GLACIER FLOW

ESS 431 PRINCIPLES OF GLACIOLOGY
ESS 505 THE CRYOSPHERE
GLACIER FLOW
OCTOBER 15, 2014
T.J. Fudge
721 ATG
tjfudge @ uw.edu
Homework
• This week – Harry and Gwen estimate the speed
of a glacier.
Friday Discussion Session
• Melting ice sheets
• Avalanches (last week’s homework)
• Glacier flow
Sources for Today
Paterson, W.S.B. 1994. The Physics of Glaciers.
3rd ed. Pergamon.
Lecture notes from C.F. Raymond
International Glaciological Society
Annals of Glaciology 20
I. Joughin and S. Tulazyk, 2002. Science 295,
476
How does a glacier move?
•The ice can deform as a viscous fluid.
•The ice can slide over its substrate.
• (It can also fracture, but that’s not so
important for the big picture)
That’s our next 2 lectures in a nut shell.
• Time to start silly-putty “ice cap”
Glacier video
Historic glacier footage shot by UW researchers
• at Blue Glacier (Mt Olympus)
• at Nisqually Glacier (Mt Rainier)
• at South Cascade Glacier (North Cascades)
Original 16mm movie footage has been
transferred to VHS, and then to DVD ...
What do we mean by “evidence”?
Watch for
• evidence of viscous flow
• evidence of basal sliding
• Evidence of brittle failure
Blue Glacier Video
VTS_2_1.VOB
Kinematics vs Dynamics
Kinematic description of flow
In a steady state,
• Flow is what it needs to be, to carry away upstream
accumulation.
• Glacier has adjusted its shape to make this flow happen.
• Glacier will grow or shrink if adjustment hasn’t happened (yet)
Dynamic description
• Ice is a material with certain rheological properties (stiffness).
• Flow is determined by forces (stresses) applied to it.
Rheological properties don’t figure in kinematic description.
Accumulation and ablation don’t figure in dynamic description.
• To figure out how a glacier changes over time, we need to use
both descriptions.
Essentials
of a Glacier
Income: snowfall
(accumulation)
Source
Region
Sink
Region
Down-slope
transport of
ice mass
Loss: melting, calving
(ablation)
Glacier
“Flow” Chart
Climate
Meteorological
Processes
Mass
Balance
Here is where
we are today
Ice Dynamics
Processes
Glacier
Geometry
Some Important Questions
• How big are glaciers? (Area and Volume,
sea-level equivalent)
• How fast do glaciers move?
• How much water runs off? (Hydrology)
• How do glaciers erode old landscapes and build up
new landscapes? (Geomorphology)
• How do glaciers change with climate? (Sensitivity
to climate change, and response times)
What determines glacier mass balance?
Accumulation – snow deposition
• Air-mass characteristics
• Topography
• Wind redistribution
Ablation - melting from heat
• Solar input
• Surface reflectivity
• Clouds
• Wind
• Temperature
• Humidity
Net Mass
Balance and
Altitude
Net Mass
Balance for
13 Glaciers
• Why do the curves have
similar shapes (sort of)?
• Why are they separated
so much in elevation?
Net Mass Balance for more Glaciers
• Why do the curves have similar shapes (sort of)?
• Why are they separated so much in elevation?
From Oerlemans, J. (2011) Minimal glacier models.
Measuring the Motion of a Glacier
Traditional Surveying
On surface:
• Measuring angles (with Theodolite) and distances
(with Electronic Distance Meter or EDM) from
fixed stations on the glacier margins
At depth:
• Measuring tilt in boreholes
Traditional Surveys
Angles and distances
with theodolite and
EDM (“total station”)
measure locations of
markers on/in the ice.
Map view
Now GPS receivers
measure
Ice
displacements of
flow
markers between
surveys.
Initial marker
locations
Final
locations
Flow variation across a glacier
Raymond, C.F. 1971,
J.Glaciol.10(58), 55-83.
Flow variation across a glacier
Mer de glace, glacierchange.wordpress.com
Borehole Tilting – the third dimension
We want to get velocity as a function of depth
What do you need to observe?
• xs(t) positions
measured by any
standard survey
method
• Dx(z) from angle of
tilt meter lowered
into hole
Flow variation with depth
t2
t1
t0
Borehole tilting
Measuring the Motion of a Glacier
• GPS (Global Positioning System)
• Feature tracking in repeated satellite images or
aerial photography
• Interferometric Synthetic Aperture radar (inSAR)
inSAR
Interferometric
Synthetic
Aperture Radar
To get speed,
count fringes
from a
stationary
point on
bedrock.
Rutford Ice Stream,
West Antarctica,
flows toward the
bottom of the image
Velocities of
Ross Ice Streams
inferred with inSAR
Joughin and Tulaczyk.
2002. Science 295, 476.
What Determines Glacier Motion?
ICE DEFORMATION
MOTION OVER BED
•
•
•
•
• Sliding
depends on …?
• Bed deformation
depends on …?
Ice “flow law”
temperature
fabric, grain size
chemistry, impurities
