Cambridge Austerdalsbre Expedition, Paper No. 5
1. Introduction
It is commonly observed that when a glacier passes down an ice fall there are at the bottom a series of undulations curving transversely across its surface. These wave ogives, as they are sometimes called, were first noticed in 1843 by J. D. ForbesReference Forbes 1 on the upper part of the Mer de Glace, at the foot of the ice fall of the Glacier du Géant. Further downstream the undulations give place to a parallel series of dark bands which are often very conspicuous; they are convex downstream, like the undulations, and give the observer a vivid impression of the differential flow in the ice. The dark bands were seen by Forbes on the Mer de Glace and elsewhere in 1842 and are usually known as Forbes bands, although the nomenclature of the subject is in some confusion. The causes of both sets of features and the nature of the connexion between them have been much debated. Leighton has given a reviewReference Leighton 2 of the observational evidence and of the theories put forward to account for it, and some later work is described in Reference Lliboutryreferences 3, Reference Ives and King4 and Reference King and Ives4a.
The purpose of the Cambridge Austerdalsbre Expedition was to learn as much as possible, by quantitative study, about the motion of a glacier at the foot of an ice fall. The immediate aims were to understand the formation of wave ogives and Forbes bands, and it was also hoped that the investigation would throw light on the problem of erosion at the foot of an ice fall. It had been proposed, originally by Forbes,Reference Forbes 1 and later by Streiff-BeckerReference Streiff-Becker 5 and by Haefeli,Reference Haefeli 6 and it was widely believed, that wave ogives were essentially pressure waves, caused by the pressure of the ice in the ice fall acting on the ice at the bottom. We were therefore particularly concerned to test this theory by measuring the plastic distortion of the ice which it implies.
Austerdalsbreen is one of the distributary glaciers from Jostedalsbreen, the ice cap in Western Norway (Fig. 1). Two separate ice falls, Odinsbreen and Thorsbreen, drop some 700 m. from the ice cap to a low valley where they join together to form the single glacier known as Austerdalsbreen. This glacier, which is 4 km. long and 0.7 to 1.0 km. wide, exhibits both wave ogives and Forbes bands very well. A good description of the glacier, and probably the earliest, was given by de SeueReference Seue 16 in 1870 in a book which contains many measurements of glacier movement and which deserves to be better known; SlingsbyReference Slingsby 17 explored the glacier in 1894 and EversReference Evers 18 gives an account of its appearance in 1934. The velocity of the lower part of the glacier was measured photogrammetrically in 1937 by Finsterwalder, Evers and Pillewizer.Reference Ig. Pillewizer 19 , Reference Finsterwalder 20
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Fig. 1. Map of the upper part of Austerdalsbreen, showing the wave ogives from the foot of the Odinsbre ice fall and the stake pattern of August 1956. The arrows show the velocity at the marked points (August 1956). The velocity is plotted to the same scale as the map. The inset shows the position of Austerdalshreen on a map of southern Norway
The present paper describes one aspect of the work of the Cambridge expedition, but it may be useful to take the opportunity of summarizing the whole venture. In 1954 a party of the Brathay Exploration Group led by A. B. Ware set up permanent painted reference marks and mapped the main features of the glacier. The Cambridge expedition started work in 1955 led by W. V. Lewis. C. Bull and J. R. Hardy carried out a gravity survey,Reference Bull and Hardy 7 and from it calculated the thickness of the ice—this is believed to be the first recorded use of the method on a valley glacier. The main effort in man-power, however, in 1955 was devoted to the excavation of a tunnel in the ice at the foot of the Odinsbre ice fall, by parties of Cambridge undergraduates and others, led by G. de Boer. The subsequent deformation of the tunnel was measured and interpreted by J. W. Glen.Reference Glen 8 Further survey work and some preliminary velocity measurements were done under the direction of J. E. Jackson, and the writer mapped the wave ogives. There were also parties of the Brathay Exploration Group and from Cambridge at work above the ice fall and below the snout.
In 1956 there were two main projects: (a) the insertion of a pipe, by W. H. Ward, to measure glacier flow down to the bed of the glacier; (b) a detailed study by myself of the motion and deformation of the wave ogives, which is the subject of the present paper. This last was coupled with mapping of the Forbes bands by parties from the Brathay Exploration Group and from the Perse School, Cambridge. Dr. Glen also measured the rate of slip of the ice past the rock walls.Reference Glen 9 In January 1957 Mr. Ward led a small winter party to the glacier. They were not successful in finding the pipe, which was, and still is, buried under avalanche debris, but they did succeed in resurveying some of the markers left in the waves during the previous summer.
We continued observations on the wave ogives in the summer of 1957; the object was to survey the progress of the waves and the markers, and to test the working hypothesis which had been set up to explain the pattern of deformation measured in 1956. There were two visits: the first in early July by a party of the expedition led by Mr. Ward, and the second in August by a party of the Brathay Exploration Group led by M. F. Robins, who resurveyed the stakes we had left in July.
