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 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 4 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) 5 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 6 Budd and Jacka (1989)

Rheology: (Combination of) Elastic, Viscous and Plastic Responses elastic ε ∝ σ viscous ˙ ε ∝ σ plastic ε = F ( σ − σ th ) • immediate response • constant rate of deformation • threshold yield stress σ th • reversible: all strain • continuous response • immediate response is recovered above threshold • strain is irreversible • strain is irreversible • often constant viscosity • strain hardening or • or rate-dependent viscosity softening • various yield surfaces • various flow rules 7 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) ε ij = A τ n − 1 σ ( d ) ˙ (5.1) ij • power law exponent n ∼ 3 • rate factor A ( T ) depends strongly on temperature • rate factor A also depends on ice water content, fabric, . . . • τ = σ e is the effective shear stress, i.e. the second invariant of the deviatioric stress tensor from Equation (4.14) � 1 � 1 2 2 σ ( d ) ij σ ( d ) τ = σ e = ij 8

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 x ∂x + ∂v y ∂y + ∂v z ε ii = 0 ˙ ⇐ ⇒ ∂z = 0 9

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 x ∂x + ∂v y ∂y + ∂v z ε ii = 0 ˙ ⇐ ⇒ ∂z = 0 • a Newtonian viscous fluid (like water) is characterized by the shear viscosity η ε ij = 1 2 η σ ( d ) ˙ (5.2) ij . Comparison with Equation (5.1) gives the viscosity of glacier ice 1 η = 2 Aτ n − 1 . 9

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: • 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) 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 ( multiply by 1 ε ij = A 2 τ 2( n − 1) σ ( d ) ij σ ( d ) ε ij ˙ ˙ 2) ij 1 = A 2 τ 2( n − 1) 1 2 σ ( d ) ij σ ( d ) 2 ˙ ε ij ˙ ε ij ij � �� � � �� � ǫ 2 ˙ τ 2 • with effective strain rate ˙ ǫ = ˙ ε e (analogous to τ = σ e ) � 1 ǫ := ˙ 2 ˙ ε ij ˙ ε ij . (5.3) 11

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) n . ǫ xz = Aσ ( d ) ˙ (5.5) xz • the flow relation Equation (5.1) can be inverted using Eq. (5.4) to replace τ ij = A − 1 τ 1 − n ˙ σ ( d ) ε ij σ ( d ) n − 1 ǫ − n − 1 ij = A − 1 A ˙ ε ij ˙ n n σ ( d ) ij = A − 1 ǫ − n − 1 n ˙ ε ij . ˙ (5.6) n • comparison with Equation (5.2) shows that the shear viscosity is η = 1 2 A − 1 ǫ − n − 1 n ˙ n . (5.7) 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 ˙ ǫ • this is unphysical • fix the problem by adding a small quantity η o ⇒ finite viscosity � 1 � − 1 η − 1 = ǫ − n − 1 2 A − 1 + η − 1 n ˙ (5.8) 0 . n 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 : σ xz xz ) 3 = Aσ 3 ε xz = A ( σ ( d ) ˙ (5.9) xz • 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 : σ zz σ xx = σ yy = 0 zz = 2 yy = − 1 σ ( d ) σ ( d ) xx = σ ( d ) 3 σ zz ; 3 σ zz ε yy = − 1 ε zz = − 1 9 Aσ 3 ε xx = ˙ ˙ 2 ˙ zz ε zz = 2 9 Aσ 3 ˙ (5.10) 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 (c) Uniaxial compression confined in the y -direction forcing : σ zz σ xx = 0; ε yy = 0; ˙ ε xx = − ˙ ˙ ε zz yy = 1 σ yy = 1 σ ( d ) 3 (2 σ yy − σ zz ) = 0; 2 σ zz zz = − 1 3 ( σ yy + σ zz ) = − 1 σ ( d ) xx = − σ ( d ) 2 σ zz ε zz = 1 8 Aσ 3 ˙ (5.11) zz • typical for the near-surface layers of a valley glacier • ice shelf occupying a bay 17