GLACIAL EROSION. J.F. NYE and P.C.S. MARTIN H.H. Wills Physics Laboratory, University of Bristol, England

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1 RÉSUMÉ GLACIAL EROSION J.F. NYE and P.C.S. MARTIN H.H. Wills Physics Laboratory, University of Bristol, England Des champs de lignes de glissement sont discutés pour un glacier parfaitement plastique se déplaçant sur un lit irrégulier. Il est avancé que l'érosion s'exercera de telle manière que le lit devient une ligne de glissement ou une enveloppe de lignes de glissement. Ceci impliquequ'ilexiste une limite supérieure aux courbures longitudinales concaves du lit, mais qu'il n'y a pas de limite correspondante pour les courbures convexes. Le mécanisme fournit une explication naturelle pour les longs profils typiques des glaciers de vallées. Des exemples sont donnés pour montrer que ce qu'on appelle des bares rocheuses peuvent se former localement à des endroits où il n'y a pas une roche clairement plus dure et que, de même, une roche clairement plus dure ne forme pas nécessairement une barre rocheuse. ABSTRACT Slip-line fields are discussed for a perfectly plastic glacier moving over an irregular bed. It is argued that erosion will proceed in such a way that the bed becomes either a slip-line or an envelope of slip-lines. This implies that there is an upper limit to the longitudinal concave curvatures of the bed, but no corresponding limit to convex curvatures. The mechanism provides a natural explanation for the typical long profiles of glacier valleys. Examples are given to show that so-called rock bars can form at piaces where there is no obviously harder r>ck, and that, conversely, an obviously harder rock does not necessarily form a rock bar. With the help of a computer, we have been working out slip-line fields for glacier flow, using the perfect plasticity approximation with constant flow stress.* One of the results seems as if it may throw light on the problem of glacial erosion, and it may therefore be of interest to report it here separately from the main investigation. Figure 1 a shows a slip-linefieldin a region of compressing flow, with the ice moving from left to right. It may be built up by numerical integration from left to right essentially in the way described in (Nye 1967). The bed is a rough irregular surface made up of parts of three sloping planes, this particular shape being chosen purely for purposes of illustration. Up to point A the bed is a envelope of slip-lines (a-lines), but the a-line through A meets the bed tangentially again at B. The slip-line AB then becomes a boundary of the plastically deforming region and, as may be verified by considering the velocity solution, the material below AB is stationary. Between B and C the bed is again an envelope of a-lines. A fan of a-lines is generated at the sharp edge C, and CD is again an envelope. There is a discontinuity in tangential velocity across AB. Thus part of the bed is in contact with material that is moving with a definite non-zero velocity and part is in contact with stationary material. We assume to begin with that erosion will not occur under the stationary material, but that it will proceed at a uniform rate under the moving material (regardless of the precise ice velocity, the normal pressure, the type of rock and other factors). We shall relax this restriction later, but by making it here it The perfectly plastic solid with constant flow stress (Hill, 1950, p. 128) is an idealisation of the flow properties of ice. For a plane-strain problem, as here, the material is assumed not toflowat all under shear stresses below a critical value, called the flow stress. When the material is flowing the maximum shear stress at each point is equal to theflowstress. The slip-lines are two orthogonal sets of intersecting curves, called a-lines and /?-lines, whose directions at each point in the body are the directions of maximum shear stress. Mathematically, the slip-lines are the characteristics of the hyperbolic differential equations of the theory, and for this reason the determination of their shape is the essence of the problem. 78

2 allows us to concentrate on the main point. Thus erosion will not, at first, occur along the bed section AB. As erosion progresses the slip-line field will change slightly, and finally a stage will be reached when the bed has the shape shown in figure 16. To the left of it follows an envelope; 7=" is a slip-line. The sharp edge Chas become the edge G (the erosion being assumed to proceed at a uniform rate normal to the surface). FG and GH are still envelopes of slip-lines. (a) Fig. 1 Slip-line fields (not to scale): (a) Compressing flow; before erosion. (b) Compressing flow; after erosion. (c) Extending flow; before erosion. (d) Extending flow; after erosion. A similar argument applies to a region of extending flow, except that to build up the slip-line field one integrates in a direction against theflow*rather than with it. Thus figure lc shows the same bed as figure la but with the ice in extending flow still moving from left to right. The diagram, which is self-explanatory, is built up from right to left. Figure I d shows the bed after erosion. Our numerical method of constructing slip-line fields works for a bed of any shape whatever the bed does not have to be made up of straight segments. The general result is that if the bed is sufficiently concave a stationary region is formed. Less concave parts of the bed, plane parts and convex parts all form envelopes of the slip-line field, unless they happen to occur within a region which has already been rendered stationary by a larger-scale concavity. For example, if a small convex bump existed on * Note added in proof : But subsequent work has revealed problems of numerical instability when integrating in this direction that we have not managed to overcome yet. 79