AVALANCHING
• Fracture/failure
Transport Processes
• Internal deformation
• Basal sliding
• [Avalanching]
speed at
surface due
to internal
deformation
Sliding speed
How is Silly Putty Relevant?
• Viscous flow
• Elastic response
• Brittle behavior
• Check out silly-putty “ice cap”
Vertical sections
Along valley Across valley
Idealized Flow Patterns
in a Valley Glacier
Accumulation
Area
Equilibrium
Line
Ablation
Area
Map
View
Longitudinal Flow Patterns in a Valley Glacier
What determines Ice Transport?
Q is ice flux (m3 a-1) through a cross-section.
Q  thickness  width  speed
(m)
(m) (m a-1)
Q(x)= ice velocity u(x,y,z) integrated over the cross section
What controls ice flow speed u?
• Ice thickness
• Surface slope
• Ice properties (temperature  viscosity)
• Bed properties (temperature, slipperiness)
Forces
• Forces exist inside continuous bodies, (e.g. a glacier).
• These forces can cause the material to deform.
• Understanding how this works is our next goal.
Are you stressed? Let’s Define Stress
Ice inside a glacier is subjected to
forces due to
• the weight of the overlying ice,
• the shape of the glacier surface.
F
Stress = force per unit Area
Area
Types of Stress
As a force/unit area, stress also has a direction.
Force can be directed normal to the area.
• Result is pressure if the force is the same
on all faces of a cube.
• Result is Normal stress if forces are
different on different faces
Force can be directed parallel to the area.
• Result is shear stress. Shear stress on
internal areas parallel to the sloping
glacier surface at depth z drives ice flow t = r g z sin(q)
Magnitudes of Stress
Stress = Force/unit area.
Units are Pa = N/m2
N = Force = M a = 1 kg  1 m s-2
t=0
t=1 s
t=2 s
This is the “push” that must be
continually applied to a 1 kg mass to
make it accelerate at 1 m s-2 on a
frictionless table.
u (m s-1)
1 kg
4
2
0
0
2
4
Time (s)
Ooops …! Just dropped your water bottle!
Bottle is in free fall …
• Volume ~ 1 liter, what is its Mass M?
• Gravitational acceleration g = 9.8 m s-2
(for purposes of quick-and-dirty
estimations, g = 10 m s-2)
• F = M g = …? 1 kg * 10 m s-2 = 10 N
Stretching a Rubber Band
Mass of lead weight:
M = 6 oz ~ 0.2 kg
Force stretching band:
F = Mg
~ 0.2 kg 10 m s-2 = 2 N
Stress in a Rubber Band
F
t=
Area
Cross-section area of rubber band
~ 1mm  5 mm ~ 510-6 m2
2N
5
t=
= 4 10 Pa
6
2
5 10 m
For comparison, note that 1 atmosphere ~ 105 Pa
Pressure in a Glacier
F
Mg
P=
=
Area Area
Mass M = r V
• r = ice density = 900 kg m-3.
• V = volume = Area  depth (z)
So Pressure at depth z is:
M g r  Area  z  g
P=
=
= r gz
Area
Area
How deep do we have to drill into a glacier
before the ice pressure is 1 atmosphere?
Area
z
Depth for 1 Atmosphere Pressure?
P
z=
rg
5
10 Pa
=
 11m
3
2
900 kg m  9.8 m s
So pressure rises by 1 atmos ~ 1 bar = 105 Pa
for every 11 meters of depth in a glacier.
Does ice deform in response to this pressure?
Shear Stress t
Total force from ice column:
F = M g = rVg = r g  Area  h
How much of this
weight will contribute to
q
shear deformation?
Mg
Mg cos(q) pushes
ice onto bed
q
t = F/Area
M g sin(q ) rVg sin(q )
t=
=
Area
Area
=
r ( Area  h) g sin(q )
Area
= r g h sin(q )
Mg sin(q) causes
shear deformation
Units are Pa, just
like pressure
How is Shear Stress related to Depth?
Shear stress increases
• with depth z, and
• with glacier slope q
t = r g z sin(q)
0
Glacier with
steep slope q
Depth
z
0
Shear stress t
Glacier with
low slope q
An Example of Shear Stress in a Glacier
q
t = rg z sin(q)
r gz
If z = h = 130 m
q = 5o
r gz sin(q)
kg
m
o
t b  900 3  10 2  130 m  sin(5 )
m
s
 10 Pa = 1bar
5
at the base.
This is a typical value for basal shear
stress under a glacier.
Are Glacier Thickness and Slope Related?
Suppose a glacier has become steeper or
thicker (due to mass imbalance) • It flows faster
• It quickly reduces thickness h or slope q,
until t b~ 1 bar again.
Can we estimate glacier thickness (z=h) from
its (known) slope, if we know tb~ 105 Pa?
t = r g z sin(q)
h ~ tb / [r g sin(q)]
.
Strain Rate e
At start …
after a short time …
D
D zz
and later …
Dz
Dx
 Dx 
Shear strain
1
e = 2  
(deformation)
 Dz 
Shear strain RATE
(rate of deformation)
Dx
(What are its dimensions?)
(Units are time-1)
 Dx 
 Dx 
 De  1 
Dz  1  Dx  1 
Dt  1  Du 
 =


e =   = 2 
=
 = 2 


2 Dz
2 Dz
D
t
D
z
D
t
 Dt 












Constitutive Relations
Different materials respond differently to
applied forces and stresses.
A constitutive relation is a relationship
between stress and strain rate for a given
material.
Let’s look at some different responses to
stress, using silly putty.
Constitutive Behavior of Ice
Du
n

