Elmer/Ice advanced Workshopelmerfem.org/elmerice/wiki/lib/exe/fetch.php?media=...Slip Coefficient 2...

Elmer/Ice Course– 22-24 November 2017 - Grenoble

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Elmer/Ice advanced Workshop

22-24 November 2017

Olivier GAGLIARDINI

IGE - Grenoble - France

Basal Conditions (Friction laws & Hydrology)


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Basal Conditions ü  The Physics

- Sliding at the base of glacier - The role of basal water - Different drainage systems

ü  Friction laws and Hydrology - Linear friction law - Weertman type friction law - Water-pressure dependant friction laws - Double continuum hydrology model - GlaDS model

ü  Implementation in Elmer/Ice - Various friction laws

ü  Examples


subglacial hydrology

surface mass balance accumulation / ablation

basal sliding

ice flow

intraglacial hydrology

lakes supraglacial hydrology

runoff

ice deformation

cavities channels

free surface

moulins

Coupling water / friction and more...


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Relationship between velocity and water

@deFleurian


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Scale of interest z z

Friction law for Glacier flow modeling

Friction law ?

x Bedrock roughness

⌧b + f(un, N, ..) = 0

N = pi � pw(= ��nn � pw)


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Concept of friction law Ice

Bed

[Weertman (1957), Fowler (1981), Gudmundsson (1997)]

film of water

⌧b + f(un, N, ..) = 0


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Ice

Bed

How water enhances glacier sliding

Water network under pressure

Low normal stress High normal stress

If water pressure and/or velocity increase

How does it affect the form of the sliding law ?


Ocean Film of water

~ m

Lakes

~ km

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Water at the base of glaciers

Effective pressure:

⇥b = t · � · nub = u · t�nn = n · � · nN = ��nn � pw


geothermal heat flux

negative surface temperature

Increase of T melting point [Zwally et al., 2002]

positive surface temperature

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Why is there (liquid) water?

friction heat

Deformational heat


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Two types of drainage systems

@deFleurian


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@deFleurian


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@deFleurian


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@deFleurian


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@deFleurian


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@deFleurian


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Two tightly-related systems

@deFleurian

From [Hewitt, 2011]


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Two tightly-related systems

@deFleurian


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ü  Examples


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Friction laws A friction law is a relation that gives the basal shear stress as a function of the sliding velocity and other variables (effective pressure, ...):


Linear friction laws:

Weertman type friction law (non-linear):

� Drag factor or friction parameter

As Sliding parameter

C Sliding parameter

n Glen’s flow law exponent

⌧b + �ub = 0

ub + C⌧b = 0

⌧b + f(ub, N, ..) = 0

⌧b + (ub/As)1/n = 0


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Friction laws – water pressure dependant The friction should depend on the water pressure


Raymond and Harrison, 1987, Bindschadler (1983), Budd et al. (1984) :

Iken's bound, 1981:

�b/N < mmax

mmax

the maximum up-slope of the bed

not fulfilled by the previous law

ub + k⌧pb N�q = 0 e.g. p = n = 3, q = 1


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Illustration of Iken's bound

Iken’s bound :

[Iken (1981), Fowler (1986), Schoof (2005)]

[Gagliardini et al., 2007]


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Coulomb-type friction law

3 parameters:

Schoof (2005), Gagliardini et al., 2007:

where

Fulfills the Iken's bound:

Sliding parameter in absence of cavitation

Maximum value of �b/N

Post-peak exponent

⌧bN

+ C

✓�

1 + ↵�m

◆1/n

= 0


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Coulomb-type friction law

[Schoof (2005), Gagliardini et al., 2007]

where


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Tsai Coulomb law

[Tsai et al., 2015] ⌧b +min((ub/As)1/n ; fN) = 0

2 parameters:

Fulfills the Iken's bound:

Sliding parameter in absence of cavitation

friction coefficient

0 <⌧bN

f

f


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Comparison

[Brondex et al., 2017]