3 the bed between A and B in figure la, it could be entirely surrounded by stationary material. Eventually, however, it would protrude into moving material and would begin to be eroded. This model of the perfectly plastic glacier, on the bed which erodes uniformly under moving ice, leads us, then, to two main conclusions : (1) after a certain amount of erosion the bed either follows a slip-line or is an envelope of slip-lines; (2) as a consequence there is, at this stage, a limit to the concavity of the bed, set by the curvature of the a-lines, but no limit to its convexity. The profile thus reached is not completely unvarying, since erosion takes place perpendicular to the bed surface rather than vertically, but it is much more stable in form than the original one. We have assumed that the type of flow, extending or compressing, remains the same as erosion proceeds. It may be that long-term changes in the accumulation-ablation pattern, in the length of the glacier, or in the shape of the bed, will occasionally change the type of flow at a particular locality. Even if this does happen, however, the conclusion that the bed becomes either a slip-line or an envelope of slip-lines remains valid. A key question is, obviously, what determines the curvature of the a-lincs? The curvature seems to be decided in these solutions mainly by the thickness of the ice, but also to some extent by the slope of the bed and by the behaviour of other parts of the slip-line field. But it is apparently not possible in any circumstances for the a-lines to curve upwards faster than a critical amount. Thus, a stable profile under erosion will not contain concave longitudinal curvatures exceeding a certain value. Our main reason for thinking that there is a maximum curvature is in the slip-line field computed for a horizontal bed in (Nye 1967). Under thick ice the a-lines are cycloids with a radius of curvature of 4 h, where h is the thickness. But as the ice becomes thinner, towards the snout, or the edge of the icecap, the a-lines are no longer cycloids and this expression breaks down. Finally, very near the edge the radius of curvature becomes infinite. It follows that there is a minimum radius of curvature. Its value is in fact Ao 100 m, and it occurs a distance 1.5 Ao Ü 15 m from the edge, (ho is a characteristic length, about 10 m, defined as klpg, where k is the flow stress in shear, p is the density and g is the gravitational acceleration.) On a sloping bed the radii of curvature of the a-lines are in general rather smaller, but we have verified that there is still a minimum, which, for moderate slopes, is probably not very different from the value quoted. How far do these conclusions apply to the real situation? We have idealised both the ice and the bed. First let us relax the restriction on the bed that erosion proceeds at a uniform rate under the moving ice. Provided we simply retain the proposition that no erosion can occur under stationary material the conclusion that the bed becomes a slipline or an envelope of slip-lines remains valid. If, for example, a part of the bed were made of softer rock so that it eroded faster than elsewhere, it still could not form a basin of more than a certain depth. For, if it did, a stationary piece of ice would form in the highly concave place, which would prevent further erosion there. The glacier may be thought of as a giant sanding block. The long profile of the block is flexible, but whereas it can have any curvature in the concave sense it can have only limited curvature in the convex sense. This block is covered by sandpaper of non-uniform roughness. It is then set to work on a rock surface of non-uniform hardness and is pressed down with a non-uniform distribution of pressure. The sandpaper is also endowed with a property of extensibility so that its velocity over the abraded material is non-uniform. The result would surely be to produce a smooth surface with the typical phugoid* shape of the long profile of a glacier containing smoothly concave basins, with more A term used in aerodynamics to describe a curve of flight (Lanchester 1910, frontispiece and Chapter 2). 80

