7 Basal Motion

Home , Basal sliding, Glaciology, Regelation

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 |∆p| a = |τb| 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.

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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

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 =ε ˙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 1 !n+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.

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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. For small roughness R the nominator is 1 + π2e2R2 ≈ 1 (Eq. (7.2) with m = 3 and r = 0).

2. limiting case: Very small wavelength, sliding motion by regelation only (not very realistic). 1 τb uR = (7.13) 2πχR2 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 = 0).

91 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 20 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 2π function zb(x, y) with zero average (for example zb(x) = a sin λ x , if bedrock elevation only varies in the downslope direction, Fig. 7.6). Over a representative area of the glacier bed, the basal drag τb has to balance the driving stress τd = ρgH sin α.

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) 4 -1

2 (mm h

C) 5 o (

-5

200

150

100 (m) WC 50

30 200 vert disp (cm) disp vert 20 100 10

0 0 hor disp (cm)

-100 -10

24 d (cm vel vert ) -1 20 2

16 0 12 -2 8 -1 ) hor vel (cm d 4 -4

1.10.00 8.10.00 15.10.00 22.10.00 29.10.00

Figure 7.2: A mini-surge in autumn 2000 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 vertical93 (green) velocities of a pole in the vicinity of the borehole where the water pressure was recorded. From Helbing (2006). Chapter 7 Basal motion

0.20 Station A Station B Station C 0.10 Station Z

[m] 0.00

-0.10 (a)

-0.20 27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

159 30 100 158 Z 2300 C B A

[km] 2100 157

2500 20 156 656 657 658 659 660 661 662 663 [m/a] [km] 50 [cm/Tag]

10 (b)

0 0 60 27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15

40 10

[m/a] 20

5 [cm/Tag] (c)

0 0

27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 April Mai 1996

Figure 7.3: A speedup event measured at four permanent GPS stations on Unteraargletscher in 1996. 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, VAW).94 Physics of Glaciers HS 2020

1 1 a b c u =76.4 m a− d e f u =71.7 m a− 0 s s

Site FOXX Site GULL 100

th L13 with APaterson ice deformation th th L13 with ACuffey from L13 (ACuffey) 200 th L13 string1 from L13 (APaterson) L13 string2 from L13 (string1 & 2)

300

ice temp.

400 CBC Depth below surface (m) 500 HWT

600 CTS HWT //////////// ///////////// 1 1 ub=55.8 m a− ud=20.6 m a− CTS 1 1 u =31.5 m a− u =40.2 m a− 700 b d //////////////////////////// 10 5 0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 60 70 15 10 5 0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 60 70 80 1 1 1 1 Temperature ( ◦ C) du/dz (a− ) Velocity (m a− ) Temperature ( ◦ C) du/dz (a− ) Velocity (m a− )

1 1 a b c u =76.4 m a− d e f u =71.7 m a− 0 s s

Site FOXX Site GULL 100 th L13 with APaterson ice deformation th th L13 with ACuffey from L13 (ACuffey) 200 th L13 string1 from L13 (APaterson) L13 string2 from L13 (string1 & 2)

300 ice temp.

400 CBC Depth below surface (m) 500 HWT

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. (2014). 95 Chapter 7 Basal motion

a FOXX

) 150 1 − a

m ( 100 y

t surface velocity u i s c o l e

V 50 basal motion ub

ice deformation ud

b GULL

) 150 1 − a

m ( 100 y

t surface velocity u i s c o l e

V 50 basal motion ub

ice deformation ud 100 c 80 FOXX 60 40 GULL 20

Sep Nov Jan Mar May Jul Sep Basal motion ratio (%) 2011 2012 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. (2014).

96 Physics of Glaciers HS 2020

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).

As water pressure pw 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 ps when this happens, we consider an inclined bedrock with periodical obstacles so that the local normal stress on the bedrock is

σn(x, y) = σm + ∆σ(x, y) where mean stress σm = −po = −ρigH 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 Ar, weighted with the direction of the face normal of the bedrock dzb (we only consider the along-ﬂowline part dx ) Z 1 dzb τb = σn(x, y) dxdy . (7.14) Ar Ar dx

A sinusoidal bed of amplitude a and wavelength λ, varying only in ﬂow direction 2π (Fig. 7.6a) is described by zb(x) = a sin λ x . The stress variation is ∆σ(x) = 2π −∆σmax cos λ x (Iken and Bindschadler, 1986). Therefore the average basal drag is Z λ Z λ 1 dzb a 2 2π τb = ∆σn(x) dx = − ∆σmax cos x dx . (7.15) λ 0 dx λ 0 λ As expected, the basal drag increases with bed roughness R = a/λ. Using the R 2 1 integral cos t dt = 2 (t + sin t cos t) evaluated between 0 and λ gives

λτb λ tan α ∆σmax = = −σm . (7.16) aπ a π

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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 λτb λ tan α ps = (−σn)min = −(σm + ∆σmax) = −σm − = −σm 1 − , (7.17) aπ a π

where σm = −ρigH cos α and τb = ρigH sin α have been used. The separation pressure decreases with increasing basal shear stress τb, 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 pc 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 F1 = ρigHλ 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 Fc = ρigHλ sin(β − α) (7.18) The pressure on face c with size c = λ sin β therefore is

