WATER AT THE BED OF A GLACIER. By J, F. NYE (H. H. Wills Physics Laboratory, University of Bristol, Bristol, England)

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1 WATER AT THE BED OF A GLACIER By J, F. NYE (H. H. Wills Physics Laboratory, University of Bristol, Bristol, England) ABSTRACT. In the Weertman (1962) theory the water film at the bottom of a temperate glacier, or at the bottom of a cold glacier or ice-sheet that has a temperate base, is required by the regelation process to transport the water produced by melting at places of high pressure to neighbouring places of low pressure where it freezes. The film is also required to act as the agent for transferring melt water down the glacier bed. These two roles appear to conflict, for the first demands a thickness of about 1 am while the second requires a thickness about times greater. One cannot, it seems, have a steady state where there exists both regelation and a film thicker than a few micrometres. It is suggested that the solution to the difficulty is to retain the regelation process so that the glacier can overcome the fine-scale obstacles in its path, and to drain the melt water at the bed of the glacier by a channel system. Channels are vulnerable to closure by purely geometrical processes arising from the ice movement, quite apart from closure by over-burden pressure. Channels incised downwards into the rock are expected to be much longer lived, for this reason, than channels incised upwards into the ice, and thereby the necessary time is provided for erosion to produce well-developed channels in the rock. RÉSUMÉ. Veau sur le Ut d'un glacier. Dans la théorie de Weertman (1962) le film d'eau au fond d'un glacier tempéré, ou au fond d'un glacier froid ou d'une calotte qui a une base tempérée, est censé transporter par un processus de regel l'eau produite par la fusion aux points de haute pression vers les points voisins à basse pression où il regèle. Ce film est également censé agir comme l'agent du transfert de l'eau de fusion vers le bas du lit du glacier. Ces deux rôles apparaissent contradictoires car le premier demande une épaisseur d'environ un micron tandis que le second suppose une épaisseur environ mille fois supérieure. On ne peut, semble-t-il, avoir un état stable lorsqu'il existe à la fois du regel et un film plus épais que quelques microns. On suggère que la solution à cette difficulté est de retenir le processus de regel pour expliquer que la glace puisse franchir les obstacles de petite dimension sur son chemin et de faire collecter l'eau de fusion îe long du lit du glacier par un système de canaux. Les canaux sont susceptibles d'être obstrués par des effets purement géométriques dus au mouvement de la glace, phénomène tout différent de l'obstruction par surpression. C'est pourquoi on peut s'attendre à ce que ies canaux creusés dans le rocher soient de beaucoup plus longue durée que les canaux creusés vers le haut dans la glace, l'érosion dispose ainsi du temps nécessaire pour créer des canaux bien développés dans le roc. The purpose of this contribution is to initiate discussion at the Symposium on the problem presented by the water film at the bottom of a temperate glacier, or at the bottom of any glacier or ice-sheet that is at the melting point at its base. In the Weertman (1962) theory the water film has three different roles, (1) In the regelation process it transports the water produced by melting at places of high pressure to places of low pressure. (2) The presence of a water film means that molecular adhesion of the ice to the rock, which might otherwise contribute to the drag, is absent. (3) The water film acts as the agent for transferring melt water down the glacier bed. We shall be concerned with roles (1) and (3). Let us look first at role (1). In the Weertman- Lliboutry model (Weertman, 1957, 1964; Lliboutry, 1968) of ice sliding over a sinusoidal bed without cavitation the resistance to flow arises partly from the regelation process and partly from plastic deformation. In order to calculate the thickness of the water film necessary for the regelation process when the regelation is taking place in the presence of deformation, let us make the simplifications that the deformation in the ice is by Newtonian viscous flow and that the amplitude of the bed corrugations is small. It is shown in the Appendix that the thickness t of the water film is then independent of the viscosity of the ice. In other words, t is the same whether the ice is deforming or not. This rather remarkable result only holds when the bed is sinusoidal and there is no reason to believe that it holds when the deformation takes place by a non-linear flow law. Nevertheless, it does suggest that the thickness of the water layer is only weakly dependent on the details of the deformation taking place in the ice.

