Basal processes and geomorphology Ian Hewitt, University of Oxford - - PowerPoint PPT Presentation

Basal processes and geomorphology

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

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Sediments and sliding

• Till rheology
• Deformation

Drainage in sediments

• Darcy flow
• Canals

Geomorphology

• Meltwater deposits
• Deformational deposits

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Sediments and sliding

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Basal sediments

Many glaciers and ice sheets have a layer of sediments (till) at the bed

• eg. Siple coast ice streams, Iceland, Svalbard

Samples suggest a poorly sorted spectrum of grain sizes Sediments result from glacial erosion (eg. abrasion, plucking)

Hooke & Iverson 1995

Sometimes the water may be frozen (it may be hard to say where ‘ice-till’ interface is!) Till is often water saturated Deformation (fractal dimension ~2.9) Midtre Lovenbreen, Glaciers Online

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Ub τb τb Ub

Sliding over sediments

β ub τ b β ub τ b β ub τ b hT β ub τ b hT

‘Sliding’ could involve:

• Shear deformation of sediment

layer

• Slip at the ice-till interface
• Slip on slip-planes within the

sediment layer

• Shear of a finite horizon of the

sediment

β ub τ b

Macroscopic resistance may come from flow around sediment landforms

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Till rheology

Yield stress depends on effective stress

Iverson et al 1998, Clarke 2005 Hooke & Iverson 1998, Kamb 1991, Iverson 2011

Laboratory experiments on samples show that till has a yield stress

τf = c0 + µσe ⇥ σe = P pw pw ⇥ N

Effective pressure at ice-till interface

• µ = tan ψ ⇥ 0.4

Coefficient of friction

⇥ c0 ⇥ 3 kPa

Cohesion Experiments suggest stress is largely independent

• f strain rate (i.e. perfectly plastic).

Till rheology has been a subject of some controversy (effective stress increases with depth into the till - it is weakest at the top).

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Till rheology

Boulton & Hindmarsh popularized a power law rheology It is questionable whether there is enough data to confirm such a relationship.

• ˙

ε = A(τ τf)aσ−b

e

˙ ε ⇥ τ

Strain rate

τf σe

Stress

Kamb suggested an exponential relationship (following critical-state soil mechanics theory)

˙ ε = A exp (ατ/τf) τf = µσe ˙ ε ⇥ τ

Strain rate

τf

Stress

τf = µσe σe

Boulton & Hindmarsh 1987, Kamb 1991

Experiments suggest

• ie. almost perfectly

plastic.

⇥ a ⇥ α ⇥ 75 a = 1.33 b = 1.8

It is desirable to have some way of describing plastic flow when the yield stress is exceeded.

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β ub τ b

Sliding over till

σe = N + ∆ρswg(Zb − z) hT

Effective stress increases with depth through till Pore water pressure roughly hydrostatic Deformation only if Deforming horizon

• hT = [τb µN]+/µ∆ρswg

Viscous rheology

τ ⇥ τf = µσe σe ≤ τb/µ

Sliding law

τb = µN + CU1/a

b

Nb/a

• ˙

ε = A(τ τf)aσ−b

e

8 Sliding speed

τf

Basal shear stress

σe Ub τb

Numerical models often use a `pseudo-plastic’ law

τb = CUq

b

q ⌧ 1

β ub τ b

A similar law applies to describe ice flow

• ver sediments with topography

τb = µN + RU1/n

b

Meltwater drainage

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Drainage through till

Estimates of till conductivity vary considerably, but it is generally thought to be low. Although water seeps vertically into the till, horizontal transport through the till is insufficient to evacuate the water produced from melting.

q ⌅ 10−7 · 10 103 · 10 · 10 m s−1 m kg m−3 m s−2 Pa m ⌅ 10−10 m2 s−1 q = KhT ρwg ⇧φ ·

− −

⇤ m dx ⌅ 5 · 103 mm y−1 km ⌅ 1.6 ⇥ 10−4 m2 s−1

• eg. in Antarctica

Water flows in a patchy film at the ice-till interface, or in some form of channels or canals. Canals Patchy sheet

Alley 1989, Creyts & Schoof 2009 Walder & Fowler 1994 10

Canals

Walder & Fowler 1994, Ng 2000 Creep Melting Erosion Creep

˜ N ⇤ 0.8 MPa ⇧ G N > ˜ N N < ˜ N

Channels - mostly melted into ice Canals - mostly eroded into sediment Walder & Fowler suggested two possibilities for steady states: Canals are favoured when the potential gradient is small (e.g. interior of ice sheets). Effective pressure in canals DECREASES with increasing discharge Gravitational potential gradient The crucial difference is that sediment erosion tends to produce a wide cross-section.

