karthaus sept. 2007 – slide number 1 numerical modelling of ice sheets and ice shelves tony payne...

Karthaus Sept. 2007 – Slide Number 1

Numerical modelling of ice sheets and ice shelves

Tony PayneCentre for Polar Observation and Modelling

University of Bristol, U.K.


Reasons for numerical modelling

o Underlying equations cannot be solved analytically.

o Several different equations that are coupled.

o Irregular geometries when modelling real ice masses.


Process of numerical modelling

o Continuum mechanics description of problem

o Scaling the equations

o Discretization (quantities no longer in continuous form but represented at finite locations)

o Model coding

o Validation of the model

o Use of the model


Validating of numerical modelling

o Results from the numerical model are only useful if they can be shown to be independent of the discretization process

o Need to show that results do not vary with time step and grid sizes

– Repeat experiments using a range of grid and time steps

o Numerical models should be tested against any analytical solutions available

o EISMINT benchmarks also very useful in model development


Types of numerical modelling

o Finite element

– Great freedom in dealing with irregular geometries

– Fairly complicated to develop although software now available to do many of standard tasks

– Computationally demanding

o Finite difference (finite volume)

– Simple to use and develop

– Problems in coping with highly irregular domains


Basis of finite differences

o Taylor series give a relation between continuous and finite representations

o The series is truncated after a certain number of terms

o The terms omitted give rise to the truncation error of the approximation

o When equations implemented also incur rounding error (due to finite representation of number in memory: use dp)

2 32 3

2 3( ) ( ) ...

2! 3!

t tdH d H d HH t t H t t

dt dt dt


Taylor series

2 32 3

2 3( ) ( ) ...

2! 3!


dt dt dt

t t+∆t

( )dH

H t tdt

H(t)

H(t +∆t)

22

2( )

2!

tdH d HH t t

dt dt



2 32 3

2 3

22 3

2 3

2 32 3

2 3

( ) ( ) ...2! 3!

( ) ( )...

2! 3!

( ) ( )( )

( ) ( ) ...2! 3!

( ) ( )( )


dt dt dt

tdH H t t H t d H t d H

dt t dt dt

dH H t t H tO t

dt t


dt dt dt

dH H t H t tO t

dt t

The order of the approximation is the order of the first omitted term



o In general second-order approximations are sufficient

o In certain circumstances more accurate higher orders may be needed, e.g. near abrupt transitions

2 32 3

2 3

2 32 3

2 3

22

2

2

( ) ( ) ...2! 3!

( ) ( ) ...2! 3!

( ) 2 ( ) ( )( )

3 ( ) 4 ( ) ( 2 )( )

23 ( ) 4 ( ) ( 2 )

2


dt dt dt


dt dt dt

d H H t t H t H t tO t

dt tdH H t H t t H t t

O tdt t

dH H t H t t H t t

dt t

2( )O t


Stability of finite differences

o Linear instability: Courant-Friedlichs-Lewy constraint on grid size for explicit techniques

o Non-linear instability arises when we try to solve non-linear problems using linear techniques

o In general it manifests itself as high-frequency peaks which grow rapidly

0 1t

cx


Potted history of numerical modelling

o 1970’s Modelling of glaciers and ice sheets

– 1- and 2-d. models

– zero-order shallow ice approximation

– Mahaffy, Budd, Oerlmans, Pollard, …

– Key techniques: solving non-linear parabolic eqns. in 1 and 2d.



o 1980’s Ice temperature evolution

– 3d. models

– still zero-order flow models

– Jensen, Huybrechts, Ritz, Greve …

– Key techniques: stretched vertical coordinate



o 1980’s Modelling of ice shelves and streams

– 2-d. models

– first-order shallow ice approximation but vertically integrated

– MacAyeal, Determann, Thomas …

– Key techniques: solving coupled non-linear elliptic eqns. in 2d.



o 1990’s Coupled system evolution

– 2d. Ice shelf models

– 3d. Ice sheet, temperature models

– Marshall, Huybrechts, Hulbe …

– Key techniques: internal boundaries



o 2000’s General ice flow models

– 3-d. models

– first-order shallow ice approximation with no vertically integration

– Pattyn, Blatter, Gudmundsson …

– Key techniques: solving coupled non-linear elliptic eqns. in 3d.


