Parameter estimation for grounding-line transition · 2013-02-22 · Parameter estimation for grounding-line transition Gunter Leguy, Xylar Asay-Davis, William Lipscomb Los Alamos

Parameter estimation for grounding-line transition

Gunter Leguy, Xylar Asay-Davis, William Lipscomb

Los Alamos National Laboratory

February 15th, 2013

LANL (Los Alamos National Laboratory) Parameter estimation for grounding-line transition February 15th, 2013 1 / 20

Introduction

We are exploring how basal physics influences resolution in a 1-Dvertically integrated flowline model.

Several models of the form τb = β2u have been investigated:I Schoof (2007): β2 = - C|u| 1n−1I Pattyn (2006): β2 = - exp[β0(xg − x)]

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.


Model equations

Valid in both, the ice sheet and the ice shelf:

conservation of mass : Ht + (uH)x = a,

stress− balance equation : τl + τb + τl = 0.

τl =[2A− 1

nH|ux |1n−1|ux

]x,

τd =− ρigHsx ,τb = basal shear stress.


Assumption and Boundary conditions

Symmetric ice sheet at the ice divide

u = 0sx = (H − b)x = 0

}at x = 0,

At the grounding line, we have the flotation condition and the balancebetween τl and τd .

H = ρwρib,

2A− 1n |ux |

1n−1ux = 1

2ρig(

1− ρiρw

)H

}at x = xg .


Basal stress

We adopt the formulation from Schoof (2005):

τb =− γN

(u

u + λmaxmmax

ANn

) 1n

.

λmax = wavelength of bedrock bumps

mmax = maximum bed obstacle slope

Effective pressure: N ≡ pi − pw

I 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)


Basal stress (continued)

N(p) = η(p;H, b)ρi g H, η non-dimensional function

η ∈ [0, 1]

p ∈ [0, 1]

We want the following limit for η:

When p = 0, η = 1: no water-pressure support.

When p = 1, η = 0: subglacial water has a free connection to the ocean,Paterson(2010)

At the grounding line, η = 0 (effective pressure goes to zero to allowcontinuity).

One expression with the right limit: N(p) = ρigH(

1− HfH

)p, where

Hf = max(0, ρwρi b)


Basal stress (continued)Right limits:

I if u � κN(p)n and τb ≈ −γ

κ1/n|u| 1n−1u (frozen ice at the bed)

I if u � κN(p)n and τb ≈ −CN(p) u|u| . (hydrological connection)

Physically motivatedTransitions smoothly from non-zero basal stress in the ice sheet tozero basal sliding in the ice shelf

−60 −40 −20 0

−200

−150

−100

−50

0

−60 −40 −20 0

−200

−150

−100

−50

0

Basal stress term (kPa)

Distance to grounding line (km)

a) p = 0 b) p=1

Figure:LANL (Los Alamos National Laboratory) Parameter estimation for grounding-line transition February 15th, 2013 7 / 20

Numerics

Moving-grid:

Numerically more accurate

GL must lie on a grid point

Could be used as a benchmark solution for fixed-grid model

Harder to implement in 3-D models

Fixed-grid

Suitable for 3-D model

Constant resolution

Numerically less accurate

GL usually falls between two grid points leading to interpolation error


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 schemeI Moving grid with high enough resolution used as a benchmark for

fixed-grid model


Numerics (end)Experimental set-up:

Moving grid resolution benchmark: 156m ≤ Res ≤ 316 m.

Fixed-grid resolution comparison: 1.6 km and 0.8 km.

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



0 0.5 1 1.5 2−1

0

1

2

3

4

5

6




Results linear bedThe fixed-grid solution is inaccurate in capturing the retreat.

