[PPT] - Introduce the basic relationship for viscous flow of rock and ice PowerPoint Presentation

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Intro to Quantitative Geology www.helsinki.fi/yliopisto

Introduction to Quantitative Geology

Rock and ice as viscous materials

Lecturer: David Whipp david.whipp@helsinki.fi 27.11.2017

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Goals of this lecture

Introduce the basic relationship for viscous flow of rock and

ice

Explore two different end-member types of viscous flow in a

channel

Discuss the effects of temperature on viscosity and

nonlinear viscosity

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Examples of viscous flow: Alpine glaciers

Alpine glaciers flow downhill under their own weight

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Riggs Glacier, Alaska, USA

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Modern uplift rates are relatively rapid,

especially beneath the Gulf of Bothnia

Helsingin Sanomat, 19.3.2012 Turcotte and Schubert, 2002

Glacio isostatic adjustment

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Surface uplift due to glacio isostatic adjustment  is controlled by flow of the underlying asthenosphere

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Fluid: Any material that flows in response to an applied stress
Deformation is continuous
Stress is proportional to strain rate

      where ! is the shear stress, "#⁄"$ is the velocity gradient (equivalent to strain rate) and # is the velocity in the  %-direction

τ ∝ du dz

What is a fluid?

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

τ = η du dz

Viscosity, defined

Constant of proportionality & is known as the dynamic

viscosity, or often simply viscosity     

Viscosity has units of Pa s (Pascal seconds) or kg m-1 s-1
You can think of viscosity as a resistance to flow
Higher viscosity → more resistant to flow, and vice versa
The terms kinematic viscosity and bulk viscosity (or

compressibility) are not the same thing as the dynamic viscosity

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1-D:

http://en.wikipedia.org Low viscosity High viscosity

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

τ = η du dz

Viscosity, defined

Constant of proportionality & is known as the dynamic

viscosity, or often simply viscosity     

Viscosity has units of Pa s (Pascal seconds) or kg m-1 s-1
You can think of viscosity as a resistance to flow
Higher viscosity → more resistant to flow, and vice versa
The terms kinematic viscosity and bulk viscosity (or

compressibility) are not the same thing as the dynamic viscosity

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1-D:

http://en.wikipedia.org Low viscosity High viscosity

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Approximate viscosities of common materials

Viscosity of natural materials is hugely variable
Range of almost 20 orders of magnitude for

rocks and lava

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Material Viscosity [Pa s] Air 10-5 Water 10-3 Honey 101 Basaltic lava 103 Ice 1010 Rhyolite lava 1012 Rock salt 1017 Granite 1020

A honey dipper works because of the viscosity of honey

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

A Newtonian material has a linear relationship between

shear stress and strain rate

In other words, & is a constant value that does not depend
n the stress state or flow velocity
Air, water and thin motor oil are practically Newtonian fluids
Rocks rarely deform as Newtonian fluids

τ = η du dz

Newtonian (linear) viscosity

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u = 1 2η dp dx(z2 − hz) − u0z h + u0

$ $ $

Linear viscous flow in a channel

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Fig. 6.2a, Turcotte and Schubert, 2014
The general solution for the 1-D velocity of a fluid across a

channel with boundary conditions (1) # = 0 at $ = ℎ and  (2) # = #0 at $ = 0 is        where "(⁄"% is the applied pressure gradient

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Styles of linear viscous flow: Couette flow

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Couette flow occurs when there is (1) a difference in velocity

between the channel boundaries and (2) effectively no pressure gradient    $ $ $

Fig. 6.2a, Turcotte and Schubert, 2002

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u = u0 ⇣ 1 − z h ⌘ u = 1 2η dp dx(z2 − hz) − u0z h + u0

$ $ $

Couette flow solution

If we assume "(⁄"% = 0,

      reduces to   

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Fig. 6.2a, Turcotte and Schubert, 2002

