Ondřej Souček supervizor: Prof. RNDr. Zdeněk Martinec, DrSc

Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier

Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique

Benchmarks

Numerical modeling of ice-sheet dynamics

Ondřej Souček

supervizor: Prof. RNDr. Zdeněk Martinec, DrSc.

Department of Geophysics, Faculty of Mathematics and Physics, Charles

University in Prague

Research Institute of Geodesy, Topography and Cartography, Zdiby

1.6.2010

Ondřej Souček Ph.D. defense



Benchmarks

Overview

Ice-sheets and their dynamics

Continuum thermo-mechanical model of a glacier

The Shallow Ice approximation

The SIA-I iterative algorithm

Numerical benchmarks

ISMIP-HOM A exp. - steady-stateISMIP-HOM F exp. - prognosticEISMINT - Greenland Ice Sheet models




Benchmarks

Scheme of an ice sheet




Benchmarks

Transport processes in an ice sheet




Benchmarks

Simplified ice sheet model

Cold ice

Lithosphere

Temp. ice

z

x,y

Atmospheres

CTS

b

F (x,t) = 0

F (x,t) = 0F (x,t) = 0

Cold-ice zone (pure ice only)

Temperate-ice zone (liquid water present)

Free surface and glacier base represented by differentiable surfaces




Benchmarks

Modelled problem - cold ice zone

Stokes’ flow problem

divτ + ρ~g = ~0

Equation of continuity (Constanthomogeneous ice density)

div~v = 0

Energy balance – Heat transportequation

ρc(T )T = div(κ(T )gradT )+τ.. ε

Rheology

τij = −pδij + σij

σij = 2ηεij

εij =1

2

(

∂vi

∂xj

+∂vj

∂xi

)

η =1

2A(T ′)−

13 ε

− 23

II

A(T ′) = A0 exp

(

−

Q

R(T0 + T ′)

)

εII =

√

˙εkl ˙εkl

2




Benchmarks

Boundary conditions

Free surface

Kinematic boundary condition

∂fs

∂t+ ~v · gradfs = as

Dynamic boundary condition

Traction-free

τ · ~ns = ~0

Surface temperature

T = Ts(~x, t)

Accumulation-ablation function

as= a

s(~x, t)

Glacier base

Kinematic boundary condition fb (~x, t) given

Dynamic boundary conditions

Frozen-bed conditions, T < TM :

~v = ~0

Sliding law T = TM :

~v = ~g(τ · ~n)

Geothermal heat flux

~q+= ~q

geo(~x, t)




Benchmarks

Scaling Approximation – plausibility and motivation

Glaciers and ice sheets are typically very flat features, with the aspect(height:horizontal) ratio less than 1

10for small valley glaciers and one

order less for big ice sheets ( 1100

)




Benchmarks


Natural scaling may be introduced:kinematic quantities

(x1, x2, x3) = ([L]x1, [L]x2, [H]x3)

(v1, v2, v3) = ([V↔]v1, [V↔]v2, [Vl]v3)

(fs(x1, x2), fb(x1, x2)) = [H](fs(x1, x2), fb(x1, x2))

ε =[H]

[L]=

[Vl]

[V↔]

dynamic quantities -

p = ρg [H]p

(σ13, σ23) = ρg [H](σ13, σ23)

(σ11, σ22, σ12) = ρg [H](σ11, σ22, σ12)




Benchmarks


Natural scaling may be introduced:kinematic quantities

(x1, x2, x3) = ([L]x1, [L]x2, [H]x3)

(v1, v2, v3) = ([V↔]v1, [V↔]v2, [Vl]v3)

(fs(x1, x2), fb(x1, x2)) = [H](fs(x1, x2), fb(x1, x2))

ε =[H]

[L]=

[Vl]

[V↔]

dynamic quantities - Shallow Ice Approximation SIA

p = ρg [H]p

(σ13, σ23) = ερg [H](σ13, σ23)

(σ11, σ22, σ12) = ε2ρg [H](σ11, σ22, σ12)




Benchmarks


Standard scaling perturbation series constructed: all field unknowns(vi , σij , fs , fb) are expressed by a power series in the aspect ratio ε

q = q(0) + εq

(1) + ε2q(2) + . . .

and inserting these expressions into governing equations (eq. of motion,continuity, rheology) leads to separation of these equations according to theorder of ε. (Implicit assumption: not only q is now scaled to unity but also itsgradient (???))




