frictional sliding of the ice over solid bedrock or sediment, internal deformation of the subglacial sediment,

Transcription

1 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 u b to shear stress along the base 77

2 Chapter 7 Basal motion τ b, the stress normal to the interface σ n (or the overburden pressure p o ), and water pressure p w u b = B(material, bed roughness,... ) F(τ b, σ n, p w ). (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 := σ n p w p o p w. Many authors have traditionally assumed a power law for F (e.g. Budd et al., 1979; Paterson, 1999) u b = B τb m N r (7.2) The exponents are assumed to be m = 1, 2 or 3 and r =... 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 τ b = F (u b, 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 λ 2. The pressure is increased by p on the upstream face of an obstacle, and decreased by the same amount on the downstream face 2 p a 2 = τ b λ 2 = p = 1 λ 2 2 a τ 2 b = τ b 2R 2. (7.4) In the last equality we used the bed roughness R := a/λ. The shear stress τ b (basal drag) averaged over the area λ 2 has to be equal to the driving stress τ d and therefore is τ b = ρgh sin α. The overburden pressure is σ m = p o = ρgh cos α >. 78

3 Physics of Glaciers I HS Sliding speed ub (m a 1 ) u r u d u tot.2 a c 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 u R decreases with wave length while ice deformation u D is becoming dominant. Values for R =.5 and τ b =.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 T m = dt m p = γ p (7.5) dp where γ = K kpa 1 is the Clausius-Clapeyron constant. The temperature difference induces a heat flux through the obstacle of Q = k rock 2 T m a = 2k rock γ p a = k rockγ λ2 a 3 τ b. (7.6) 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) u R = Q ρl = k rockγ ρl λ 2 a τ 3 b = k rockγ aρl 1 R 2 τ b. (7.7) 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 u D = ε xx a = Aτ n 1 σ (d) xx a. (7.8) 79

4 Chapter 7 Basal motion Inserting the mean stress and the shear stress we get σ xx = ρgh cos α p σ yy = σ zz = ρgh cos α σ (d) yy σ m = ρgh cos α 1 3 p σ (d) xx = σ xx σ m = 2 3 p = σ (d) zz = ± 1 3 p τ 2 = 1 2 ( σ (d)2 xx + σ (d)2 yy Inserting this in Equation (7.8) we arrive at u D = 2aA ( ) n+1 ( ) + σ zz (d)2 1 + ( ) = 3 p2 τ b R 2 ) n. (7.9) 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 u b = u R + u D. For big obstacles u R is small (proportional to 1/a), while for small obstacles u D (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) a c = 2 n 1 n ( ) 1 krock γ 2 R n 1 τ 1 n 2 b. (7.1) ρla 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 τ d =.1 MPa (which has to be compensated by a τ b of equal magnitude) we get a controlling obstacle size of.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 u b = 2 3 n 2 3 n+1 krock γa 2 τ 1 n 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 =. 8

5 Physics of Glaciers I HS 217 With the same values as above, the sliding speed for the controlling obstacles is u b.625 m a 1 ( τb ) 2. (7.11) R 4 MPa For a bedrock roughness of R = 1 and a typical driving stress of τ d =.1 MPa the sliding speed is u b m a 1. 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, 197). 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 (197) 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. u D = (1 + π2 e 2 R 2 ) n 1 2 4π n+2 e n 1 R Aλ τ n n+1 b (7.12) A and n are the parameters of Glen s flow law. For small roughness R the nominator is 1 + π 2 e 2 R 2 1 (Eq. (7.2) with m = 3 and r = ). 2. limiting case: Very small wavelength, sliding motion by regelation only (not very realistic). 1 τ b u R = (7.13) 2πχR 2 λ The quantity χ contains all material parameters of ice and rock and is χ = 12 MPa a m 2 (Eq. (7.2) with m = 1 and r = ). 81