Fc sin(β − α) pc = = ρigH (7.19) c sin β

With the identity sin(β − α) = cos α sin β − cos β sin α, and using the overburden pressure po = ρigH cos α = −σo, we obtain the critical pressure sin(β − α) tan α τb pc = po = po 1 − = po − (7.20) cos α sin β tan β tan β

where τb = ρgH sin α is the average basal shear stress, equal to the driving stress. The water pressure pw has to remain below pc such that the following condition, called the Iken’s bound (Iken, 1981), always holds

τb τb = ≤ tan β . (7.21) N po − pc

Field measurements from Findelgletscher, indicating both ps and pc, 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 ps and pc.

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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). 99 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 1982. ps is separation pressure, pc the critical pressure, and p0 is overburden pressure. After Iken and Bindschadler (1986).

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 := po − pw of the form −0.4 us = CN (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)

n−1 n+1 2 n λ N ub = kλ τb . (7.23) a 10 (pc − pw)

Inversion of the sliding law A physically more meaningful sliding relation had been proposed by (Lliboutry, 1968; Fowler, 1986; Lliboutry, 1987), but was only recently revived by Schoof (2005)

100 Physics of Glaciers HS 2020 and Gagliardini et al. (2007). Instead of expressing basal velocity as function of basal shear stress, the average basal resistance (shear stress) τb is expressed in terms of basal speed τ u b = f b N 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 7.10

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 ub/N against τb/N (which corresponds to the ‘classical’ notion of a sliding law as determining ub as a function of τb and N) simply amounts to ﬂipping this graph on its side. The domain of τb/N is then clearly limited (as required by Iken’s bound), and the mapping τb/N → ub/N for τb/N in that domain is in general multivalued. From Schoof (2005)

101 Chapter 7 Basal motion

Figure 7.10: Cavity conﬁgurations for the bed with α = 0.39 whose sliding law is shown in Figure 7.9. Sliding velocities ub/N are (a) 1.5, (b) 3, (c) 12. Contact areas extend over some fairly steep upstream faces even when ub/N = 12. From Schoof (2005)

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 10 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 40 a−1 and reach 1000 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

102 Physics of Glaciers HS 2020

The coeﬃcient of internal friction µ is (Lambe and Whitman, 1979)

µ ≈ 0.8 to 1.2 for dense packing of the grains, µ ≈ 0.6 to 0.8 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 = 0.443 has been measured.

Viscous till Subglacial till is not a viscous material. However in several studies a quasi-viscous constitutive equation of the form

m −r ε˙ = k(τ − τc) N (7.25) has been used. Here k is a material constant and τc 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., 2000a; Iverson et al., 1998). These authors found that the sediments followed a Coulomb failure criterion with almost negligible cohesion co

τf = µ N + c0 = 0.443 N + 1.3 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 ˙0 (7.27) µ N with an exponent of n = 40 ± 20. Notice that even if the form is similar to relation (7.25), the very high exponent and the till shear strength τf make it behave quite diﬀerently. The till shear strength is strongly dependent on the void ratio e := Vw/Vs between water volume Vw and sediment volume Vs. Figure 7.13 summarizes data from in situ tests.

103 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 (2001)

UPB Model

The UPB (undrained plasctic bed) model was conceived by Tulaczyk et al. (2001) 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 τb, sediment water content µ, and void space e. The following assumptions are made

• The glacier bed is at the pressure melting temperature Tm.

• The basal shear stress τb is always equal to the sediment yield strength τf .

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0 Figure 7.12: Till shear strength as function of eﬀective pressure σn (N in our no- tation) measured on till samples from Whillans Ice Stream. From Tulaczyk et al. (2001)

Figure 7.13: Dependence of shear stress at failure τf on void ratio e. From Tulaczyk et al. (2001).

105 Chapter 7 Basal motion

• The speed of the ice stream of width W and thickness H depends on the driving stress τd and the basal shear stress τb

A 3 4 −3 ub = (τd − τb) W H (7.28) 2

• The melt rate at the glacier base is controlled by frictional dissipation and the heat fluxes QG (geothermal heat flux) and QI (heat flux from base into ice)

−1 µ˙ = (ubτb + QG − QI ) L (7.29)

1 • The change rate in void space within the sediment layer of thickness Hs is

−1 e˙ = (1 + e)µ ˙ Hs (7.30)

• From the experimental relation between basal shear strength and void ratio τf = a exp(−b e) (with a = 944 kPa and b = 21.7; Fig. 7.13), and the assumption τb = τf , it follows that τ˙b = −b e˙ τb . (7.31)

⊕ µ ub ⊕ ⊕ ⊖

e ⊖ τb ⊕ ⊖

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 = Vw/Vs, and noting that Vs is constant

V˙w V˙w e˙ = = Vs + Vw Vs+Vw 1 + e Vs Vs

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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 τb 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. (2000b))

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