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Influence of Friction law on GL

[Brondex et al., 2017]

[Tsai et al., 2015]


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ü  Examples


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Two approaches in Elmer/Ice Double continuum approach

- Implemented by Basile de Fleurian - in the distribution - http://elmerice.elmerfem.org/wiki/doku.php?id=solvers:hydrologydc

Cavity sheet and discrete channels

- Model developed by Mauro Werder (Werder et al., 2013) - Implemented in Elmer by O. Gagliardini - In the distribution - http://elmerice.elmerfem.org/wiki/doku.php?id=solvers:glads


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The double continuum approach

@deFleurian


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Computation of the water load


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3 states of the Channel Equivalent Layer


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GlaDS model

[Creyts et al., 2010]


Two components system, 3 variables: Network of cavities Channels

Cavity thickness (nodal variable)

Channel cross-sectional area : (edge variable)

h S

X

i

Z

⌦i

"✓ev

⇢wg

@�

@t�r✓ · q + ✓ (w �mb � v)

#d⌦

+X

j

Z

�j

"�@✓

@sQ+ ✓

⌅�⇧

L

1

⇢i�

1

⇢w

!� vc

!#d�

+

Z

@⌦N

✓qN d��X

k

✓

�Ak

m

⇢wg

@�

@t+Qk

s

!= 0 ,

�i, hi

Si

�i, hi

Si

�, h, S

(Werder et al., 2013)


Discharge (Darcy-Weisbach law) : Cavity thickness evolution :

with

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GlaDS model, cavities


q = �kh↵| grad�|��2 grad�

@h

@t= w(h)� v(h,�)

v(h,�) = Ah|N |n�1N


creep, closing (opening)

opening term w(h) = max(0;

ub

lr(hr � h))


Channel cross-sectional area evolution :

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GlaDS model, Channels


Q = �kcS↵c

��@�

@s

��c�2 @�

@s

vc(S,�) = AcS|N |n�1N


⌅(�) =

��Q@�

@s

��+

��lcqc@�

@s

��

@S

@t=

⌅(S,�)�⇧(S,�)

⇢iL� vc(S,�)

Creep, closing (opening)

Energy dissipated

⇧(S,�) = �ctcw⇢w(Q+ flcqc)@�� m

@sSensible heat change

with

Discharge (Darcy-Weisbach law) :


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Comparison

Double Continuum

GlaDS

Cavity only = = Channels continuous discrete Coupling

Channels closing No Yes N ! u N $ u


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ü  Implementation in Elmer/Ice - Various friction laws - GlaDS solvers

ü  Examples


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! Bedrock BC Boundary Condition 1 Target Boundaries = 1

Flow Force BC = Logical True Normal-Tangential Velocity = Logical True

Velocity 1 = Real 0.0e0 Slip Coefficient 2 = Real 0.1 Slip Coefficient 3 = Real 0.1 End

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Friction laws in Elmer/Ice Friction law in Elmer:

where is the surface normal vector

Ciui = �ijnj with i = 1, 2, 3

In Normal-Tangential coordinate : n = (1, 0, 0)

and Cnun = �nn

Ct1ut1 = �nt1

Ct2ut2 = �nt2

n

t

Friction law applied through the two Slip Coefficients 2 and 3


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Linear friction laws: ⇥b = �ub

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Friction laws in Elmer/Ice

$beta = 0.1 Slip Coefficient 2 = Real $ beta Slip Coefficient 3 = Real $ beta

Non-Linear friction laws: Need a User Function to evaluate the Slip Coefficient

Rewrite the friction law in the form where is the Slip Coefficient estimated through a user function

Weertman:

Schoof, 2005 Gagliardini et al., 2007

with

�b = Ct(ub)ub

Ct(ub)

Ct(ub) = u(1�n)/nb /A1�n

s

� =ub

CnNnAs

Ct(ub) = CN

✓�u�n

b

1 + �m

◆1/n


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Example of a call (File USF_Sliding.f90):

