[PPT] - Glacier Sliding Ian Hewitt, University of Oxford PowerPoint Presentation

SLIDE 1

Glacier Sliding

Ian Hewitt, University of Oxford hewitt@maths.ox.ac.uk

SLIDE 2

Classical sliding theory ( hard bed )

Regelation
Viscous deformation

Soft bed sliding

Till strength
Cavitation
Bed deformation

Sliding / friction laws

Numerical models

SLIDE 3

Rignot Morlighem 2012 Rignot et al 2010

Satellite-derived ice surface speeds

SLIDE 4

Ub τb

Sliding law / Friction law

To calculate ice flow, we need a basal boundary condition.

τb Ub

⇤ · u = 0

0 = ⇥p + ⇥ · τ + ρig

Stokes flow Friction law relates basal shear stress and basal speed

⇤b = f(Ub, . . .)

This is a parameterization of unresolved processes close to the bed.

Ub = |ub| τb = |τ b|

Historically thought of as ‘sliding’ law

Ub = F(⇤b, . . .)

Modern view point

⇤b = f(Ub, . . .)

May be multi-valued

τb ⇡ ρigh ∂s ∂x

Shallow ice approximation

z = s(x, t) z = b(x) h

SLIDE 5

Isaac et al 2015

Numerical ice-sheet models

Most numerical models use a friction law of the form τb = CUm

b

τ b = CUm

b

ub Ub C = C(x, y)

r in vector form

The ‘slipperiness’ is usually treated as a fitting parameter(s), chosen to achieve a good fit with observations of surface velocities. The slipperiness reflects unresolved properties of the bed that may vary with time. We want to understand what physical processes govern variations of the slipperiness.

SLIDE 6

Numerical ice-sheet models

10 20 30 40 50 60 0.00 0.01 0.02 0.03 0.04 0.05 Sea level equivalent (cm) Density Plastic Weertman + Plastic Viscous + Weertman + Plastic a 0.020

Probability density Ritz et al 2016 Sea-level-equivalent mass loss by 2100 (cm) Models show different results depending on the form of the basal friction law.

SLIDE 7

τb N

Sliding with cavitation Power law

ub τb ub τb N

Coulomb plastic

ub/N n

Weertman 1957, Budd et al 1979 Lliboutry 1968, Schoof 2005, Gagliardini et al 2007, Budd et al 1979, Zoet & Iverson 2015 Kamb 1991, Clarke 2005, Iverson 2010

Example friction laws

SLIDE 8

Hard-bedded sliding

SLIDE 9

Hard-bedded sliding

A film of water exists between ice and the underlying bedrock (a few microns thick). Microscopically, free slip is allowed (i.e. ). Macroscopic resistance comes from the roughness of the bedrock ( ).

β ub τ b

Flow over roughness occurs via regelation and viscous (plastic) deformation.

Weertman 1957

τb micro = 0 τb macro = f(Ub)

SLIDE 10

Dimensional analysis, using Glen’s flow law

⇤ = a

‘roughness’

a ⇧

ice rock The ice deforms viscously around obstacles in the bed ice rock

high p low T low p high T

heat flow water flow Regelation: pressure difference across obstacles causes a temperature difference

results in upstream melting and downstream freezing

Balance of conductive / latent heat flow

refreezing melting

Viscous flow and regelation

⇥

UV aA 2n ⇥ ⇧ n

b

⇤2n ⇤

UR = ✓ kΓ ρiLa ◆ τb ν2

SLIDE 11

⇤ = a

a

⇧

ice rock

Viscous flow and regelation

Combining these two mechanisms: There is a ‘controlling obstacle size’ for which stress / speed cross over:

⇥

UV aA 2n ⇥ ⇧ n

b

⇤2n ⇤ ⇤ a ∝ U −(n−1)/(n+1)

b

effective for LARGE bumps effective for SMALL bumps

⇥ ⇧b = ⇤2 R U 2/(n+1)

b

Weertman sliding law

UR = ✓ kΓ ρiLa ◆ τb ν2 − R = ✓ ρiL 2kΓA ◆1/(n+1)

SLIDE 12

2T = 0

4ψ = 0

⇤

Stokes flow Heat equation

ˆ

Zb(k) = lim

M→∞

1 M

⌅ M

−M

Zb(x)eikx dx

A more sophisticated approach to (Newtonian) viscous flow and regelation

via Fourier transform effective ice viscosity

k∗ = ✓ ρiL 4kΓηi ◆1/2 ∼ 2π/50 cm ✓ ◆ ηi ∼ 1/Aτ n−1

b

transitional wavenumber power spectrum of bed profile regelation important for short wavelengths

∼ Zb(x)

