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Parameterization of basal hydrology near groundinglines in a one-dimensional ice sheet model
Gunter Leguy, Xylar Asay-Davis, William Lipscomb
Los Alamos National Laboratory
January 31th, 2014
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 1 / 19
Introduction
We are exploring how basal physics influences resolution in a 1-Dvertically integrated flowline model. Goals:
Present a new parametrization of the basal shear stress (in the caseof a Marine ice sheet):
I Physically motivated (ocean connection)I Transitions smoothly between finite basal friction in the ice sheet
and zero basal friction in the ice shelf
Hope for ≈ 1km resolution near the grounding line.
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 2 / 19
Model equations
Valid in both, the ice sheet and the ice shelf:
conservation of mass : Ht + (uH)x = a,
vertically integrated stress− balance equation : τl + τb + τd = 0.
τl =[2Ā−
1nH|ux |
1n−1|ux
]x,
τd =− ρigHsx ,τb = basal shear stress,
s =
{H − b x < xg(
1− ρiρw)H x ≥ xg
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 3 / 19
Assumption and Boundary conditions
Symmetric ice sheet at the ice divide
u = 0sx = (H − b)x = 0
}at x = 0,
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
At the grounding line, we have the flotation condition and the balancebetween τl and τd .
H = ρwρi b,
2Ā−1n |ux |
1n−1ux =
12ρig
(1− ρiρw
)H
}at x = xg .
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 4 / 19
Basal stress
We adopt the formulation from Schoof (2005):
τb =− C |u|1n−1u
(Nn
κu + Nn
) 1n
.
κ =mmaxλmaxAb
λmax = wavelength of bedrock bumps
mmax = maximum bed obstacle slope
Ab = average ice temperature at the bed
Effective pressure: N ≡ pi − pwI pi ≡ ρigHI pw?
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)Surfaceelevation(km)
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 5 / 19
Basal stress (continued)
N(p) =ρigH
(1− Hf
H
)p,
where Hf = max(0,ρwρib), and p ∈ [0, 1].
N(p) satisfies the following limits:
When p = 0, N(p) = ρigH (no water-pressure support).
When p = 1, N(p) = ρig(H − Hf ) (full water-pressure support from theocean wherever the ice-sheet base is below sea-level).
At the grounding line when p > 0, N(p) = 0 (τb is continuous across thegrounding line).
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 6 / 19
Ice sheet domain (103km)
N (
kPa)
Hf/H
H (
km)
-b,
s, z
b (
km)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2
0
2
4
0.7 0.8 0.9 1 1.1 1.20
2
4
Hf
0.7 0.8 0.9 1 1.1 1.20
0.5
1
0.7 0.8 0.9 1 1.1 1.20
10
20
30
p=0
p=0.25
p=0.5p=0.75
p=1
−b
a
b
c
d
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 7 / 19
Basal stress (continued): τb = −C |u|1n−1u
(N(p)n
κu+N(p)n
) 1n
.
2 asymptotic limits:I if κu � N(p)n and τb ≈ −
γ
κ1/n|u| 1n−1u (frozen ice at the bed)
I if κu � N(p)n and τb ≈ −CN(p) u|u| . (hydrological connection)Transitions smoothly from non-zero basal stress in the ice sheet tozero basal sliding in the ice shelf
Basal stress term (kPa)
Distance to grounding line (km)
a) p = 0 c) p=1b) p=0.5
−40 −30 −20 −10 0−25
−20
−15
−10
−5
0
−40 −30 −20 −10 0−25
−20
−15
−10
−5
0
−40 −30 −20 −10 0−25
−20
−15
−10
−5
0
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 8 / 19
Numerics
Chebyshev polynomials:
Spectrally accurate
GL lies on a grid point
Used as a benchmark solution forfixed-grid model
Harder to implement in 3-D models
Δx~2km Δx~2m
x=0 x=xg
u1, H1
Fixed-grid
Suitable for 3-D model
Constant resolution
Numerically less accurate
GL usually falls between two gridpoints leading to interpolation error
I Possibility to use GLP
H1 H2u3/2HN uN+1/2
x=0 x=xc
Δx
HN+MuN+M+1/2
Ice Sheet Ice shelf
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 9 / 19
Numerics: Fix grid GLP
In our code we make sure that the grounding line is located in the lastgrounded cell.
sub-grid-scale interpolation of the grounding line
use a GLP similar to PA GB1 in Gladstone et al. (20120a):1 Determine GL using linear interpolation of the function f ≡ Hf /H2 Compute basal and driving stresses once each assuming that the cell
is entirely grounded and then entirely floating.3 The stresses are linearly interpolated between their grounded and
floating values.
