Physics of Glaciers, Chapter 5: Glacier Flow Martin Lthi HS 2020 - - PowerPoint PPT Presentation

Physics of Glaciers, Chapter 5: Glacier Flow

Martin Lüthi HS 2020

Introduction: Description of Glacier Flow

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Flow of Glaciers

Martin Lüthi

Introduction: Flow of Glaciers, Deformation of Ice and Tensors

Day Topic C.P. P. H. Lecturer 21.9. Ice sheets, sea level, shallow ice equation 14 1, 2 2 fw 28.9. Mass balance, time scales 3, 4 3 3, 14 fw 5.10. Glacier seismology 11.5 13 14 fw 12.10. Deformation of ice, stress, strain 3 5 9 ml 19.10. Flow of glaciers 8 11 4, 5 ml 26.10. Flow of glaciers, crevasses 8 12 10,12 ml 1.11. Temperatures, heat flow 9 10 6 ml 9.11. Advection, polythermal glaciers 9 7 7 ml 16.11. Basal motion, subglacial till 7 14 (3, 8) ml 23.11. Glacier hydraulics 12, 8.8 14 (3, 8) mw 30.11. Glacier hydraulics 12, 8.8 14 (3, 8) mw 7.12. Glacier hydraulics, Jökulhlaups 6 6 8 mw 14.12. Tidewater glaciers and calving 6 6 8 gj

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Ice Physics

Properties of Ice

• Solid ice has twelve (!) different phases
• fundamentally different crystal structure
• two amorphous states
• phases depend on temperature, pressure

and crystallization history

• under atmospheric conditions only ice Ih

(phase I in a hexagonal lattice)

• evidence of ice Ic (cubic lattice) in very

cold, high altitude clouds x

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Polycrystalline Glacier Ice

• Glacier ice is a polycrystalline material
• hexagonal ice crystals deform readily on

their basal plane (lines in right graphic)

• crystal orientations are initially random
• during deformation and growth of new

crystals, fabric formation occurs

• new crystals are oriented favourably to

stress regime

• Model of ice deformation
• (a) axial shortening by 29 %
• (b) axial extension by 33 %
• (c) pure shear by 38 %
• (d) simple shearing γ = 0.72

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Zhang (1994)

Deformation of Polycrystalline Glacier Ice

A polycrystalline material deforms due to many processes which change the structure, average grain size and polycrystal fabric:

• dislocation climb/glide
• grain boundary migration
• grain rotation
• dynamic recrystallization
• polygonization (subdivision

into independent grains)

• subgrain formation
• nucleation (creation of

new grains)

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• Ch. Wilson

Deformation of Polycrystalline Glacier Ice

• A-B: under stress the ice immediately

deforms elastically

• elastic strain can be recovered
• B-C-D-E: creep / viscous flow continues

as long as stress is applied

• creep / viscous flow is dissipative and

strains are permanent

• crystals are completely recrystallized after

1-3 % strain

• only secondary and tertiary creep is

considered for glacier flow

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Budd and Jacka (1989)

Rheology: (Combination of) Elastic, Viscous and Plastic Responses

elastic ε ∝ σ

• immediate response
• reversible: all strain

is recovered viscous ˙ ε ∝ σ

• constant rate of deformation
• continuous response
• strain is irreversible
• often constant viscosity
• or rate-dependent viscosity

plastic ε = F(σ − σth)

• threshold yield stress σth
• immediate response

above threshold

• strain is irreversible
• strain hardening or

softening

• various yield surfaces
• various flow rules

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• G. Jouvet

Flow Relation for Polycrystalline Ice: Viscous Flow

The widely used flow relation (Glen’s flow law) for glacier ice is (Glen, 1952; Nye, 1957) ˙ εij = A τ n−1 σ(d)

ij

(5.1)

• power law exponent n ∼ 3
• rate factor A(T) depends strongly on temperature
• rate factor A also depends on ice water content, fabric, . . .
• τ = σe is the effective shear stress,

i.e. the second invariant of the deviatioric stress tensor from Equation (4.14) τ = σe = 1 2σ(d)

ij σ(d) ij

1

2

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Important Properties of Glen’s Flow Law

• elastic effects are neglected. Good approximation for time scales of days and longer
• stress and strain rate are collinear: shear stress leads to shearing strain rate
• only deviatoric stresses lead to deformation rates
• isotropic pressure induces no deformation.
• glacier ice is incompressible (no volume change, except for elastic compression)

˙ εii = 0 ⇐ ⇒ ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 0

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Important Properties of Glen’s Flow Law

• elastic effects are neglected. Good approximation for time scales of days and longer
• stress and strain rate are collinear: shear stress leads to shearing strain rate
• only deviatoric stresses lead to deformation rates
• isotropic pressure induces no deformation.
• glacier ice is incompressible (no volume change, except for elastic compression)

˙ εii = 0 ⇐ ⇒ ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 0

• a Newtonian viscous fluid (like water) is characterized by the shear viscosity η

˙ εij = 1 2η σ(d)

ij .

(5.2) Comparison with Equation (5.1) gives the viscosity of glacier ice η = 1 2Aτ n−1 .

