Page 1
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
Page 2
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 3
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scheme of an ice sheet
Ondřej Souček Ph.D. defense
Page 4
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Transport processes in an ice sheet
Ondřej Souček Ph.D. defense
Page 5
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 6
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 7
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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)
Ondřej Souček Ph.D. defense
Page 8
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
)
Ondřej Souček Ph.D. defense
Page 9
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
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)
Ondřej Souček Ph.D. defense
Page 10
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
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)
Ondřej Souček Ph.D. defense
Page 11
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
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 (???))
Ondřej Souček Ph.D. defense
Page 12
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
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)
Ondřej Souček Ph.D. defense
Page 13
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Scaling Approximation – plausibility and motivation
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
Ondřej Souček Ph.D. defense
Page 14
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Problems of SIA
Zeroth order model – looses validity in regions where higher-order terms
become important:
Regions of high curvature of the surface
Ondřej Souček Ph.D. defense
Page 15
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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)
Ondřej Souček Ph.D. defense
Page 16
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 17
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 18
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 19
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 20
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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)
Ondřej Souček Ph.D. defense
Page 21
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Page 22
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Page 23
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP - HOM experiment A
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
Ondřej Souček Ph.D. defense
Page 24
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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)
Ondřej Souček Ph.D. defense
Page 25
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment F
Free surface (left - Elmer, right- SIA-I)
Ondřej Souček Ph.D. defense
Page 26
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
ISMIP-HOM experiment F
Surface velocities
Numerical performance
Figure: Gagliardini & Zwinger, 2008
Ondřej Souček Ph.D. defense
Page 27
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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’
Ondřej Souček Ph.D. defense
Page 28
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 29
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Extension for non-linear rheology
t = 50a (left - Elmer, right - SIA-I) t = 100a (left - Elmer, right - SIA-I)
Ondřej Souček Ph.D. defense
Page 30
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT benchmark - Greenland Ice Sheet Models
Paleoclimatic simulation
Prognostic experiment - global warming scenario
Ondřej Souček Ph.D. defense
Page 31
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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)
Ondřej Souček Ph.D. defense
Page 32
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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
Ondřej Souček Ph.D. defense
Page 33
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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
Ondřej Souček Ph.D. defense
Page 34
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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)
Surface topography (km), age = 125 ka
Ondřej Souček Ph.D. defense
Page 35
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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)
Surface topography (km), age = 115 ka
Ondřej Souček Ph.D. defense
Page 36
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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)
Surface topography (km), age = 75 ka
Ondřej Souček Ph.D. defense
Page 37
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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
Ondřej Souček Ph.D. defense
Page 38
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
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)
Ondřej Souček Ph.D. defense
Page 39
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
EISMINT Greenland - Paleoclimatic experiment
(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)
Ondřej Souček Ph.D. defense
Page 40
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
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
Ondřej Souček Ph.D. defense
Page 41
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
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
Ondřej Souček Ph.D. defense
Page 42
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
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
−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)
Ondřej Souček Ph.D. defense
Page 43
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
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
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 = 500 a
−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)
Ondřej Souček Ph.D. defense
Page 44
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Prognostic experiment
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
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 = 500 a
−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
Ondřej Souček Ph.D. defense
Page 45
Ice sheets and their dynamicsContinuum thermo-mechanical model of a glacier
Shallow Ice Approximation (SIA)SIA-I Iterative Improvement Technique
Benchmarks
Thank you for your attention!
Ondřej Souček Ph.D. defense