SLIDING MOTION OF GLACIERS 679'

melting. This process

f melting

and refreezing is here called 'regelation,' and

its contribution to the sliding process is called 'regelation sliding' (the use

f

the term 'regelation' in this context is discussed by Kamb and LaChapelle [1964,

p. 160]). (2) The

ice responds plastically to the increased normal pressure

n

the upstream faces, arching upward at these points and thereby permit- ting the ice to move forward; correspondingly, it closes down behind the irreg-

ularities, in response to the reduced normal pressure

n the downstream

faces.

m• • ..... * '•"*:^• ^f this process is called 'plastic-flow sliding.' The plastic- flow response

f

the ice is assumed to be that

f a generalized

Newtonian fluid,

rather than plasticity in the strict sense.

The sliding process defined by these specifications is illustrated schematically in Figure 1. We seek a relationship between the sliding velocity v and the drag

Surface z = Zo- FX, y) '"•,•,,,•//' Regelation

BEDROCK layer

Fig. la. Schematic

representation

f ice sliding

with velocity v over an arbitrary bedrock topographic surface z -- Zo (x, y). Coordinate system is as shown, with the x axis in the sliding direction. A cross section

f the

bed in the y direction (perpendicular to the direction

f sliding)

would

be ge{erally similar to the x cross section shown. The regelation layer

consists

f ice formed

by the refreezing

f v•ater

that has migrated along the ice-rock contact from areas

f high normal

pressure. Warping

f the

ice flow vectors near the bedrock surface is an indication

f plastic

de-

formation taking place in the ice.

stress

r basal

shear stress r, as a function

f bed roughness,

appropriately defined. For this purpose the bedrock topography zo(x, y) is Fourier ana- lyzed, and the regelation and plastic-flow contributions to the sliding motion are calculated separately for the individual Fourier components. This analysis

can be carried

ut

rigorously wheil the amplitudes

f

the Fourier components

are small compared to their wavelengths, that is, when the roughness is low enough that the heat and plastic-flow problems can be treated as problems in a half-space. For the plastic-flow problem, a rigorous analysis is possible in the case

f linear rheology

(Newtonian viscosity), and a practical approximation to nonlinearity can'be d•v•loped frbn•.• this,: as a starting point.

T'he

results can

then be appropriately combined to obtain r as a function

f v for sliding

under

the simultaneous

peration
f regelation

and plastic flow. In making this com-

bination, there emerges a eharaeteris•tie length .X•, here called the transition

Kamb 1970

Nye-Kamb theory

Nye 1969, Kamb 1970

τb = ηiUb

k2

∗

π ⇤ ∞ ˆ Zb(k)k3 k2 + k2

∗

dk

⇥

SLIDE 13

β ub τ b

Sliding with cavitation

Lliboutry 1968

Cavitation occurs when pressure on downstream face of bumps reduces to critical level

≈ pc

Sliding law becomes dependent on effective pressure

⇥b = f(Ub, N) N = pi pc = pi (macroscopic) ice

normal stress

≈ pc ≈ pc ≈ pc

Increased

≈ pc

Decreased

≈ pc

Bed Ice High Low

N N

SLIDE 14

⇧b Ub

Lliboutry suggested the sliding relationship was non-monotonic - a ‘multivalued’ sliding law

Sliding with cavitation

Iken suggested there should be a maximum shear stress associated with cavities ‘drowning’ the bed roughness.

Lliboutry 1968, Iken 1981,1983

⇧b = µN Ub

SLIDE 15

Sliding with cavitation

⇥ ⇤ τb = Nf ⇥ Ub N n ⇤

Fowler suggests cavities never really ‘drown’ bed - stress is just transferred to larger bumps

⇥ τb/N Ub/N n τb = CU p

b N q

0 < p, q < 1

‘Generalized’ Weertman law Some experimental support for this law with p = q = 1

3

(Budd et al 1979)

Schoof suggests an alternative with a maximum shear stress

⌅b N = µ

Ub

Ub + ⇥AN n ⇥1/n ⇥ τb/N Ub/N n = µN

Fowler 1986, Schoof 2005, Gagliardini et al 2007

Regularised ‘Coulomb’ law

SLIDE 16

[ken and Bindschadler: Subglacial water pressure and surface velocity

[mm/h] Horizontal velocity

Horizontal velocity [mm/h]

3o-.------------------------------------1 1

------------------------------------------------.-30

a

20 ฀฀

.

::

: ....... ,

..

!

20 10

June

฀฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀

b

I

30 [mm/h]

[mm Ih] r30 Horizontal Velocity Horizontal Velocity i

25 251

!

