Introduction

At the Obergurgl Symposium I presented a brief criticism of Lliboutry’s theory of the sliding of glaciers (Reference WeertmanWeertman, 1962). I pointed out that Lliboutry’s theory would lead to sliding velocities 10⁸ to 10¹² times larger than are normaIIy observed. I concluded that the most likely application of Lliboutry’s theory is to the problem of the avalanching of thin ice slabs where high sliding velocities are encountered. I obtained these fast velocities from an adaptation of Lliboutry’s theory to the case in which the glacier bed is rough and irregular both in the direction of sliding and in the direction perpendicular to sliding. (This was the type of bed I had used in my theory of sliding.)

Lliboutry’s answer to my criticism ([Union Géodésique et Géophysique Internationale], 1963, p. 62) was that he felt the bed model he used is more realistic. This model (the washboard model) is rough only in the direction of sliding and is perfectly smooth in the perpendicular direction. Moreover, he pointed out that his analysis is influenced by the pressure of the water trapped in the troughs or hollows of his type of glacier bed. (For a bed rough in two directions such as I considered, water entrapment is relatively unimportant.) My rejoinder to this reply ([Union Géodésique et Géophysique Internationale], 1963, p. 67) was:

Professor Lliboutry has not answered my criticism in any of his papers, including his recently published book (Reference LliboutryLliboutry, 1965[a], p. 649). Moreover in a popular article (Reference LliboutryLliboutry, 1965[b]; see however, Reference WeertmanWeertman, 1966) he implies that there are no difficulties attached to his theory and that it satisfactorily explains glacier sliding. However in the latest account of his sliding theory, which is given in his book, he again finds it necessary to assume the presence of a water pressure whose value is picked in a completely arbitrary manner (p. 651). He gives no jusgification whatsoever for his choice of this particular value.

It is the purpose of this paper to present my criticism of Lliboutry’s theory in a more detailed form. These criticisms will be based on his theory as he has developed it in his book (Reference LliboutryLliboutry, 1965[a], p.694–52).

Review of Lliboutry’s Theory

Lliboutry’s theory is the result of adding a third mechanism of sliding to the two I had considered in my sliding theory (Reference WeertmanWeertman, 1957, Reference Weertman1962, Reference Weertman1964). We shall now review this mechanism as it is given in his book (Reference LliboutryLliboutry, 1965[a]). The notation employed is identical to that used by Lliboutry.

Figure 1 shows Lliboutry’s washboard model of a glacier bed. The surface of the bed is sinusoidal with wave-length λ and amplitude a. The glacier is moving to the right with velocity v. The average overburden pressure is ρ g h, where ρ is the ice density, h is the glacier thickness, and g is the gravitational acceleration. Water is trapped in the troughs, and the water pressure there is equal to p. The ice velocity is taken to be sufficiently fast so that the ice rests only on the tops of the undulations. Ice is in contact with rock over a distance X. It extends down from the tops of the crests a distance Z. Since X is assumed to be small compared to λ the distance Z is given by

Fig. 1.

The Lliboutry washboard model of a glacier bed, after figure 16.21 of Reference LliboutryLliboutry (1965[a], p. 650)

The vertical pressure on the ice over the distance X is taken to be σ ₁ The rock in contact with the ice pushes up with a force σ ₁ X over a distance X and the water in the hollow pushes up with a force p(λ−X) over a distance (λ−X). The overburden of ice produces a downward force ρ g h λ over a distance λ. By balancing the upward and the downward forces Lliboutry obtains the equation

The same type of argument is applied to the forces in the horizontal direction. Let f represent the shear stress at the bed causing ice flow to the right in Figure 1. (Lliboutry calls this shear stress the frictional stress.) The resultant equation is

Lliboutry assumes that σ ₁ will be much greater than p and therefore uses the approximate equation:

Ice in the regions where contact is made with rock creeps under an effective stress which is taken to be equal to

According to Glen’s creep law, the creep rate in these regions must be:

where B is a constant and n is another constant approximately equal to 3. Since ice in contact with rock moves up a distance Z in a time interval X/v, Lliboutry considers that the creep rate

thus must approximateFootnote *

Following Lliboutry’s notation, we now let v _c instead of v represent the sliding velocity. If Equation (4) is put equal to (5), the following value is obtained for v _c :

where r = λ/a is a measured roughness (rugosité) of the bed. (The smaller is r the rougher is the bed.)

