Physics of Glaciers, Chapter 5: Glacier Flow, Part 2

Martin Lüthi HS 2020 Glacier Flow, Side Friction (Mer de Glace 1908)

E. Spelterini 1 Glacier Flow Through a Cylindrical Channel

• glaciers usually flow through bedrock channels which provide resistive forces • we investigate how valley walls affect the velocity field • consider a glacier in a semi-circular valley with radius R and slope α • using a cylindrical coordinate system with coordinates x, r and ϕ

• main stress components: σxx, σrr, σϕϕ

• shearing stress components: σrx = σxr, σxϕ = σϕx, σrϕ = σϕr R ∆x ϕ r x

2 Force Balance

• the body force from a slice is balanced by tractions acting on the circumference • for an ice volume of length ∆x at distance r from the center the force balance is

πr2 σ πr∆x = −ρg ∆x sin α , rx 2 1 σ = − r ρg sin α . (5.28) rx 2 R ∆x ϕ r x

3 Velocity and Stress Components

• the only non-zero velocity component is in x-direction • therefore most components of the strain rate tensor vanish • assumption of no stress gradients in x-direction • assumption of equal shear stress along the cylindrical perimeter (possibly inaccurate)

(d) (d) (d) ε˙rr =ε ˙xx =ε ˙ϕϕ = 0 =⇒ σrr = σxx = σϕϕ = 0 (5.29) (d) (d) and also σrϕ = σxϕ = 0 (5.30)

• second invariant of the deviatoric stress tensor 1 τ = σ = |σ | = rρg sin α e rx 2

4 Glacier Flow Velocity

• with Glen’s ﬂow law, the shear deformation rates are 1 du 1 n =ε ˙ = Aτ n−1σ = −A ρg sin α rn 2 dr rx rx 2

• integration with respect to r between 0 and R, with boundary condition u(R) = 0

1 n Rn+1 u(0) − u(R) = 2A ρg sin α (5.31) 2 n + 1

• the channel radius R is equivalent to the ice thickness H, therefore

1n u = u (5.32) def, channel 2 def, slab

The ﬂow velocity on the center line of a cylindrical channel is eight times slower than in an ice sheet of the same ice thickness.

5 Velocity and Ice Flux

• the velocity at any radius is

1 n rn+1 u(r) = u(0) − 2A ρg sin α (5.33) 2 n + 1

• the ice ﬂux through a cross section is (for uR = 0)

Z R πR2 2A 1 n Z R q = u(r) πr dr = u(0) − ρg sin α π rn+2 dr 0 2 n + 1 2 0 πR2 2 πR2 = u(0) − u(0) 2 n + 3 2 πR2 2 πR2 n + 1 = u(0) 1 − = u(0) (5.34) 2 n + 3 2 n + 3

6 Mean Velocity

¯ πR2 • the mean velocity within the cross section is deﬁned using q = u¯ 2 and is n + 1 u¯ = u(0) (5.35) n + 3

• the mean velocity at the glacier surface is

1 Z R 2A 1 n Rn+1 n + 1 u¯ = u(r) dr = u(0) ρg sin α = u(0) (5.36) R 0 n + 1 2 n + 2 n + 2

R ∆x ϕ r x

7 Flow Through a Parabolic or Elliptic Channel

(d) (d) (d) (d) • we assume no stress gradients in x-direction, and σxx = σyy = σzz = σyz = 0. • the equation system (5.22c) reduces to

∂σ ∂σ xy + xz = −ρg sin α ∂y ∂z ∂σ zz = −ρg cos α ∂z

q 2 2 • from τ = σxy + σxz and the ﬂow law the stresses and the velocity u is calculated • numerical results for ﬂow velocity and stress from Nye (1965) in dimensionless quantities

8 Flow Through a Parabolic or Elliptic Channel

• Z is across glacier, Y is along z (opposite sign), H maximum channel depth

z σxz σxz vertical Y = TY = = H σxz, slab(z = 0) ρgH sin α H − y σ across Z = T = xy H Z ρgH sin α half width u(y, z) W = U = H 4uslab