201-h-..-t=1

฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀

15

Fig. 2. a. Velocity records for poles near the centre line of the glacier. Broken lilies indicate mean

velocities over periods when no short-interval measurements were made.

b. Top: velocity 0/ two poles 0/ profile c. Broken lines indicate mean velocities over periods whell no

short-interval measurements were made. Upper middle: depth of the water level below surface in four bore holes. Lower middle: discharge of the terminal stream (by courtesy of Grande Dixence. S.A .). Broken line: a dam had broken alld discharge data refer to only part of the outlet stream. Bol/om: thick line: air temperature near the glacier terminus ( by courtesy of Grande Dixence. S.A. ). Thin line: temperature of the free atmosphere at the 700 mbar level near Payerne (by courtesy 0/ Schwei;;erische Meteorologische Anstalt). the water level was very roughly, that is within approximately 10 m, at the same depth below surface in all

f them. This applies also to holes 8 and 9 during the brief

period when they had connections.

In the two semi-marginal holes, I and 5, where the ice is only 110 and 85 m thick, respectively, the water levels

tended to be somewhat deeper. That is, the piezometric surface, which approximately paralleled the ice surface in the central part of the glacier, appeared to drop slightly towards the margin; transverse profiles over the ice surface were al- most horizontal in the study area. The record of hole 5,

Fig. 3. Longitudinal sections through the part of glacier
studied. (a) Vertical section along a straight line through

bore h07es

1. 7, and 10. Poles or bore holes which were

located at a significant distance from this section are indicated with broken lines. (b) Section following approximately the Talweg of the glacier and passing through bore hole 4.

m

'""!

2700 2600 2800 2700

2600

2500

A3 D2

10 7 b) I I

:11

I I

I

. ""-7,C7'c7'0

I

500m

103

Iken & Bindschadler 1986

Speed Water pressure

Field measurements

Iken and Bindschadler: Subglacial water pressure and surface velocity

40

35 30 25

20

15

10

[m]

Horizontal Velocity (pole C31

[mm/h]

I

r /.

O'D

.A ;tAr:.:

1.

) :

-
velocity fall 1982

I I

I

180 140 100 60

Depth of water level below glacier

25 [mm / h]

I

1

T/.

C

20

฀฀฀฀

JjC.

..

t x ...

.0.

x

฀฀฀

or

1 5 /11

I I

100 60

C

1 5 - 30 May 1982

฀฀

30 May - 4 June 1982

4 - 20 June 1982

x

1980

aD

I

20

surface

I

20

[m]

Fig. 6. Velocity of pole C3 as a function of the subglacial

water pressure (shown as depth of water level below surface). The water pressure. equal to the ice-overburden pressure 00 at the centre line. corresponds to a depth of water level of 18 m below the surface. Different symbols refer to different periods:

large open symbols indicate that the scatter of depths of

water levels in different bore holes was small. as well as the scatter of velocity values at profiles B to D.

small. full symbols indicate a larger scatter.

upward pointing arrows indicate that cavities were presumably shrinking (water levels were dropping). In this case. the sliding is too small compared to the steady state. downward pointing arrows indicate that cavities were presumably growing. In this case. the sliding velocity is too large compared to the steady state. The lower part of the figure is an enlarged section. c is shown at the top of Figure 2b. (The method of calcul- ation of short-term velocities and of the estimation of tolerances is described in Appendix I.) Velocity variations

f

longer duration and larger amplitude were in general similar at profiles B to D. (An

bvious exception is the marked velocity peak occurring at

profile

D on 7 June.) Velocities measured at profile c,

which was in the centre of the study area, correlate best with the water-level data, also shown in Figure 2b. In Fig- ure 6, the velocity of pole C3 is plotted against the mean depth of water levels in bore holes 3, 4, 6, and 11, which were all between 160 and 175 m deep. The points of this plot were selected subject to certain conditions, as explained in Appendix 11. The plot shows some remarkable features from which important conclusions can be drawn: (I) The points cluster along a distinct curve which appears to have an asymptote where the subglacial water pressure approaches the ice-overburden pressure

r a value close to that. (At its centre line, the glacier

is 180 m deep and

the ice-overburden

pressure is equivalent to that of a column of water 162 m in height; the corresponding water level is 18 m below the surface). (2) A functional relationship is still distinct down to water levels as low as 80 m below the surface. (3) Data points of different periods, indicated by different symbols, are approximately on the same curve. This applies even to the data from the pilot study in June 1980. (Exceptions are the low-pressure points of the period 30 May-4 June 1982.) The first feature will be discussed in a later paragraph. Feature (2) permits a definite conclusion to be drawn on the mechanism of water-pressure dependent motion rele- vant in the study area.