This then is Lliboutry’s derivation of his additional mechanism of sliding. He further develops his theory by combining this mechanism with the sliding mechanisms I considered.

In the original version of his sliding theory Lliboutry found the sliding velocity by setting the velocity v _c given by Equation (6) equal to the sliding velocity calculated from my creep rate enhancement mechanism. This latter mechanism, which he calls mechanism B, gives a velocity v _B = C _B a f ³, where C _B is a constant containing such factors as the roughness. (He calls the mechanism which leads to Equation (6) mechanism C.) The velocity given in Equation (6) can be rewritten as v _c = C _c a(ρ g h−p)⁶/f ³, where C _c is another constant. Setting v _B = v _c results in the equations f = (C _c/C _B)^1/6(ρ g h−p) and v = 2v _c = 2v _B = 2(C _B C _c)^1/2 a(ρ g h−p)³. The shear stress f is determined by the term (ρ g h−p). The sliding velocity depends on both a and (ρ g h−p). Neither of these quantities can be evaluated from the theory.

Lliboutry also claims that the shear stress f is independent of sliding velocity. However an examination of these last equations does not appear to substantiate his claim since v, a, f and (ρ g h−p) are all related to each other. In addition to mechanism B, Lliboutry, in the later version of his theory, discusses the regelation sliding mechanism, which he calls mechanism A. Mechanism A gives a sliding velocity v _A = C _A f/a,where C _A is still another constant. There is a minor mathematical error in the last equation on p. 651 of his book. A factor

was omitted. If this factor is inserted in this equation and his subsequent equations are corrected, it is found that the velocity at which mechanism C first becomes more important than mechanism A is v = 2v _c = 2v _A (The uncorrected equations are v = 4v _c/3 = 4v _A.) The sliding velocity still is given, however, byv = 2v _B = 2v _c.

It is seen that Lliboutry makes no attempt in this analysis to evaluate the water pressure or to decide what determines it. In comparing his theory with sliding velocities he merely picks an arbitrary value for the term (ρ g h−p), and because he can pick a value of (ρ g h−p) that will lead to a reasonable sliding velocity he claims his theory to be correct. It is my contention that because in the theory no reasoning or analysis is presented by which the water pressure (as well as the thickness of the water in the troughs) may be estimated the theory is not a finished one and it cannot be used as it stands to make meaningful predictions of sliding velocities. Footnote †

Another difficulty with the Lliboutry theory arises because no account is taken of the fact that water must be continuously removed from the bottom of a glacier. The geothermal heat alone melts each year approximately 0.5 cm.³ of ice per cm.² of area from the bottom of a glacier. This water and water produced from other sources must be removed through flow down pressure gradients.

Comments Concerning the Water Pressure

Let us investigate how the water pressure might be estimated by using Lliboutry’s model. Consider again Figure 1. Suppose the water trapped in the troughs is truly isolated. If the amount of water is specified, the distance X is uniquely determined by simple geometry. Therefore, by Equation (3b) the water pressure p is equal to ρ g h−f λ ²/π ² a X. Since X can have any value p can have any value from 0 to ρ g h. Thus unless there is some way to predict how much water is in the hollows the theory cannot yield a sliding velocity.

There is an additional difficulty. Figure 1 represents irregularity of only one particular wavelength and amplitude. In actuality a spectrum of wavelengths and amplitudes is present. For simplicity consider a bed made up with irregularities of only two wavelengths, as shown in Figure 2. If the troughs of the smaller wavelength are filled with water the troughs of the large wavelength are essentially empty of water. The sliding past the larger obstacles thus is controlled by mechanisms I have proposed in my theory and Lliboutry’s theory does not apply.

Fig. 2.