1024

1025

1026

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

analyticnumeric (fix)numeric (moving)

1024

1025

1026

analyticnumeric (fix)numeric (moving)

1/A

Advance Retreat

Grounding lineposition (km)p = 0, Res = 0.8 km

Figure:LANL (Los Alamos National Laboratory) Parameter estimation for grounding-line transition February 15th, 2013 11 / 20

Results linear bed (continued)MISMIP-type experiments 1 & 2

Ice sheet domain (103 km) 1/A

Gro

un

din

g lin

e p

osi

tion

(1

03

km

)p=0

p=0.5

p=1

Ice s

heet

surf

ace

ele

vati

on

(km

)

Res = 0.8 km

0

2

4

6

1

1.5

2

0

2

4

6

0.5

1

1.5

2

0 0.5 1 1.5 2

0

2

4

6

1023

1024

1025

1026

0.5

1

1.5

fix

moving

moving(advance)

moving(retreat)

fix (advance)

fix (retreat)


Results linear bed (continued)MISMIP-type experiments 1 & 2 (note the different scales for thedifferent p-values)

0

200

400

0

200

400

0

20

40

60

80

0

20

40

60

80

1023

1024

1025

1026

0

5

10

15

1023

1024

1025

1026

0

5

10

15

advance

Retreat

1/A 1/A

p=0

p=0.5

p=1

Res = 1.6 km Res = 0.8 km

Grounding line position error estimateErr

or

est

imate

(km

)


Results linear bed (end)

MISMIP-type experiments 1 & 2 (note the different scales for thedifferent p-values)

1/A 1/A

p=0

p=0.5

p=1


SLR volume error estimate

Err

or

est

imate

(km

2)

1023

1024

1025

1026

0

1000

2000

1023

1024

1025

1026

0

1000

2000

1023

1024

1025

1026

0

100

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1023

1024

1025

1026

0

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300

1023

1024

1025

1026

0

20

40

60

1023

1024

1025

1026

0

20

40

60

advance

retreat

Element ofcomparison: 2850km3 ≈ 7.2 mm.


Results poly bedMISMIP-type experiment 3

Ice sheet domain (103 km) 1/A

Gro

un

din

g lin

e p

osi

tion

(1

03

km

)p=0

p=0.5

p=1

Ice s

heet

surf

ace

ele

vati

on

(km

)Res = 0.8 km

0 0.5 1 1.5 2

0

2

4

6

1024

1025

1026

0.5

1

1.5

0 0.5 1 1.5 2

0

2

4

6

1024

1025

1026

0.5

1

1.5

0 0.5 1 1.5 2

0

2

4

6

1024

1025

1026

1027

0.5

1

1.5

fixed

moving

movingadvance

movingretreat

fixedadvance

fixedretreat


Results poly bed (continued)MISMIP-type experiment 3 (note the different scales for the differentp-values)

1/A 1/A

p=0

p=0.5

p=1


Grounding line position error estimateErr

or

est

imate

(km

)

1024

1025

1026

0

50

100

150

1024

1025

1026

0

50

100

150

1024

1025

1026

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1027

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1024

1025

1026

0

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60

Advance

Retreat

1024

1025

1026

1027

0

5

10

15

20


Results poly bed (continued)MISMIP-type experiment 3 (note the different scales for the differentp-values)

1/A 1/A

p=0

p=0.5

p=1


SLR volume error estimateErr

or

est

imate

(km

2)

1024

1025

1026

0

200

400

600

800

1024

1025

1026

0

200

400

600

800

1024

1025

1026

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1024

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1027

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1024

1025

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Advance

Retreat


conclusion

For low p-values the need of high resolution in the vicinity of theGL remains.

Error estimate decreases substantially as the value of p increasesand therefore the required model resolution decreases as pincreases.

p does not play any role in the bulk of the ice sheet. It impacts arelatively small distance (no more than 25 km from the groundingline in our experiments) which is enough to impact the solution.

With p ≈ or = 1, a resolution of ≈ 1km is sufficient to capture thesolution.

Paper in preparation for the cryosphere


Future work

3-D implementation

Add the missing physics (lateral drag, buttressing)

Data comparison for admissible values of p (any value besides 0and 1 valid?)



Documents

Parameter estimation for grounding-line transition · 2013-02-22 · Parameter estimation for grounding-line transition Gunter Leguy, Xylar Asay-Davis, William Lipscomb Los Alamos