Glacier Flow

Physics of Glaciers, Chapter 5: Glacier Flow

Martin Lüthi HS 2020 Introduction: Description of Glacier Flow

Flow of Glaciers Martin Lüthi

1 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 2 Ice Physics

Properties of Ice x • 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

3 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

Zhang (1994) 4 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)

Ch. Wilson 5 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

Budd and Jacka (1989) 6 Rheology: (Combination of) Elastic, Viscous and Plastic Responses

elastic ε ∝ σ viscous ε˙ ∝ σ plastic ε = F(σ − σth)

• immediate response • constant rate of deformation • threshold yield stress σth • reversible: all strain • continuous response • immediate response is recovered • strain is irreversible above threshold • often constant viscosity • strain is irreversible • or rate-dependent viscosity • strain hardening or softening • various yield surfaces • various ﬂow rules

G. Jouvet 7 Flow Relation for Polycrystalline Ice: Viscous Flow

The widely used ﬂow relation (Glen’s ﬂow law) for glacier ice is (Glen, 1952; Nye, 1957)

n−1 (d) ε˙ij = A τ σij (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)

1 1 2 τ = σ = σ(d)σ(d) e 2 ij ij

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

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

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) ∂v ∂v ∂v ε˙ = 0 ⇐⇒ x + y + z = 0 ii ∂x ∂y ∂z

9 Important Properties of Glen’s Flow Law

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

9 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)

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)

10 More Important Properties of Glen’s Flow Law

Implicit assumptions and approximations of Glen’s ﬂow law:

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)

10 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

2 2(n−1) (d) (d) 1 ε˙ijε˙ij = A τ σ σ (multiply by ) ij ij 2 1 1 ε˙ ε˙ = A2τ 2(n−1) σ(d)σ(d) 2 ij ij 2 ij ij | {z } | {z } ˙2 τ 2

• with eﬀective strain rate ˙ =ε ˙e (analogous to τ = σe) r1 ˙ := ε˙ijε˙ij . (5.3) 2

11 Inversion of the ﬂow relation (2)

• this leads to a relation between tensor invariants ˙ = Aτ n (5.4) • this is the equation for simple shear (most important deformation mode in glaciers) (d)n ˙xz = Aσxz . (5.5) • the ﬂow relation Equation (5.1) can be inverted using Eq. (5.4) to replace τ

(d) −1 1−n σij = A τ ε˙ij (d) n−1 n−1 −1 n − n σij = A A ˙ ε˙ij (d) 1 n−1 − n − n σij = A ˙ ε˙ij . (5.6) • comparison with Equation (5.2) shows that the shear viscosity is

1 − 1 − n−1 η = A n ˙ n . (5.7) 2 12 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

13 Finite viscosity

• the shear viscosity in Equation (5.7) becomes inﬁnite for small strain rates due to negative power of ˙ • this is unphysical

• ﬁx the problem by adding a small quantity ηo ⇒ ﬁnite viscosity −1 −1 1 − 1 − n−1 −1 η = A n ˙ n + η . (5.8) 2 0

14 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 (d) 3 3 ε˙xz = A(σxz ) = Aσxz (5.9)

• This stress regime applies near the base of a glacier. 15 Simple Stress States: Unconﬁned Uniaxial Compression

(b) Unconﬁned uniaxial compression along the z-axis

forcing : σzz

σxx = σyy = 0 2 1 σ(d) = σ ; σ(d) = σ(d) = − σ zz 3 zz xx yy 3 zz 1 1 ε˙ =ε ˙ = − ε˙ = − Aσ3 xx yy 2 zz 9 zz 2 3 ε˙zz = Aσ (5.10) 9 zz

• 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) 16 Simple Stress States: Conﬁned Uniaxial Compression

forcing : σzz

σxx = 0;ε ˙yy = 0;ε ˙xx = −ε˙zz 1 1 σ(d) = (2σ − σ ) = 0; σ = σ yy 3 yy zz yy 2 zz 1 1 σ(d) = −σ(d) = − (σ + σ ) = − σ xx zz 3 yy zz 2 zz 1 3 ε˙zz = Aσ (5.11) 8 zz

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

17 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 2 σ(d) = σ = −2σ(d) = −2σ(d) zz 3 zz xx yy 1 τ 2 = σ2 + σ2 3 zz xz 2 ε˙ = −2ε ˙ = −2ε ˙ = Aτ 2σ zz xx yy 3 zz 2 ε˙xz = Aτ σxz (5.12)

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