A washboard glacier bed containing undulations of two wavelengths

Another approach to the question of the water pressure is to assume that the pockets of water in the troughs of Figure 1 are not isolated but are connected to a source or reservoir of water which is maintained somehow at a pressure p. The amount of water in the troughs thus is not fixed. In this case it is easy to show that Figure 1 corresponds to a physical situation in unstable equilibrium. According to Equations (3b) and (6), if the sliding velocity increases the pressure p is reduced and the amount of water in the troughs is increased. The reverse holds if the sliding velocity is reduced. Suppose p is fixed and the glacier is sliding with the velocity given by Equation (6). If the sliding velocity were to be reduced slightly from this value the water pressure in the cavity would increase. Thus water would be forced from the troughs into the reservoir, which in turn would cause a further reduction in the velocity until all the water is squeezed out of the troughs. The same argument shows that an increase in velocity would cause the troughs in Figure 1 to fill completely with water. Once filled, the larger troughs such as shown in Figure 2 would commence being filled with water and eventually the sliding velocity would reach avalanche speeds provided the water pressure p is not itself affected by this catastrophic behavior. Obviously the water pressure will be affected and therefore a careful analysis must be made as to what factors determine p. Clearly much remains to be done with Lliboutry’s theory before it can be claimed to be complete.

In particular Reference LliboutryLliboutry (1959, p. 260) estimated that the critical thickness of the water-filled cavities is 21 cm. and the newer version of the theory appears to require an even larger thickness. Thus, according to his theory, an extremely large separation exists between the bottom of a glacier and its bed. If the cavities are interconnected so that water can flow freely between them it would be difficult to maintain a water pressure greater than atmospheric. The rate of flow of water in a water layer subjected to a pressure differential is proportional to the cube of the thickness of the water layer. A water layer only about a millimeter in thickness suffices to remove the water produced by the geothermal heat (see Equation (9) of Reference WeertmanWeertman (1962)) Water flow through a layer 21 cm. in thickness would be approximately 10⁹ greater! This amount of melt water is orders of magnitude larger than is normally available to any glacier,

If, as in my theory, a glacier bed is used which is rough in two dimensions rather than only one, the water pressure difficulties just discussed disappear. The water pressure is the overburden pressure. Water is free to flow along the bed. The thickness of the water film is determined by the amount of water that is flowing through a bed and the pressure gradient that is driving this flow. The pressure gradient is determined primarily by the slope of the upper ice surface and secondarily by the slope of the bed itself.Footnote *

Summary

As I see it, the difference between Lliboutry’s and my theory of the sliding of glaciers reduces to the following. 1 maintain that Lliboutry has not derived a meaningful equation for the sliding velocity because his equation contains two unknown parameters, the water pressure, and the thickness of the water layer, whose values cannot even be estimated from the theory developed so far, Therefore as the theory now stands it is impossible (unless the theory is reduced to the status of a phenomenological theory) to compare theory with experiment. Another criticism of his theory is that he assumes a glacier bed which is rough in only one direction whereas actual beds are rough in both directions. Also no account is made in the theory for the flow of water at the bed.

Lliboutry’s main critism of my theory is his claim that it leads to too slow sliding velocities. He claims it can predict velocities no larger than 1 meter per year and cannot predict velocities of the order of 100 meters per year. He arrives at this conclusion in a curious way. My sliding equation depends sensitively on the roughness of the bed. The rougher the bed the slower is the sliding velocity. In his criticism, Lliboutry says, in effect, let us take a particular value for the roughness and calculate the sliding velocity. He chooses an arbitrary value of the roughness and comes up with a sliding rate of 1 meter per year. From this calculation he concludes the theory must be wrong. At this level of logic the obvious answer to his criticism is that the theory is not wrong, he is wrong because he picked an incorrect value for the roughness. Unfortunately no one has yet made extensive measurements of the roughness of beds of glaciers. Until this is done, obviously one cannot conclude that my theory is either correct or incorrect.

The roughness of a glacier bed obviously is determined from an interaction of the erosional processes occurring at the bottom of a sliding glacier and the properties of the rock making up the bed. Both my theory and Lliboutry’s could be criticized as not being completely developed since they do not lead to a prediction of the actual roughness. Because of the obvious difficulties in developing any sliding theory to this degree of refinement I hope that I have the reader’s indulgence throughout this paper in regarding the roughness of the bed as a fixed parameter and not one that must be determined by theory.

The opinions expressed in this paper are those of the author and do not necessarily express the official views of the U.S. Army.