Glacier Hydrology Ian Hewitt, University of Oxford - - PowerPoint PPT Presentation

Glacier Hydrology

Ian Hewitt, University of Oxford hewitt@maths.ox.ac.uk

Subglacial hydrology

• Tunnels / channels
• Cavities
• Canals
• Sheet flow

Water sources

• Surface melting, precipitation
• Basal melting
• Lakes

Large-scale models

Accumulation Surface melting (runoff)

Internal creep

Basal melting / freezing

‘Sliding’

ub us

Glacier hydrology

⌧b ⇠ 100 kPa ub ⇠ 30 m y1 ⇠ Tm T ⇠ Tm T ⇠ Tm T

Basal melting Frozen bed (no sliding?) Basal freezing Geothermal heating Frictional heating

= ⌧bub ⇠ 0.1 W m2 ⇠ m ⇠ 10 mm y1 ) G ⇠ 0.06 W m2

Basal melting

Thermal setting

Pattyn 2010

Basal melting ~ 10 mm/y

Water sources in Antarctica

mm/y

Aschwanden et al 2012

mm yr-1

Basal melting ~ 10 mm/y Surface runoff ~ 1 m/y

van den Broeke et al 2016

Water sources in Greenland

Zs Zb H α ⇤ φ = ρwgZb + pw φ = ρwgZb + ρig(Zs Zb) N N = pi − pw

Hydraulic potential

Direction of subglacial water flow

in terms of effective pressure Potential gradient if water pressure were at overburden

− N = 0

Predominant control on water flow direction from surface slope Potential gradient

∂φ ∂x = Ψ + ∂N ∂x ∂x ∂x Ψ = ρig tan α + (ρw ρi)g tan θ

Increasing water flow

Kamb & LaChapelle 1964, Lliboutry 1968, Walder & Hallet 1979, Alley et al 1986, Creyts & Schoof 2009 Röthlisberger 1972, Nye 1976 Walder & Fowler 1994

Subglacial drainage systems

Weertman film

Weertman 1972, Walder 1982

Poiseuille flux

h

Weertman suggested water could flow as a film Leads to an instability Larger

• h

Water flow dissipates energy through heating Larger flux Melting of ice roof Flow wants to concentrate in localized channels / tunnels However, a patchy film may still exist

• eg. Alley 1989, Creyts & Schoof 2009

✓ ◆ Q = h3 12µ ✓ Ψ + ∂N ∂x ◆

Röthlisberger channels

Ice wall melting is counteracted by viscous creep

Röthlisberger 1972, Nye 1976

N = pi − pw = pi pw

Creep Melting

Neighbouring channels compete with one another leads to an arterial network

w w

Effective pressure INCREASES with discharge Steady state

N ⇡ K3/4

c

ρiL ˜ A !1/n Ψ11/8n Q1/4n

Röthlisberger/Nye theory (ignoring pressure dependence of melting temperature) water mass conservation wall evolution local energy conservation momentum conservation (turbulent flow parameterization)

∂t ∂s ρw ∂S ∂t + ∂Q ∂x = m ρw + M mL = Q ✓ Ψ + ∂N ∂x ◆ Q = KcS4/3 ✓ Ψ + ∂N ∂x ◆1/2 ∂S ∂t = m ρi ˜ ASN n ✓ ◆

Distance km m

5 10 15 20 25 30 35 40 45 −500 500 5 10

m3 s−1

1500

Hydraulic potential Discharge

N

Q

Röthlisberger channels

| Q = Qin pw = pout + Min

Jökulhlaups (GLOFs)

Nye 1976, Spring & Hutter 1981, Clarke 2003

• A success of the Röthlisberger channel theory is the application to ice dammed floods.

AL ρwg ∂N ∂t = mL Q

at Combine channel evolution equation

x = 0

with a lake filling equation

Fowler 2009

∂S ∂t = S4/3Ψ3/2 ρiL ˜ ASN n

Creep Sliding Walder 1986, Kamb 1987

Cavities grow through sliding over bedrock hr

Linked cavities

10 m

Approximate steady-state relationship

N Q < 0 N(Q)

Cavity size is controlled by parameter Λ = Ub

N n

i.e. depends on effective pressure and sliding speed Flow is distributed Smaller ‘orifices’ control the flow Model Effective pressure DECREASES with discharge

∂ ˆ S ∂t = Ubhr ˜ A ˆ SN n

Drainage system stability

Creep Sliding Melting Walder 1986, Kamb 1987, Schoof 2010, Hewitt 2011

A linked cavity system can become unstable to produce channels hr

• eg. if discharge becomes sufficiently large, or sliding speed sufficiently low

Conversely, a channel can become unstable to cavities

• eg. if discharge low, or sliding speed sufficiently high

Energy is still dissipated by water flow

∂S ∂t = m ρi + Ubhr ˜ ASN n

• 10

y (km) y (km) x (km) x (km) 5 10 5

5 10 15 20 0 5 10 15 20

b c e d

Seasonal evolution of drainage system

Schoof 2010

Ice flow Time Network of ‘conduits’ forced by prescribed surface runoff

a

Melt Creep closure

b

Sliding Creep closure

j i

• Conduits

Drainage through sediments

ThT ⌃⇧ ⌃t = ⌅ · (KThT⌅⇧) + m

Water can infiltrate sediment layers and moves according to a diffusion equation Till is relatively impermeable, so easily saturates

Compressibility Melting / freezing

Film flow is unstable (Walder 1982) Canals Patchy sheet

Alley 1989, Creyts & Schoof 2009 Walder & Fowler 1994

Locally deep water film leads to locally more melting

hT

Subglacial lakes

Hundreds of lakes have been detected using radar and satellite observations.

Siegert 2005, Wingham et al 2006, Fricker et al 2007, Stearns et al 2008

‘Active’ lakes grow and drain quite frequently

• through a jokulhlaup-like instability?

Wright & Siegert 2012

The formation and drainage of lakes may be important for ice-stream dynamics. Subglacial lakes are also of great interest to biologists.

On a large scale, distributed systems are described as a ‘sheet’ flow + some additional ingredients to determine water pressure

h q = Khα⌅⇧

Average water depth Average water flux Average water pressure

pw h

Mass conservation

Λ = Λc ∂h ∂t + ⇤ · q = m ρw + M

Ice-sheet modelling

Basal melting Englacial/supraglacial source

y [ km ] x [ km ]

• 230
• 220
• 210
• 200
• 190
• 180
• 170
• 160
• 150
• 140
• 2550
• 2540
• 2530
• 2520
• 2510
• 2500
• 2490

z [ m ]

• 500

500 1000

Combined sheet / channel modelling

Hewitt 2013, Werder et al 2013

Time Ice speed Subglacial discharge (areal m2/s)

Hewitt 2013, EPSL

Summary

Uniform water film is unstable. Röthlisberger channels form arterial networks. Distributed flow in linked cavities or patchy films is possible. Evolution of the drainage system has important consequences for ice dynamics (surges, ice streams, seasonal/diurnal velocity changes)