In 1958 a party again led by Mr. Ward visited the glacier at the beginning of July and resurveyed the wave ogives. We intend to continue observations of the year-to-year movements in the area of the waves, but the work we set out to do at the foot of the ice fall is in other respects now completed. In 1958 the main attention of the expedition was transferred to the middle section of the glacier and to the snout, and these new projects will be reported elsewhere.
This paper contains the results from the study of the wave ogives made in 1956 and 1957, namely, the geometrical features of the waves, their size and shape, and the distribution pattern of velocity, rate of deformation and rate of ablation in the wave area. Other observations on the glacier made in 1956 and 1957, including detailed analysis of the annual movements and observations on the Forbes bands, will be published separately. The leading results of the 1956 work, but not those of 1957, have already been briefly reported. 10
The work on Austerdalsbreen in 1956 prompted G. R. Elliston, who took part in it, to undertake an identical series of measurements in 1957 on Svínafellsjökull with the Cambridge University South-East Iceland Expedition (leader: R. E. Lawrence). The results (private communication) are in many respects strikingly similar to those found on Austerdalsbreen.
The outcome of the investigation of the wave ogives on Austerdalsbreen was unexpected. We set out to find plastic distortions which should represent the process of wave formation by pressure, since the pressure theory seemed at that time to be the most plausible explanation of the waves. Instead we found that the pattern of deformation occurring over the wave area had no obvious connexion with the waves themselves. It can, in fact, be explained as an effect which is independent of the presence of the waves—being due, as it appears, to the distortion which must necessarily take place in the ice as it meets slight changes in the gradient and curvature of its bed. The theory of this effect, which is discussed in Section 5, is of general application to any valley glacier and gives essentially a correction term to the existing flow equations—a correction which becomes important in regions of rapidly changing bed curvature such as we are concerned with here.
At this point we were faced with the following situation: the plastic deformation in the wave system had been measured and appeared to show that the waves were not forming in that area by pressure in August 1956 or between July and August 1957. The present paper leaves the argument there. For completeness, however, we may mention the next development, which is given in full in Reference Nyereference 11.
In view of the Iack of connexion between the measured deformations in the wave area and the waves themselves, it must be concluded that, if a pressure mechanism of wave production does operate on this glacier, it must do so at some other season of the year—or higher up the ice fall—for we could not rule out these possibilities directly from our observations. At the same time, the evidence that a pressure mechanism was not operating over the wave area in summer led to a search for some other process which might be producing the waves. Such a process was found and is very simple. All elements of ice are stretched out longitudinally as they pass down the ice fall, owing to the high local velocity, and they therefore present greater surface area. Those passing through the ice fall in the summer therefore lose more ice by ablation than those which spend the summer in regions of lower velocity. Waves are thus produced by a combination of ablation and plastic deformation—by the increased volume of ablation made possible by the plastic stretching in the ice fall. When the process is examined analytically,Reference Nye 11 it is found that the theory predicts correctly both the positions and amplitudes of the observed waves on Austerdalsbreen. That is not necessarily to say that a pressure mechanism does not operate on other glaciers. It does show, however, that the effect on Austerdalsbreen is fully accounted for, within the uncertainty of present data, without invoking the additional hypothesis of a pressure mechanism. The formation of waves by ablation in this way is something that must occur, it would seem, in any ice fall below the firn line, and there may be a related process in an ice fall above the firn line, although this case is more difficult to analyse.
2. The Stake System and the Measurements
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(i) The Stake System. Fig. 1 shows the Odinsbre and Thorsbre ice falls. Wave ogives emanate from the bottom of both ice falls, but the situation on the Thorsbre side of the glacier is complicated by the presence of the avalanche fan shown in the figure. We therefore chose the comparatively straight Odinsbre ice fall and its associated waves for our study. Some seven or eight crests and troughs can be distinguished before the waves die away; the positions of the troughs on 3 July 1957 are shown in the figure. The maximum amplitude (one half of the total swing) is 6 m. and the wave length decreases from 240 m. to 75 m. as one goes down the glacier.Footnote * To a good approximation the waves are spaced at annual distances apart (the spacing may differ from an annual movement by 5 or 10 per cent, but discussion of this possible difference is being reserved for the time being).
To measure the plastic distortion occurring in the wave system, we set up, at the end of July 1956, the pattern of stakes shown in Figs. 1 and 2, starting in the lower part of the ice fall and extending through the first three waves, a total distance of 623 m. Figs. 13, 14 and 15 (p. 386) give some idea of the terrain covered. The pattern consisted of five key stakes lettered A, B, C, D, E, which were to be theodolite stations, and a series of intermediate numbered stakes arranged to form a single line which changed direction at A, B, C and D. This will be referred to as the “main line”. The intervals between successive stakes were about 30 m. Near the upper end the main line was arranged to pass accurately (±1 m.) through the position occupied in 1955 by the mouth of the tunnel. The course of the line was dictated to some extent by the need to avoid a very heavily crevassed area on its western side. In addition to the main line it was desired to have four stakes at the corners of a square centred on each of the key stakes, and the other stakes shown in Fig. 2 were inserted for this purpose (for practical reasons the topmost square was centred on Stake 3 rather than Stake A). There was a further single stake, lettered CO, at the point where the waves died out.