6 Chapter 7 Basal motion 7.3 Sliding and water pressure Basal motion of Alpine glaciers is highest in spring and early summer (May-June) when meltwater production is increasing. Despite higher meltwater input, sliding speeds are generally lower in summer (July-August). Röthlisberger and Aellen (1967) explained this observation with the principle that not the amount of meltwater, but the water pressure at the glacier base is important. The water pressure is highest in spring when the subglacial drainage system is not yet well developed (Röthlisberger, 1972). In the summer months sudden motion events sometimes called mini-surges happen, usually coinciding with raising water pressure at the glacier base. The measurements presented in Figure 7.2 are an instructive example of such an event on Unteraargletscher. After a big rainfall and warm temperatures water enters the subglacial drainage system and locally raises the water pressure to above overburden pressure. The glacier is lifted by 2 cm and at the same time accelerates considerably. This in turn leads to a reduction of water pressure, the glacier decelerates and couples to the bedrock again. Figure 7.3 shows another speedup event that was measured on two of four permanent GPS stations. While station B and C show a strong reaction, A and D remain quiet. The velocity event seems to move down glacier, first affecting C and a little later also B. Vertical velocity peaks before the horizontal velocity, especially at C. Cavities and separation from the bedrock Elevated water pressure at the base affects the sliding speed in several ways by increasing the area of separation of ice from the bed, therefore increasing the shear traction on the parts still in contact with the bed, by exerting a net downglacier force on the roofs of subglacial cavities, by weakening deforming subglacial till over which the glacier is moving. Like in the Weertman theory above, we assume that the contact between ice and glacier bed is frictionless, so that tangential forces cannot be transferred. Think of the ice being separated from the bedrock by a thin water film everywhere. All basal drag has to be provided by form drag of ice pushing on upstream facing parts of the bedrock. We consider a glacier with surface inclination α, and a tilted coordinate system so that the x-axis follows the mean bedrock. The bedrock is described by an oscillating function z b (x, y) with zero average (for example z b (x) = a sin ( 2π λ x), if bedrock 82

7 Physics of Glaciers I HS (mm h -1 ) ( o C) (m) WC hor disp (cm) vert disp (cm) -1-1 hor vel (cm d -1 ) vert vel (cm d -1 ) Figure 7.2: A mini-surge in autumn 2 on Unteraargletscher. (a) Hourly precipitation and (b) air temperature measured at a nearby weather station. (c) Water pressure measured in a borehole. (d) Horizontal (red for x-direction, blue for y-direction) and vertical (green) displacements and (e) horizontal (red) and vertical 83 (green) velocities of a pole in the vicinity of the borehole where the water pressure was recorded. From Helbing (26).

8 Chapter 7 Basal motion.2.1 Station A Station B Station C Station Z [m]. [m/a] (a) [km] Z 23 C B A [km] (b) [cm/tag] [m/a] 2 5 [cm/tag] (c) April Mai 1996 Figure 7.3: A speedup event measured at four permanent GPS stations on Unteraargletscher in Shown are (a) vertical displacement, (b) horizontal velocity and (c) vertical velocity. Notice that the acceleration takes place during the uplift of the glacier surface. (Unpublished data from Gudmundsson, 84 VAW).

9 Physics of Glaciers I HS 217 a b c u s =76.4 m a 1 d Depth below surface (m) ice temp. CBC Site FOXX L th 13 with A Paterson L th 13 with A Cuffey L 13 string1 L 13 string2 HWT ice deformation from L13 th (A Cuffey ) from L13 th (A Paterson ) from L 13 (string1 & 2) e Site GUL 6 //////////// ///////////// CTS u b =55.8 m a 1 u d =2.6 m a Temperature ( C) du/dz (a 1 ) Velocity (m a 1 ) Temperature ( C) ////////////////////////////.1.2 du/dz c u s =76.4 m a 1 d e f u s =71.7 m a 1 X Site GULL A Paterson A Cuffey g1 g2 ice deformation from L13 th (A Cuffey ) from L13 th (A Paterson ) from L 13 (string1 & 2) HWT.3.4 (a 1 ) CTS u b =55.8 m a 1 u d =2.6 m a Velocity (m a 1 ) Temperature ( C) //////////////////////////// du/dz (a 1 ) HWT CTS u u d =4.2 m a 1 b =31.5 m a Velocity (m a 1 ) Figure 7.4: Measured ice deformation (panels b, e) and basal motion (panels c, f) at two sites on the Greenland ice sheet. Notice that measured deformation profiles do not agree with theory. This is due to horizontal stress transfer. From Ryser et al. (214). 85

