Comments on Dr. Loewe’s Letter and Notes on Crevasses

Dr. Loewe’s letter is a stimulus to improve our theories. The present theory of the depth of crevasses is so rudimentary that one should not expect more than a rough agreement with experience. Orowan and Ward put forward a simple theory ¹ which connected the maximum depth of crevasses, d, with the “yield stress” of ice. If the yield stress of ice in compression is taken to be 2 bars (approximately 2 kg./cm.²), d is calculated as 20 m. The calculation is successful in that it shows how plasticity theory can give a result which is in general quantitative agreement with observation. But the result cannot be regarded as anything more than a first approximation, because the calculation takes as its model not a crevasse in a flowing glacier, but a tall, slender and stationary column of ice.

The last paragraph of Dr. Loewe’s letter seems to spring from a misunderstanding. If Orowan’s tall, slender column model is extended so that only one horizontal dimension is infinite, the stresses and critical height do not change by more than 15 per cent. This follows from the well-known relation in plasticity between the yield stress in plane stress and in plane strain. Orowan’s second model (which he applied to an ice cap) refers to a case where both horizontal dimensions are large; it is therefore not relevant to the crevasse problem.

The following calculation on crevasse depths gives the possibility of a more detailed comparison between theory and experiment than has been possible hitherto. Let us first assume that transverse crevasses can be formed wherever a longitudinal tensile stress (greater than zero) exists in a glacier. This is equivalent to saying that there will always be a sufficient number of points where the tensile strength of the ice is effectively zero. The stress analysis of a perfectly plastic glacier ² shows that, in “extending flow,” a longitudinal tensile stress can exist down to a depth of approximately d = σ/pg, where σ is the yield stress in extension and p is the density of ice. Atmospheric pressure is neglected in this calculation. The longitudinal stress is σ on the surface of the glacier; it decreases with depth and reaches zero at approximately this critical depth d. The question then arises, what is the value of σ? Previously it was guessed that σ is about z bars. Now, a better estimate of a could be obtained by measuring the longitudinal rate of extension, ϵ̇, on the glacier surface and using this in conjunction with Glen’s creep formula for ice. ³ Glen’s formula was established for compression, but we assume that it also holds for extension. Thus we write

(1)

where B and n are constants; and so we have, for the maximum depth of the crevasses,

(2)

Let us now consider the effect of atmospheric pressure.* It is, to a good approximation, the maximum difference of principal stresses that determines the rate of strain. Therefore, an atmospheric pressure, p ₀, say, acting normally on the upper surface of the glacier, reduces by p ₀ the longitudinal tensile stress needed to produce a given ϵ̇. The longitudinal tensile stress at the surface is thus (σ − p ₀), where σ is still given by (1). The longitudinal stress now remains tensile down to a smaller depth than before, namely, (σ − p ₀)/ρg. (If σ = 2 bars and p ₀ = 1 bar, the thickness of the tensile layer is seen to be halved by the presence of atmospheric pressure.) A tensile crack would be expected to propagate to this depth, if, as before, we take the tensile strength of the ice to be zero. The crack will, of course, disturb the stress distribution in its neighbourhood as it propagates. If we neglect this change we have the following situation: a crack of depth (σ − p ₀)/ρg, with its tip in a region of zero tensile stress, and with atmospheric pressure acting within it. In these circumstances the crack will continue to propagate downwards until its tip reaches a depth where the longitudinal pressure in the ice just balances the internal pressure in the crack. This depth is σ/ρg, and is independent of p ₀. The conclusion is, that while atmospheric pressure alters the precise stresses under which the crack propagates, it does not alter the final depth that the crack reaches; for this depth is determined not by the condition of zero tensile stress but by the condition that the longitudinal stress in the glacier should be equal to atmospheric pressure. Formula (2) therefore holds independently of atmospheric pressure. Perhaps the weakest assumption made in its derivation is that the stress distribution below the level to which cracks penetrate is unaffected by the presence of the cracks. This assumption is legitimate when the crevasses are closely spaced but can easily be shown to be false when they are widely separated.

With ρ = 0.90 gm./cm.², g = 980 cm./sec.² and, as found by Glen for a temperature of 0° C., n = 3.3 and B = 0.10 years⁻¹ bars⁻ⁿ, we have, from (2),

where ϵ̇ is in years⁻¹ and d is in metres.

An experimental check of this formula would be as follows. A glacier showing transverse crevasses is selected (for example, the Mer de Glace). The surface velocity is measured at a number of points along a line perpendicular to the crevasses. The velocity measurements should extend over a sufficient time and distance for irregularities to be averaged out. Hence ϵ̇ is calculated. (For example, if the surface velocities of two points separated by a distance x are v ₁ and v ₂ then

.) At the same time d is found by measuring the maximum depth of the crevasses.

The relation between the above calculation based on a glacier model and Orowan’s calculation based on a tall column model may be seen as follows. Neglect the very small amount of shearing parallel to the bed that takes place in the upper layers of a glacier, and imagine a series of closely spaced crevasses of depth given by (z). The ice between them will tend to spread out under its own weight in the way that Orowan envisaged for an isolated column. But it must be remembered that this ice is not resting on a rigid base as in Orowan’s model, but is resting on the lower layers of the glacier, which are themselves extending longitudinally at the rate ϵ̇. Now the vertical compressive stress (ρgd + p ₀) producing the “Orowan spreading” of the material just above the depth d exceeds the horizontal compressive stress (p ₀) by ρgd. The rate of vertical compression so produced must equal the rate of longitudinal spreading (constancy of volume and plane strain), and we have seen that such a stress difference ρgd produces a strain-rate ϵ̇. Thus, the “Orowan spreading” at the bottoms of the crevasses is exactly sufficient to ensure continuity with the longitudinal extension rate of the underlying layers of the glacier. Higher up the crevasse walls the Orowan spreading is smaller, by reason of the smaller depth, and so the ice here does not extend longitudinally as fast as the underlying ice of the glacier. This is why the crevasses open up. The argument is strictly correct for very closely spaced crevasses. For a wider spacing, further analysis is necessary.