Chapter 7 Basal motion “On comprend que ces masses glacées, entranes par la pente du fond sur lequel elles reposent, dégagées par les eaux de la liaison qu’elles pourraient contracter avec ce même fond, souleveées même quelquefois par ces eaux, doivent peu à peu glisser et descendre en suivant la pente des vallées ou des croupes qu’elles couvrent.” (H.-B. de Saussure, 1779) Already in 1779 H.-B. de Saussure postulated that glaciers slide over their beds. He also reasoned that melt water influences the amount of sliding motion, and eventually lifts up the glacier. Many observations support these ideas. The movement of a glacier over its substrate is an important contribution to the velocity of a glacier or ice sheet with temperate base. The substrate can vary from solid, glacially polished granite to gravel or very fine sediments such as sand, clay and silt. The latter are commonly referred to as glacial till. We use the generic term basal motion to refer to one of the following processes • frictional sliding of the ice over solid bedrock or sediment, • internal deformation of the subglacial sediment, • ice flow around obstacles at the glacier base, • regelation: pressure melting and refreezing at obstacles. Until today there is no theory that describes all of these processes consistently. Basal motion is certainly dependent on basal shear stress and overburden pressure. Water pressure in the drainage system under the glacier and within the sediments is the key factor that controls the amount of basal motion. Since it can vary rapidly, it induces many sudden changes in glacier motion. Many approaches have been taken to quantify the velocity at the glacier base. A sliding law relates the velocity at the glacier base ub to shear stress along the base 87 Chapter 7 Basal motion τb, the stress normal to the interface σn (or the overburden pressure po), and water pressure pw ub = B(material, bed roughness, . ) ·F(τb; σn; pw) : (7.1) An important assumption commonly made is that the sliding relation does not depend on the absolute pressure but only on effective pressure N := σn − pw ∼ po − pw. Many authors have traditionally assumed a power law for F (e.g. Budd et al., 1979; Paterson, 1999) m −r ub = B τb N (7.2) The exponents are assumed to be m = 1; 2 or 3 and r = 0 ::: 1, based on simple theories and measurements. Sliding relations of this kind are usually implemented in flow models of ice sheet and glaciers. A physically more meaningful way to write the sliding relation is 00 τb = F (ub;N) : (7.3) This form is harder to implement in numerical models, especially in the shallow ice approximation. Also, this kind of relation usually allows several values of sliding speed to produce the same basal shear stress, such that no unique solution exists. 7.1 Regelation sliding The first physical theory of glacier sliding was published by Weertman (1957). The theory explains how the glacier can move around solid obstacles at the glacier base. Ice deformation and regelation are the two mechanisms considered. Assume that the glacier base is a plane with little cubes of side length a that are arranged in a square pattern. The distance between two cubes is λ (the wavelength). It is assumed that a thin water film is everywhere between glacier and bedrock, so that no tangential forces (by friction) can be transmitted. Force balance The form drag induced by the obstacles provides the net basal drag that opposes the driving stress. The pressure (normal stress) on an obstacle is calculated with a force balance for one bump and the surrounding area λ2. The pressure is increased by ∆p on the upstream face of an obstacle, and decreased by the same amount on the downstream face 2 2 2 1 λ τb 2 j∆pj a = jτbj λ =) ∆p = τb = : (7.4) 2 a2 2R2 In the last equality we used the bed roughness R := a/λ. The shear stress τb (basal 2 drag) averaged over the area λ has to be equal to the driving stress τd and therefore is τb = ρgH sin α. The overburden pressure is −σm = po = ρgH cos α > 0. 