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More Important Properties of Glen’s Flow Law

Implicit assumptions and approximations of Glen’s flow law:

• polycrystalline glacier ice is a viscous fluid with a stress dependent viscosity
• or, equivalently, a strain rate dependent viscosity
• such a material is called a non-Newtonian fluid, or more specifically a power-law fluid
• polycrystalline glacier ice is treated as isotropic fluid:

no preferred deformation direction due to crystal orientation fabric

• crude approximation: in reality glacier ice is anisotropic (although to varying degrees)

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More Important Properties of Glen’s Flow Law

Implicit assumptions and approximations of Glen’s flow law:

• polycrystalline glacier ice is a viscous fluid with a stress dependent viscosity
• or, equivalently, a strain rate dependent viscosity
• such a material is called a non-Newtonian fluid, or more specifically a power-law fluid
• polycrystalline glacier ice is treated as isotropic fluid:

no preferred deformation direction due to crystal orientation fabric

• crude approximation: in reality glacier ice is anisotropic (although to varying degrees)

More accurate flow laws exist, but they are

• more complex, difficult to implement in models
• underconstrained by measurements
• strongly deformation history dependent (e.g. crystal fabric)

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Inversion of the flow relation

• Glen’s flow law (Eq. 5.1) can be inverted
• stresses are expressed in terms of strain rates
• multiplying Equation (5.1) with itself gives

˙ εij ˙ εij = A2τ 2(n−1)σ(d)

ij σ(d) ij

(multiply by 1 2) 1 2 ˙ εij ˙ εij

˙ ǫ2

= A2τ 2(n−1) 1 2σ(d)

ij σ(d) ij

• τ 2
• with effective strain rate ˙

ǫ = ˙ εe (analogous to τ = σe) ˙ ǫ :=

• 1

2 ˙ εij ˙ εij . (5.3)

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Inversion of the flow relation (2)

• this leads to a relation between tensor invariants

˙ ǫ = Aτ n (5.4)

• this is the equation for simple shear (most important deformation mode in glaciers)

˙ ǫxz = Aσ(d)

xz n .

(5.5)

• the flow relation Equation (5.1) can be inverted using Eq. (5.4) to replace τ

σ(d)

ij = A−1τ 1−n ˙

εij σ(d)

ij = A−1A

n−1 n

˙ ǫ− n−1

n

˙ εij σ(d)

ij = A− 1

n ˙

ǫ− n−1

n

˙ εij . (5.6)

• comparison with Equation (5.2) shows that the shear viscosity is

η = 1 2A− 1

n ˙

ǫ− n−1

n .

(5.7)

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Inversion of the flow relation (3)

• polycrystalline ice is a strain rate softening material:

viscosity decreases as the strain rate increases

• calculation of stress state from the strain rates possible

e.g. from field measurements

• only deviatoric stresses can be calculated from deformation rates
• the mean stress (pressure) cannot be determined

because of the incompressibility of ice

• the mean stress will be determined by solving the full

continuum force balance equations for a given geometry

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Finite viscosity

• the shear viscosity in Equation (5.7) becomes infinite for small strain rates

due to negative power of ˙ ǫ

• this is unphysical
• fix the problem by adding a small quantity ηo ⇒ finite viscosity

η−1 = 1 2A− 1

n ˙

ǫ− n−1

n

−1 + η−1

0 .

(5.8)

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Simple Stress States: Simple Shear

• play with Glen’s flow law (Eq. 5.1)
• investigate simple, yet important stress states
• homogeneous stress state on small samples of ice

e.g.~in the laboratory

• only external surface forces
• neglecting body forces

(a) Simple shear in the xz-plane forcing : σxz ˙ εxz = A(σ(d)

xz )3 = Aσ3 xz

(5.9)

• This stress regime applies near the base of a glacier.

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Simple Stress States: Unconfined Uniaxial Compression

(b) Unconfined uniaxial compression along the z-axis forcing : σzz σxx = σyy = 0 σ(d)

zz = 2

3σzz; σ(d)

xx = σ(d) yy = −1

3σzz ˙ εxx = ˙ εyy = −1 2 ˙ εzz = −1 9Aσ3

zz

˙ εzz = 2 9Aσ3

zz

(5.10)

• easy to investigate in laboratory experiments
• applies in the near-surface layers of an ice sheet
• deformation rate is 22 % of that at a shear stress of equal

magnitude (Eq. 5.9)

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Simple Stress States: Confined Uniaxial Compression

(c) Uniaxial compression confined in the y-direction forcing : σzz σxx = 0; ˙ εyy = 0; ˙ εxx = − ˙ εzz σ(d)

yy = 1

3 (2σyy − σzz) = 0; σyy = 1 2σzz σ(d)

xx = −σ(d) zz = −1

3 (σyy + σzz) = −1 2σzz ˙ εzz = 1 8Aσ3

zz

(5.11)

• typical for the near-surface layers of a valley glacier
• ice shelf occupying a bay

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Simple Stress States: Shear Combined with Unconfined Uniaxial Compression

(d) Shear combined with unconfined uniaxial compression forcing : σzz and σxz σxx = σyy = σxy = σyz = 0 σ(d)

zz = 2

3σzz = −2σ(d)

xx = −2σ(d) yy

τ 2 = 1 3σ2

zz + σ2 xz

˙ εzz = −2 ˙ εxx = −2 ˙ εyy = 2 3Aτ 2σzz ˙ εxz = Aτ 2σxz (5.12)

• This stress configuration applies at many places in ice sheets.

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