We used square stakes in round holes—the stakes, which were of 2.5 cm. (1 inch) square section, 3.05 m. (l0 feet) long and made of hardwood, being set in the holes made by 3.2 cm. (
inch) diameter drills. This technique, developed by Ward,Reference Ward 12 ensured that the stakes were firmly gripped by the ice and did not float. -
(ii) Theodolite Resections. Fixed marks on the rock walls were surveyed by a combination of theodolite triangulation and resection. The positions of Stakes A, B, C, D, E, horizontally and vertically, were measured by theodolite resection to the marks on three occasions: between 28 and 30 July, between 11 and 14 August, and between 21 and 24 August.
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(iii) Taping. Each of the stake intervals shown in Fig. 2 by a full line was measured by steel tape on four occasions: 30 July, 12, 21 and 27 August (except for five of the transverse intervals which were measured on three occasions only). Normally the measurement was made between corresponding points of the tops of the two stakes (e.g. the top N.E. corners of both stakes), with the tape freely suspended between the measuring points at a tension judged by the observer to be constant; sometimes, however, the ice surface was such that the tape had to touch it. If the top of a stake could not be reached easily, the measurement was made to a mark 30.5 cm. (1 foot) or 61 cm. (2 feet) from the top. Each measurement was normally made three times.
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(iv) Levelling. The difference of height between the tops of each successive pair of stakes was measured by level and staff on three occasions: between 31 July and 4 August, between 13 and 17 August, and between 22 and 23 August.
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(v) Tilt of Stakes. The stakes were bored in as near the vertical as possible. The angle to the vertical and the direction of tilt were measured on 30 July, 12 August and 21 August by using a simple plumb-line and a foot rule.
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(vi) Ablation and Reboring. The height of the top of each stake above the ice surface was measured three or more times over the period 24 July to 27 August. The melting of the ice made it necessary to rebore the holes of all the stakes at least once during this period, and some were rebored twice. The height of the top of the stake was measured before and after each reboring.
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(vii) Offset Measurements. The extent to which the stakes of any one leg of the system were out of line was measured three times over the whole period; the measurements are available but have not been used.
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Fig. 2. Plan of the stake pattern of August 1956. The points where the line crossed the crests (Cr) and the troughs (Tr) of the wave ogives are shown
3. Results
The next step is to find out from the above measurements as much as possible about the absolute and relative motion of the ice at each stake. For the sake of precision the calculation of movement at each stake has been made to refer to that element of ice which was situated at the bottom end of the stake when it was first inserted. We shall call this element of ice the “reference element”.
(i) Longitudinal and Transverse Strain-rates. The fact that the stakes were not exactly vertical and that their holes had to be rebored means that the distances measured between the tops of the stakes are not quite the same as the distances between the reference elements. The necessary corrections, which could all be calculated with the data available, involve the following: the tilts of the stakes, the amount of the reboring, the angle between the line of measurement and the horizontal, and an allowance for the occasions when the measurement could not be made to the top of the stake. The corrections were typically about 0.15 m. (the maximum correction of 0.48 m. arose in a case where the holes of the two stakes had been rebored very unequally, and where the line of measurement ran at 26 degrees to the horizontal). The distances between reference elements having thus been found, the average rate of change of each distance was calculated over the period, and hence the strain-rate
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where Δl is the increase in the distance between the two stakes in the time interval Δt, and
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where l 1 and l 2 are the initial and final distances.
For the stake intervals on the main line all observations were given equal weight in the calculations, except for a few rejected in the field. The measurements on the stake intervals transverse to the main line were slightly more erratic and were therefore treated graphically; it was noticed that when a down-glacier wind had been recorded, which blew the tape sideways, the readings were abnormally high.
Let x denote the distance down the glacier. For definiteness, x is measured along the line defined by the reference elements on 12 August, with the origin taken at the top of the ice fall, to be consistent with the notation of Reference Nyereference 11. Let z denote the transverse coordinate, Oz being horizontal and perpendicular to Ox. Oy is perpendicular to Ox and Oz and hence is roughly normal to the glacier surface. The strains measured along the main line then give the longitudinal component of strain-rate
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Fig. 3. The longitudinal strain-rate,
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The strains measured on the stake intervals perpendicular to the main line (3—AE, 3—AW, B—BE, B—BW, etc.) give the transverse component of strain-rate
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Fig. 4. The transverse strain-rate
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The strains measured on the stake intervals at 45 degrees to the main line, shown in Fig. 2, enable the shear strain-rate component
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(ii) Vertical Velocities. The theodolite resections of Stakes A, B, C, D, E, CO gave the vertical component of the movement of these stakes. The relative vertical velocity of each pair of neighbouring stakes was also known from the levelling surveys. The absolute vertical velocity V of each stake of the main line could therefore be calculated; the values are shown in Table I and are plotted in Fig. 5.
