical with the rate of strain ε _y of the cross-tunnel. If we substitute the value of ε _y which was taken during the second period of measurement (see Part I, p. 578), while σ ₁ is estimated as the overburden pressure of 28 m. of ice and firn at an estimated mean density of ρ _i = 850 kg./m.³, then it follows that:

(b)The measurements in circular profile K₂ (~2.6 m. diameter) do not cover a sufficiently long period of time to provide reliable results. An initial measurement period of 90 days showed that the deformed circular profile resembled an ellipse, from which it follows—contrary to our expectations—that the state of stress in the plane considered is not hydrostatic. The orientation of the two axes of the ellipse indicates the direction of the standard principal stresses, while the ratio of the principal stresses can be deduced from the change in length of these axes. A rough application of the relations given under (c) showed in the main that the apparent viscosity of the ice may here reach 6×10Reference Haefeli and Bendel ¹⁴ poise.

For the sake of comparison, it should be mentioned that in the circular profile of the Z’Mutt tunnel, i.e. in temperate ice, the following values for the apparent viscosity were discovered (see Ref. 1, Part I, p. 580):

The dependence of the μ value upon the pressure from above, or the shear stress, is conditioned by the exponential character of the stress-strain curve (cf. Fig. 12.)Reference Nye ^{10 ,} Reference Steinmann ^{11 ,} Reference Glen ²⁴ .

(c) In the deformation of the circular profile K₁ (2.9 m. dia.) observed in the longitudinal tunnel, the horizontal measurement increased somewhat to begin with, but then shrank again approximately to its original value. From this, a certain time variation of horizontal side pressure is indicated (cf. Fig. 8., p. 579 of Part I).

The general calculation of deformation in a circular profile under the influence of a vertical pressure from above p ₁ and a horizontal pressure p ₂ gives the following radial alterations in length for an elastic materialReference Föppl ¹³ :

Vertical:

where G is the shear modulus.

Horizontal:

for p ₁=p ₂

For the horizontal measurement to remain constant (Δr ₂ = 0), p ₂ must be 25% OF p ₁ With this assumption it follows that the specific reduction of the vertical cross-section is:

To arrive at the viscous-fluid creep from the elastic deformation, we put:

If this value of G is introduced into equation (7), and the states of stress of the elastic and plastic deformations identified, then the specific rate of contraction v ₁of the vertical diameter is calculated as follows:

Measuring period 20/2/51 to 8/1/53; v ₁ = 0.0333% per day; p ₁ = 27 t./m.²; ΔT = 687 days.

This relatively low value of the apparent viscosity μ, which is not far from the approximate value obtained for the circular profile K₂ in (b) above, also lies within the range of the values for the Z’Mutt tunnel in temperate iceReference Haefeli ²⁰ . It should be noted that the latest laboratory experiments of S. Steinemann revealed no sudden alteration or increase of the ice plasticity on passing the point of pressure melting, i.e. on crossing from temperate to cold ice (oral communication). The recent measurements in the ice tunnel recorded up to the present time seem to confirm this point of view, in the sense of a continual increase of the apparent viscosity with lower temperatures as stated by Glen in the form of a “minimum creep rate” for polycrystalline iceReference Glen ²⁴ .

VI. Comparison of Greenland and Alpine Relationships

Without embarking at this stage on the conclusions which can be drawn from the above examination for alpine glaciers in both scientific and technical aspects, we will now limit ourselves to a short discussion of some theoretical relations concerning the movement of ice caps with plane deformation, with special reference to the Greenland ice sheet above the firn line (central Greenland).

If the length of the cap in relation to its breadth is very great, the glacial movement may in the first approximation be assumed to be an even laminar flow with some discontinuities (slipping movement): in this case, the apparent viscosity of the medium varies from point to point. For the division of movement (M) on a horizontal rock base we arrive, by reason of the equation of continuity, at the following mean value for the stress deviator Δσ=σ ₁−σ ₃ in the stationary state of the cap:

The definition of the individual symbols can be seen in Fig. 13. (p. 628). From this, it follows that the mean difference Δσ between horizontal and vertical principal stress, upon which depends, among other things, the danger of longitudinal cracks forming, is directly proportional to the annual accumulation a and inversely proportional to the height H of the ice cap.

Fig. 13.

Diagrammatic representation of the relations in the Greenland ice sheet. (Profile after Expeditions Polaires Françaises, Missions Paul-Emile Victor. Vertical exaggeration ×100).

For example, if one calculates for Greenland with an ice thickness of 3200 m., an accumulation a = 0.30 m. of ice per year and a mean apparent viscosity of the ice of 10Reference Robin ¹⁵ poise, there is a

mean deviator of about 60 kg./m.², which has no great practical significance. In the ice centre, therefore, there exists almost a hydrostatic state of stress.

On the other hand, it follows that a very small deviator Δσ suffices to make possible a stationary condition in the ice centre.

