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Elmer/Ice Course– 22-24 November 2017 - Grenoble
1
Elmer/Ice advanced Workshop
22-24 November 2017
Olivier GAGLIARDINI
IGE - Grenoble - France
Basal Conditions (Friction laws & Hydrology)
Elmer/Ice Course– 22-24 November 2017 - Grenoble
2
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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...
Elmer/Ice Course– 22-24 November 2017 - Grenoble
4
Relationship between velocity and water
@deFleurian
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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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)
Elmer/Ice Course– 22-24 November 2017 - Grenoble
6
Concept of friction law Ice
Bed
[Weertman (1957), Fowler (1981), Gudmundsson (1997)]
film of water
⌧b + f(un, N, ..) = 0
Elmer/Ice Course– 22-24 November 2017 - Grenoble
7
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 ?
Elmer/Ice Course– 22-24 November 2017 - Grenoble
Ocean Film of water
~ m
Lakes
~ km
8
Water at the base of glaciers
Effective pressure:
⇥b = t · � · nub = u · t�nn = n · � · nN = ��nn � pw
Elmer/Ice Course– 22-24 November 2017 - Grenoble
geothermal heat flux
negative surface temperature
Increase of T melting point [Zwally et al., 2002]
positive surface temperature
9
Why is there (liquid) water?
friction heat
Deformational heat
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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Two tightly-related systems
@deFleurian
From [Hewitt, 2011]
Elmer/Ice Course– 22-24 November 2017 - Grenoble
18
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
19
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, ...):
⇥b = t · � · nub = u · t�nn = n · � · nN = ��nn � pw
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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Friction laws – water pressure dependant The friction should depend on the water pressure
⇥b = t · � · nub = u · t�nn = n · � · nN = ��nn � pw
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
21
Illustration of Iken's bound
Iken’s bound :
[Iken (1981), Fowler (1986), Schoof (2005)]
[Gagliardini et al., 2007]
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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Coulomb-type friction law
[Schoof (2005), Gagliardini et al., 2007]
where
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
26
Influence of Friction law on GL
[Brondex et al., 2017]
[Tsai et al., 2015]
Elmer/Ice Course– 22-24 November 2017 - Grenoble
27
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
28
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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GlaDS model
[Creyts et al., 2010]
[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)
Elmer/Ice Course– 22-24 November 2017 - Grenoble
Discharge (Darcy-Weisbach law) : Cavity thickness evolution :
with
33
GlaDS model, cavities
[Creyts et al., 2010]
q = �kh↵| grad�|��2 grad�
@h
@t= w(h)� v(h,�)
v(h,�) = Ah|N |n�1N
(Werder et al., 2013)
creep, closing (opening)
opening term w(h) = max(0;
ub
lr(hr � h))
Elmer/Ice Course– 22-24 November 2017 - Grenoble
Channel cross-sectional area evolution :
34
GlaDS model, Channels
[Creyts et al., 2010]
Q = �kcS↵c
����@�
@s
�����c�2 @�
@s
vc(S,�) = AcS|N |n�1N
(Werder et al., 2013)
⌅(�) =
�����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) :
Elmer/Ice Course– 22-24 November 2017 - Grenoble
35
Comparison
Double Continuum
GlaDS
Cavity only = = Channels continuous discrete Coupling
Channels closing No Yes N ! u N $ u
Elmer/Ice Course– 22-24 November 2017 - Grenoble
36
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 - GlaDS solvers
ü Examples
Elmer/Ice Course– 22-24 November 2017 - Grenoble
37
! 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
37
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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Linear friction laws: ⇥b = �ub
38
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
39
Example of a call (File USF_Sliding.f90):
39
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
40 40
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
41
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 law - Double continuum hydrology model - GlaDS model
ü Implementation in Elmer/Ice - Various friction laws
ü Examples
Elmer/Ice Course– 22-24 November 2017 - Grenoble
42
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
Elmer/Ice Course– 22-24 November 2017 - Grenoble
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