10 Chapter 7 Basal motion Velocity (m a 1 ) a surface velocity u s basal motion u b FOXX ice deformation u d Velocity (m a 1 ) b surface velocity u s basal motion u b GULL Basal motion ratio (%) c Sep 211 ice deformation u d Nov Jan 212 FOXX GULL Mar May Jul Sep Time Figure 7.5: Measured contributions of ice deformation and basal motion to the surface velocity at two sites on the Greenland ice sheet throughout a year. Notice that ice deformation is reduced in summer when basal motion is high. From Ryser et al. (214). 86

11 Physics of Glaciers I HS 217 Figure 7.6: Left: The geometry used to calculate the separation pressure. Right: The force components used to calculate the critical water pressure. From Iken (1981). elevation only varies in the downslope direction, Fig. 7.6). Over a representative area of the glacier bed, the basal drag τ b has to balance the driving stress τ d = ρgh sin α. As water pressure p w rises, the glacier first separates from the bedrock in the lee of obstacles because the normal stress is lowest there. To calculate the separation water pressure p s when this happens, we consider an inclined bedrock with periodical obstacles so that the local normal stress on the bedrock is σ n (x, y) = σ m + σ(x, y) where mean stress σ m = p o = ρ i gh cos α is equal to the (negative) overburden pressure, and σ is the fluctuating contribution that is negative on the stoss faces of the bumps and positive on the lee faces (the signs are due to the convention that compression ˆ= negative). The basal drag is the sum of all normal forces over a representative area A r, weighted with the direction of the face normal of the bedrock (we only consider the along-flowline part dz b ) dx τ b = 1 σ n (x, y) dz b dxdy. (7.14) A r A r dx A sinusoidal bed of amplitude a and wavelength λ, varying only in flow direction (Fig. 7.6a) is described by z b (x) = a sin ( 2π x). The stress variation is σ(x) = λ σ max cos ( 2π x) (Iken and Bindschadler, 1986). Therefore the average basal drag λ is τ b = 1 λ λ σ n (x) dz b dx dx = a λ λ σ max cos 2 ( 2π λ x ) dx. (7.15) As expected, the basal drag increases with bed roughness R = a/λ. Using the integral cos 2 t dt = 1 (t + sin t cos t) evaluated between and λ gives 2 σ max = λτ b aπ = σ λ tan α m a π. (7.16) 87

12 Chapter 7 Basal motion Separation from the bed occurs if the water pressure exceeds the lowest normal stress, occurring at the lee sides of the bumps. The separation pressure is p s = ( σ n ) min = (σ m + σ max ) = σ m λτ ( b aπ = σ m 1 λ ) tan α, (7.17) a π where σ m = ρ i gh cos α and τ b = ρ i gh sin α have been used. The separation pressure decreases with increasing basal shear stress τ b, and increases with increasing roughness R or obstacle size a. The process of bed separation is illustrated in Figure 7.7. During formation of the cavity a considerable amount of water is stored under the glacier. Consequently the shear traction increases in those parts of the glacier sole still in contact with the bedrock, which leads to increased ice deformation rates there (eightfold for double shear stress). If the subglacial water pressure rises further, a critical pressure p c will be reached above which the glacier accelerates without bounds. Such acceleration will occur if the water pressure is sufficient to push the ice up the steepest slope of the bedrock. On the simplified geometry of (Fig. 7.6b) the weight force of the ice block F 1 = ρ i ghλ is split into its components along and perpendicular to the steepest bedrock restraining bedrock face (labeled b in the Figure) with inclination angle (β α). The force that the ice exerts on the steepest bedrock face (labeled c in the Figure) is F c = ρ i ghλ sin(β α) (7.18) The pressure on face c with size c = λ sin β therefore is p c = F c c = ρ sin(β α) igh sin β (7.19) With the identity sin(β α) = cos α sin β cos β sin α, and using the overburden pressure p o = ρ i gh cos α = σ o, we obtain the critical pressure ( sin(β α) p c = p o cos α sin β = p o 1 tan α ) = p o τ b (7.2) tan β tan β where τ b = ρgh sin α is the average basal shear stress, equal to the driving stress. The water pressure p w has to remain below p c such that the following condition, called the Iken s bound (Iken, 1981), always holds τ b N = τ b p o p c tan β. (7.21) Field measurements from Findelgletscher, indicating both p s and p c, are shown in Figure 7.8. The mismatch between theory and observation may be explained by the nonuniform sizes of bedrock obstacles that would raise the value of p s and 88

