glacier dynamics slides 2011 - Glaciers Group

Glacier DynamicsGlaciers 617

Andy Aschwanden

Geophysical Institute

University of Alaska Fairbanks, USA

October 2011

1 / 81

“The tradition of glacier studies that we in-herit draws upon two great legacies of theeighteenth and nineteenth centuries: classicalphysics and romatic enthusiasm for Nature.. . . It is easy to undervalue the romantic con-tribution, but in glaciology, it would be a mis-take to do so”Garry Clark, J. Glaciol., 1987)

Course Material

lecture is based on lectures by Ed Waddington & Martin Truffer Martin’s “Continuum Mechanics” script book recommendation: Greve and Blatter (2009)

Greve, R. and H. Blatter (2009). Dynamics of Ice Sheets and Glaciers.Advances in Geophysical and Enviornmental Mechanics andMathematics. Springer Verlag.

3 / 81

Introduction to glacier flow

4 / 81

How does a glacier move?

In a nut shell: The ice can deform as a viscous fluid The ice can slide over its substrate Well-described by continuum mechanics

Measurements & Observations 5 / 81

Measurements & Observations

1779 Gravitation theory by de SaussureH. B. de Saussure observes sliding

“. . . the weight of the ice might be sufficient to urge it down the slopeof the valley, if the sliding motion were aided by the water flowing atthe bottom.”

1827-1836 Hugi BlockJ. Hugi observed that a boulder moved 1315 m downstream between 1827and 1836

we would interpret this as clear evidence of glacier flow but back then, some people argued that a boulder slides on the

glacier surface, the glacier itself is motionless



1840-1846 Dilatation theory by L. Agassiz

glacier ice contains innumerable fissures and capillary tubes during the day, these tubes absorb the water and during the night, the water freezes this distension exerts a force and propels the glacier in the direction of

least resistance



1864-1930 Viscous flow theory by J. Forbes

made his own observations on Mer deGlace, France

glacier flows fastest in the center opposes Agassi’s theory if the dilatation theory were true then flow would be greatest at sunset and near the glacier margins


Measuring surface velocities

Measuring angles (with Theodolite) anddistances (with Electronic Distance Meteror EDM) from fixed stations on theglacier margins

GPS (global positioning system) feature tracking in repeated aerial

photography or satellite images


Measuring surface velocities

credit: USGS, Joughin et al., 2010

interferometric synthetic apertureradar (InSAR)

count fringes from a stationarypoint on bedrock


Measuring borehole tilt

with tiltmeter(inclinometer)


Kinematic vs. Dynamics

Kinematic description of flowIn a steady state,

Flow required to transport away upstream accumulation Glacier has to adjust its shape to make this flow happen

⇒ Rheological properties don’t figure in kinematic description

Dynamic description of flow

ice is a material with certain rheological properties (e.g. viscosity) Flow is determined by stresses applied to it

⇒ Accumulation/ablation don’t figure in dynamic description


“Glacier Flow” Chart

climate

meteorological process

massbalance

ice dynamics processesthis lecture

glaciergeometry


Why is it so hard to predict the future of an icesheet?

It’s easy because

composed of a single, largely homogenous material viscous flow is governed by the Navier-Stokes equations (19th century

physics) move very slowly (turbulence, Coriolis force, and other inertial effects

can be ignored)

It’s so hard because the stress resisting ice flow at the base can vary by orders of

magnitude ocean interactions could trigger instabilities


Forces

a force is any influence that causes an object to undergo a change inspeed, a change in direction, or a change in shape

forces exist inside continuous bodies such as a glacier these forces can cause a glacier to deform

Forces, Stress & Strain 15 / 81

What is stress?

“Stress” is the force per unit area acting on a material

σ =F

A

Inside a glacier, stresses are due to weight of the overlying ice (overburden pressure) shape of the glacier surface (pressure gradients)


Types of stress

As a force per unit area, stress has a direction

Force can be directed normal to thearea

Result is pressure if the force is thesame on all faces of a cube.