∂x ∂x Ψ = ρig tan α + (ρw ρi)g tan θ N / Ψ7/15Q1/15 / N / Ψ−1/3Q−1/3

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Q Q

Interaction of sliding and drainage

Initiation of sliding

Ub m ∝ τbUb/L ∝ Q N

Increased melting

τb = CU p

b N q

• Lower effective pressure

Positive feedback Increased discharge

∂N ∂Q < 0

A consequence of is the potential for a positive feedback

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Ub τb = CU p

b N q

N = c/Q1/3 Q = G + τbUb − kU 1/2

b

ρwL A ⌅

Model

τb ≈ −ρigh∂s ∂x

The relationship between ice thickness and speed can become multivalued

Fast moving, lots of water Slow moving, not much water

Surges and ice streaming

accumulation ice flux

∝ − dh dt ∝ A − Qi − Qi ∝ h

Temporal variability Heinrich events, surges Spatial variability Ice streams

\

MacAyeal 1992 13

Meltwater deposits

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Meltwater deposits

Deposition of sediments in Röthlisberger channels can build eskers

Creep Erosion

Erosion of sediments from canals can create tunnel valleys

Creep Melting Erosion Creep Creep Melting Deposition

• Most likely under falling water speed, near margin
• Sediment is flushed from the surrounding bed
• Esker size depends on margin retreat rate

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Eskers

Canadian eskers form predominantly

• n crystalline bedrock.

Storrar et al 2014 Clark & Walder 1994

Probably reflects channelised drainage

16 Bridgenorth Esker

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Ice Esker Melting Sediments

Eskers

Subglacial channel Retreat Model An extended version of the Röthlisberger channel model that incorporates sediment transport.

Hewitt & Creyts 2019 Geophys. Res. Lett.

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m V Q S ! Vm

A

x Qs

A

S

Sediment deposition acts to clog the channel:

∂S ∂t = QΨ ρiL ˜ ASNn D ∂A ∂t = D

Eskers

Esker size increases roughly quadratically with sediment flux, decreases roughly linearly with meltwater flux, and decreases linearly with ice-sheet retreat rate.

Hewitt & Creyts 2019 Geophys. Res. Lett.

Boundary-layer analysis leads (….) to an approximation of the cross-sectional area: Meltwater flux

A = C Q−4/5

m

Q29/15

s m

V −1

m

Sediment flux Retreat rate

Deformational deposits

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Drumlins

Subglacial waveforms formed by deformation/erosion by ice flow

Drumlins

Inferred drumlin beneath Rutford ice stream from seismic survey

King et al 2007 21 Smith et al 2007

¬2,100 Bed elevation (m) ¬1,900 ¬1,700 5,000 m W E 5 , m 5,000 m 500 m Ice flow direction Stiff till/basal sliding Dilatant till/ deforming bed

Mega-scale glacial lineations

King et al 2009

Radar profile of bed beneath Rutford ice stream (West Antarctica)

I c e f l

• w

d i r e c t i

• n

I c e f l

• w

d i r e c t i

• n

Rutford Ice Stream Dubawnt Lake, Canada 5 km 5 km

compared with de-glaciated bed of former ice stream.

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Instability theory Hindmarsh 1998, Schoof 2007, Fowler 2000, 2001, 2002, 2009, 2010, 2010, Fowler & Chapwanya 2014

• N

∂s ∂t + ∂q ∂x = 0

Conservation of sediment Sediment flux Sliding law Instability if

s q U

• U

∂τ ∂N

• U

> ∂τ ∂N

• q

τ = τ(U, N)

• q = q(τ, N)

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σ(h3)x = V.[h3Vψ], Ψ = s − N + Φ, A = 1 2 f(¯ u, N) µ − N

• +

, εrht = 1 − ΠhN, b = s − δh, bt + V.[A¯ ui] + σγ [B(τe)h]x = βV.[A3VN] + γV.[B(τe){hVΨ + θVb}], τ e = σhi − {hVΨ + θVb} αst + ¯ usx = w(Φ, N).                                            (2.59)

Instability theory

Modified theory may explain evolution of ribbed moraine, drumlins, and mega-scale lineations

Fowler & Chapwanya 2014

• Ice flow over sediments causes transverse dune-like instability.
• Water flow carrying sediments causes longitudinal rill-like instability.

10 8 6 4 2 k2 8 7 6 5 4 3 2 1 –1 2 (a) 4 ribs, P = 0.7 6 8 10 k1 10 8 –1 5 4 3 2 1 –1 6 4 2 2 4 6 8 10 (b) MSGL, P = 10 drumlins, P = 2 k2 k1 (c) MSGL, P = 10 k2 5 10 15 20 5 10 15 20 –1.0 –0.5 0.5 1.0 1.5 2.0 2.5 k1

Growth rates

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Barchyn et al 2016

Instability theory

Barchyn et al 2016

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Summary

Friction laws for soft beds are similar to hard-bed sliding laws (even though the local slip / deformation mechanism may be different). Drainage over till may occur through films, cavities and canals. Ice flow over deforming till can be unstable and produce ribbed moraine, drumlins, and mega-scale glacial lineations. An important (unsolved) question is how the development of these bed- forms controls/affects ice dynamics. Eskers form through deposition in Röthlisberger channels. Tunnel valleys form through excess erosion in canals.

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