13 Physics of Glaciers I HS 217 Figure 7.7: Demonstration of bed separation with a numerical model (only the bottom part of the geometry is shown). Top: Water pressure is lower than separation pressure, no cavities form. Middle: Water pressure exceeds separation pressure, a cavity starts forming and the whole ice mass is pushed downslope. Bottom: The water-filled cavity reaches its steady size. Notice that a considerable part of the glacier loses the contact to the bedrock. From Iken (1981). 89

14 Chapter 7 Basal motion Figure 7.8: Measurements of the speed of a stake on the surface of Findelgletscher as a function of water pressure: (a) theoretical values, (b) measured in p s is separation pressure, p c the critical pressure, and p is overburden pressure. After Iken and Bindschadler (1986). p c. Iken and Bindschadler (1986) also emphasize the lack of rock-to-rock friction in the theory. A later study at the same location (Iken and Truffer, 1997) observed a very different behavior with a much lower susceptibility of the glacier to pressure changes. A study by Jansson (1995) finds that a relation between surface speed and effective pressure N := p o p w of the form u s = C N.4 (7.22) fits the observations on Findelgletscher and similar measurements on Storglaciären (Sweden). A more elaborate formula which extends Equation (7.12) to include water pressure was proposed by Truffer and Iken (1998) u b = kλ τ n b Inversion of the sliding law ( ) n+1 ( λ a N 1 (p c p w ) ) n 1 2. (7.23) A physically more meaningful sliding relation had been proposed by (Lliboutry, 1968; Fowler, 1986; Lliboutry, 1987), but was only recently revived by Schoof (25) and 9

15 Physics of Glaciers I HS 217 Gagliardini et al. (27). Instead of expressing basal velocity as function of basal shear stress, the average basal resistance (shear stress) τ b is expressed in terms of basal speed τ ( b N = f ub ) N The resulting sliding law is shown in Figure 7.9, the configuration of the water-filled subglacial cavities for different sliding speeds in Figure 7.1 Figure 7.9: Sliding laws calculated for different beds. Sliding laws are only shown after the onset of cavitation. Note that smaller values of α correspond to more irregular beds. α = in the legend denotes a sinusoidal bed. Note also that plotting u b /N against τ b /N (which corresponds to the classical notion of a sliding law as determining u b as a function of τ b and N) simply amounts to flipping this graph on its side. The domain of τ b /N is then clearly limited (as required by Iken s bound), and the mapping τ b /N u b /N for τ b /N in that domain is in general multivalued. From Schoof (25) 91

16 Chapter 7 Basal motion Figure 7.1: Cavity configurations for the bed with α =.39 whose sliding law is shown in Figure 7.9. Sliding velocities u b /N are (a) 1.5, (b) 3, (c) 12. Contact areas extend over some fairly steep upstream faces even when u b /N = 12. From Schoof (25) 7.4 Movement over and within subglacial sediments Granular sediments such as gravels and sand with an important fraction of clay can be found under most glaciers. Under some glaciers and ice streams sediment layers of up to 1 m have been found (e.g. Black Rapids Glacier, Whillans Ice Stream, Breiðamerkurjökull). These sediments deform under applied shear stress if they contain a large volume fraction of water at very high pressure. Strain rates in till often exceed 4 a 1 and reach 1 a 1 in the Siple Coast ice streams. Till rheology Subglacial till is an almost cohesion-less frictional material. It can deform only as individual grains slide past each other. For this to happen the material dilates and additional pore space is created that is filled by pressurized water. Friction between sediment particles becomes lower as the effective pressure between the particles decreases (i.e. the water pressure increases). Grainy sediments can only be sheared if the ratio between shear stress and effective pressure reaches a threshold value τ µ. (7.24) N 92

