Models of ice sheet dynamics and meltwater lubrication Ian Hewitt, - - PowerPoint PPT Presentation

Models of ice sheet dynamics and meltwater lubrication

Ian Hewitt, University of Oxford Thanks to: Christian Schoof, Mauro Werder, Gwenn Flowers, John Fell Fund, ERC

30°W 60°W 60°W 60°W 90°W 120°W 120°E 120°E 120°E 90°E 90°E 90°E 7 ° S 8 ° S 60°E 60°E 60°E 30°E 0°E 150°W 150°E 150°E 150°E 180°E

Ice velocity (m year–1)

1,000 100 10 <1.5

B B' C' C

Ocean Ice Land 4,000 2,000 (MSL) 0

• 2,000

Elevation (m) West Antarctica

B' B

Ronne Ice Shelf Ellsworth Mountains Bentley Subglacial Trench Ross Ice Shelf Vertical exaggeration x80 Gunnerus Bank

• 2,000

(MSL) 0 2,000 4,000

Elevation (m)

C C'

Gamburtsev Subglacial Mountains Vincennes Subglacial Basin Astrolabe Subglacial Basin

East Antarctica

Vertical exaggeration x80 Vostok Subglacial Highlands Aurora Subglacial Basin

Antarctica

Ice sheets in climate models

Most climate models have a static ice sheet The ice sheet stores and releases water according to surface energy balance Some interesting feedbacks are (at least potentially!) not captured

Models of ice sheet dynamics

Stokes flow

r · u ∂τij ∂xj ∂p ∂xi = ρgi r · u = 0

Ice Bedrock

pni + τijnj = 0 ◆

x z Dij = 1 2 ✓∂ui ∂xj + ∂uj ∂xi ◆ n ⇡ 3 ⇡ A = A(T)

Rheology

q τij = A−1/nD1/n−1Dij ✓ D = q

1 2DijDij

τ b = f(Ub)ub Ub

surface bed Boundary conditions bed

pni + τijnj = 0 ◆

surface

(δij ninj)τjknk = f(Ub)(δij ninj)ui Ub z = s z = b ∂s ∂t + u∂s ∂x + v ∂s ∂y = w + a u ∂b ∂x + v ∂b ∂y = w − m

Reduced models (depth-integrated)

• ∂H

∂t + r · (Hu) = a m Z ∂s ∂t r ·

• (s b)n+2|rs|n−1rs
• = a m

Shallow ice approximation Shallow stream approximation

z = s z = b z = b z = s f(Ub) u Ub = ρgH ∂s ∂x + ∂ ∂x  ηH ✓ 4∂u ∂x + 2∂v ∂y ◆ + ∂ ∂y  ηH ✓∂u ∂y + ∂v ∂x ◆ f(Ub) v Ub = ρgH ∂s ∂y + ∂ ∂x  ηH ✓∂u ∂y + ∂v ∂x ◆ + ∂ ∂y  ηH ✓ 2∂u ∂x + 4∂v ∂y ◆

Inverse methods

Inversion for basal slipperiness (frozen-time problem)

f(Ub) = βUb

Assume a linear friction law Basal boundary condition Forward problem: given geometry, temperature & basal slipperiness, determine ice velocity

τ b = β(x)ub

Inverse problem: use observed ice velocity to infer basal slipperiness

J (β) = 1 2 Z |u − uobs|2 dV + R(β)

Stokes flow

r · u ∂τij ∂xj ∂p ∂xi = ρgi r · u = 0

Isaac et al 2015

Inferred basal slipperiness Observations Model Surface velocity

Inversion for basal slipperiness

−500 500 1,000 2,000 1,500 Bed elevation (m) 6 ° N 65° N 70° N 75° N 80° N 70° W 60° W 50° W 40° W 3 ° W 300 km 20 km 20 km 50 km

c b a d e f

30 km 30 km 20 km

Inversion for bed topography (assumes steady state)

Morlighem et al 2014

Future

Assimilate time-dependent observations Make use of more observations - eg. internal radar layers

Duncan Young / UTIG

Sliding

Microscopic view of a ‘hard’ bed

β ub τ b

A film of water exists between ice and the underlying bedrock, allowing slip Resistance comes from the roughness of the bedrock