Stresses

o Easiest to think of as a force per unit area.

o Distinct from pressure in that they have a direction (are vector quantities).

o Units are Pa or bar (1 bar = 100 kPa)

o Normal stresses (usually denoted by s) act perpendicular to a surface.

o They can be tensile (positive, extensional) or compressive (negative).

o Shear stresses (t) act parallel to a surface.


Stress tensor

o Imagine a cube.

o There are three normal stresses corresponding to the three axes

o There are six possible shear stresses

, ,

, ,

, ,

x y z

xy xz yz

yx zx zy


Stress tensor

o Subscripts denote direction for normal stresses

o Direction perpendicular to surface and then direction acting for shears

o However, reduces to three shears if assume body in equilibrium and can not rotate

yx

xy xy

yx

y

x

yx xy


Stress equilibrium or force balance

o In glaciology we can neglect acceleration so that Newton’s second law reduces to a force balance

o If there were a force imbalance, acceleration would operate ‘instantaneously’ to restore equilibrium

o Balance is between internal stresses and body forces such as gravity


Equations of stress equilibrium

Assume static balance of forces by ignoring acceleration

0

0

xyx xz

y xy yz

zy xzz

x y z

y x z

gz y x

g

z

x

y



( ) ( ) ( ) ( )x xx x x xx x x x x x x

x x

x x+∆x

x



0

0

xyx xz

xyx xz

y z x x y z x z yx z y

x z y

xxx

∆x

∆y

∆z



0

0

xyx xz

xyx xz


x z y

∆x

∆y

∆z

zx xzz zz z



0

0

xyx xz

xyx xz


x z y

∆x∆y

∆z

yx xyy yy y




0

0

xyx xz

y xy yz

zy xzz

x y z

y x z

gz y x

g

z

x

y


Strain rates

o Strain is a measure of the deformation of an object

o Typically ratio of a length before deformation to one after (proportional change) and has no units (i.e. is a ratio)

o A strain rate is simply the rate at which this happens.

o Denoted by epsilon dot (dot shows that it is rate) and has units of yr-1


Strain rates

o Defined as gradients in velocity

o Strain can be normal (simple extension or contraction)

o Or shear which changes shape

o Six strain rates that are equivalent to six stresses

1

2

x

xy

u

x

u v

y x

dx

v1

u1

v2

u2dy

y

x


Glen’s flow law

A is related to temperature and impurities

n is a constant equal to 3

is effective stress (measure of overall stress regime)

is effective strain rate

nA 1nij ijA

relates to strain rates to stress deviators


Definition of stress deviators

o in the case of normal stress, the deviator equals the stress minus the pressure (mean of three normal stresses)

o in the case of shear stress, the deviator simply equals the shear stress

1

3i i x y z

ij ij


Effective stresses

o Measure of total stress environment

o Also for strains

2 2 2 2 2 2 22 2x y z xy xz yz


Shallow-ice approximation(s)

o Provide a rigorous simplification of the full stress equilibrium equations.

o Based on use of scaling and in particular on the aspect ratio of an ice mass (thickness to span).

o The aspect ratio is small O(0.005) hence shallow ice.

o Three are 3 flavours (or orders) depending on what effects are left in.