There is no doubt that it is the sliding motion which is affected by the water pressure, and

that the short-term variations of measured surface velocity are related to those of the sliding velocity. Possible mechanisms are: (a) Growth of water-filled cavities at the glacier sole as a function

f

subglacial water pressure (e.g . L1iboutry, 1968, 1978, 1979; Iken, 1981)

r,

in Weertman's (1979) terminology with some modification non-uniform growth of a water layer which has

฀฀

non-uniform and locally substantial thickness. (b) Decoupling of glacier sole and bed where the glacier is afloat. (c) Increased sliding when the pore pressure in subglacial sediments is large enough that deformation of subglacial sediments can take place (Boulton, 1979). In the study area the glacier is up to 180 m thick. It can only be afloat if the water level is not deeper than

18 m below the surface. Clearly, mechanism (b) above is ID-

adequate to explain most of the velocity variations. Deformation of granular sediments (c) is possible if (I) where T is the shear stress, 09 the overburden pressure, Pw the pore pressure in the sediment, and F r is the friction factor, a constant. For densely packed granular sediments, F r is in the range of 0.8-1.1; for loose packing Fr is between 0.6 and 0.7 (Lambe and Whitman, 1979). Near the centre line, at the glacier sole,

00 ฀฀ 15.9 bar - 15.9 x 105 Pa

and T ฀฀I bar ฀฀ !OS Pa. (The latter figure was estimated from glacier geometry. Assuming that the sliding velocity is constant along a transverse profile, one finds with a geometric shape factor of 0.58 and a mean surface slope, taken over 2 km, of 6.5 0 that T ฀฀1.05 bar. Assuming, alternatively, that the basal 105

Iken & Bindschadler 1986

Speed Water pressure Some field work shows a definite relationship between ice speed and borehole water pressure However, a consistent relationship is not always observed

eg. Bindschadler 1983, Iken & Bindschadler 1986
eg. Sugiyama & Gudmundsson 2004, Harper et al 2007, Howat et al 2008, Fudge et al 2009

SLIDE 17

0.4 0.2 0.0 50 Ablation rate, m d–1 Hydraulic head (m), relative to floatation elevation –50 –100 –150 –200 –250 150 160 170 180 190 200 210 220 230 240 Day of year 2012 Weakly connected Channelized Modelled BH4 BH6 BH7 Modelled Moulin3 300 200 100 Sliding speed, m yr–1

a b

Field measurements

Measurements from west Greenland suggest diurnal variations in ice velocity correlate with water pressure in moulins, but are out of phase with pressure in boreholes. Moulin Borehole Ice speed Hoffman et al 2016

SLIDE 18

Soft-bedded sliding

SLIDE 19

Soft-bedded sliding

Subglacial till has a complicated rheology (more complicated than ice)

Boulton & Hindmarsh 1987, Kamb 1991, Tulaczyk 2000, Clarke 2005

Laboratory experiments suggest plastic behaviour, i.e. no deformation beneath a yield stress

τf = c0 + σe tan ψ ⇥ σe = P pw

effective stress

pw ⇥ N tan ψ ⇥ 0.44 ˙ ε = A(τ τf)aσb

e

Visco-plastic model
τ = τf

Stress must be transferred laterally to sticky spots No ‘local’ sliding law in this case When yield stress exceeded, there are two main possibilities: Till layer depth hT

Ub = hTAτ a

b N b

Ub Ub τb = µN

Perfect plasticity

SLIDE 20

Iverson & Zoet 2015

Laboratory experiments

SLIDE 21

iSTARt1 (Bedmap2) 75.4°S, 94.5°W iSTARt1 75.4°S, 94.5°W Outer PI Bay 73.4°S,107.0°W iSTARt9 (Bedmap2) 75.6°S, 99.2°W iSTARt9 75.6°S, 99.2°W Inner PI Bay 74.8°S,102.5°W 5 km 5 k m xy scale for all panels Ice flow all panels

–1500 –950 Elevation (m.a.s.l.) –1200 –500 Elevation (m.a.s.l.) –1500 –950 Elevation (m.a.s.l.) –1100 –700 Elevation (m.a.s.l.) –950 –750 Elevation (m.a.s.l.) –1000 –550 Elevation (m.a.s.l.)

a b c d e f

Bingham et al 2017, Kyrke-Smith et al 2018

High-resolution radar measurements of bed topography

The ‘correct’ friction law and coefficients depend on the resolution of your model (the friction law is to describe unresolved processes!)

SLIDE 22

Summary

Cavities

β ub τ b β ub

Hard bedrock Soft sediments

β ub τ b τ b

⇥

τb = RU 1/m

b

τb = CU p

b N q

τb = µN

⇥

⇧b = µN

Ub

Ub + ⇤AN n ⇥1/n

⇥ τb/N Ub/N n ⇥ τb/N Ub/N n τb Ub