17 Physics of Glaciers I HS 217 The coefficient of internal friction µ is (Lambe and Whitman, 1979) µ.8 to 1.2 for dense packing of the grains, µ.6 to.8 for loose packing of the grains. Sediments with plate structure (clay) are easier deformable than grainy sediments. For the exceptionally soft sediments that underlay the Siple Coast ice streams (Antarctica) a value of µ =.443 has been measured. Viscous till Subglacial till is not a viscous material. However in several studies a quasi-viscous constitutive equation of the form ε = k(τ τ c ) m N r (7.25) has been used. Here k is a material constant and τ c is a critical shear stress below which no deformation occurs. Usually 1 < m < 2 is assumed so that the sensitivity of the strain rate to stress is even lower than for ice. Relation (7.25) has been proposed to interpret sediment deformation data from Breiðamerkurjökull, Iceland. Figure 7.11 shows that the interpretation can also be accomplished with a plastic till rheology. Important: The till flow relation (7.25) is entirely intuitive and has not been justified by field or laboratory data. Plastic till Extensive field and laboratory studies have shown that the subglacial till under the Siple Coast ice streams is a very soft treiboplastic (frictional plastic) material (Tulaczyk et al., 2a; Iverson et al., 1998). These authors found that the sediments followed a Coulomb failure criterion with almost negligible cohesion c o τ f = µ N + c =.443 N kpa. (7.26) The till behaves almost like a perfectly plastic material, but a rate of deformation that depends on stress level was detected. This can be expressed as a quasi-viscous flow law ( ) n τf ε = ε (7.27) µ N with an exponent of n = 4 ± 2. Notice that even if the form is similar to relation (7.25), the very high exponent and the till shear strength τ f make it behave quite differently. The till shear strength is strongly dependent on the void ratio e := V w /V s between water volume V w and sediment volume V s. Figure 7.13 summarizes data from in situ tests. 93

18 Chapter 7 Basal motion Figure 7.11: a) Till deformation measured in a tunnel under Breiðamerkurjökull, Iceland (Boulton and Hindmarsh, 1987). b) The data can also be explained with a plastic till rheology. From Iverson and Iverson (21) UPB Model The UPB (undrained plasctic bed) model was conceived by Tulaczyk et al. (21) to explain some important aspects of the Siple Coast ice streams in Antarctica. These ice streams are underlain by a thick layer of fine-grained sediment which is at a water pressure close to overburden pressure. The model is a nice example of a dynamical system with quite some predictive power. The model predicts that the and ice stream resting on weak till can be in one of two possible steady state configurations: fast motion, or almost stagnant. The UPB model is formulated as dynamical system (evolution equations) in the variables basal shear stress τ b, sediment water content µ, and void space e. The following assumptions are made The glacier bed is at the pressure melting temperature T m. The basal shear stress τ b is always equal to the sediment yield strength τ f. 94

19 Physics of Glaciers I HS 217 Figure 7.12: Till shear strength as function of effective pressure σ n (N in our notation) measured on till samples from Whillans Ice Stream. From Tulaczyk et al. (21) Figure 7.13: Dependence of shear stress at failure τ f on void ratio e. From Tulaczyk et al. (21). 95

20 Chapter 7 Basal motion The speed of the ice stream of width W and thickness H depends on the driving stress τ d and the basal shear stress τ b u b = A 2 (τ d τ b ) 3 W 4 H 3 (7.28) The melt rate at the glacier base is controlled by frictional dissipation and the heat fluxes Q G (geothermal heat flux) and Q I (heat flux from base into ice) µ = (u b τ b + Q G Q I ) L 1 (7.29) The change rate in void space within the sediment layer of thickness H s is 1 ė = (1 + e) µ H 1 s (7.3) From the experimental relation between basal shear strength and void ratio τ f = a exp( b e) (with a = 944 kpa and b = 21.7; Fig. 7.13), and the assumption τ b = τ f, it follows that τ b = b ė τ b. (7.31) µ u b e τ b Figure 7.14: Relations between the quantities of the UPB model. Solid lines indicate evolution equations, dashed lines algebraic equations, and dotted lines dependence on the change rate. 1 with help of the definition e = V w /V s, and noting that V s is constant V w V s + V w = V ( w ) = V V s+v w s V s ė 1 + e 96