Whereas the horizontal shear stresses at the division of movement are τ _xz = 0, this is not the case in any general vertical section (P) through the cap because of the tilting of the surface. Here τ _xz is the more decisive factor for the progress of movement, and with a horizontal rock base and a small angle of slope α of the firn surface, the following equation is validReference Haefeli and Bendel ¹⁴ :

If one supposes the average firn accumulation a from M to P to be known, then from considering the balance it follows in terms of the quantities defined in Fig. 13. that:

This equation for the average cross-section velocity v _m in the stationary state of the cap is generally valid, i.e. it is wholly independent of the particular form of the velocity profile or the question of whether or not slipping on the base takes place (pure condition of continuity).

For the simple assumption of a parabolic velocity profile, which would correspond to a constant viscosity of the medium without slipping on the base (v _u = 0), the following relationships are valid:

Thanks to the Greenland expedition of Paul Victor we are to-day adequately informed about the contours of the rock base in Greenland (see Ref. 4, Pt. I, p. 580). The following numerical example is based on the profile contained in Rapport Préliminaire No. 15, Série Scientifique (Campagne 1950). For a hypothetically selected mean annual accumulation of a = 0.3 m of ice, the average values set out in Table II work out as follows (Equations 13 and 14):

Table II.

Ice Sheet Movement and Viscosity Relations (for slipping velocity on the rock bed v _u = 0)

With regard to magnitude, these apparent viscosities agree with the values reached for the cold ice cap on the Jungfraujoch. From this it follows that the movement in the firn area of inland ice can also be conceived as occurring without slipping on the base, i.e. only as a result of the plastic deformation (creep) of the cap.

We are fully aware that the comparison of the small ice cap on the Jungfraujoch with the enormous inland ice mass, whose greatest thickness is 60–70 times and breadth up to 3000 times larger, must be regarded at least as daring. But the following considerations may be borne in mind to assist such a comparison.

In the light of recent glaciological research, the plastic properties of polycrystalline glacier ice, or its apparent viscosity, seen from a crystallographical point of view, depends chiefly upon the following factors: on the one hand upon the significant shear stress and on the other upon the temperature of the ice. As far as the shear stresses are concerned, the difference between Greenland and the Jungfraujoch is of no consequence, for the greatest shear stress on the top of the ice tunnel, which, with no side pressure, corresponds to half the pressure from above, is of the same magnitude as the shear stress along the rock base of the inland ice (Table II).Footnote * On the other hand, the normal stresses, or hydrostatic pressures, differ widely; yet up to the present, no resulting influence of hydrostatic pressure upon the plastic behaviour of cold ice can, according to Steinmann, be confirmed.

With respect to ice temperature, we are in both cases concerned with cold ice, although the mean temperature of the ice on the Jungfraujoch is appreciably higher than in central Greenland. The relatively high value of viscosity for Point 3 of the inland ice (Table II) indicates a low mean ice temperature, together with small shear stresses.

Following the seismic examination of the ice sheet by the Expéditions Polaires Françaises (Missions Paul-Emile Victor) and the relatively high speed of seismic waves recorded thereby (ca. 3800 m. /sec.), J.-J. Holtzscherer also comes to the conclusion that the central inland ice is probably cold down to the base: that is to say, its temperature throughout lies below the point of pressure melting and frost penetrates down to the bottom (see Ref. 4, Part I, p. 580). Robin reached a similar conclusion, proving theoretically by a study of heat flow that with an annual accumulation of 30 cm. of ice, the temperature of the rock base in the central zone of the Greenland ice sheet lies well below the point of pressure melting, which under a thickness of 3000 m. of ice is about −2° C.Reference Robin ¹⁵ , while for a surface temperature of −29° C. the calculated bottom temperature according to Robin is about −12° C. From seismic shooting information Holtzscherer estimates the basal temperature near the centre of the ice sheet at around −10° C.

Quite independently of the above-mentioned seismic and thermodynamic examinations, the results gained from the ice cap on the Jungfraujoch offered the possibility of shedding light upon the Greenland problem from a third angle, namely on the basis of glacier movement. The attempted extrapolation of the small ice cap onto the Greenland ice sheet leads one to presume that in central Greenland there exists a core in which the ice does not slip on the base but is frozen hard to it, where the resultant shear stresses lie well below the shear strength. This would mean that the 0° isotherm in the core dips below the glacier bottom (Permafrost) corresponding to the above-mentioned hypotheses based on seismic and thermodynamic examination⁴, Reference Robin ¹⁵ .

The fact that three different methods lead to basically the same result should not allow one to overlook the provisional nature of the extrapolation of the results obtained from the Jungfraujoch. The deviation of the actual three-dimensional motion from the above, where a plane creep motion has been assumed, requires further clarification. The question of the influence of pressure from all sides upon the ability of the ice to deform is vital for the justification of such an extrapolation. In this connection, mention must be made of the latest experiments of S. Steinmann, where superposition on a shear stress of a hydrostatic pressure up to 90 kg./cm.² reveals practically no influence of this pressure upon the plastic behaviour of the cold ice.Footnote * On the other hand we must remember that in central Greenland the overburden pressure of the ice reaches 300 kg./cm.².

The observation and measurement of smaller and more easily accessible natural objects, such as for example the ice cap of the Jungfraujoch, offers the possibility, among other things, of acting as a link between arctic exploration on the one hand and research in the ice laboratory on the other, so as to further our knowledge of nature by direct observationReference Glen and Perutz ^{16 —} Reference Bader ²³ .