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Fig. 5. The vertical velocity component, V, and the velocity component normal to the surface, v, as functions of x
The errors in the theodolite observations appeared greater than those from the levelling. Consequently the figures quoted for V agree exactly with the relative velocities obtained by levelling, and the zero for velocity is fixed by using an averaging process on all the theodolite measurements. The average adjustment necessary to the velocity V measured on any one stake by theodolite was 4.3 m./yr., and it is therefore thought that V has been measured to about ±2 m./yr. The relative values of V for neighbouring stakes, on the other hand, are thought to be correct, in most cases but not in all, to ±0.5 m./yr.
Table 1 Velocities of the Main Line of Stakes
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We may mention that the relative vertical velocity of successive stakes changed slightly over the period of measurement in a fairly consistent way: in 13 cases out of 17 the changes were in the directions that would be expected if the stakes were moving through a fixed velocity field.
(iii) Long Profile. Fig. 6a shows the longitudinal profile of the glacier from the approximate top of the ice fall O to CO. In the part from Stake 1 to CO, which was measured by levelling, the waves are clearly seen and the positions of eight crests are indicated. The section covered by the main line of stakes is also shown separately in Fig. 6b. Crests are seen near C, near D and near E and a less prominent crest appears just below A.
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Fig. 6a Profile (August 1956) down the line 0-CO shown in Fig. 1. The bed is calculakd from the surface velocities and the measured depth at the pipe (1956). The position of the tunnel on 27 August 1955 is also shown. The Zero on the height scale is chosen arbitrarily
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Fig. 6b The part of the profile of Fig. 6a containing the main line of stakes, 1 to 18, on a larger scale
Table I gives the values of θ and θ s , calculated from the levelling measurements. θ for each stake interval is the angle between the horizontal and the line joining the reference elements on 13–17 August, and is positive when the slope is downhill in the downstream direction. θ s , is the same as θ except that it refers to the ice surface at each stake rather than to the reference element. θ s , is plotted against x in Fig. 7 (p. 399).
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Fig. 7. The surface slope θ s , the angle of dip θ* of the streamline at the surface, and the “average” surface slope
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The exact positions of the crests and troughs can be defined in various ways. The positions shown in the figures are, in most cases, those which show the greatest departure from an estimated average surface. A less subjective criterion, not entirely free from objection, is to take the crests and troughs as the points of inflexion of the curve of θ s : x—that is, the places of greatest curvature of the surface. It is then clear from the θ s curve that the crest near Stake C seen on the profile has a fine structure, and is made up of two crests with a very small intervening trough. This fine structure is indicated in Figs. 7 and 9, where it seems relevant, but is omitted in the other figures.
(iv) Horizontal Velocities. The theodolite resections of Stakes A, B, C, D, E, CO also gave the horizontal components of the movements of these stakes (Fig. 1). The results from the taping of the main line do not, however, give directly the relative horizontal movements of all the stakes because (a) the taped distances are not horizontal, and (b) the movements, as Fig. 1 shows, are not exactly along the line of the stake system. We have adopted the following procedure. Each leg of the main line defines a vertical plane p. Let U denote the horizontal component of the velocity of a reference element projected on to this vertical plane. Then by combining the results from the taping and the levelling, as indicated in Fig. 8, we have calculated the relative values of U for the stakes along each leg of the main line. These relative values were used to check the theodolite observations. It again appeared that most of the error was in the theodolite observations, and so a correction to the absolute velocities was made on that basis. The values of U in Table I agree exactly with the relative values obtained from the taping and levelling. The average adjustment necessary to the value of U measured on any one stake by theodolite was 20 m./yr., and it is therefore thought that U has been measured to an accuracy of about ±10 m./yr. The relative values of U for neighbouring stakes, on the other hand, are probably correct to ±0.5 m./yr. or better.
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Fig. 8. For a given stake interval the diagram shows the velocity component OA, in the plane p, of one stake relative to the other. The vector OB is calculated from measurements by tape and the vector OC from measurements by level. Hence the relative horizontal and vertical components OD and OC are known
Table I also shows the horizontal velocity (not projected on to the plane p), the angle β which this velocity makes with the plane p, and its grid bearing (the local grid north used was oriented roughly to geographic north). It will be noticed that where the line changes direction there are two values of U and β, one corresponding to each of the two possible planes p.
The values of θ* shown in Table I are defined by
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θ* thus gives the downward inclination, in the plane p, of the velocity vector at each reference element. It is double-valued where there are two possible planes p. θ* is plotted against x in Fig. 7.
(v) Ablation. The mean rate of ablation over the period 30 July to 20 August is plotted against x in Fig. 9. Some results for other periods, notably 30 July to 7 August (faster ablation) and 7 August to 20 August (slower ablation) are also available; they are not shown but have been used to help in drawing the curve, maxima and minima only being inserted if they appear in at least two different periods of observation. It will be seen that there is a close correlation with the positions of the crests and troughs, the ablation rate at the crests being some 30 per cent higher than the rate at the troughs. One also notices that the fine structure in the crest near Stake C, small though it is, is faithfully reflected in the ablation curve.