= 2e = 2 At
Dz
Glen Constitutive Law for Ice
A, n are characteristic
of the material.
z
Note that A here is not
an Area.
A is a constitutive
parameter describing
the softness of ice.
For Ice
• n~3
• A~210-16 Pa-3 yr-1 at 0oC
• A~610-18 Pa-3 yr-1 at -20oC
These numbers distinguish ice
from motor oil, silly putty, or
honey.
For comparison, a viscous fluid (n =1)
Du
t
= 2e =
Dz

 = viscosity, which
is different for
• Water
• Cold engine oil
Effect of n=3 in ice:
• Ice becomes softer, the higher the shear stress t
Du
t
n
= 2e = 2 At =
Dz
 1

 (2 At n 1 )


Viscosity is smaller
when stress is larger
Velocity Profile in a Temperate Glacier
If you do the
calculus:
A
u( z ) = ( r g sin(q ) )3 (h 4  z 4 )
2
h = ice
thickness
z = depth
At the surface
A
A 3
3
(z=0): usurf = h ( r gh sin(q ) ) = ht b
2
Averaged over the depth h
2
u
usurf
( )
 n 1 
u =
 usurf = 4 5 usurf z
n 2
A is constitutive parameter, not Area.
u
Ice Flux in a Temperate Glacier
Flux Q is total discharge rate
across the channel in
volume/time (m3 a-1)
•We need to average velocity
over both depth and width

u ( x) =
h( x, y ) W ( x )
0

o
u( x, y, z )dzdy
Area
Q( x) = u ( x)  Area
Ice Flux in a wide glacier
When the depth and velocity do not vary much
laterally (e.g. on a very wide glacier, or on a big ice
sheet) we may sometime talk about
ice flux q(x) per unit width of channel.
(units of m2 a-1)
q( x) = u ( x)  h( x)
Velocity Profiles in a Glacier: n=3 or n=1?
n=1 viscous
fluid
n=3 Glen Law
for ice
How can a Crystalline Solid Flow?
Basal planes are held together by Hydrogen bonds
• Ice can deform along
basal planes like a deck
of cards
• Bonds break, shift and
re-connect
Dislocations Help Deformation
Do we need to break all the bonds at
once right across the crystal?
• No. All crystals have defects or
imperfections called dislocations.
• Need to break only a few bonds at
a time.
• We recover the energy when bonds
reconnect at new sites.
• Dislocation has moved.
• Dislocations act as catalyst for
easy deformation.
Measuring Constitutive Parameters
Deform samples of
Uniaxial Compression
initially isotropic ice
under constant stress.
Isotropic means there is
no preferred direction for
the c axes.
Simplest Experiment
• apply known stress
• measure strain rate
Simple Shear
Creep Behavior of Ice
Strain
rate
Steady-state
creep
e
Secondary
creep
Primary
creep
Tertiary
creep
0.01
0.1
Strain e (or time t)
Primary Creep
Some crystals are “Hard” and others are “Soft” due
to the orientation of their basal planes relative to the
applied stress
•Soft grains deform easily at first
•After some deformation has occurred, hard grains
start to block deformation of soft grains
Ice gets harder
with time
Secondary-Tertiary Creep
•Hard crystals get bent elastically, acquiring
strain energy like a stretched spring
•This is an energetically unfavorable state
•They lose mass to neighbors that are able to
creep without picking up strain energy
As hard grains shrink
and disappear, ice
becomes softer again
Temperature Dependence of Ice Flow
Deformation Rate is
proportional to
exp(-Q/RT)
Q = 60 kJ mol-1 activation energy for creep
R = 8.314 J mol-1 K-1 gas constant
T temperature (Kelvin)
Summary: Stress and Strain Rate
Shear stress t
(Force/Area)
Shear strain rate (time-1)
Summary: Mechanical Behavior
Linear (Newtonian) fluid
• Viscosity = 
Ice is a nonlinear fluid
• softness = A
• stress exponent n=3
Both t and du/dz
increase with depth z
Du
t
= 2e =
Dz

Du
n
= 2e = 2 At
Dz
Summary: Glacier Motion
Summary: Deformation velocity
2A
(rg sin q )n (hn1  z n1 )
u( z ) =
(n  1)
n 1


2A
z


n n 1
(rg sin q ) h 1    
=
(n  1)
 h 
n 1


2A
z


n
(rgh sin q ) h1    
=
(n  1)
 h 
n 1

2 A n   z  
=
tb h 1  
 h 
(n  1)


Some
Typical
Speeds
Crevasse Formation
Like silly putty, ice breaks at right angles to the
direction of greatest stretching
So where does the crevasse form in this situation?
Crevasses are features that can be tracked
in repeat imagery.
Equilibrium
Length of a
Glacier

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