Journal of Glaciology, Vol. 56, No. 200, 2010 965 Weertman, Lliboutry and the development of sliding theory A.C. FOWLER MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Republic of Ireland E-mail: [email protected] ABSTRACT. This paper reviews the seminal contributions (and tussles) of Weertman and Lliboutry to the theory of glacier sliding in the 1950s and 1960s, and summarizes later developments up to the present day. THE SLIDING LAW WEERTMAN: REGELATION AND ENHANCED CREEP Solid ice, in the form of glaciers and ice sheets, flows on In an early book on the dynamics of glaciers, Koechlin long timescales, due to solid-state creep processes, and the (1944) discusses the flow of a glacier, together with the resulting rheology is usually described in terms of a necessary boundary condition at the base. This he takes to generalized Newtonian fluid. Most commonly, this takes be that the stress exerted on the bed roughness elements the form of Glen’s law, although other variants are possible. should be equal to the strength of ice, and thus his basal The formulation of a viscous glacier flow in this way condition is that the shear stress is constant.* The first leads to an elliptic partial differential equation for the flow paper in the subject which properly addresses the physics is velocity, which requires the posing of boundary data at that of Hans Weertman (1957b). In a mere six pages he both the upper (free) surface and the base, or bed, of the enunciates the basics of the phenomena which are still glacier. Mostly, viscous liquids are thought to obey a no- considered to explain the mechanism of sliding. That slip boundary condition at a solid boundary, on the glaciers slide at their base was well known (e.g. McCall, intuitive notion that the fluid molecules effectively ‘adhere’ 1952; Ward, 1955); Weertman’s contribution was to to the boundary, but there is no intrinsic reason why this explicitly describe the physics of the process. This he should be so, and indeed the veracity of the condition was does by quantifying two processes: regelation and a subject of controversy in the 19th century (Goldstein, ‘enhanced creep’. 1938). A generalization, called the Navier slip condition, Regelation, otherwise known as pressure melting, is the allows a nonzero slip velocity proportional to stress, and it process whereby ice can move past obstacles without is thought that such a condition is necessary to alleviate an deformation; as (temperate) ice approaches the upstream unphysical singularity (discontinuous velocity) which face of an obstacle, the pressure is higher, and therefore would otherwise occur in the motion of a contact line (because of the Clapeyron effect) the melting temperature is where a two-fluid interface intersects a wall (Dussan and lower than that far away. The ice therefore melts, forming a Davis, 1974). layer of water with a thickness of the order of microns It has long been understood that glaciers also slip at the (Nye, 1967), and squirts round the obstacle to where the bed, and the resulting boundary condition for the flow pressure is lower, where it refreezes, as the melting which relates basal velocity, u, to basal stress, , has come temperature is now higher. Because of the higher tempera- to be known as the sliding law, and it is the efforts that ture in the lee of the obstacle, there is a compensating heat have been put into its determination that form the subject flow back through the obstacle, and it is this which delivers of this paper. An issue which needs immediate clarification the latent heat necessary to melt the ice. Weertman’s model is what is meant by the basal stress. The quantity refers to of the bed is that of an array of cubical obstacles of side a the shear stress in the ice near the bed (on the scale of the separated by a distance l. Weertman’s choice of notation ice depth, di) but far from the bed (on the scale, l, of the differs from that used here. Thus the parameter bed roughness). In the language of matched asymptotic a expansions, the basal shear stress is the limiting value at the ¼ ð2Þ l bed of the outer solution in an asymptotic expansion in the small parameter represents the aspect ratio of the bed roughness, and provides a second such parameter in addition to .If is l ¼ 1 ð1Þ the basal shear stress, as discussed above, then, since the di lubricated bed between the obstacles offers no resistance, (Fowler 1981). each obstacle bears a stress of =32, which causes a 2 difference of temperature of ÁTa ¼ C=3 across the { obstacles, where C is the slope of the Clapeyron curve. kÁT Equating the heat flow through the obstacle, a a2,to a *This almost uncited book is worth a look. He appears to discuss 2 the latent heat flux, iLuRa , where k is thermal conductivity deformation of subglacial till (p. 68, see also p. 152), and also refers to various formerly surging glaciers in the European Alps apart from (of both ice and rock), i is ice density, L is latent heat and uR Vernagtferner, such as Oberer Grindelwaldgletscher, Aletschgletscher and is the sliding velocity due to regelation alone, he derives the Rhonegletscher. The discussion on basal friction is on pp. 98f. regelative sliding law { The factor 3 occurs in Weertman’s discussion, because he divides the ¼ 2 ð Þ excess normal stress by 3 to find the excess longitudinal stress. RrauR, 3 966 Fowler: Weertman, Lliboutry and the development of sliding theory where the regelative roughness coefficient, uV, for a given a are comparable. Equating these two determines the controlling obstacle size as 3iL Rr ¼ , ð4Þ ð À Þ= 1=2 kC n 1 2 Rr à ¼ ð Þ Â 3 À3 a 2 , 11 is a material constant. Using values i = 0.9 10 kg m , Rv  5 À1 À1 À1 À1 L = 3.3 10 Jkg , k =2Wm K , C = 0.0074 K bar = and now selecting only this obstacle size, we find  À7 À1 0.74 10 KPa , we find Weertman’s law in the form À À  3 2 ¼ :  16 2 : ð Þ =ð þ Þ Rr 2 10 bar m a 0 63 10 Pa m s 5 ¼ 2Ru2 n 1 , ð12Þ To give some idea, a driving stress of 1 bar over obstacles of where elevation 1 m and asperity ¼ 0:2 provides a regelative 1=ðnþ1Þ velocity of little more than a centimetre a year! Only very RvRr R ¼ : ð13Þ small obstacles or very smooth beds can support a signifi- 4 cant regelative sliding velocity. (Weertman omitted the factor 4 because he took u ¼ u ¼ Next, Weertman considers the effect of differing sizes of R u , whereas Equation (13) assumes u ¼ u þ u .) With asperities, but his argument (to this reader) is somewhat V R V n ¼ 3 and the values in Equations (5) and (9), we have mysterious. A possible paraphrase is the following. If we À = = À = = suppose that an array of obstacles of asymptotically different R 15:1 bar m 1 2 a1 2 ¼ 0:85  1010 Pa m 1 2 s1 2: ð14Þ sizes, a a ÁÁÁ (but with the same aspect ratio, ) are 1 2 Taking ¼ 1 bar ¼ 105 Pa, ¼ 0:2, we find a sliding superimposed, then we can suppose that the effects of the À velocity of 1.4 m a 1. Given the crudeness of the model, small-scale roughness act effectively as a surface drag to this gives a laudable estimate of the sliding velocity. the flow past the largest-scale roughness. With a fairly Particularly, the choice of is rather arbitrary, and since obvious notation, ¼ 2R u a þ , ¼ 2R u a þ , r R 1 2 2 r R 2 3 the sliding velocity is inversely proportional to its fourth etc., and thus X power, higher sliding velocities are easily obtained. For 2 À1 ¼ RruR ai: ð6Þ example, a sliding velocity of 44 m a is obtained with a i choice of ¼ 0:1. Consequently the largest obstacles control the rate of sliding, Weertman’s heuristic argument can be slightly tightened which for reasonable sizes will then be negligible. in the following way. Considering the flow past obstacles of Weertman next considers the viscous creep of ice past an size a, it seems reasonable (because the regelative velocity is obstacle. The differential stress past an obstacle is =22, a plug flow) to consider the velocity to be the sum of uR and = uV, thus while the strain rate is uV a, where uV is the creep- controlled sliding velocity. Assuming Glen’s law in the form T T n n ¼ þ ð Þ "_ ¼ B , where "_ is the strain rate and B and n are contants, u 2 2 a, 15 Rra Rv this suggests the viscous sliding law which implicitly defines the stress, Tðu, aÞ, past this size of 1=n 2 uV ¼ Rv , ð7Þ obstacle. Then the total stress is, following our previous a superposition argument, where the viscous roughness coefficient is X ¼ Tðu, aiÞ: ð16Þ ¼ 2 : ð Þ i Rv 1=n 8 B We define dimensionless partial stress, velocity and obstacle Weertman used a version of Glen’s law with n ¼ 4:2, but if size as we take Cuffey and Paterson’s (2010) recommended value of ð þ Þ= À À À n 1 2 B ¼ 2:4  10 24 Pa 3 s 1 with n ¼ 3, then we have T ¼ , u ¼ U, a ¼ aÃA; ð17Þ 2R :  8 1=3 ¼ : 1=3: ð Þ Rv 1 49 10 Pa s 4 7 bar a 9 then is defined by Repeating our earlier estimate for ¼ 0:2 and a ¼ 1m, 5 U ¼ þ nA, ð18Þ we find that a shear stress of 1 bar (10 Pa) produces a A viscous sliding velocity of 150 m aÀ1, but now (with and the sliding law is constant ) the sliding velocity decreases as obstacle size X is reduced.