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Fig. 9. The average rate of ablation between 30 July and 20 August 1956 as a function of x
4. Examination of the Pressure Wave Hypothesis
Fig. 3 shows that there is a longitudinal compression of between 0 and 1.5 yr.−1 over practically the entire line of stakes, and that it oscillates considerably from point to point. It is perhaps especially remarkable that
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The oscillations in
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But there is another test of the hypothesis that the waves are forming by plastic deformation. If this were so, the crests would be rising and the troughs falling relative to their surroundings. Some test of this is given by the curve V: x in Fig. 5, which came from the levelling results and is thus independent of the curve of
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There is another way of examining the rate of rise and fall of the waves. Since the surface is inclined to the horizontal at angles up to 30 degrees one might think of trying a correlation with the velocity component, v, say, perpendicular to the average surface, rather than with the component V, which is vertical. There is, however, a serious difficulty—namely, to define the direction of the “average surface” without prejudicing the issue. The difficulty arises not merely because of the relief of the surface, but because of the very high value of the forward velocity of the ice, which is about 10 times the value of v. This means that the value of v obtained depends considerably on one’s judgement of what is properly to be taken as the slope of the average surface. It must be emphasized, therefore, that conclusions about rise and fall relative to an average surface are reached with this important reservation.
By taking the curve shown in Fig. 7 as the slope
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We have now arrived at the following position. Three correlations were expected if the waves were forming by longitudinal pressure: (1) between the longitudinal strain-rate
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Nevertheless, the observed distribution of velocities, both vertical and horizontal, and especially the very considerable oscillations in
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5. Theory of the Strain-Rate Distribution
In previous papers the writer has formulated a theory of glacier flow which leads to the following equation for
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn38.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-15390-mediumThumb-S0022143000017081_eqn39.jpg?pub-status=live)
ϕ′ is the rate of accumulation (negative if ablation), h the thickness (shortest distance from bed to surface), u the forward velocity of the ice averaged over the thickness, κ the curvature of the bed (positive when convex), and α the slope of the surface.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn40.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn41.gif?pub-status=live)
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Fig. 10 shows a longitudinal section through a parallel-sided glacier of thickness h. Suppose the bed at P has curvature κ and at Q, a distance dx from P, the curvature has changed to κ+dκ. The shaded element of glacier at P, if it is to remain conformable to the bed, must therefore bend about a horizontal axis as it moves from P to Q. For a first approximation we shall assume simple bending, but we do not know a priori which will be the neutral plane. If the origin is taken on the neutral plane the strain increment of the fibre y = constant will be
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn43.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-40305-mediumThumb-S0022143000017081_fig44g.jpg?pub-status=live)
Fig. 10 Illustrating the bending of a glacier as the curvature of the bed changes
Let the forward velocity of the glacier be u, so that dx = u dt, and divide both sides of (3) by dt. Then
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn45.gif?pub-status=live)
If the neutral plane is half-way through the thickness, the strain-rate at the upper surface
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn46.gif?pub-status=live)
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We note, first, that this bending strain-rate is proportional not to the curvature of the bed but to its rate of change dκ/dx, and, second, that it changes sign through the thickness of the glacier. In both these respects it differs from the strain-rate given by equation (2), which is proportional to κ and is uniform with depth. The bending strain does not appear in the theory which leads to equation (2) because it is assumed there that |dκ/dx| is much less than |κ/h|, and therefore that the bending strain-rate is negligibly small compared with the strain-rate given by the term uκ cot α. A rough calculation shows that at the foot of the Odinsbre ice fall this is not the case, for |dκ/dx| and |κ/h| are of comparable magnitude.
In the absence of a comprehensive theory it seems plausible to take the following view. Equation (2) cannot take account of rapid fluctuations in κ, and is therefore only to be applied to deduce average values of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn48.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn49.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-06675-mediumThumb-S0022143000017081_eqn50.jpg?pub-status=live)
In a full theory there would presumably be further terms involving d 2 κ/dx 2 and higher derivatives. These terms will become important when dκ/dx is so large that it is no longer permissible to assume simple bending—just as in the theory of elasticity simple bending theory is not accurate for a beam whose length is of the same order as its thickness. In fact, in the present case κ/h is about equal in order of magnitude to dκ/dx, as already noted, both being ~10−5 m.−2—that is, the change in curvature in a distance equal to the thickness is of the same order as the curvature. By using the term (5), then, we are, in effect, applying simple bending theory to a beam whose length is of the same order of magnitude as its thickness. We can therefore only expect a rough agreement with the theory.