21 Physics of Glaciers I HS 217 Figure 7.15: (a) Basal melt rate m r (our µ) as a function of the basal strength calculated for the cross section of Ice Stream B in the UpB area (Eq. 7.29). The open and solid circle are plotted where the condition of basal melt rate equal to zero is met (undrained bed model). (b) Ice stream velocity bed strength curve calculated from Equation (7.28) for the same cross section of Ice Stream B. The open and solid circles are drawn for the same value of the bed strength τ b as in panel (a). They indicate the stable equilibrium and the linearly unstable equilibrium of the UPB ice stream model. The arrows in both panels show the directions in which the UPB system migrates between the equilibrium states. (From Tulaczyk et al. (2b)) 97

22

23 Bibliography Benn, D. and Evans, D. (1998). Glacier and Glaciations. Arnold, a member of the Hodder Headline Group, 338 Euston Road, London NW1 3BH. Boulton, G. S. and Hindmarsh, R. C. A. (1987). Sediment deformation beneath glaciers: Rheology and geological consequences. Journal of Geophysical Research, 92(B9): Budd, W. F., Keage, P. L., and Blundy, N. A. (1979). sliding. Journal of Glaciology, 23(89): Empirical studies of ice Cuffey, K. and Paterson, W. (21). The Physics of Glaciers. Elsevier, Burlington, MA, USA. ISBN Dahl-Jensen, D., Mosegaard, K., Gundestrup, N., Clow, G. D., Johnsen, S. J., Hansen, A. W., and Balling, N. (1998). Past temperatures directly from the greenland ice sheet. Science, 282: Duval, P. (1977). The role of water content on the creep rate of polycristalline ice. In Isotopes and impurities in snow and ice, pages International Association of Hydrological Sciences. Publication No Fowler, A. C. (1986). A sliding law for glaciers of constant viscosity in the presence of subglacial cavitation. Proceedings of the Royal Society of London, Ser. A, 47: Funk, M., Echelmeyer, K., and Iken, A. (1994). Mechanisms of fast flow in Jakobshavns Isbrae, Greenland; Part II: Modeling of englacial temperatures. Journal of Glaciology, 4(136): Gagliardini, O., Cohen, D., Raback, P., and Zwinger, T. (27). Finite-element modeling of subglacial cavities and related friction law. Journal of Geophysical Research, 112(F2). Harrison, W. D. (1975). Temperature measurements in a temperate glacier. Journal of Glaciology, 14(7):

24 Chapter 7 BIBLIOGRAPHY Helbing, J. (26). Glacier dynamics of Unteraargletscher: Verifying theoretical concepts through flow modeling. Mitteilungen 196, Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie der ETH Zürich, Gloriastrasse 37-39, ETH Zürich, CH-892 Zürich. pp Hutter, K. (1983). Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets. D. Reidel Publishing Company/Tokyo, Terra Scientific Publishing Company. Iken, A. (1981). The effect of the subglacial water pressure on the sliding velocity of a glacier in an idealized numerical model. Journal of Glaciology, 27(97): Iken, A. and Bindschadler, R. A. (1986). Combined measurements of subglacial water pressure and surface velocity of Findelengletscher, Switzerland: Conclusions about drainage system and sliding mechanism. Journal of Glaciology, 32(11): Iken, A., Echelmeyer, K., Harrison, W. D., and Funk, M. (1993). Mechanisms of fast flow in Jakobshavns Isbrae, Greenland, Part I: Measurements of temperature and water level in deep boreholes. Journal of Glaciology, 39(131): Iken, A. and Truffer, M. (1997). The relationship between subglacial water pressure and velocity of Findelengletscher, Switzerland, during its advance and retreat. Journal of Glaciology, 43(144): Iverson, N. R., Hooyer, T. S., and Baker, R. W. (1998). Ring-shear studies of till deformation: Coulomb-plastic behavior and distributed strain in glacier beds. Journal of Glaciology, 44(148): Iverson, N. R. and Iverson, R. M. (21). Distributed shear of subglacial till due to coulomb slip. Journal of Glaciology, 47(158): Jansson, P. (1995). Water pressure and basal sliding on Storglaciären, northern Sweden. Journal of Glaciology, 41(138): Kamb, B. (197). Sliding motion of glaciers: Theory and observation. Reviews of Geophysics and Space Physics, 8(4): Lambe, T. W. and Whitman, R. V. (1979). Soil mechanics. John Wiley and Sons, New York. Lliboutry, L. A. (1968). General theory of subglacial cavitation and sliding of temperate glaciers. Journal of Glaciology, 7(49): Lliboutry, L. A. (1987). Realistic, yet simple bottom boundary conditions for glaciers and ice sheets. Journal of Geophysical Research, 92(B9):