β ub τ b =

‘Cavitation’ occurs in the lee of bumps

Nye’s sliding theory

Take Fourier transform

⇤ k∗ = ρiL 4kCη ⇥1/2

• 4ψ = 0

⇤

• τb = ηiUb

k2

∗

π ⇤ ∞ ˆ Zb(k)k3 k2 + k2

∗

dk

• ⇥

Stokes flow

ν 1

Fourier transform of bed profile, i.e. power spectrum

• ˆ

Zb(k) = lim

M→∞

1 M

• ⌅ M

−M

Zb(x)eikx dx

• Nye 1969, Fowler 1986, Schoof 2005
• 4ψ = 0

⇤

Stokes flow

ν 1 ⌅

Theory can be extended to account for cavitation Riemann-Hilbert problem

→∞

• ⌅

−

τb = Nf ⇥Ub N ⇤ ⇥ ⇤

Newtonian ice

2T = 0

Heat equation

β ub τ b β ub

Sliding laws

Hard bedrock Cavities Soft sediments

β ub τ b τ b

• ⇥

τb = RU 1/m

b

τb = µN

• ⇥

⇧b = µN

• Ub

Ub + ⇤AN n ⇥1/n

⇥ τb/N Ub/N n ⇥ τb/N Ub/N n τb Ub

(plastic)

τb = CN qU p

b

Subglacial water

A distributed drainage model

h ∂h ∂t + r · q = m + r φ = ρgb + pw q = K(h)rφ z = s z = b φ ⇡ ρgs + (ρw ρ)gb

Basal water ‘Water table’ Ice surface

10 m

Turbulent dissipation causes channelisation

S

h

‘Wormholing’ Mechanical erosion Chemical erosion (eg limestone caves) Analogues

Cordillera Blanca, Peru

Creep Melting

Subglacial conduits

Röthlisberger 1972, Nye 1976

Cross-sectional area evolution

M = 1 ρwL ⇤ ⇤ ⇤ ⇤Q∂φ ∂s ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ∂S ∂t = ρw ρi M − 2A nn S|N|n−1N ⇤ ⇤

Melting

∂S ∂t + ∂Q ∂s = M + qin

Mass conservation Energy

T ≈ 2.5 K

Flux

⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ Q = −KcSα ⇤ ⇤ ⇤ ⇤ ∂φ ∂s ⇤ ⇤ ⇤ ⇤

−1/2 ∂φ

∂s

z = s z = b ⇡

• φ = ρgs + (ρw ρ)gb N

gH ⇠ cpT ⇠ gH ⇠ φL ⇠ H ⇡ 1 km ⇡ φ ⇡ 0.03

Creep Melting

∂S ∂t C2S α > 1

Unstable equilibrium Mass conservation prevents unbounded growth ... but neighbouring channels compete with one other

Q ∂S ∂t = C1SαΨ3/2 C2SN n

Subglacial conduits

Schoof 2010

a

Melt Creep closure

b

Sliding Creep closure j i

• Conduits

Conduit network

• Water flow

Increasing steady input

∂t

• ∂S

∂t = C1SαΨ3/2 C2SN n + C3Ub

S

h

A hybrid sheet-conduit drainage model

5 10 15 20

t = 50d

a

4 8 8

10 20 30 40 50 5 10 15 20

t = 500d

c

4 8 x (km)

5 10 15 20

t = 150d

b

4 8 y (km)

60 10 20 30 40 50 60

Werder et al 2013 Water flow

A hybrid sheet-conduit drainage model

Meltwater lubrication

Seasonal ice velocity modulation

Sole et al 2013

Time Ice speed

Runoff [ mm / d ] Time J A J O J

Subglacial drainage model coupled to ice flow model

Annual cycle of surface melt runoff routed into moulins τb = CUbN

Time Ice speed Subglacial discharge (areal m2/s)

Hewitt 2013, EPSL

Landforms

Bridgenorth Esker

Storrar et al 2014

Eskers beneath Laurentide ice sheet

Summary Lots of interesting fluid phenomena going on in ice sheets! Inverse methods increasingly useful for fitting models to data

• assimilating over time is the next step

Forecasting models still do not resolve important feedbacks

• detailed studies of these needed

Temperature

Ice temperature

✓ ◆ ρwL ✓∂φ ∂t + u · rφ ◆ + ρwLr · j = τij ˙ εij, T = Tm, φ > 0 ρc ✓∂T ∂t + u · rT ◆ = r · (krT) + τij ˙ εij, φ = 0, T  Tm ✓ ◆

T [ C ] φ [ % ]

• 10
• 5

2 4

T [ C ] φ [ % ]

• 10
• 5

2 4

‘Standard’ enthalpy gradient model Compaction pressure model

Polythermal ice