2 190 'J.F.Nye The result derived in the Appendix is that t 127] WCKX 7tQ w L 0 where rj w is the viscosity of water, C is the constant relating the depression of the melting point to the pressure ( deg bar" 1 ), if is the mean thermal conductivity of ice and rock, Q W is the density of water, L 0 is the latent heat of ice per unit mass, and /. is the wavelength of the sinusoidal bed. It is interesting to notice what this formula does not contain. Besides being independent of the viscosity of the ice, t is independent of the coordinate x along the bed : thus the water film is of uniform thickness,just as in the traditional regelation experiment where a wire is pulled through ice (Nye, 1967). t is also independent of the ice velocity U (again like the wire experiment) and of the amplitude A of the sine wave. Furthermore, because t is proportional to?i lls it does not change much over a large range of wavelength. Putting rj w = 1.8 X10~ 2 poise, K = 2 Jm 1 deg" 1 s" 1, Q W = 1 g cm" 3, L 0 = 3.3 X 10 5 J kg- 1, we find: for X =5cm,t = 0.5 jim. and for A=5m, t 2-5jxm When the bed is not a single sine wave the expression for t is complicated; t does depend on the viscosity of the ice and on x, but it is still independent of the ice velocity and independent of the absolute magnitude of the displacement of the bed provided Fig. rock /. Water layer due to regelation on a sinusoidal rock bed. Melting at P, freezing at Q. Broken arrows show heat flow, full arrows show ice and water flow. Water flow takes place equally in the up~glacier and the down-glacier directions. the bed slopes remain small. Without a detailed analysis of this more general case we may tentatively argue from the result for a single sine wave, that since the range of wavelengths relevant for drag is from a few centimetres to a few metres (Nye, 1970) the thickness of the water layer is about 1 txm. Now Weertman (1962) has concluded that the water layer, acting in role (3) as the agent for transferring melt water down the glacier bed, is much thicker than this, for instance 1 mm (other large sources of melt water not considered by Weertman would make it thicker still). One may therefore ask: what then becomes of the regelation process? To try to answer this let us consider a sinusoidal bed (Fig. 1) with a uniform water layer of the thickness just calculated and let the thickness be artificially increased by a small uniform amount with the applied shear stress held constant. To obtain a steady state under these new conditions let there be artificial heat sources and sinks present along the ice-water surface; we shall first work out the signs of these heat sources and then deduce what happens when they are absent (this is an adaptation of a method used by Frank (1967) in the related wire-pulling problem). Since the applied stress is the same as before, the pressures are the same. The ice-water interface being everywhere at the melting point appropriate to the pressure, the temperatures will be the same as before. The flow of heat will therefore be the same. But because the water layer is thicker, but has the same pressure gradients in it, the flow of water will be faster. The flow of water is governed by the rate of melting at places like P and freezing at places like Q. These rates will thus be faster than before. To preserve a steady state we therefore need to supply extra heat at P and to extract heat at Q by means of our artificial heat sources and sinks. If these are now switched off there

3 Water at the bed of a glacier 191 will be freezing at P and melting at Q, so the layer will get thinner around P and thicker around Q. What happens next takes place outside the range of small perturbation theory and is speculative. It seems likely that the thickness near P returns to its original value and the excess water near Q becomes entrained within the ice (or, changing the model, flows in some way out of the plane of the diagram), thus allowing the thickness at Q to return to its former value and so restoring the original conditions. If the original perturbation were in the opposite direction, namely a uniform thinning, a similar argument shows that the thickness will initially increase at P and decrease further at Q. Steady-state regelation cannot proceed until more water is provided by melting or otherwise. Thus there is a tendency to stability of film thickness at P and instability at Q. But the essential point is that steady-state regelation demands a definite film thickness, and this is about 1 izm. There seems then to bé a fundamental difficulty in having the water-layer play roles (1) and (3) simultaneously. Another way of making the same point is to remark that in a case like that of Figure 1 the water flows up-glacier and down-glacier in equal amounts (and not predominantly down-glacier as is sometimes erroneously supposed). The flows are associated with the oscillations of pressure that, acting normally on the undulating ice surface, combine to produce the drag. On the other hand, in role (3) the water flows down-glacier only and the pressure does not oscillate. It does not seem possible to reconcile these two requirements by superposing pressure oscillations (to give drag) on a steady pressure gradient (to give drainage) because the film thickness requirements are totally different in magnitude. In a temperate glacier, there has to be some mechanism at or near the bottom of a glacier for dealing with the water that arrives from the surface (where it is produced by rainfall or melting), and with the melt water produced by geothermal heat, by heat of internal deformation and by frictional heat. In a cold glacier or ice sheet with a temperate bottom there is still water produced by the last three processes to be dealt with. The problem is that if one places this water in the form of a thin sheet it stops the regelation process. But without regelation there is no means for the ice to overcome small-scale obstacles on the bed plastic deformation overcomes the large-scale obstacles but, as Weertman has shown, it will not suffice for the smaller obstacles. So one must retain the regelation process. As a solution, I propose, contrary to Weertman's view (1962), that the drainage water is in a separate channel system, and that the continuous water layer is thus relieved of role (3) and has the thickness demanded by regelation. The weight of the overlying ice would tend to close the channel drainage system, but this tendency is resisted by the water pressure and also, as Weertman suggests, by the melting that will occur if the water is initially warmer than the ice; there is also the melting that comes from frictional heating in the water. But there is another process that tends to close the channels; it arises from the forward motion of the ice over the bed and is purely geometrical in nature. Figure 2 shows an example. This process, like the action of turning off a valve, is quite distinct from a pinching-off due to overburden pressure, for it can occur even when the water pressure balances the over-burden pressure; in Figure 2, for example, the water pressure could be equal to the ice over-burden pressure throughout the closing-off process. If the drainage system is to be continuous, as seems likely, there must, at any one time, be a sufficient length of channel that is stable against both sorts of closure process. The channels must not only have a sufficiently high water pressure in them, but Fig, 2. Closure of channel by ice movement. r0ck