Benchmarks


Keeping only lowest-order terms, we arrive at the Shallow Ice Approximation,which can be explicitly solved

p(0)(·, x3) = f

(0)s (·)− x3

σ(0)13 (·, x3) = −

∂f(0)s (·)

∂x1(f (0)

s (·)− x3)

σ(0)23 (·, x3) = −

∂f(0)s (·)

∂x2(f (0)

s (·)− x3)

where (·) ≡ (x1, x2)




Benchmarks


Also the velocities may be expressed semi-analytically as

v(0)1 (·, x3) = v

(0)1 (·)sl +

∫ x3

f(0)

b(·)

A(T ′(·, x ′3))(

σ(0)13

2+ σ

(0)23

2)

σ(0)13 (·, x

′3)dx

′3

v(0)2 (·, x3) = v

(0)2 (·)sl +

∫ x3

f(0)

b(·)

A(T ′(·, x ′3))(

σ(0)13

2+ σ

(0)23

2)

σ(0)23 (·, x

′3)dx

′3

v3 from equation of continuity

v(0)3 (·, x3) = v

(0)3 (·)sl −

∫ x3

f(0)

b(·)

(

∂v(0)1

∂x1(·, x ′

3) +∂v

(0)2

∂x2(·, x ′

3)

)

dx′3




Benchmarks

Problems of SIA

Zeroth order model – looses validity in regions where higher-order terms

become important:

Regions of high curvature of the surface




Benchmarks

Problems of SIA

Zeroth order model – looses validity in regions where higher-order terms

become important:

Regions of high curvature of the surfaceRegions where a-priori dynamic scaling assumptions areviolated (floating ice – SSA, ice streams)




Benchmarks

Problems of SIAZeroth order model – looses validity in regions where higher-order terms

become important:

Regions of high curvature of the surfaceRegions where a-priori dynamic scaling assumptions areviolated (floating ice – SSA, ice streams)

Figure: RADARSAT Antarctic Mapping Project




Benchmarks

Solution?

Higher-order models – continuation of the expansion procedure andsolving for higher-order corrections (Pattyn F., Rybak O.)

Full Stokes Solvers – Finite Elements Methods (Elmer - Gagliardini O.),Spectral methods (Hindmarsch R.)

PROBLEM is the speed of full-Stokes and higher-order techniques, thecomputational demands disable usage of these techniques in large-scaleevolutionary models




Benchmarks

Any ideas?

Aim:

Find an "intermediate" technique that would exploit the scalingassumptions of flatness of ice but would provide "better" solution thanSIA:

KEY IDEA: Apply the scaling assumptions of smallness not on theparticular stress components but only on their deviations from thefull-Stokes solution




Benchmarks

SIA-I

~un //δ~un+ 1

2// ~un+ 1

2 = ~un + ǫ1δun+ 1

2

��

~un+1 = (1 − ǫ2)~un+ 1

2 + ǫ2~u∗n+ 1

2 ~u∗n+ 12

oo~vn+ 1

2oo




Benchmarks

Ice-Sheet Model Intercomparison Project - Higher Order

Models (ISMIP-HOM) - experiment A

fs (x1, x2) = −x1 tanα , α = 0.5◦,

fb(x1, x2) = fs (x1, x2) − 1000

+ 500 sin(ωx1) sin(ωx2)

with

ω =2π

[L].