Result is normal stress if forces aredifferent on different faces

Force can be directed parallel to thearea

Result is shear stress


Magnitudes of stress & forces

Again, stress is force / (unit area) frictionless table water bottle in free fall stretching a rubber band


Pressure in a glacier

mass m = ρV

ρ = ice density ≈ 900 kg m−3

V = Volume = Area × depth = A · zSo pressure p at depth z is

p =m · g

A=

ρ ·A · z · gA

= ρgz

How deep do we have to drill into a glacier before the ice pressure is 1atmosphere?


Depth for 1 atm pressure?

z =p

ρ · g =?

So pressure rises by 1 atm for every x meters of depth in a glacier Does ice deform in response to this pressure?


Shear stress τ

Total stress t from ice column:

t = ρVg = ρ g h

How much of this weight will contribute to shear deformation? shear stress τ = ρ g h sin α

normal stress σ = ρ g h cos α


Shear stress in a glacier

h = 130 m α = 5

τ = ρ g h sin α

Shear stress at the glacier base, τb, is ≈ 1 bar, which is a typical value forbasal shear stress under a glacier


Are glacier thickness and slope related?

Suppose a glacier becomes thicker or steeper due to mass imbalance: it flows faster it quickly reduces thickness h or slope α, until τb ≈ 1 bar again

Can we then estimate glacier thickness (z = h) from its slope if we knowτb ≈ 1 bar?

τ = ρ g z sin α ⇒ h ∼ τbρ g sin α

How are shear stress and shear strain rate (velocity) related?


Strain rate γ

shear strain γ = 1

2

∆x∆z

= 1

2γ

shear strain rate γ = 1/2∆x/∆z

∆t= 1/2

∆u∆z


Material Science

Rheology

is the study of the flow of complex liquids or the deformation of softsolids.

different materials respond differently to applied stresses We are not going through this in great detail.

⇒ For further information, consult the handout or take Martin Truffers “Ice Physics 614” class

We need to define a relationship between shear stress and shear strain rate, τ = f (γ)

Material Science 25 / 81

Example: Shear Experiment

apply constant shear stress τ

τ is the applied shear stressγ is the shear angleγ is the shear rate

initial elastic deformation primary creep: shear rate γ decreases with time secondary creep: shear rate remains constant acceleration phase tertiary creep: constant shear rate (but higher)




This suggests that the shear rate is a function of the applied shear stress, temperature

T , and pressure p

γ = γ (τ, T , p)




The dynamic viscosity η relates shear stress and shear rate

τ = η (T , p, |γ|) γ


Constitutive behavior of ice

γ = 2 A(T , p) τn−1τ

A(T , p) = A0e−Q/(RT ) Arrhenius law

A is called rate factor (ice softness) Q and R are the activation energy and the gas constant, respectively n is the exponent of the flow law, usually taken as n = 3

This is known as Glen’s flow law or Glen-Steinemann flow law but similarto Norton’s flow law for the flow of steel at high temperatures


Temperature dependence

With some emperically-derived values for Q we get:

Temperature T [ C]

Rat

e fa

ctor

A [s

Pa

]

ice at 0 C is about 1000× softer than ice at -50 C


Thin-film flow

∂u∂z = 2 A(T , p) τn−1τ ⇒

uh − ub = h

02 A (ρ g sin α z)n

dz =

We assume ub = 0, i.e. glacier is frozen to the bed


Basal sliding

Despite decades of intensive research, basal sliding remains one of thegrand unsolved problems. We know:

sliding is a function of basal water pressure and bed roughness if ice is underlain by till, experiments suggest a plastic or

nearly-plastic behavior

Beyond the scope of this lecture


Recap

so far, we have only considered shear stresses while they are often the most important ones in glacier flow, there are other stresses (e.g. normal stresses) which can be important

too we have assumed ice surface and bed surface are parallel let’s step back and look at the bigger picture yep, we will need basic calculus and some continuum mechanics