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Poiseuille flow occurs when (1) there is no velocity difference

between the walls of the channel and (2) a pressure gradient is applied   

Poiseuille flow

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$ʹ $ $ $ $ $ʹ $ʹ $ʹ

Fig. 6.2b, Turcotte and Schubert, 2002

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u = 1 2η dp dx(z2 − hz) u = 1 2η dp dx(z2 − hz) − u0z h + u0

Poiseuille flow solution

Using the same equation as we have previously, we can start

with the general solution

If we set #0 = 0, the velocity solution becomes

 

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$ $ $ $ $ʹ $ʹ $ʹ $ʹ

Fig. 6.2b, Turcotte and Schubert, 2002

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Salt tectonics

One example of a geological system that can exhibit both

Couette and Poiseuille flow behavior is the flow of rock salt beneath sedimentary overburden

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http://commons.wikimedia.org Finlay Point  Cape Breton Island, Nova Scotia, Canada

Head of salt diapir

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

η = A0eQ/RTK

Temperature dependence

In general, rock viscosity depends strongly temperature

      where *0 and + are material properties known as the  pre-exponent constant and activation energy, , is the universal gas constant and -K is temperature in Kelvins

What happens to rock viscosity at -K approaches

absolute zero?

What happens as -K approaches infinity?

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Temperature-dependent viscosity

The viscous strength of quartz, for

example, rapidly decreases with increasing temperature

Note that the viscous strength is simply

the viscosity & multiplied by a nominal strain rate

How might temperature-

dependent viscosity be important in the Earth?

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Viscous strength of quartz

σd z

← Increasing Temperature

Fig. 5.13, Stüwe, 2007

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Temperature-dependent viscosity

The viscous strength of quartz, for

example, rapidly decreases with increasing temperature

Note that the viscous strength is simply

the viscosity & multiplied by a nominal strain rate

How might temperature-dependent

viscosity be important in the Earth?

18

Viscous strength of quartz

σd z

← Increasing Temperature

Fig. 5.13, Stüwe, 2007

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

τn = Aeff du dz

Nonlinear viscosity

In general, rocks will deform about 8 times as quickly when the

applied force is doubled

Relationship between shear stress and strain rate is thus

NOT linear

Mathematically, we can say

      where . is the power law exponent and *eff is a material constant

The power law exponent for many rocks is 2-4
*eff is similar to &, but has units of Pan s

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Flow of glaciers

Gravity drives the flow of alpine

glaciers from higher elevation zones

f accumulation to lower elevation

zones of ablation

Depending on the temperature of the

region and the ice itself, the glacier may either be frozen to the bedrock (cold-based) or sliding along the bedrock (warm-based)

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Zone of accumulation Zone of ablation Equilibrium line

Fig. 9.14, Ritter et al., 2002

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How do glaciers move?

Basal sliding
Bottom of the glacier sliding along the

substrate

Can occur as a result of slip atop a thin

water layer, melting/re-freezing or slip atop water-saturated sediment

Internal deformation
Ice flow is nonlinear viscous and sensitive

to temperature

Deformation is concentrated near the

bed

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Briksdal Glacier, Norway

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Flow of glaciers

In the exercise this week, we will look more closely at glacial

flow

Velocity across a glacial valley
Down an incline

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$ $ $

Fig. 6.3, Turcotte and Schubert, 2014

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Recap

Viscous flow is a common deformation behavior for rock and

ice, where the deformation rate is proportional to the applied shear stress

Couette and Poiseuille flows refer to end-member behaviors
f linear viscous channel flows, and depend on the channel

boundary velocities and pressure changes along the channel

Most rocks do not exhibit a linear relationship between stress

and strain rate (nonlinear viscosity), and their viscosity is strongly temperature-dependent

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References

Ritter, D. F., Kochel, R. C., & Miller, J. R. (2002). Process Geomorphology (4 ed.). MgGraw-Hill Higher Education. Stüwe, K. (2007). Geodynamics of the Lithosphere: An Introduction (2nd ed.). Berlin: Springer. Turcotte, D. L., & Schubert, G. (2014). Geodynamics (2nd ed.). Cambridge, UK: Cambridge University Press.