0

0

xyx xz

y xy yz

zy xzz

x y z

y x z

gz y x

g

z

x

y


Scaling

o Replace variable with a constant scale multiplied by a scaled variable

o allows size of different terms to be compared without further analysis

o Scaled variable should always be O(1)

0

0

original variable

constant scale

scaled variable

H H H

H

H

H


Scaling

0 0 0

0

0

20

00

z H z x X x y X y

H

X

gHgH

X


Word on stress deviator scaling

o Some justification for this scale comes from the condition at the ice mass’ upper/lower boundary

o At the upper surface, stress components must balance

00 0

0

20

00

0

0

xz x

xz x

s

xH s

T gHX x

gHT

X


“Glaciology is a ONE BAR subject” (or 100 kPa in S.I.)


Vertical equation

Scale equations (note have dropped tildas from scaled variables)

2 20 0 0

2 20 0 0

2 2 1

zy xzz

zy xzz

H H Hg g

H z X y X x

z y x


Horizontal equations

Scale equations (note have dropped tildas from scaled variables)

2 20 0 0

20 0 0 0

2

2

0

0

0

xyx xz

xyx xz

y xy yz

H H Hg

X x X y H X z

x y z

y x z


Zero-order approximation

Retain highest order (biggest) terms in each equation

2

2

2 2

0

0

1

xyx xz

y xy yz

zy xzz

x y z

y x z

z y x



Retain highest order (biggest) terms in each equation

0

0

1

x xz

y yz

z

x z

y z

z



Easier to move back to unscaled version of equations now

xz x

yz y

z

z x

z y

gz


Higher-order approximations

Successively include smaller terms

2

2

2 2

0

0

1

xyx xz

y xy yz

zy xzz

x y z

y x z

z y x


First-order approximation


2

2

2

0

0

xyx xz

y xy yz

zyz

x y z

y x z

z y

2 xz

x

1


Second-order approximation


2

2

2 2

0

0

1

xyx xz

y xy yz

zy xzz

x y z

y x z

z y x


Quick zero-order derivation

o assume on basis of aspect ratio that

normal stress dominates in vertical

vertical shear dominates horizontal shear

xyx

x y

0xz

y xy

z

y x

0yz

zyz

z

z y

xz

x

g



o Integrate vertical stress balance

o It can be shown that all normal stresses are equal (i.e. that the pressure is hydrostatic)

0

0

( ) ( )

x xz

y yz

z

z x y

x z

y z

z g s z



0

0

xz

yz

g s zx z

g s zy z

o Substitute horizontal normal stresses with vertical normal stress



0

0

( ) ( )

( ) ( )

xz

yz

xz

yz

sg

x z

sg

y z

sz g s z

xs

z g s zy

o Tidy up

o Gravitational driving stress (RHS) balanced entirely by vertical shear (LHS)

o For whole ice column gravitational driving is balanced locally by basal drag alone


Quick zero-order approximation

Normally used in combination with Glen’s flow law

1

2xz

u w

z x

1

2 22

nxz

x

A

2y 2

z 22 xy 2 2xz yz



Shown that vertical shear stresses are only important ones

1

2 2 2

2

ˆ

ˆ

nxz

xz yz

xz

yz

uA

z

sg s z

xs

g s zy



Integrate from bed to some height within ice to get velocity

12 2 2

1ˆ2 2 2

ˆ2

ˆ ˆ2

n

n

nz

n n

h

u s s sA g s z

z x y x

s s su z g A s z dz

x y x



For A assumed constant with depth

12 2 2

1

12 2 2

2

2( )

1

2( )

2

n

n n

n

n n

A s s su s g H u h

n x y x

A s s suH g H u h H

n x y x


Simple glacier flow model: 1-d. equations

1

22( )

2

nn nA s s

uH g H u h Hn x x



o Ice thickness (H) evolution

o Zero order model of ice flow (u)

o Mass balance model (b)

o Here simplified by assuming flat bed

112 ( )

2

nnn

H Hub

t x

A g s su H

n x x



o Mass balance model (b)

o Nonlinear (D) parabolic equation

o Here simplified by assuming flat bed 1

22 ( )