Field equations for ice flow


Recap

so far, we have only considered shear stresses while they are often the most important ones in glacier flow, there are other stresses (e.g. normal stresses) which can be important

too we have assumed ice surface and bed surface are parallel let’s step back and look at the bigger picture yep, we will need basic calculus and some continuum mechanics


Continuum Mechanics

Continuum mechanics is the application of classical mechanics tocontinuous media. So

What is classical mechanics? What are continous media


Classical MechanicsIn classical mechanics, we deal with two type of equations

conservation (balance) equations (fundamental to physics) material equations (less fundamental)

Balance Laws balance of mass balance of linear momentum balance of angular momentum balance of energy

Material Laws / Constitutive Equations / Closures

often obtained from experiments and theoretical considerations (suchas material objectivity)


Classical Mechanics

Densities classical mechnics has concept of point mass determine velocity (and acceleration) by looking at changes (and rate

of changes) in position ⇒ Kinematics how do forces affect a mass? ⇒ Dynamics

Volume Ω has mass m, then

mass m =Ω ρ dv , ρ mass density

linear momentum mv =Ω ρv dv , ρv linear momentum density

internal energy U =Ω ρu dv , ρu energy density

(1)


General Balance Laws

Imagine a volume Ω enclosed by a boundary ∂Ω

G(t) =

Ω

g(x , t) dv (2)

S(t) =

Ω

s(x , t) dv (3)

G , g

S, s

physical quantity and densitysupply and density



G can only change if if there is a supply S within Ω

if there is a flux F of G through the boundary ∂Ω

F (t) =

∂Ω

(gv + φ(x , t)) ·n da (4)

gvφ(x , t)

transport of g with velocity v through ∂Ωflux density (defined later)



General balance law is:

dG

dt= S − F (5)

d

dt

Ω

g(t) dv =

Ω

s(x , t) dv −

∂Ω

(gv + φ) ·n da (6)



Assume that G = G(t) thus integral and differential can be exchanged Use Gauss’ Theorem:

∂Ω

(gv + φ) ·n da =

Ω

∇ · (gv + φ) dv

We can then write

Ω

∂g

∂tdv =

Ω

s(x , t) dv −

Ω

∇ · (gv + φ) dv (7)



We made no assumption about the shape and size of Ω We can make Ω infinitely small

∂g

∂t= s(t) − ∇ · (gv + φ) (8)

ordg

dt=

∂g

∂t+ v ·∇g = s(t) − ∇ · (v + φ) (9)


Mass Balance

the mass of a material volume cannot change, dM/dt = 0g = ρφ = 0s = 0

mass densityflux of g through the boundarysupply of g

from the above we get∂ρ

∂t+ ∇ · (ρv) = 0 (10)

⇒ known as the continuity equationWe assume ice is incompressible (density does not change), then

∇ · v = 0 (11)

Mass Balance 44 / 81

Momentum Balance

Newton’s Second Law total rate of change of momentum P equals sum of all forces F

dPdt

= F =

∂ω

tn da

internal

+

ω

f dv

external

(12)

P and F are vector quantities identify the internal stress vector tn = Tn, see p. 31 T is the Cauchy stress tensor external forces f are, e.g. gravity or the Coriolis force (forces are a

source of momentum)

Momentum Balance 45 / 81

The Cauchy Stress Tensor

We’ve already introduced shear stress, but forces can be applied ondifferent faces and in different directions

T =

txx txy txztyx tyy tyztzx tzx tzz

We identify τ = txz

For example tez =

txztyztzz

is the stress vector of a cut along the xy-plane


Momentum Balance

∂ (ρv)∂t

= −∇ · (ρv ⊗ v) + ∇ ·T + f (13)

g = ρvφ = −Ts = f

momentum densityflux of g through the boundarysupply of g

By using the mass balance, we can write

d (ρv)dt

= ∇ ·T + f (14)