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Introduction to Quantitative Geology

Rock and ice as viscous materials

Lecturer: David Whipp david.whipp@helsinki.fi 27.11.2017

Goals of this lecture

ice

channel

nonlinear viscosity

Examples of viscous flow: Alpine glaciers

especially beneath the Gulf of Bothnia

Glacio isostatic adjustment

Surface uplift due to glacio isostatic adjustment is controlled by flow of the underlying asthenosphere

where ! is the shear stress, "#⁄"$ is the velocity gradient (equivalent to strain rate) and # is the velocity in the %-direction

τ ∝ du dz

What is a fluid?

τ = η du dz

Viscosity, defined

viscosity, or often simply viscosity

compressibility) are not the same thing as the dynamic viscosity

1-D:

τ = η du dz

Viscosity, defined

viscosity, or often simply viscosity

compressibility) are not the same thing as the dynamic viscosity

1-D:

Approximate viscosities of common materials

rocks and lava

Material Viscosity [Pa s] Air 10-5 Water 10-3 Honey 101 Basaltic lava 103 Ice 1010 Rhyolite lava 1012 Rock salt 1017 Granite 1020

A honey dipper works because of the viscosity of honey

shear stress and strain rate

τ = η du dz

Newtonian (linear) viscosity

u = 1 2η dp dx(z2 − hz) − u0z h + u0

$ $ $

Linear viscous flow in a channel

channel with boundary conditions (1) # = 0 at $ = ℎ and (2) # = #0 at $ = 0 is where "(⁄"% is the applied pressure gradient

Styles of linear viscous flow: Couette flow

between the channel boundaries and (2) effectively no pressure gradient $ $ $

u = u0 ⇣ 1 − z h ⌘ u = 1 2η dp dx(z2 − hz) − u0z h + u0

$ $ $

Couette flow solution

reduces to

between the walls of the channel and (2) a pressure gradient is applied

Poiseuille flow

$ʹ $ $ $ $ $ʹ $ʹ $ʹ

u = 1 2η dp dx(z2 − hz) u = 1 2η dp dx(z2 − hz) − u0z h + u0

Poiseuille flow solution

with the general solution

$ $ $ $ $ʹ $ʹ $ʹ $ʹ

Salt tectonics

Couette and Poiseuille flow behavior is the flow of rock salt beneath sedimentary overburden

Head of salt diapir

η = A0eQ/RTK

Temperature dependence

where *0 and + are material properties known as the pre-exponent constant and activation energy, , is the universal gas constant and -K is temperature in Kelvins

absolute zero?

Temperature-dependent viscosity

example, rapidly decreases with increasing temperature

the viscosity & multiplied by a nominal strain rate

dependent viscosity be important in the Earth?

Viscous strength of quartz

σd z

← Increasing Temperature

Temperature-dependent viscosity

example, rapidly decreases with increasing temperature

the viscosity & multiplied by a nominal strain rate

viscosity be important in the Earth?

Viscous strength of quartz

σd z

← Increasing Temperature

τn = Aeff du dz

Nonlinear viscosity

applied force is doubled

NOT linear

where . is the power law exponent and *eff is a material constant

Flow of glaciers

glaciers from higher elevation zones

zones of ablation

region and the ice itself, the glacier may either be frozen to the bedrock (cold-based) or sliding along the bedrock (warm-based)

How do glaciers move?

substrate

Surface uplift due to glacio isostatic adjustment  is controlled by flow of the underlying asthenosphere

      where ! is the shear stress, "#⁄"$ is the velocity gradient (equivalent to strain rate) and # is the velocity in the  %-direction

viscosity, or often simply viscosity     

viscosity, or often simply viscosity     

channel with boundary conditions (1) # = 0 at $ = ℎ and  (2) # = #0 at $ = 0 is        where "(⁄"% is the applied pressure gradient

between the channel boundaries and (2) effectively no pressure gradient    $ $ $

      reduces to   

between the walls of the channel and (2) a pressure gradient is applied   

      where *0 and + are material properties known as the  pre-exponent constant and activation energy, , is the universal gas constant and -K is temperature in Kelvins

      where . is the power law exponent and *eff is a material constant