4 sharply convex places where either the rock is harder, or where there were pre-existing convexities. The main point is that abrasion at a given place does not proceed at a rate given simply by the local conditions (hardness of rock, amount of rock fragments carried by the ice, velocity of ice, pressure and so on). It depends also on the rates of abrasion occurring at other places just as a sanding block sands down the soft places at a rate determined mainly by the hard places. Thus, a local theory of erosion is not adequate; erosion is a co-operative process. The present theory recognises this by placing a restriction on the relative rates of erosion occurring at different points; the restriction is on the curvature of the bed. It seems therefore that it is hardly necessary to idealise the bed at all to obtain the conclusion. But we must now ask whether the idealisation of the ice is a serious restriction. The fact that ice is not perfectly plastic with constant flow stress means that, in reality, boundaries such as AB in figure \a will not be sharp, but will mark regions of high shear strain-rate. For the same reason, the velocity in the "stationary " regions will not really be zero, but will be markedly less than it is outside these regions. So the main qualitative feature of the perfectly plastic solutions, that parts of the bed may be protected from erosion by sluggish ice, seems likely to remain valid in real glaciers. One can only suppose that the limitation in curvature of the a-lines also has its analogue in real ice. (a) B (b) Fig. 2(o) Three dykes of hard rock A, B, C embedded in a softer matrix. B is not quite so hard as A and C. Eventually erosion produces a single basin with no rock bar at B. Fig. 2(6) Two dykes of rock A, B which are only slightly harder than the rock surrounding them. Eventually erosion produces a basin with rock bars at A and B.

5 To examine the consequences of the argument in a little more detail consider the bed shown in figure 2a by the line 1 where there are three dykes of hard rock A, B, C embedded in a softer matrix. Let A and C be of equal hardness and let B be a little softer. Successive stages of erosion, according to our model, are shown by lines 2, 3 and 4. At first basins appear between A and B and between B and C; but after a sufficient time, depending on the relative hardnesses of B and A (or C), they are converted into a single basin between A and C. Thus, rock bars are seen at A and C and are associated with harder rock at these places; but an almost equally hard dyke at B.has produced, eventually, no rock bar. The reason is that the shape of the basin is determined not only by the relative hardness of the rocks, but also by the large scale geometry of the slip-line field. Alternatively, one might have the situation of figure 26 where A and B are dykes of rock which is only slightly harder than the matrix material. After a sufficient time a basin will form between them. The eventual depth of this basin is limited by the curvature of the slip-line between A and B. If erosion continues for long enough there will be a basin of this limiting depth even though A and B arc only imperceptibly harder than the surrounding rock. This example is intended to show, in principle, that an observable rock bar need not be associated with an obviously harder rock. Figure 2a, on the other hand, shows that an obviously harder rock (B) need not be associated with a rock bar. Moreover, the eventual depth of the basins in figures la and 2b, as distinct from the time taken to make them, bears no relation whatever to the relative hardnesses of the rocks. Could this confusing situation possibly account for some of the confusing field evidence on the question of what determines the location of basins and overdeepened places in glacier valleys? The process described has something in common with the notion of rotational slip, mentioned by Forbes as early as 1848 (Forbes 1859) and by Lorcnzi in 1889 (see Clark and Lewis (1951), p. 550, and Lliboutry (1965), p. 681) and forcefully advocated by Lewis (1947, 1949) and Clark and Lewis (1951) (see also Ward (1947)). It extends this notion in two ways: first, by suggesting the mechanism by which the hollows in the bed attain the shape of slip-lines, and, second, by suggesting why the curvature of the hollows is limited. It may well be that what we have said applies primarily, but not exclusively, to erosion underneath thick ice, for we have assumed that the ice always makes contact with the bed; underneath thin ice this will not be true. Cavities are seen to exist on the downstream side of obstacles and they are thought, for example, by Carol (1947) and Lewis (1947), to play an essential part in the formation of steps and of roches moutonnées. Thus, the rather sharp concavity on the downstream side of a roche moutonnée is probably the result of a process supplementary to the one we are describing here. It is possible that the ideas we have described may also be useful in studying the flow of ice around the obstacles that are contemplated in current theories of glacier sliding. But we should emphasise that the model usually employed there is a completely smooth bed provided with some definite irregularity. In contrast, our model assumes from the start that there is some mechanism for producing roughness, and then proceeds to study the large-scale flow pattern that results from irregularities in an already rough bed. We should like to thank Professor R.F. Peel and Mr. H.A. Osmaston for their helpful comments on an earlier draft of this paper. REFERENCES CAROL, H., (1947): Journ. Glaciol., /, CLARK, J. M., and LEWIS, W.V., (1951): Journ. Geol., 59, FORBES, J.D., (1859) Occasional papers on the theory of glaciers (Black: Edinburgh) pp