What’s new? shear stress τ replaced with Cauchy stress tensor T


Momentum Balance

For glacier ice, we assume ice is incompressible acceleration term d(ρv)

dt can be neglected supply of momentum is through gravity only, f = ρg

The momentum balance for glacier flow is

∇ ·T + ρg = 0 (15)


Angular Momentum Balance

no, I don’t want to do the whole derivation here if you’re interested (or bored), go through Equations 3.77 – 3.83 in

Greve and Blatter (2009) here, we just use the result the stress tensor T is symmetric

T = TT (16)

Angular Momentum Balance 49 / 81

Energy Balance

only the total energy (sum of kinetic and internal energy) is aconserved quantity

balance of kinetic energy is not an independent statement but a mereconsequence of the momentum balance

here, we formulate the balance of specific inner energy, u:

ρdu

dt= −∇ ·q + tr (T ·D) + ρr (17)

g = u

φ = qp = tr (T ·D)s = ρr

D = 1/2vT + v

specific internal energyheat fluxdissipation powerspecific radiation powerstrain rate or velocity gradient tensor

Energy Balance 50 / 81

Balance Equations

mass dρdt = −ρ∇ · v (1)

momentum ρdvdt = ∇ ·T + f (3)

internal energy ρdudt = −∇ ·q + tr (T ·D) + ρr (1)

left-hand sideρ (1)v (3)u (1)

right-hand sideT (6)q (3)

so we have 5 equations for 14 unknown fields the system is highly under-determined

⇒ closure relations required


Closure Relations

balance equations are universally valid closure relations describe the specific behavior of a material closure relations are often called constitutive equations


Material Science

Rheology

is the study of the flow of complex liquids or the deformation of softsolids.

We are not going through this in great detail.⇒ For further information, consult the handout or take Martin Truffers “Ice Physics 614” class

We need to define a constitutive equation for the heat flux q and for the internal energy u

a relationship between stress and strain, T = f (D)

Constitutive Equations 53 / 81

Generalization

Assume secondary creep incompressibility of ice ⇒ uniform pressure does not lead to

deformation T = −pIisotropic

+ T

deviatoric

with the pressure p = −1/3trT

The only non-straightforward step from

τ = η (T , p, |γ|) γ (18)

toT = η (T , p, | • |)D (19)

is the generalization of |γ|


Generalization

T = η (T , p, | • |)D (20)

is a relation between two tensors (i.e. T and D) material objectivity tells us that such a relationship must be

independent of the chosen vector basis a second rank tensor has 3 scalar invariants

IT = trT (21)IIT = 1/2

trT2 − tr

T2

(22)

IIIT = det T (23)

trT = 0 for incompressible media σe =

√IIT is the effective stress

role of IIIT is not clear, but lab experiments indicate that the thirdinvariant is not important


Generalization

Numerous lab experiments and field measurements suggest

1η (T , p, σe)

= 2A (T , p) f (σe) (24)

whereA(T , p) = A0e

−Q/(RT ) Arrhenius lawf (σe) = σn−1

e power law


Generalization

What’s new?

We have replacedγ = η (T , p, |τ |) τ (25)

byD = η (T , p, |σeff |)T (26)


Calorimetric Equation of State

Internal energy, u depends linearly on temperature, T ,

u = u0 + c (T − T0) (27)

u0 is a reference value at a reference temperature T0

c is the heat capacity (a measure how much heat is needed to raisethe temperature by a certain amount)

we obtaindu

dt= c

dT

dt(28)


Heat Flux

Fourier’s law of heat conduction

Tj-1 Tj Tj+1

t

modified after Christophe Dang Ngoc Chan, wikicommons image

heat flows from higher to lowertemperature

this is the second law ofthermodynamics

q = −k∇T

the thermal conductivity k is ameasure for how fast the heatspreads


Recap

Balance Equations

mass 0 = ∇ · v (1)momentum ρdv

dt = −∇ ·T + ∇p + f (3)temperature ρc

dTdt = −∇ ·q + tr (T ·D) + ρr (1)