Ice considered isothermal

Boundary conditions

Free surfaceFrozen bedAt the sides, periodic boundaryconditions are prescribed:

~v(x1, 0, x3) = ~v(x1, [L], x3)

~v(0, x2, x3) = ~v([L], x2, x3)




Benchmarks

ISMIP - HOM experiment A

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

v x (

m a

-1)

at f

s

x

30

40

50

60

70

80

90

100

110

120

0 0.2 0.4 0.6 0.8 1

σ xz

[kPa

] at

fb

x

Figure: Pattyn F., ISMIP-HOM results preliminary reportOndřej Souček Ph.D. defense



Benchmarks


0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

v x (

m a

-1)

at f

s

x

30

40

50

60

70

80

90

100

110

120

0 0.2 0.4 0.6 0.8 1

σ xz

[kPa

] at

fb

x

Figure: Pattyn F., ISMIP-HOM results preliminary reportOndřej Souček Ph.D. defense



Benchmarks


Computational detailsResolution: 41 × 41 × 41

Number of iterations: ε = 120 ∼ 40 iter., ε = 1

10 ∼ 100 iter.

Each iter. step took approximately 0.22 s at Pentium 4, 3.2.GHz

For comparison Elmer (Gagliardini) Full-Stokes solver ∼ 104 CPU s




Benchmarks

ISMIP-HOM experiment F

Experiment description - ice slab flowingdownslope (3◦) over a Gaussian bump

Linear rheology!

Output: Steady state free surface profileand velocities

Comparison with ISMIP-HOMfull-Stokes finite-element solution(Olivier Gagliardini computing by Elmer)




Benchmarks


Free surface (left - Elmer, right- SIA-I)




Benchmarks


Surface velocities

Numerical performance

Figure: Gagliardini & Zwinger, 2008




Benchmarks

ISMIP-HOM experiment FSurface velocities (left-Elmer,right-SIA-I)

Numerical performance

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160 180 200C

PU-t

ime/

time-

step

[s]

Iteration

’time_demands.dat’




Benchmarks

Extension for non-linear rheology

We take setting from ISMIP-HOMexperiment A (L=80) - flow of aninclined ice slab over a sinusoidal bump,periodically elongated

Evolution of free surface, steady state

Impossible to compare with evolution inElmer (too CPU time demanding)

Time demands of the SIA-I algorithmpractically the same as for the linearcase !!!

Let’s compare only velocities atparticular time instants




Benchmarks

Extension for non-linear rheology

t = 50a (left - Elmer, right - SIA-I) t = 100a (left - Elmer, right - SIA-I)




Benchmarks

EISMINT benchmark - Greenland Ice Sheet Models

Paleoclimatic simulation

Prognostic experiment - global warming scenario




Benchmarks

EISMINT Greenland - Paleoclimatic experiment

Two glacial cycles (cca 250 ka)

Temperature + sea-level forcing based on ice-core δ18O isotope record

-14-12-10-8-6-4-2 0 2 4 6

-250 -200 -150 -100 -50 0

∆ T

(K

)

time (ka)

-140

-120

-100

-80

-60

-40

-20

0

20

-250 -200 -150 -100 -50 0

∆ Se

a le

vel (

m)

time (ka)




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

3

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Surface topography (km), age = 150 ka




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

3

3

0.0

0.5

1.0

1.5

2.0

2.5

y [1

03 km

]

0.0 0.5 1.0 1.5

x [103 km]

Surface topography [km], age = 135 ka




Benchmarks


0

0

0

0

0

1

1

1

1

2

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)





Benchmarks


0

0

0

0

0

1

11

1

2

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)





Benchmarks


0

0

00

1 1

1

1

1

2

2

2

33

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)





Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

3

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Surface topography (km), age = 0 a




Benchmarks


1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

-250 -200 -150 -100 -50 0

Gla

ciat

ed a

rea

(106 k

m2 )

time (ka)

1.5

2

2.5

3

3.5

4

4.5

5

-250 -200 -150 -100 -50 0

Ice-

shee

t vol

ume

(106 k

m3 )

time (ka)




Benchmarks


(Huybrechts, 1998)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

-250 -200 -150 -100 -50 0

Gla

ciat

ed a

rea

(106 k

m2 )

time (ka)




