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Parameterization of basal hydrology near grounding lines 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 mo January 31th, 2014 1 / 19

Parameterization of basal hydrology near grounding lines ...I Transitions smoothly between nite basal friction in the ice sheet and zero basal friction in the ice shelf Hope for ˇ1km

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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