88 Physics of Glaciers HS 2020 0.12 0.10 ) 1 a 0.08 m ( b ur u d 0.06 ud e e u p tot s g n i d 0.04 i l S 0.02 ac 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Obstacle size a (m) Figure 7.1: Left: A sketch to explain the quantities considered in Weertman’s sliding theory. Right: The importance of regelation uR decreases with wave length while ice deformation uD is becoming dominant. Values for R = 0:5 and τb = 0:1 MPa. Regelation Pressure melting occurs at the upstream face of each obstacle, freezing at the downstream face. The difference of melting point temperature between the two faces is dTm ∆Tm = ∆p = γ ∆p (7.5) dp where γ = 7:42 · 10−5 K kPa−1 is the Clausius-Clapeyron constant. The temperature difference induces a heat flux through the obstacle of 2 2∆Tm ∆p λ Q = krock = 2krockγ = krockγ τb : (7.6) a a a3 The speed of basal motion is then determined by the amount of ice that melts in a time unit in front of the obstacles (and refreezes on the lee face) 2 Q krockγ λ krockγ 1 uR = = τb = τb: (7.7) ρL ρL a3 aρL R2 The meltwater flows around the obstacle and releases heat as it refreezes on the low-pressure side. The first equality of Equation (7.7) shows that the flow velocity is controlled by the heat flux. The second equality shows that smaller obstacles, or greater wavelength (i.e. less obstacles) increase the regelation sliding speed. Ice deformation The stress increase in front of the obstacles diminishes with distance from the obstacle. For simplicity it is assumed that the additional pressure −∆p influences ice deformation over a distance a. The velocity due to compression of ice in front of the obstacle is n−1 (d) uD =" _xxa = Aτ σxx a : (7.8) 89 Chapter 7 Basal motion Inserting the mean stress and the shear stress we get σxx = −ρgH cos α ∓ ∆p σyy = σzz = −ρgH cos α 1 σ = −ρgH cos α ∓ ∆p m 3 2 σ(d) = σ − σ = ∓ ∆p xx xx m 3 1 σ(d) = σ(d) = ± ∆p yy zz 3 1 1 τ 2 = σ(d)2 + σ(d)2 + σ(d)2 + (∼ 0) = ∆p2 2 xx yy zz 3 Inserting this in Equation (7.8) we arrive at n+1 2 n 1 1 τb uD = 2aA : (7.9) 3 2 R2 The mean basal shear stress enters in the n-th power as we would expect from a process controlled by ice flow. Sliding velocity The sliding velocity is the sum of both contributions ub = uR + uD. For big obstacles uR is small (proportional to 1=a), while for small obstacles uD (proportional to a) is small. Figure 7.1 (right) shows this relationship qualitatively. The highest resistance to downslope movement of the glacier is obtained when both contributions are equal. The size of the controlling obstacles can be obtained by setting equal Equations (7.7) and (7.9) 1 n−1 n+1 2 1−n krockγ n−1 2 ac = 2 2 3 4 R τ : (7.10) ρLA b Inserting values for temperate ice and for granite (k = 2:1 W m−1 K−1), a rather high bedrock roughness R = 1, and a typical driving stress of τd = 0:1 MPa (which has to be compensated by a τb of equal magnitude) we get a controlling obstacle size of 0:5 m. For lower roughness (i.e. higher wavelength), sliding is dominated by ice deformation. If we assume that the sliding velocity is determined by the controlling obstacles we obtain 1 n+1 1 ! 2 2 3−n − n+1 krockγA τb u = 2 2 3 4 : b ρL R Since n is about 3, the sliding velocity varies as the square of the basal shear stress and inversely as the fourth power of the roughness. The Weertman regelation sliding law is a special case of Equation (7.2) with m = 2 and r = 0. 90 Physics of Glaciers HS 2020 With the same values as above, the sliding speed for the controlling obstacles is 2 m 1 τb ub ∼ 0:625 : (7.11) a R4 MPa For a bedrock roughness of R = 1 and a typical driving stress of τd = 0:1 MPa −3 −1 the sliding speed is ub ∼ 6 · 10 m a . For lower roughness the contribution of regelation sliding is small, and the sliding is dominated by form drag due to ice deformation. 7.2 Sliding over sinusoidal bedrock The simple and beautiful sliding theory of Weertman has been extended to more realistic shapes of the bedrock by Nye (1969, 1970). Of special importance are the sliding over a sinusoidal bed, over spherical bumps and over arbitrary bedrock with a constant wavelength contributions (white roughness spectrum). Here we cite the most important results from Kamb (1970) for two limiting cases. These, and the solutions for any wavelength, can be derived from his equations (92) and (96). 1. limiting case: Great wavelength, regelation negligible. This case is a realistic simplification for a typical smooth glacier bed with wave length of a few meters. n−1 2 2 2 2 (1 + π e R ) n uD = Aλ τ (7.12) 4πn+2en−1Rn+1 b A and n are the parameters of Glen’s flow law.