Nye (1965) 9 Flow Through a Parabolic or Elliptic Channel

• parametrization of the side friction with shape factor Sf 2A u = (S ρg sin α)n zn+1 + u (5.38) center n + 1 f b

• comparison with the velocity of a circular channel (Eq. 5.33) gives Sf = 0.5 • this is an ellipse with W = 1 • values for diﬀerent shapes are tabulated

SF W parabola semi-ellipse rectangle 1 0.445 0.500 0.558 2 0.646 0.709 0.789 3 0.746 0.799 0.884 4 0.806 0.849 ∞ 1 1 1 Nye (1965) 10 Flow Through Parabolic Channels

Nye (1965) 11 Black Rapids Glacier: measurements/modeling along transsect

Velocity variations Modeled basal stress • along a transsect • winter: dotted • during melt the season • summer: dashed

Amundson and Truﬀer (2006) 12 Black Rapids Glacier: model results

left: velocity (a/b/c) • modeled velocity in • winter • spring • summer • units: m/a

right stress (d/e/f) • modeled octahedral stress τ • units: 100 kPa = 1 bar

13 Black Rapids Glacier: stress transfer during summer

Stress diﬀerences Stress diﬀerences • spring stress state • summer stress state • octahedral stress wrt. winter • octahedral stress wrt. winter • units: 100 kPa • shaded areas: lower stress than in winter

Amundson and Truﬀer (2006) 14 Crevasses: Types

Colgan (2016) 15 Crevasses: Processes for Formation

Crevasse formation • caused by divergent ﬂow (extension) • longitudianl fractures • crossing fractures • shearing fractures • crevasse formation if critical extension rate or critical stress is reached • increasing ice velocity → extension cracks • decreasing ice velocity → crevasse closure • convex surface or large bedrock ondulations lead to crevasse formation

Hansbreen 16 Crevasses: Conditions for Formation

Bedingungen für Spalten • crevasses open perpendicular to direction of maximum extensional stress • critical stress for crevasse formation

τc ' 50 − 120 kPa (temperate ice) • additional inﬂuence of normal stresses and shear stresses • crevasses are advected with glacier ﬂow

Aletschgletscher 17 Crevasses: Deformation Fields / Stress Fields

Colgan (2016) 18 Longitudinal and Radial Crevasses

Spannungen • longitunial crevasses by ﬂow over bump • radial crevasses at glacier terminus (paw shaped) • concentric crevasses above cavities

Morteratsch- / Rhonegletscher 19 Marginal Crevasses

Direction of Marginal Crevasses • Marginal Crevasses from from margin upward • rapid ﬂow in glacier center leads to extensional stress at angle of 45 degrees • crevasse opening perpendicular to extensional stress → crevasse orientation 45 degrees upstream • downstream rotated of marginal crevasses by glacier ﬂow

Rhonegletscher, 1899 20 Bergschrund

Bergschrund • formation of a upper marginal crevasse bergschrund where ice ﬂow starts • above: thin ice, often frozen to ground • below: thick ice, glacier ﬂow • avalanche snow often collects in bergschrund

21 Crevasses: Overview of formation processes

22 Crevasse Depth

• crevasses are tension cracks that usually form from the surface down • crevasse depth is limited by compressive stress at depth

• stretching rate ε˙xx at glacier surface corresponds to a deviatoric stress (Nye, 1955, 1957)

1 ε˙ n σ(d) = xx xx A

• vertical normal stress at depth h is σzz = −ρgh • bottom of crevasse at depth with no horizontal tension σxx = 0 (d) 1 1 • using the mean stress we also ﬁnd σxx = − 2 σzz = 2 ρgh • equating these expressions gives

1 2 ε˙ n h = xx (5.39) cr ρg A

• deepest crevasses occur in areas of rapid extension • very deep crevasses in cold ice since A decreases with temperature 23 Water-ﬁlled Crevasses and Critical Strain Rate

• improved relationships by introducing a critical strain rate ε˙crit • ponding water in crevasses changes stress ﬁeld (Benn, 2007)

" 1 # n 2 ε˙xx − ε˙crit hcr = + ρwghw (5.40) ρig A

• further approaches use Linear Elastic Fracture Mechanics (LEFM)