2

nnn

s sb D

t x x

A g sD H

n x


Glacier model: spatial discretization

o Regular 1-d. grid

o Staggered so that fluxes evaluated halfway between cells

1

1/ 2

1

1/ 2

1/ 21/ 2

2

,

i i

i

i i

i

ii

s ss

x x

D D

Ds

D Hx

GRID i-1 i-½ i i+½ i+1

H,s,b ○ ○ ○

∂s/∂x ● ●

D(1) ○ ● ○ ● ○

D(2) ● ●



o Substitute FD fluxes

1 11/ 2 1/ 2

1 1/ 2 1 1/ 2 1/ 2 1/ 22

1

1

i i i ii i i

i i i i i i i i

s s s ssb D b D D

x x x x x

b s D s D s D Dx



o Forms a tridiagonal matrix

1.5 2.5 1.5 2.5 2 2

2.5 3.5 2.5 3.5 3 32

3.5 4.5 3.5 4.5 4 4

. . . .

1

. . . .

D D D D s b

D D D D s bx

D D D D s b


Glacier model: explicit time discretization

o First-order, forward approximation to time derivative

o Disadvantages: first-order error (Taylor’s) and not very stable (CFD limit)

i-1 i-½ i i+½ i+1

time ○ ● ○ ● ○

time + dt ○

1i i

i

s ss

t t


Glacier model: fully implicit discretization

o First-order, backward approximation to time derivative

o still first-order error

o but now far more stable

o however need to evaluate flows a new time step

i-1 i-½ i i+½ i+1

time ○

time + dt ○ ● ○ ○ ●

1

1

i i

i

s ss

t t


Glacier model: implicit solution

o Use direct Gaussian solver on tridiagonal (NR)

o No additional computing cost compared to explicit

o Stable as long as diagonally dominant (i.e. always in this case)

1.5 2.5 1.5 2.5 2 2 2

2.5 3.5 2.5 3.5 3 3 3

3.5 4.5 3.5 4.5 4 4 4

1/ 2 1/ 2 2

. . . .

1

1

1

. . . .

i i

E E E E s s b

E E E E s s b

E E E E s s b

tE D

x


Glacier model: semi-implicit discretization

o Second-order, centred approximation to time derivative (Crank-Nicolson)

o need to evaluate flows at both time steps

i-1 i-½ i i+½ i+1

time ○ ● ○ ○ ●

time + dt ○ ● ○ ○ ●

1

1/ 2

i i

i

H HH

t t


Glacier model: non-linear instability

o Diffusivity (D) is a highly non-linear function of thickness (power 5) and surface slope (power 2)

o Good type problem for methods later on

o Options:

– Ignore the problem (just use D from previous time step)

– Picard iteration

– Under-relaxation, Hindmarsh schemes

– Newton-Raphson iteration


Glacier model: Picard iteration

Calculate Dfrom H, s

Solve for new Sgiven D

DONE

Check forconvergence


Glacier model: relaxation and Hindmarsh

o Non-linear instability manifests itself as high-frequency oscillation as numerical solution over corrects past the ‘true’ solution but never finds it

o Under-relaxation reduces the correction applied in each iteration and makes convergence more likely

o The scheme published by Hindmarsh and Payne (1996) recognises the (chaotic) oscillatory behaviour and aims for the ‘mid-point’

iterationsH


Glacier model: Newton-Raphson solver

o Recognises the non-linear nature of the problem fully

o Very fast convergence

o More complicated to programme but MAPLE etc makes evaluation of Jacobian easy

1

1 1

FD ( )

( ) ( ) ( )

( )

. . .. .