6 HILL, R., (1950): The mathematical theory of plasticity (Clarendon Press: Oxford). LANCHESTER, F.W., (1910): Aerodonetics, 2nd ed. (Constable: London). LEWIS, W.V., (1947): Trans, and Papers, Inst. of Brit. Ceogr., 13, LEWIS, W.V., (1949): Geografiska Annaler, LLIBOUTRY, L., (1965): Traité de Glaciologie, Vol. 2 (Masson: Paris). NYE, J.F., (1967): Journal of Glaciology, 6, WARD, W.H., (1947): Journ. Glaciol., 1, DISCUSSION L. LLIBOUTRY II convient de noter que les auteurs font 3 hypothèses implicites, qui peuvent prêter à discussion. 1. Le bedrock constitue une ligne de glissement ou une enveloppe de lignes de glissement. Mais on peut concevoir que, dans une région où l'écoulement est très compressif ou très extensif, le bedrock ne constitue plus une direction de cisaillement maximum. 2. Il y a toujours continuité dans la déformation. Or après un verrou, le glacier peut se casser et se ressouder instantanément, après un mouvement de faille. Ceci doit changer tout à fait l'allure des courbes de glissement sur la face aval entre 2 ruptures. 3. Mais la plus importante objection est que de nombreux géomorphologues et glaciologues pensent que l'érosion proglaciaire est beaucoup plus importante que l'érosion glaciaire. Le glacier ne ferait que polir la roche et l'évacuer, mais son débitage (quarrying) aurait surtout lieu à la faveur des reculs du front qui laissent le bedrock à découvert. La formation de bassins surcreusés serait alors liée au fait que les fluctations du front, sont beaucoup plus nombreuses dans les parties peu inclinées de la vallée que dans les parties à forte pente. J.F. NYE I. The question of whether the bed is necessarily parallel to a direction of maximum shear stress is not simple. If one actually measured the direction of maximum shear stress in a two-dimensional experiment at the bottom of a glacier, one would almost certainly find that it was not parallel to the bed. The higher the longitudinal strain-rate, the further from a direction of maximum shear stress the bed would be. But this is not quite the issue. The real question that has to be answered concerns the best interpretation, in the real world, of a theoretical model. We have, for example, this solution in the theory of perfect plasticity for a parallel-sided slab of a certain fixed slope. In theory, the lower plane boundary can be placed anywhere between the top surface and the envelope of the slip-lines. It is usually taken as the envelope of the slip-lines, on the grounds that it is there exerting the maximum possible shear traction and that this is appropriate for a "rough" boundary. But there is no decisive reason why one should not place it higher up, as Prof. Lliboutry suggests; it would then, of course, be exerting a traction less than k. I should like to give two arguments why the conclusions of the paper are not weakened by this ambiguity in the physical interpretation of the model. Suppose, by extension from the case of the parallel-sided slab, one drew in a lower boundary in figure 1 (a) which was intermediate between the top and bottom surfaces shown in the figure. In order to draw it one would need to postulate some such boundary condition as rip = constant, or r = mk, where r is the tangential traction on the boundary, p is the normal pressure on the boundary and m is a fraction between 0 and 1. It pro- 83

7 bably would not make much difference which condition was chosen. Now, without detailed calculation, it seems very plausible that such an intermediate boundary would be less sharply curved than the lower boundary drawn in the figure, made up of slip-lines and their envelope. In other words, with such a boundary condition, the flowing medium would be even less able to get down into sharp hollows in the bed. Thus, stagnant regions would'be more readily formed and the conclusions of the paper would be strengthened. This brings me to the second argument, which considers a very deep and narrow pothole on the bed, filled with ice. The ice in the pothole cannot move while the ice of the glacier above it must move. Some form of shear fracture across the top of the pothole is inevitable. A model of this process in a perfectly plastic substance would certainly have to have a slip-line (or, less likely, an envelope of slip-lines) across the top of the pothole. The pothole is just an extreme form of a concave bed curvature. Once a stagnant region develops it would seem that its upper boundary must be a slip-line (or an envelope of slip-lines). Thus, in summary, 1 agree that one could contemplate having a lower boundary that was not parallel to a slip-line if the bed were gently undulating; but hollows beyond a certain size will lead to stagnant regions, and then it would seem necessary that the lower boundary of the deforming material should be a parallel to a slip-line. 2. As Prof. Lliboutry says, the paper assumes continuity of deformation. In a crevassed zone one would, of course, expect the slip-line field to be quite different, since the geometry of the free surfaces is quite different. But I would expect that below the penetration depth of the crevasses the field would not be very different from the one sketched. There will presumably be a perturbation in it extending to a depth of the order of the crevasse spacing. (One remembers that in plasticity theory velocity discontinuities can propagate far from their point of origin, but I am here thinking of a real strainrate hardening medium where this cannot happen.) 3. I must leave this question of whether the dominant erosive process occurs under the ice or in front of it to the gcomorphologists. We certainly postulate that basins can eroded under the ice, and our reasoning is different to whether this takes place mainly by abrasion or by quarrying. The "soft" and "hard" places referred to in the paper mean places that are less resistant and more resistant to erosion, in the widest sense, and, if quarrying were the dominant process, they should perhaps be interpreted as welljointed and poorly-jointed rocks respectively. If the formation if overdeepencd basins were conclusively shown to be connected essentially with fluctuations of the glacier front, rather than a process that can take place under steady conditions, our theory would have to be discarded. E. LA CHAPELLE Field evidence supports the existence of non-deforming regions. A tunnel into the Blue Glacier (Washington) ice fall this summer encountered such a region. It was bounded by a zone of high shear strain. H. LISTER Riegels, the gcomorphological feature of glaciated valleys have no adequate explanation, glibly being explained as harder rock bands. Your flow geometry explanation seems much more acceptable,but the spacing along the valley must surely be a function of flow and hence of temperature? 84