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Friction laws in Elmer/ice

Normal-Tangential Velocity = Logical True Flow Force BC = Logical True

!! Water pressure given through the Stokes 'External Pressure' parameter !! (Negative = Compressive) External Pressure = Equals Water Pressure Velocity 1 = Real 0.0 Slip Coefficient 2 = Variable Coordinate 1 Real Procedure ”ElmerIceUSF" "Friction_Coulomb”

!! PARAMETERS NEEDED FOR THE BASAL SLIDING LAW Friction Law Sliding Coefficient = Real $As Friction Law Post-Peak Exponent = Real $m Friction Law Maximum Value = Real $C Friction Law PowerLaw Exponent = Real $n Friction Law Linear Velocity = Real $ut0

Problem when The law is linearized for small velocity:

ub ! 0

Ct(ub) = Ct(ub) for ub > ut0

Ct(ub) = Ct(ut0) for ub ut0


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GlaDS solvers Three solvers: •  GlaDSCoupledSolver: main solver, all 3 variables are solved in a

coupled way

•  GlaDSsheetThickDummy: just here to declare (to save previous values)

•  GlaDSchannelOut: just here to export in VTU format edge type variables (not accounted for by ResultOutput solver).

Coupling with SSA: user functions HorizontalVelo and OverburdenPressure. Including Moulins (101 boundary elements) in serial or parallel meshes : python tool makemoulin.py in Meshers\

�, h, S

�, h, S


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ü  Friction laws and Hydrology - Linear friction law - Weertman type friction law - Water-pressure dependant friction law - Double continuum hydrology model - GlaDS model


ü  Examples


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Examples Friction :

Weertman - Tests/GL_MISMIP, Tests/Contact, Tests/Friction_Weertman. - http://elmerice.elmerfem.org/wiki/doku.php?id=userfunctions:weertman Coulomb - Tests : Tests/Friction_Coulomb and Tests/Friction_Coulomb_Pw - http://elmerice.elmerfem.org/wiki/doku.php?id=userfunctions:coulomb Budd - http://elmerice.elmerfem.org/wiki/doku.php?id=userfunctions:budd#general_description

Hydrology :

Double continuum approach - Tests/Hydro_SedOnly and Tests/Hydro_Coupled - http://elmerice.elmerfem.org/wiki/doku.php?id=solvers:hydrologydc

Cavity sheet and discrete channels - Tests/GlaDS and Tests/GlaDS_SSA - http://elmerice.elmerfem.org/wiki/doku.php?id=solvers:glads


Gladstone, R.M., R.C. Warner, B.K. Galton-Fenzi, O. Gagliardini, T. Zwinger and R. Greve, 2017. Marine ice sheet model performance depends on basal sliding physics and sub-shelf melting, The Cryosphere, 11, 319-329, doi:10.5194/tc-11-319-2017 de Fleurian, B., O. Gagliardini, T. Zwinger, G. Durand, E. Le Meur, D. Mair, and P. Råback, 2014. A double continuum hydrological model for glacier applications, The Cryosphere, 8, 137-153, doi:10.5194/tc-8-137-2014. Gagliardini, O., T. Zwinger, F. Gillet-Chaulet, G. Durand, L. Favier, B. de Fleurian, R. Greve, M. Malinen, C. Martín, P. Råback, J. Ruokolainen, M. Sacchettini, M. Schäfer, H. Seddik, and J. Thies, 2013. Capabilities and performance of Elmer/Ice, a new-generation ice sheet model, Geosci. Model Dev., 6, 1299-1318, doi:10.5194/gmd-6-1299-2013. Gagliardini O., D. Cohen, P. Råback and T. Zwinger, 2007. Finite-Element Modeling of Subglacial Cavities and Related Friction Law. J. of Geophys. Res., Earth Surface, 112, F02027.

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References

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