, ,

. .. . .

k k kk k ki i ii i ik k k

i i i

s sf b D s

t x x

f s ds f s f sf s

ds dsf s

dsf s

f f fs s f

s s s


Glacier model: final comments

o Can insert expressions for basal slip but these typically make assumption about all gravitational driving balanced locally

o Strictly only appropriate at grid scales 10-20 times ice thickness

o Easily coupled to simple mass balance – climate models (also isostasy)

o Very successful in explaining features of Ice Ages

o VERIFY numerics by comparing the Nye-Vialov steady-state profiles


Examples

This type of model was very useful in understanding causes of Ice Ages and relation to Milankovitch forcing

snowline


Examples

Coupled via snowline to simple climate models.

Used to explain disparity between forcing and response


Examples

Range of experiments showed that 100 kyr cycles could be reproduced if extra physics added to cause ‘fast’ deglaciation

Pollard, 1982


Simple ice sheet model: 2-d. equations

o 2-d. version of previous non-linear parabolic equation

o Very similar derivation

o 2-d. nature of problem increases complexity of numerics

1

2 2 22

.

2 ( )

2

nn

n

H s sb D s b D D

t x x y y

A g s sD H

n x y


Simple ice sheet model: spatial discretization

o Number of choices for how quantities calculated on staggered grid

o Problem originally studied in GCM by Arakawa

o C-grid traditionally used by glaciologists although B may have advantages


Simple ice sheet model: spatial discretization

o Problems with evaluating all derivatives on C-grid

o B-grid more symmetrical

● flux

○ thickness

i-1 i-½ i i+½ i+1

j+1 ○ ○ ○

j+½ ●

j ○ ● ○ ● ○

j-½ ●

j-1 ○ ○ ○

i-1 i-½ i i+½ i+1

j+1 ○ ○ ○

j+½ ● ●

j ○ ○ ○

j-½ ● ●

j-1 ○ ○ ○


Simple ice sheet model: implicit solution

o No longer have simple tridiagonal matrix

o This means direct (Gaussian) methods of solution will be very, very expensive

o Need to look towards iterative methods …

1 1 1 1

1 1

1 1 1 1

1 1

1 1

1 1

1

1 1

. .. .

. .. .

i j i j

ij ij

i j i j

i j i j

ij ij

i j i j

i j i

ij

i j

H bD

H bD

H bD

H bU D L D U L D U

H bL D U L D U L D U

H bL D U L D U D U

H bL D U

HL D U

HL D

1 1

1

1 1

j

ij

i j

b

b

A.x=b


Simple ice sheet model: matrix solvers

o The traditional method (i.e., explicit) uses a point iteration

o The convergence of the technique can be improved by over-relaxation (SOR)

o Could use line inversion (alternating direct implicit, ADI)


Simple ice sheet model: matrix solvers

o Conjugate gradient methods are ideally suited

o Commonly available as library routines that can be easily incorporated

o Numerical recipes or SLAP library (which offers many preconditioners)

o All use sparse matrix storage

o Multigrid techniques now popular in many other fields but untried in glaciology

1( )

2f x

f

A.x b

x.A.x -b.x

A.x b


Examples

Type of model was work horse of modelling community and has produced many of the simulations on which previous IPCC reports are based (eg Huybrechts and de Wolde 1999)


Reasons for interest in temperature

o Determines whether water present at the bed

– ice frozen to bed exerts greater traction

– meltwater allows lubrication and reduced traction

– all known ice streams have water at the bed

o Determines the softness of ice

– warm is deforms more rapidly

– very large effect over natural temp. range


Ice temperature evolution : equations

o Vertical diffusion

o Vertical advection

o Horizontal advection

o Dissipation

o Local rate of change

o Upper boundary condition: air temperature

o Lower boundary condition: geothermal heat flux

2

2

z s a

z h

T k T T T Tw u v

t C z z x y C

T T

T G

z k


Potential coupling to ice flow?

o Effect of thicker ice?

o Effect of increased driving stresses?

o Effect of faster velocities?

o Effect of warmer temperatures?