8 J.P. NYE I ought to emphasise that if you put our model on a bed of completely uniform hardness there is nothing in the model to produce riegels. But, if you allow even very slight inhomogeneities in hardness of the rockbed, you do, given enough time, get riegels. Their spacing is determined by the spacing of the hard rock bands, except that, as figure2 shows, (a) a hard rock band does not necessarily produce a riegcl, and (/>) you can have a riegel even where the rock is only slightly, perhaps unobservably, harder. Instead of inhomogeneities in the hardness of the rockbed, which could include inhomogeneities in the frequency of jointing, one might also think of changes in direction of the valley or constrictions in the valley as being places which are "harder"or"softer, in the very general sense that they are more or less resistant to erosion. Thus these also form potential sites for rock-bars (see, for example, C. Embleton and C. A. M. King, Glacial and Periglacial Geomorphology, Arnold 1968, p. 171), but whether they do in fact from rock-bars would be decided by the principles illustrated infigure2. W.H. WARD Prof. Lliboutry raised the question of roughness of the rock surfaces, is there a scale attached to your diagram a)? J.F. NYE The surface of the rockbed in the model is "rough" in the sense in which the word is used in perfect plasticity theory, namely that the bed is able to exert on the ice the maximum shear traction that the ice can take. The slip-line fields in figure 1 (a) to (d) are sketched and not exactly computed, but the glacier in them has a thickness of about 1.5 ho, and, if you take ho = 10 m, this is about 15 m. The glacier in the figures has been put on a rather steep slope, to bring out the point; on a less steep slope it would, of courcc, be thicker. I.F. COLLINS In their proposed slip lime fields for the flow of ice over an uneven bed, Nye and Martin do not consider changes in flow regime. In their solutions the plastically deforming regions are either wholly extending or compressive. In practice however quite small irregularities on the bed can produce changes in regime. It should therefore be borne in mind that another type of solution may exist, involving changes from extending to compressive flow or vice versa. The exact details of the field in the change over region are very complex and involve discussion of the velocity, in addition to the stress fields. J. F. NYE Mr. Collins is quite right in pointing out that we do not consider changes from extending to compressing flow and vice versa. A fuller theory certainly ought to do so. But when he says that quite small irregularities on the bed can produce changes of regime I think one has to distinguish between different physical conditions. For example, in the Antarctic ice sheet there is an overall extending strai»-rate because the accumulation rate is positive. This is samll and rather uniform. But superimposed on it are 85

9 strain-rates arising from the irregular bed topography which, according to Robin's work, can be (but are not always) far larger and which can fluctuate in sign. Therefore the total strain-rate can fluctuate in sign too, but even then it may keep the same sign for several kilometres. At another extreme one has a glacier like Austerdalsbreen, a Norwegian valley glacier, where the strain-rate is observed to remain compressing over the whole region of the tongue, a distance of 4 km. Here the ablation rate is high and contributes a large and rather uniform negative strain-rate. The irregularities in the bed are not big enough to change the sign of the total strain-rate. I think this situation is fairly common in valley glaciers, except in the upper parts of ice falls. Therefore, although the change-over regions ought to be studied, their neglect in this first shot at an erosion theory is perhaps not too serious. 86