Flow Law

Glen’s flow law T = 2 η D = A(T )−1/nσ1−ne D (6)

now we have 15 equations for 15 unknown fields the system is now well-determined

Recap 60 / 81

Large-Scale Dynamics

Ice sheets are glaciers too! we will focus on the ice sheet part and, for now, ignore ice shelves ice is assumed to be incompressible, ∇ · v = 0

Glaciers 61 / 81

The Incompressible Navier-Stokes Equation

Momentum Balance

ρdvdt

acceleration

= − ∇ ·T

deviatoric

+ ∇pisotropic

+ ρggravity

− 2ρΩ × v Coriolis force

(29)

This is the Navier-Stokes equation for incompressible flow

still rather complicated do we really need all these terms? Scale Analysis is a power- and useful concept to assess the relative

importance of terms

Scale analysis 62 / 81

Scale Analysis

Some typical values for an ice sheet

horizontal extend [L] = 1000 kmvertical extend [H] = 1 km

horizontal velocity [U] = 100 ma−1

vertical velocity [W ] = 0.1 ma−1

pressure [P] = ρg [H] = 10 MPatime-scale [T ] = [L]/[U] = 104 a1

The aspect ratio is defined as

=[H]

[L]=

[W ]

[L]= 10−3 for an ice sheet

The scaling argument for valley glaciers is almost the same


Scale Analysis

Froude numberThe Froude number Fr is the ratio of acceleration and pressure gradient.In the horizontal we have

Fr =ρ[U]/[t]

[P]/[L]=

ρ[U]2/[L]

ρg [H]/[L]=

[U]2

g [H]≈ 10−15

and in the vertical

Fr =ρ[W ]/[t]

[P]/[L]=

ρ[W ]2/[L]

ρg [H]/[L]=

[U]2

g [H]≈ 10−21

⇒ The acceleration term is negligible


Scale Analysis

Rossby numberThe Rossby number Ro is the ratio of acceleration and Coriolis force

Ro =ρ[U]/[t]

2ρΩ[U]=

ρ[U]2/[L]

2ρΩ[U]=

[U]

2Ω[L] ≈ 2 × 10−8

and thus the Coriolis to pressure gradient is

2ρΩ[U]

[P]/[L]=

Fr

Ro≈ 5 × 10−8

⇒ The Coriolis term is also negligible


Stokes Equation

By neglecting both the acceleration term Coriolis term

the Navier-Stokes equation simplifies to

∇ ·η

∇v + ∇vT

− ∇p = −ρg (30)

gravitational force exerted on the ice is balanced by stress within theice

this is called the Stokes equation

Stokes Equation 66 / 81

Temperature Equation

Recall the equation for temperature

ρdT

dt= −∇ ·q + tr (T ·D) + ρr (31)

except for the uppermost few centimeters of ice,radiation is negligible

ρdT

dt= −∇ ·q + tr (T ·D) (32)


Boundary Conditions

Balance Equations

mass ∇ · v = 0momentum ∇ ·

η

∇v + ∇vT

− ∇p = −ρg

temperature ρcdTdt − ∇ · (k∇T ) = tr (T ·D)

Boundary ConditionsTo solve this system of equations, we need boundary conditions:

mass (kinematic) momentum (dynamic) temperature (thermodynamic)

Boundary Conditions 68 / 81

Ice-Thickness Equation

∂H

∂t= (qin − qout) + as − ab

= −∇ ·Q + as − ab (33)

as and ab are in the vertical direction the volume flux Q is defined as

Q =

QxQy

=

hb vx dz

hb vy dz

derivation on p. 70 – 72 this is the central evolution equation in ice sheet dynamics


Surface Dynamic Boundary Condition

Continuity of the stress vector

T ·n = Tatm ·n (34)