4 192 / F. Nye they must be characterized geometrically by the fact that forward motion of the ice doss not readily seal them. An example of a position that meets this last requirement would be transverse to the glacier flow in the lee of an obstacle. In general, a channel incised downwards into the bed would not be vulnerable to the sort of closure process typified by Figure 2 ; on the other hand a channel incised upwards into the ice would move with the ice and, as in Figure 3, it would be liable to become sealed off by meeting a protuberance in the bedrock. One therefore expects that any Fig. 3. Closure of channel by ice movement. channel system incised upwards into the ice will be constantly changing, with lengths constantly becoming closed off, while a channel system incised into the rock, once formed, would be much more permanent. Thus erosion of the bed by water to form a channel, once started at a particular place, will tend to continue at that place; in this way the necessary time is provided for erosion to produce a well-developed channel in the rock bed. In summary, a consideration of the regelation process leads to the conclusion that the melt water at the bottom of a temperate glacier, or at the bottom of a cold glacier or ice sheet that has a temperate base, is in channels, rather than in a continuous film; and a consideration of the relative permanence of these channels leads to the conclusion that, while there may be temporary channels incised upwards into the ice, there will be comparatively permanent channels cut downwards into the rock bed. REFERENCES FRANK, F. C Regelation: a supplementary note. Philosophical Magazine, Eighth Ser., Vol. 16, No. 144, p LLIBOUTRY, L General theory of subglaciai cavitation and sliding of temperate glaciers. Journal of Glaciology, Vol. 7, No. 49, p NYE J. F Theory of regelation. Philosophical Magazine, Eighth Ser., Vol. 16, No. 144, p NYE J. F A calculation of the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proceedings NYE, of the Royal Society, Ser. A, Vol. 311, No. 1506, p J. F Glacier sliding without cavitation in a linear viscous approximation. Proceedings of the Royal Society, Ser. A. Vol. 315, No. 1522, p WEERMAN, J On the sliding of glaciers. Journal of Glaciology, Vol. 3, No. 21, p WEERTMAN, J Catastrophic glacier advances. Union Géodësique et Géophysique Internationale. Association Internationale d'hydrologie Scientifique. Commission des Neiges et des Glaces. Colloque d'obergurgl, , p WEERTMAN, J The theory of glacier sliding. Journal of Glaciology, Vol. 5, No. 39, p APPENDIX Calculation of the thickness of the water layer necessary for regelation The ice is assumed to be moving oyer a smooth sinusoidal surface (Fig. 1) by the combined processes of regelation and Newtonian viscous flow within the ice. This problem has been treated by Nye (1969, 1970), but in those papers the purpose was to calculate the drag, which can be done without explicitly calculating the thickness of the water layer. So here we shall use the pressure distribution and the distribution of velocity normal to the surface calculated there as a means of finding the water layer thickness. The method is very similar to that used by Nye (1967)

5 Water at the bed of a glacier 193 for the motion of a wire by regelation through an ice block, except that now the ice is deforming by viscous flow. (The fact that part of the heat flow takes place in a moving medium, the ice, is ignored. The effect may become important for wavelengths greater than about InDjU, where D is the thermal diffusivity of ice (0.01 cm 2 s^" 1 ). For l/ = 50 m year ~~ x this is 4 m, which is at the upper end of the relevant wavelength range. For higher ice velocities (or wavelengths) it is not strictly justifiable to ignore the effect.) Let the bed of the glacier be given by z 0 = A sin kx, where the amplitude A is small compared with the wavelength 2n!k. Considering unit thickness perpendicular to the plane of the diagram, the volume of new ice formed by freezing within the element of bed as is w (x)ds, where w n (x) is the velocity of the ice normal to the surface. The volume of water extracted from the water layer is then (P,/D W ) w n (x)ds, Qi being the density of ice. To a sufficient approximation ds dx, and hence dq= (o i lq w )w (x)dx, (1) where q is the flow rate of the water (volume per unit time). Using the formula for viscous flow between parallel plates, as an approximation, we have t'hx) dp 1 2Î? W dx (2) where t(x) is the thickness of the water layer, rj w is the viscosit* of water and p(x) is the pressure. Equations (26) and (32) of Nye (1969) give respectively k 3 w n (x) = UA cos kx, (3) kl+k' 1 where k\ =Q i LJ4CRr] i, r\ i being the viscosity of ice, and p(x) luatji kîk 2 cos kx. (4) kî+k 2 Substitute for w (x) in (1) from (3) and integrate to obtain q. Differentiation of (4) and substitution for dpjdx in (2) gives another expression for q. Equating the two expressions gives ki f p.- t\x)n,klk\ UA sinfa{~ ^-H V=0. (5),.2,,,2 le* 6»? w Thus t(x) is a constant, independent of x, U and A and is given by /3=- 6p,??, o K, Vi klk 13

6 194 J. F. Nye or, substituting for k% and putting k =2TI/Â, t_/l2y v CKXy^ \ n QwLa J as quoted in the main part of the paper. r) t is present in (5) both explicitly and also in k\. It disappears in the final result for two reasons: because it appears in (5) in the combination r\jt\ and also because the kl which is outside the brace in (5) is part of a constant multiplying factor.

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