A final term must be added to equation (6) to take account of the transverse strain-rate in the glacier. Suppose that κ, dκ/dx and ϕ′ were all zero, and that the glacier valley, instead of being parallel-sided, widened out so as to produce a transverse strain-rate
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn51.gif?pub-status=live)
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![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn54.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn55.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn56.gif?pub-status=live)
Since in the present application
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn57.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn58.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-13531-mediumThumb-S0022143000017081_eqn59.jpg?pub-status=live)
(Dr. GIen has pointed out to me that the longitudinal bending effect has a transverse analogue, whereby fluctuations in transverse curvature produce fluctuations in transverse strain-rate. This would mean that the curves in Fig. 4 have fluctuations which have been missed, and that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn60.gif?pub-status=live)
6. Application of the Theory
(i) The Bending Term in Equation (7). Let us first consider how the values of dκ/dx may be calculated from the observations. κ in Fig. 10 was introduced as the curvature of the bed. But, since 1/κ, the radius of curvature, is about twenty times the thickness of the glacier, κ may equally well be interpreted, to this approximation, as the curvature of the upper surface, or, what is the same thing in Fig. 10, the curvature of the streamline traced out by a point on the surface of the glacier. The last interpretation is the best for our purpose, for suppose there are undulations superimposed on the surface, carried along by it and having no relation to the bed. It would then clearly be wrong to interpret κ as the curvature of the surface, but still right to interpret it as the curvature of the streamline near the surface.
A further word should be said about the streamlines in the model of Fig. 10. The strain-rates
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn61.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn62.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn63.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn64.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn65.gif?pub-status=live)
When there is an overall longitudinal compression superimposed on this bending effect, as in the present case, the streamlines at the upper and lower surfaces will, of course, diverge from one another. One would expect, however, that the values of dκ/dx would be much the same for the two streamlines. In view of these arguments we henceforth interpret κ in equation (7) as the curvature of the streamline at the upper surface.
Since we have not observed that the flow is steady with time, we must define more precisely what is to be meant by the “streamlines” in this context. It is reasonable to define them as the paths which the particles are following at the instant of observation: thus, if a time-exposure photograph of short duration were taken, the paths of the particles would appear as short lines; the family of curves which are everywhere tangential to these short lines are the “streamlines”.
With this interpretation we can use the values of θ* in Table I to calculate dκ/dx. If all the reference elements lay on one streamline, κ would simply be dθ*/dx. In fact they do not, since θ and θ* are different, and a correction is necessary. Fig. 11 shows two neighbouring reference elements A and C and the streamlines through them. Let θ 1*, θ 2*, θ 3* be the angles of dip of the streamlines at A, B, C respectively, where B is on the streamline through A and close to C. θ 1*, θ 3* are known and we wish to calculate θ 2*. One finds that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-24098-mediumThumb-S0022143000017081_eqn66.jpg?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-12179-mediumThumb-S0022143000017081_fig67g.jpg?pub-status=live)
Fig. 11 Illustrating the correction in θ* to take account of the “splay” of the streamlines
to the first order in
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn68.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn69.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-19421-mediumThumb-S0022143000017081_eqn70.jpg?pub-status=live)
where κ =(θ 2*−θ 1*)/l, κ o=(θ 3*−θ 1*)/l,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn71.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn72.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn73.gif?pub-status=live)
by using the graphs of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn74.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn75.gif?pub-status=live)
We have next calculated h, the shortest distance between the bed and the surface of the ice. The pipe inserted by Ward 10 was close to Stake 3 and is presumed to have reached bedrock at a depth of 39.3 m. on 28 August 1956. Allowing for ablation this corresponds to h = 40.1 m. on 12 August. Starting with this value at Stake 3 we have then calculated the thickness of ice at the other stakes as follows. The increase of thickness Δh in a distance Δx is made up of two terms, one due to the divergence of the streamlines and one due to the difference between θ s and θ*. Thus
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-08005-mediumThumb-S0022143000017081_eqn76.jpg?pub-status=live)
where m is the mean value of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn77.gif?pub-status=live)
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![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn79.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn80.gif?pub-status=live)
Knowing h, and dκ/dx, and taking √(U 2 + V 2) as a sufficient approximation for u, we then calculate the term
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn81.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-13230-mediumThumb-S0022143000017081_fig83g.jpg?pub-status=live)
Fig. 12. The oscillatory part of
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![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-05702-mediumThumb-S0022143000017081_fig84g.jpg?pub-status=live)
Fig. 13. View of the stake system from Stake 1. The bottoms of stakes visible in the original are marked with dots. The top of the pipe is seen to the left of Stake 2 and in line with Stake B. To the left is the avalanche fan from Thorsbreen. Note crevasses to the right of the stake line, and figures near Stake BW and to the left of Stake C.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-69657-mediumThumb-S0022143000017081_fig85g.jpg?pub-status=live)
Fig. 14. Looking up the Odinsbre ice fall and showing the upper part of the stake system. Note pipe rig to right of Stake 3.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20170923152558-35971-mediumThumb-S0022143000017081_fig86g.jpg?pub-status=live)
Fig. 15. View up-glacier from Stake E. Odinsbreen to left, Thorsbreen to right. Note the wave crest through Stake D.