Benchmarks

Prognostic experiment

Model response to an artificial warming scenario

temperature increase by 0.035◦ C per year for the first 80 years (total2.8◦ C increase)

by 0.0017◦ per year C for the remaining 420 years (0.714◦ C)

In total temperature increase of 3.514◦ C within 500 years




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Surface topography (km), t = 0 a




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)


−12−11.6

−11.2−10.8

−10.4

−10

−10

−10

−9.6

−9.6

−9.2

−9.2

−8.8

−8.8

−8.4

−8.4

−8.4

−8

−8 −8

−7.6

−7.6 −7.6

−7.2

−7.2

−7.2

−6.8

−6.8

−6.8

−6.4

−6.4

−6.4

−6

−6

−6

−6

−6

−5.6

−5.6

−5.6

−5.6

−5.6

−5.2

−5.2

−5.2

−5.2

−5.2 −5.2

−4.8

−4.8

−4.8

−4.8

−4.8 −4.8

−4.4

−4.4

−4.4

−4.4

−4.4

−4.4

−4

−4

−4

−4

−4

−4

−4

−3.6−3.6

−3.6

−3.6

−3.6−3.6

−3.6 −3.6

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2 −3.2

−2.8

−2.8

−2.8

−2.8

−2.8

−2.8

−2.8−2.8

−2.8

−2.8

−2.4−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2

−2−2

−2−2

−2

−2

−2

−2

−2

−1.6

−1.6−1.6−1.6

−1.6

−1.6

−1.6−1.6

−1.6

−1.6

−1.6

−1.2

−1.2−1.2

−1.2

−1.2

−1.2

−1.2−1.2

−1.2

−1.2

−1.2

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8−0.8

−0.8

−0.8

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

0.4

0.4

0.4

0.8

0.8

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Initial accumulation−ablation function (m a −1)




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)


−12−11.6

−11.2−10.8

−10.4

−10

−10

−10

−9.6

−9.6

−9.2

−9.2

−8.8

−8.8

−8.4

−8.4

−8.4

−8

−8 −8

−7.6

−7.6 −7.6

−7.2

−7.2

−7.2

−6.8

−6.8

−6.8

−6.4

−6.4

−6.4

−6

−6

−6

−6

−6

−5.6

−5.6

−5.6

−5.6

−5.6

−5.2

−5.2

−5.2

−5.2

−5.2 −5.2

−4.8

−4.8

−4.8

−4.8

−4.8 −4.8

−4.4

−4.4

−4.4

−4.4

−4.4

−4.4

−4

−4

−4

−4

−4

−4

−4

−3.6−3.6

−3.6

−3.6

−3.6−3.6

−3.6 −3.6

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2

−3.2 −3.2

−2.8

−2.8

−2.8

−2.8

−2.8

−2.8

−2.8−2.8

−2.8

−2.8

−2.4−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2.4

−2

−2−2

−2−2

−2

−2

−2

−2

−2

−1.6

−1.6−1.6−1.6

−1.6

−1.6

−1.6−1.6

−1.6

−1.6

−1.6

−1.2

−1.2−1.2

−1.2

−1.2

−1.2

−1.2−1.2

−1.2

−1.2

−1.2

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8

−0.8−0.8

−0.8

−0.8

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

−0.4

0.4

0.4

0.4

0.8

0.8

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Initial accumulation−ablation function (m a −1)




Benchmarks


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)


0

0

0

0

1

1

1

1

2

2

2

3

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)


−600

−600 −4

00

−200

−200

−200

−200

−200

0.0

0.5

1.0

1.5

2.0

2.5

y (1

03 km

)

0.0 0.5 1.0 1.5

x (103 km)

Topography difference (m)

−1200−1100−1000

−900−800−700−600−500−400−300−200−100

0100200300400500600700800




Benchmarks

Thank you for your attention!


Documents

Ondřej Souček supervizor: Prof. RNDr. Zdeněk Martinec, DrSc