2

2

z s a

z h

T k T T T Tw u v

t c z z x y c

T T

T G

z k


Ice temperature evolution: numerics

o Treat as column model with horizontal terms as corrections

o 1-d. diffusion equation can be solved using previous methods

o Additional techniques needed:

– Boundary conditions

– Advection terms

– Stretched vertical coordinate


Temperature evolution : upper boundary

o Dirichlet boundary condition

o Simply use air temperature at k = 1

1 1 1

11

2,

t t t tt t k k k k

k k

ta

T T T Tt kT T

z C z z

k n

T T


Temperature evolution : lower boundary

o von Neumann or flux boundary condition

o Use boundary condition to supply extra equation for n+1

o Substitute this into original approximation at n

1 1 1

1 11 1

1 11 1

112

2,

2

2

2 2

t t t tt t k k k k

k k

t tn n

t tn n

t t t tn n n n

T T T Tt kT T

z C z z

k n

T T G

z kG

T T zk

t k t GT T T T

z C z C


Ice temperature evolution: advection

o Number of options for first derivatives associated with advection

– Centred, second order

– Non-centred, first order

– Non-centred, second order

o Centred derivative is unconditional unstable (solution splits)

o Non-centred, first order introduces excessive artificial diffusion

1 1

1 1

2 1 1 2

2

,

4 3 3 4,

2 2

i i

i i i i

i i i i i i

T T

xT T T TT

x x xT T T T T T

x x


Ice temperature evolution: advection

o Non-centred, second order normally produces satisfactory results if used as part of an upwinding scheme

2 1

1 2

4 3

2 2

3 4

2 2

k k i i i

k k i i i

u u T T TTu

x x

u u T T T

x


Ice temperature evolution: stretched grid

o Ice masses have highly irregular vertical profiles

o This causes problems if a regular grid is used in the vertical

o Points will jump in and out of the domain as the ice mass thins and thickens

o Have implicitly been ignoring this problem in spatial evolution of ice mass



o Grid stretching is a technique that ensures the grid always fits to the local ice thickness

o Also ensures that boundaries always lie on grid points (hence boundary conditions easier to employ)

o However does add the complexity of equations

( )

( ) 0

( ) 1

s zz

Hs

h


Ice temperature evolution : stretched grid

o Need to add terms to deal with deformed grid

o This is easy for terms in z

2 2

2 2 2

1

1

1

T T

z zs z

z z H H

T T

z H

T T

z H



o But trickier for other terms which in t, x, and y

o If this is done with each term then an ‘apparent velocity’ for the grid can be found

*

*

1

z

T T T

x x x

s z

x x H

s H

x H x x

w u vt x y

w wT T

t H



o An irregularly spaced grid is often used in the vertical, stretched coordinate

o Points are concentrated near the bed (most deformation)

o This does increase truncation error (non centred)

o Better alternative is to use non-linear coordinate

( )exp 1

exp( ) 1

( ) 0

( ) 1

a s zHa

z s

z h


Ice temperature evolution: tidying up

o Horizontal velocities still come from zero order shallow ice model

o Vertical velocities found using incompressibility condition

o Numerical integration from bed to surface (Trapezoidal Rule, NR)

o Test of integration accuracy from surface kinematic boundary condition

( ) ( )

( ) ( )

w u v

z x y

h h hw h u h u h M

t x xs s s

w s u s u s bt x x


Ice temperature evolution: uses

o Work coupling melt rates to simple local water storage model



o Interaction between flow, temperature and basal water can lead to interesting results …


Ice streams and thermomechanics

o Coupled evolution of ice sheet flow, form and temperature

o Activation front of steep surface slopes triggers warming and sliding

o Eventual cooling through cold ice advection and loss of dissipation


Heinrich events

o Marshall and Clark (1997) simulation of Heinrich events with 3-d. thermomechanical model


EISMINT thermomechanical results

o Comparison of 10 ice sheet models with thermomechanical coupling (Payne and others 2000)

o Results show sensitivity details of numerics

o Hindmarsh (2004) uses normal modes to show that while feedback exists it is relatively weak (hence numerical problems)



o Also for Antarctica …

log10(|U|)