HfN-1, HN-1 HfN, HN
xg
f = 1
grounded: f1
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 10 / 19
Numerics (continued)
How to compare our results for all p-values?
QualitativeI Schoof (2007) boundary layer solution (in the case of frozen bed
type basal sliding)
QuantitativeI Schoof (2007) boundary layer solution (good approximation for
p=0)I Chebyshev polynomial numerical scheme
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 11 / 19
Numerics (end)Experimental set-up:
Fixed-grid resolution comparison: 3.2 km, 1.6 km and 0.8 km.
Fixed-grid with no GLP and with GLP.
We will show results for 3 values of p.
Constant accumulation rate: a = 0.3 m/yr.
MISMIP experiment: neutral equilibrium experiments.
Two different bed topographies (as in Schoof (2007a)).
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 12 / 19
Results linear bed
The fixed-grid solution is inaccurate in capturing the retreat.
Grounding-line position (103 km)p = 0, Res = 0.8 km
Posi
tion
(1
03 k
m)
1/A
1024
1025
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8analytic
numeric (no GLP)numeric (with GLP)
numeric (Cheb)
1024
1025
analytic
numeric (no GLP)
numeric (with GLP)
numeric (Cheb)
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 13 / 19
Results linear bed (continued): MISMIP-type experiments 1 & 2
Res =
0.8
km
Res =
1.6
km
Res =
3.2
km
1/A 1/A
No GLP With GLP
Difference in grounding line position
Diffe
ren
ce (
km)
−70
−50
−30
−10010
30
50
70
300
550
−70
−50
−30
−10010
30
50
70
300
550
−70
−50
−30
−10010
30
50
70300
550
−70
−50
−30
−10010
30
50
70300
550
10̂ 24 10̂ 25 10̂ 26−70
−50
−30
−10010
30
50
70
300
550
10̂ 2410̂ 25 10̂ 24 10̂ 25 10̂ 26−70
−50
−30
−10010
30
50
70
300
550
10̂ 2410̂ 25
p=0
p=0.5
p=1
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 14 / 19
Results poly bed (continued): MISMIP-type experiment 3
Res =
0.8
km
Res =
1.6
km
Res =
3.2
km
1/A 1/A
No GLP With GLP
Difference in grounding line position
Diffe
ren
ce (
km)
−750−400−30
−20−10
0
1020
30400750
−750−400−30
−20−10
0
1020
30400750
−750−400−30
−20
−100
10
2030400750
−750−400−30
−20
−100
10
2030400750
10̂ 25 10̂ 26−750−400−30
−20−10
0
1020
30400750
10̂ 2510̂ 26 10̂ 25 10̂ 26−750−400−30
−20−10
0
1020
30400750
10̂ 2510̂ 26
p=0
p=0.5
p=1 0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
0 0.5 1 1.5 2−1
0
1
2
3
4
5
6
IceSheetdomain(103km)
Surfaceelevation(km)
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 15 / 19
Results poly bed (continued): MISMIP-type experiment 3
Res =
0.8
km
Res =
1.6
km
Res =
3.2
km
Chebyshev GL position (103 km) Chebyshev GL position (103 km)
No GLP With GLP
Error estimate in grounding line position
Err
or
est
imate
(km
)
0
510
1520
2530250
500
750
0
510
1520
2530250
500
750
0
510
1520
2530250
500
750
0
510
1520
2530250
500
750
0.75 1 1.25 1.50
510
1520
2530250
500
750
0.75 1 1.25 1.50
510
1520
2530250
500
750
p=0
p=0.5
p=1
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 16 / 19
conclusion
Error estimate decreases substantially as the value of p increasesand therefore the required model resolution decreases as pincreases.
Adding a GLP is always beneficial besides sometimes for highvalues of p.
p does not play any role in the bulk of the ice sheet. It impacts arelatively small distance (no more than 20 km from the groundingline in our experiments) which is enough to impact the solution.
Resolution of ≈ 1km or coarser is sufficient to capture the solution.
Paper submitted in the cryosphere
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 17 / 19
Ratio κu/N(p=1)n for values between 0.01 and 1
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 18 / 19
THANK YOU
LANL (Los Alamos National Laboratory) Parameterization of basal hydrology near grounding lines in a one-dimensional ice sheet modelJanuary 31th, 2014 19 / 19
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