25 Physics of Glaciers I HS 217 Lüthi, M. P. and Funk, M. (21). Modelling heat flow in a cold, high altitude glacier: interpretation of measurements from Colle Gnifetti, Swiss Alps. Journal of Glaciology, 47(157): Lüthi, M. P., Funk, M., Iken, A., Gogineni, S., and Truffer, M. (22). Mechanisms of fast flow in Jakobshavns Isbrae, Greenland; Part III: measurements of ice deformation, temperature and cross-borehole conductivity in boreholes to the bedrock. Journal of Glaciology, 48(162): Nye, J. F. (1969). A calculation on the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proceedings of the Royal Society of London, Ser. A, 311(156): Nye, J. F. (197). Glacier sliding without cavitation in a linear viscous approximation. Proceedings of the Royal Society of London, Ser. A, 315(1522): Paterson, W. S. B. (1999). The Physics of Glaciers. Butterworth-Heinemann, third edition. Röthlisberger, H. (1972). Water pressure in intra and subglacial channels. Journal of Glaciology, 11(62): Röthlisberger, H. and Aellen, M. (1967). Annual and monthly velocity variatons on Aletschgletscher. In Paper presented at I.U.G.G., I.A.S.H. General Assembly, Berne, 23 September - 7 October Ryser, C. (29). The polythermal structure of Grenzgletscher, Valais, Switzerland. Masterarbeit, Abteilung für Glaziologie, VAW (unpublished), ETH-Zürich. Ryser, C., Lüthi, M., Andrews, L., Hoffman, M., Catania, G., Hawley, R., Neumann, T., and Kristensen, S. S. (214). Sustained high basal motion of the Greenland Ice Sheet revealed by borehole deformation. Journal of Glaciology, 6(222): Ryser, C., Lüthi, M., Blindow, N., Suckro, S., Funk, M., and Bauder, A. (213). Cold ice in the ablation zone: its relation to glacier hydrology and ice water content. Journal of Geophysical Research, 118(F26): Schoof, C. (25). The effect of cavitation on glacier sliding. Proc. R. Soc. A, 461: Smith, G. D. and Morland, L. W. (1981). Viscous relations for the steady creep of polycrystalline ice. Cold Regions Science and Technology, 5: Truffer, M. and Iken, A. (1998). The sliding over a sinusiodal bed at high water pressure. Journal of Glaciology, 44(147):

26 Appendix BIBLIOGRAPHY Tulaczyk, S., Kamb, B., and Engelhardt, H. F. (21). Estimates of effective stress beneath a modern West Antarctic ice stream from till preconsolidation and void ratio. Boreas, 3(2): Tulaczyk, S., Kamb, W. B., and Engelhardt, H. F. (2a). Basal mechanics of Ice Stream B, West Antarctica 1. till mechanics. Journal of Geophysical Research, 15(B1): Tulaczyk, S., Kamb, W. B., and Engelhardt, H. F. (2b). Basal mechanics of Ice Stream B, West Antarctica 2. undrained plastic bed model. Journal of Geophysical Research, 15(B1): Weertman, J. (1957). On the sliding of glaciers. Journal of Glaciology, 3(21):33 38.