Ice shelf models: derivation

o Still assume that vertical balance is dominated by normal stress gradient

o Show remainder of derivation for 1-d. confined shelf

0

0

xyx xz

y xy yz

zyz

x y z

y x z

z y

xz

x

g



o Repeat zero-order proceedure

o Left with additional stress deviator term

0

1

2

2

x xz

z

x x x z

x xz

x zg s z

sg

x z x



o Integrate through ice thickness

o Apply Leibniz rule

( ) ( )

( ) ( )

2

( , ) ( ( ), ) ( ( ), )

( ) ( )

s s sx xz

h h h

b x b x

a x a x

s sx

x x x

h h

sdz dz g dz

x z x

f b adz f z x dz f b x x f a z x

x x x x

s hdz dz s h

x x x x



o Subsititute boundary conditions for stresses

( ) ( )

( ) ( )

( ) ( )

xz x

xz x

s sx

x xz x

x

z x

h h

b

b

ss s

xh

h hx

dz dz s hx x



o Perform remaining integrations

o Terms cancel etc.

2 ( ) ( )

2

s s sxz

x xz xz bx

h h h

x bx

sdz s h dz g dz

x z x

sH gH

x x



o Two 2-d. vertically integrated equations

o also used to study ice streams with inclusion of a basal drag term

o Linear slip law normally used

o Up to this point the derivation is general

2

2

2

2

xyx y bx

xyy x by

bx

by

sH H gH

x y x

sH H gH

y x y

u

v



o Normally solved by replacing stress deviators with strain rates

o Inverse form of Glen’s flow law

o Define an effective viscosity (f)

1

1

1

12

12

1 1

2

x n

y n

xy n

u uf

A x xv v

fA y y

u v u vf

A y x y x



o Problem is that need vertically averaged quantities for stresses and velocities etc.

o No simple relation between these averages if vertical shear is present

o Must assume (e.g., MacAyeal) that no variation in the vertical

o This limits application to true ice shelves or shelfy ice streams

2xu

fx



o Effective viscosity can also be found in terms of strain rates

11

1/ (1 ) /

(1 ) / 22 221/

1

21

2

1 1

2 4

n

n n n

n n

n

f A

f A

u v u v u vf A

x y x y y x



o Final equations are coupled non-linear elliptical equations in u and v

o Non-linearity enters via effective viscosity

2 2

2 2

u v u v sf H f H gH

x x y y y x x

v u u v sf H f H gH

y y x x y x y


Ice shelf models: solution

o Solve for u given v, and vice versa

o Use previous techniques for 2-d. parabolic equations (conjugate gradients)

o Deal with non-linearity using previous techniques also

4

2

4

2

u ufH fH

x x y y

s v vgH fH fH

x x y y x

v vfH fH

y y x x

s u ugH fH fH

y y x x y


Ice shelf models: boundary conditions

o Kinematic: specify a velocity either from

– observations (if modelling shelf in isolation) or

– zero-order model (if coupling to an ice sheet model)

o Dynamic: specify a stress

– Appropriate to front of ice shelf

– Balance of forces with displaced water

Hh

12

12

xx x xy y

w

yy y xy x

w

n gHn n

n gHn n


Ice shelf models: boundary conditions

o Dynamic boundary condition messy

o Greatly simplified if implemented so that shelf front aligned along x or y

o Problem when modelling irregular shaped shelves

o Use an artificial shelf with arbitrarily low thickness to extend shelf to domain edge

1, 0

12

0

x y

xw

xy

n n

gH


Ice shelf models: thickness evolution

o Surface elevation can be found from buoyancy

o Ice thickness evolution can no longer be coupled with velocity calculation

o Must be solved separately

w i

w

s H

H uH uHb

t x y


Ice shelf models: thickness evolution

o Number of alternatives

– Convert velocities to diffusivities and use old method

– Solve using simple schemes for hyperbolic equations (e.g. staggered leapfrog etc)