Tatm is the Cauchy stress tensor in the atmosphere atmospheric pressure and wind stress are small compared to stress in

the iceT ·n = 0 (35)

this is a stress-free condition


Basal Dynamic Boundary Condition

T ·n = Tlith ·n (36)

Tlith is the Cauchy stress tensor in the lithosphere unfortunately, we have no information about the stress state of the

lithosphere this boundary condition is not very useful empirical relationship is required relate bed-parallel shear stress τb and bed-parallel basal velocity ub

ub = f (τb)


Basal Dynamic Boundary Condition

we can distinguish two cases ice is frozen to the bed if temperature below pressure melting point Tm

ice is sliding if temperature at pressure melting point Tm

ub =

0, T < Tm Dirichlet conditionf (τb), T = Tm Robin condition

despite 50 years of work (and lots of progress), this remains one ofthe grand unsolved problems in glaciology


Surface Thermodynamic Boundary Condition

the surface temperature Ts can be well approximated by the meanannual surface air temperature, as long as the latter is ≤ 0 C

we therefore simply prescribe the surface temperature (Dirichletcondition):

T = Ts


Basal Thermodynamic Boundary Condition

Cold Basethe heat flux into the ice equals the geothermal heat flux q

⊥geo

q ·n = q⊥geo

Temperate Basethe ice is at the pressure melting point Tm:

T = Tm


Thermomechanically-Coupled Glacier Flow

Balance Equations

mass ∇ · v = 0momentum ∇ ·

η

∇v + ∇vT

− ∇p = −ρg

temperature ρc dTdt − ∇ · (k∇T ) = tr (T ·D)

Boundary Conditions

mass∂h∂t + vx

∂h∂x + vy

∂h∂y − vz = as surface

∂b∂t + vx

∂b∂x + vy

∂b∂y − vz = ab base

momentum T · n = 0 surfaceub = 0, f (τb) base

temperature T = Ts surfaceT = Tm or q · n = q⊥

geo base


Numerical Modeling

the presented Stokes problem is solvable on a valley glacier scale but still very hard on a whole ice sheet scale for ice sheets, the shallowness (small aspect ratio) can be further

exploited, this leads to a hierarchy of approximations Ed Bueler’s Karthaus lecture is a good starting point


Basics and Notation We assume a fixed orthonormal basis ex , ey , ez Einstein summation convention says that double indices (here i) imply

summation

Nabla

∇ = ex∂

∂x+ ey

∂

∂y+ ez

∂

∂z= ei

∂

∂xi

Gradient the gradient of the scalar quantity g is thus

∇g = ex∂g

∂x+ ey

∂g

∂y+ ez

∂g

∂z= eig,i =

∂g/∂x

∂g/∂y

∂g/∂z

Math Stuff 77 / 81

Basics and Notation

Divergence

the divergence of the quantity g is

∇ · g =∂gx∂x

+∂gy∂y

+∂gz∂z

= gi ,i

Total Derivative or material derivative

dg

dt=

∂g

∂t+ v ·∇g

where v is the velocity

Math Stuff 78 / 81

Basics and Notation

Trace the trace of the n×n matrix A is the sum of the diagonal elements

trA = A11 + A22 + . . . + Ann = Aii

Math Stuff 79 / 81

Recall from Vector Calculus

Divergence Theorem

∂ω

φ ·n da =

ω

∇ ·φ dv (37)

The sum of all sources minus the sum of all sinks gives the net flowout of a region

Reynolds Transport Theorem

ddt

ω

φ dv =

ω

∂φ

∂tdv +

∂ω

φv ·n da (38)

What was already there plus what goes in minus what goes out iswhat is there

Math Stuff 80 / 81

The Cauchy Stress Tensor

T =

txx txy txztyx tyy tyztzx tzx tzz

For example

tez =

txztyztzz

is the stress vector of a cut along the xy-plane

Math Stuff 81 / 81

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glacier dynamics slides 2011 - Glaciers Group