If our reasoning is correct, the oscillations shown by the observed curve of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn87.gif?pub-status=live)
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In summary of this section let us first emphasize that the theoretical points come primarily, but not of course entirely, from the levelling observations. The similarity of the two sets of points in Fig. 12 thus means, roughly, that we have established an approximate connexion between (1) the levelling observations, and (2) the measurements by tape which gave
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn92.gif?pub-status=live)
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(ii) The Slowly Varying Terms in Equation (7). It remains to examine the other three, slowly varying, terms in equation (7). Taking ϕ′ as −6.5 m. of ice per year, which is the annual ablation measured from August 1956 to August 1957, and h from 39 to 88 m., we find values of ϕ′/h from −0.17 to −0.07 yr. −1. For the second term we note that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn94.gif?pub-status=live)
For the third term the mean κ, as already noted, is −0.63 × 10−3 m.−1 u runs from 409 m./yr. (with
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn95.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn96.gif?pub-status=live)
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The values of the first three terms in (7) are thus respectively −0.17 −0.7 −0.40 making a total of −1.27 yr.−1 at the upper end, and −0.07 −0.12 −0.29 making a total of −0.48 yr.−1 at the lower end. These totals are to be compared with the slowly varying part of the observed
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn99.gif?pub-status=live)
(iii) Re-examination of the v: x curve. We should add a remark about the curve of v: x (Fig. 5) in the light of the above considerations. Let us first note that the experimental data used to calculate v have now been included in the calculation of θ*, and therefore that the v curve can give no essentially new information.
We have seen that v depends to a large extent on what is regarded as the slope of the average surface. If the above theory is right we must now conclude that the variations shown by the curve are largely spurious, being caused by the choice of surface. In fact we can now see that it would have been more logical to take for
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn100.gif?pub-status=live)
Assuming our theory to be correct, let us now see how a velocity component similar to v might be calculated and how it would behave. Since we know the form of the bed we could calculate at each reference element the velocity component normal to the bed. The result would simply be a curve of mh: x. m, it may be recalled, is the mean value of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn101.gif?pub-status=live)
(iv) Conclusion. We conclude that the observed longitudinal strain-rate is predominantly due to two terms: a relatively steady term of magnitude about −0.35 yr.−1 arising from the concave nature of the bed, and an oscillatory term of magnitude about ±0.2 yr.−1 caused by the bending and unbending of the glacier in response to changes in the curvature of the bed. In addition there is a relatively steady term due to ablation of about −0.1 yr.−1, and a slowly varying term due to the changing width of the valley running from −0.7 to +0.04 to −0.12 yr.−1.
Looking at Fig. 6b, and comparing it with the curves of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn102.gif?pub-status=live)
The outstanding fact about this explanation of the observations is that it finds the reason for the oscillations of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn103.gif?pub-status=live)
7. Prediction to Verify the Theory
If the glacier bed is the primary cause of the oscillations of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn104.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn105.gif?pub-status=live)
The stakes of the 1956 main line were by that time 120 to 380 m. further down the glacier. We reconstructed the line in the position it had occupied the previous year, omitting the transverse stakes, and on 9 July we measured the distances between successive stakes by steel tape. The distances were remeasured on 18 August by a party of the Brathay Exploration Group under M. F. Robins. In this experiment, unlike that of the previous year, we avoided reboring the stake holes by inserting the stakes on 4–7 July to a depth of 5.2 m., their tops therefore being 2.1 m. below the surface. When the stakes were remeasured their tops were from 1.0 to 2.9 m. above the surface (and four of them had been lost by melting out). We also omitted the measurements of tilt made in 1956, in order to simplify the work, and being content with a somewhat lower accuracy. The mean date between the two surveys was 29 July. We took care therefore to put the stakes into the ice on 4–7 July 1957 in such a way that on 29 July 1957 the flow would have brought them into the positions occupied by the 1956 stakes on 12 August 1956 (the reference date for the 1956 measurements). This was done by theodolite resections, and a graphical system of trial and error, to an estimated accuracy of 3 m. Thus the values of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn106.gif?pub-status=live)
If the surface in 1957 had had a very different form from that of 1956 the experiment could be taken as direct confirmation that the bed rather than the surface topography is linked with the strain distribution. In fact, the surface was rather similar in the two years and so the experiment does not provide very strong evidence on this point. Nevertheless, the experiment verifies very well the prediction of the theory that the same oscillations would be found in 1957 as in 1956.