– Use more complex transport scheme (e.g. semi-lagrangian methods)

o A satisfactory general solver has yet to be found

1 1

1/ 2 1/ 2

/

2 t tt ti i i i

uD

H xH H

b Dt x x

H uHb

t xt

H H uH uHx


Ice shelf model: algorithm

Solve for u given f, v

Check forconvergence

Solve for v given f, u (old)

Calculate for f given u, v

Calculate new thickness distribution

Calculate new gravitational driving stress and boundary

conditions


Davis and others (2005) use ERS alimetry to determine change in surface elevation over last decade

Antarctic mass balance


Comparison with ERA-40 climate reanalysis for 1980 to 2001 for precipitation

Antarctic mass balance


Interest in Pine Island Glacier

o PIG has the largest discharge (66 Gt yr-1) of all WAIS ice streams

o with Thwaites Glacier, it drains 40% of the WAIS

o little studied in comparison to Siple Coast ice streams



ogrounding line retreated 8 km between 1992 and 1994

oimplies ice thinning at the grounding line of the order of 3.5 m yr-1

oradar altimetry shows widespread thinning

othinning pattern extends 150 km from grounding line

othinning maps on to template of fast flowing section of ie stream



o thinning unlikely to be related to snowfall variation

o hypothesized causes of thinning:

a) internal flow mechanics of ice stream (surging?)

b) long-term response to climate change (LGM?)

c) recent collapse of ice shelf and/or change in grounding


Outline of modelo Vertically-integrated

‘MacAyeal’ model includes

– momentum balances in x and y

– assumes vertical shear minimal

– viscous flow law

– dynamic b.c. at shelf front

o Prognostic thickness evolution but based on perturbations to ice flow

2

2

2

2

.

x y xy

y x xy

iref

sH H H u gH

x y y x

sH H H v gH

y x x y

H HUH

t t

• Same domain and grid• Ice surface now two dimensional• Time steps: 0.01 yr for H and 0.1 yr

for u/v (fast wave speeds)


Used to study Pine Island Glacier

Dynamic b.c.(stress)

Shallow ice model

‘MacAyeal’ stream

‘MacAyeal’ shelf

Kinematic b.c.(velocity)

Thickness evolution

throughout


Results: surface lowering in m after 0 yr


Results: accumulated thinning after 150 yr.

after 50 yr

Total thinning produced

by a range of 2

changes near the

grounding line


General ice-flow models: derivation

o Both the ice sheet and shelf models are limited in their applicability by the assumptions that are made in their derivation

o Ice sheet models assume local stress balance, and are only strictly applicable at 10-20 time ice thickness

o Ice shelf models assume no vertical shear

o Many problems in glaciology lie between these two extremes, e.g.

– Many (most) ice streams

– Onset areas and shear margins

– Valley glaciers

– Ice divides (coring locations)


General ice-flow models: derivation

o This motivates development of general models

o Approach is similar to ice shelf model but vertical shear terms are not discarded

o Stretched coordinates are again used

o But second derivatives in x introduce much complexity

(1 ) / 222 21/

2

2

2

2

1 12

2 2

1 1

2 2

ˆ ˆˆ ˆ

ˆ ˆˆ ˆ

xyx xz

n n

n

sg

x y z x

u u u sf f f g

x x y y z z x

u u uf A

x y z

u u uf f f

x x x x x

u uf f

x x x

ˆˆ uf

x


Dangers of numerical modelling

o Tendency to treat model as a black box or a surrogate for reality.

o Cannot simply use ‘the model says …’ without proving that the model solves the underlying equations satisfactorily and that processes being described are realistic consequences of the equations.

karthaus sept. 2007 – slide number 1 numerical modelling of ice sheets and ice shelves tony payne...

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