8. Explanation of the Motion of the 1955 Tunnel
A further test of the foregoing theory is given by the behaviour of the horizontal tunnel excavated in 1955.Reference Glen 8 The 1956 stake line was arranged to pass over the position previously occupied by the tunnel (Figs. 2, 6a, 6b) and a direct comparison between the two years is therefore possible. The horizontal component of velocity at the tunnel mouth in 1955 was measured as 320 m./yr. (mean date 18 August); this figure is of doubtful accuracy and does not differ significantly from the value at the same position (interpolated) in 1956, which was 327 m./yr. (mean date 12 August). The value of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn107.gif?pub-status=live)
GlenReference Glen 8 found in 1955 that the tunnel was (1) being carried down glacier, (2) being rotated, and (3) being bent. It is interesting to see to what extent this can be explained on the basis of the 1956 observations. As regards (1) it is sufficient to note the agreement of the velocity in the two years which has just been mentioned.
Four effects will contribute to the rotation of the tunnel. First, the change of slope, as mentioned by Glen. This gives a contribution to the angular velocity ω of −uκ, which at Stake A is 363 × 3.13 × 10−3= 1.13 radians/yr. (We shall take ω as positive when the end of the tunnel falls relative to the entrance.) Second, there is a positive rotation due to the longitudinal compression, since the tunnel, being horizontal, makes an angle θ s with the surface. This gives a contribution to ω of
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170922070254370-0344:S0022143000017081:S0022143000017081_eqn108.gif?pub-status=live)
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In a similar way we may recognize several effects which will contribute to the bending of the tunnel. First, the effect of the change of curvature of the glacier bed, which, on a simple bending theory, gives for the rate of change of curvature κ′ of the tunnel dκ′/dt = u(cos θ s + sin 2θ s sin θ s )dκ/dt is changing very fast in this region, being −0.71 × 10−5 m.−2 at Stake A and +6.73 × 10−5 m.−2 at Stake 2. Using the mean of the two values gives dκ′/dt = 0·014 m.−1 yr.−1, that is, the tunnel tends to become convex upwards from this cause. Second, the change of
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This explanation of the bending differs from that advanced by Glen,Reference Glen 8 who interpreted it as showing that the wave ogives were actually being formed in the ice surrounding the tunnel. If one confines attention to the limited region covered by the tunnel there is nothing to discriminate between the two explanations. But now that we have observations over a much larger area, and can see that similar bendings occurring elsewhere are not correlated with the wave ogives, the explanation put forward in this section is preferable; it shows that the tunnel area obeys the same principles as the rest of the wave area.
9. Final Remark on the Possibility of a Pressure Mechanism
There is one way in which a pressure mechanism of wave formation might be reconciled with the observations. It might be that the high compression measured at the extreme upper end of the 1956 stake line represents the lower edge of an extensive pressure wave which one year later would occupy the position of the crest near C. Now, first, there certainly is such a compression and photogrammetric measurementsReference Nye 11 show that it covers the whole of the lower two-thirds of the ice fall. But, second, whether this compression forms a pressure wave or not depends on whether it is greater at some seasons of the year than at others, and we have no information on this. The only reason for thinking that there is no very significant pressure wave formation of this sort is in the argument of Reference Nyereference 11—that the existing waves are fully accounted for by the ablation mechanism referred to in Section 1, which must in any case be present, and there is therefore no need to invoke a hypothetical pressure mechanism as the primary cause of the waves.
10. Acknowledgements
The Cambridge AusterdaIsbre Expedition was the conception of W. V. Lewis, who organized and led the 1955 parties. In 1956, Mr. Lewis being prevented by illness from continuing, the organization at Cambridge was done by J. E. Jackson, and the parties on the glacier were led jointly by W. H. Ward, J. I. Davidson, J. W. Glen and myself. The 1957 and 1958 parties were organized and led by Mr. Ward.
The sciengific work of 1956 and 1957 described in this paper was only made possible by much administrative work in England, and back-packing and similar tasks on the glacier, and I should like to express my thanks to all concerned. In 1956 there were at various times (not all at once) 40 to 50 people, mostly Cambridge undergraduates, working on the glacier—excluding the parties from the Perse School and the Brathay Exploration Group, who bring the total to about 80—and nearly all of them took part in one way or another in the work here described.
I should record my particular thanks to Mr. Ward, not only for the impetus he has personally given to the work, but also for his development of efficient drilling equipment and technique for inserting stakes in the ice, which has been a major factor in the success of our enterprise. I also record with gratitude the leading role which Dr. Glen has played in taking the measurements. In particular, in 1956, after the stake system had been set up and surveyed once, Dr. Glen took charge of the remaining surveys, which were meticulously performed. He is thus responsible for the greater part of the measurements used in this paper.
In the survey work we had the expert advice and assistance of Dr. Cuchlaine King, J. I. Sharp and Miss E. Peters. The reduction of the theodolite observations was done by Mr. Jackson and Mr. Ward. It is also a pleasure to mention the practical help given us on the glacier by Olav Sopp of Oslo University.
The expedition is indebted for financial help to the Royal Society, Cambridge University, the Mount Everest Foundation, Trinity College Cambridge, the Royal Geographical Society, the Tennant Fund of Cambridge University, and the University College of Swansea. I myself am indebted to Bristol University for a grant.
Dr. Glen has kindly read and commented on the manuscript of the paper.
MS. received 13 October 1958