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J E T P R O P U L S I O N L A B O R A T O R Y
ISSM Workshop 2011
Ice Sheet System modelIce flow models
Eric LAROUR1, Eric RIGNOT1,3, Mathieu MORLIGHEM1,2, Hélène SEROUSSI1,2 ChrisBORSTAD1, Feras HABBAL1,3, Daria HALKIDES1,4, Behnaz KHAKBAZ1, John SCHIER-MEIER1, Nicole SCHLEGEL1
1Jet Propulsion Laboratory - California Institute of Technology2Laboratoire MSSMat, École Centrale Paris, France3University of California, Irvine4Joint Institute for Regional Earth System Science & Engineering, UCLA
December 2011 ©Copyright 2011. All rights reserved
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Outline
1 Ice flow equationsApproximations implementedIce flow equationDiagnostic parametersBoundary conditions
2 Combining modelsMethods implemented in ISSMPenaltiesTiling methodUtilization
1/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Ice Sheet flow equationsIncompressibility
∀x ∈ Ω ∇ · v = Tr (ε) =∂u∂x
+∂v∂y
+∂w∂z
= 0 (1)
• v = (u, v ,w) ice velocity (m/yr)
• ε strain rate tensor (yr-1)
Incompressible viscous fluid
σ′ = 2µε (2)
• σ′ deviatoric stress
• µ ice viscosity
• ε strain rate tensor
Glen’s flow law
µ =B
2 εn−1
ne
(3)
• B ice hardness
• n Glen’s law coefficient (n = 3)
• εe effective strain rate (second invariant)2/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Ice Sheet flow equations
Conservation of momentum
∀x ∈ Ω ∇ · σ′ −∇P + ρg = 0 (4)
Assumptions:
1 Stokes flow (quasi-static assumption)
2 Coriolis effect negligible
Boundary conditions
Ice/Air interface: Free surface Γs σ · n = Patm n ' 0
Ice/Ocean interface: water pressure Γw σ · n = Pw n
Ice/Bedrock interface (1): lateral friction Γb (σ · n + βv)‖ = 0
Ice/Bedrock interface (2): impenetrability Γb v · n = 0
Side boundaries: Dirichlet Γu v = vobs
3/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Models description
Full-Stokes model:• Momentum balance + incompressibility• 3D model• Four unknowns (vx , vy , vz , p)
Full-Stokes (FS)
(vx, vy, vz, P )
Model equations
∂
∂x
(2µ∂vx
∂x
)+
∂
∂y
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂z
(µ∂vx
∂z+µ
∂vz
∂x
)−∂p∂x
= 0
∂
∂x
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂y
(2µ∂vy
∂y
)+
∂
∂z
(µ∂vy
∂z+µ
∂vz
∂y
)−∂p∂y
= 0
∂
∂x
(µ∂vx
∂z+ µ
∂vz
∂x
)+
∂
∂y
(µ∂vy
∂z+ µ
∂vz
∂y
)+
∂
∂z
(2µ∂vz
∂z
)−∂p∂z− ρg = 0
∂vx
∂x+∂vy
∂y+∂vz
∂z= 0
4/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Models description
Higher-order model:• [Blatter, 1995, Pattyn, 2003]• 3D model• Horizontal and vertical velocity decoupled• 2 (vx , vy ) + 1 (vz ) unknowns
Blatter-Pattyn (BP)
(vx, vy)
Model equations
∂
∂x
(2µ∂vx
∂x
)+
∂
∂y
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂z
(µ∂vx
∂z+µ
∂vz
∂x
)−∂p∂x
= 0
∂
∂x
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂y
(2µ∂vy
∂y
)+
∂
∂z
(µ∂vy
∂z+µ
∂vz
∂y
)−∂p∂y
= 0
∂
∂x
(µ∂vx
∂z+ µ
∂vz
∂x
)+
∂
∂y
(µ∂vy
∂z+ µ
∂vz
∂y
)+
∂
∂z
(2µ∂vz
∂z
)−∂p∂z− ρg = 0
∂vx
∂x+∂vy
∂y+∂vz
∂z= 0
5/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Models description
Shelfy-stream approximation:• [MacAyeal, 1989]• 2D model• Horizontal and vertical velocity decoupled• 2 (vx , vy ) + 1 (vz ) unknowns
MacAyeal-Morland (SSA)
(vx, vy)
Model equations
∂
∂x
(2µ∂vx
∂x
)+
∂
∂y
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂z
(µ∂vx
∂z+µ
∂vz
∂x
)−∂p∂x
= 0
∂
∂x
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂y
(2µ∂vy
∂y
)+
∂
∂z
(µ∂vy
∂z+µ
∂vz
∂y
)−∂p∂y
= 0
∂
∂x
(µ∂vx
∂z+ µ
∂vz
∂x
)+
∂
∂y
(µ∂vy
∂z+ µ
∂vz
∂y
)+
∂
∂z
(2µ∂vz
∂z
)−∂p∂z− ρg = 0
∂vx
∂x+∂vy
∂y+∂vz
∂z= 0
6/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Models description
Shallow ice approximation:• [Hutter, 1983]• 3D analytical model• 2 unknowns (vx , vy ) computed separately
Hutter (SIA)
(vx, vy)
Model equations
∂
∂x
(2µ∂vx
∂x
)+
∂
∂y
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂z
(µ∂vx
∂z+µ
∂vz
∂x
)−∂p∂x
= 0
∂
∂x
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂y
(2µ∂vy
∂y
)+
∂
∂z
(µ∂vy
∂z+µ
∂vz
∂y
)−∂p∂y
= 0
∂
∂x
(µ∂vx
∂z+ µ
∂vz
∂x
)+
∂
∂y
(µ∂vy
∂z+ µ
∂vz
∂y
)+
∂
∂z
(2µ∂vz
∂z
)−∂p∂z− ρg = 0
∂vx
∂x+∂vy
∂y+∂vz
∂z= 0
7/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Material non-linearity
Model equations
∂
∂x
(2µ∂vx
∂x
)+
∂
∂y
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂z
(µ∂vx
∂z+µ
∂vz
∂x
)−∂p∂x
= 0
∂
∂x
(µ∂vx
∂y+ µ
∂vy
∂x
)+
∂
∂y
(2µ∂vy
∂y
)+
∂
∂z
(µ∂vy
∂z+µ
∂vz
∂y
)−∂p∂y
= 0
∂
∂x
(µ∂vx
∂z+ µ
∂vz
∂x
)+
∂
∂y
(µ∂vy
∂z+ µ
∂vz
∂y
)+
∂
∂z
(2µ∂vz
∂z
)−∂p∂z− ρg = 0
∂vx
∂x+∂vy
∂y+∂vz
∂z= 0
Glen’s flow law
µ =B
2 εn−1
ne
(5)
• B ice hardness
• n Glen’s law coefficient (n = 3)
• εe effective strain rate (second invariant)
→ Treatment of non-linearity with fixed point
8/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Material non-linearityTreatment of non-linearity with fixed point:
Compute uh
Initial velocity u = uh+w
Compute w
a(
uh,Φ
)
= l (Φ)
Solution u = uh+w
∂w
∂z= −div
(
uh
)
Viscosity
Compute us
Initial velocity us
as (us,Φ) = ls (Φ)
Solution us
Viscosity
convergence
convergence
Vertical velocity computed with incompressibility for 2d shelfy-stream and 3dBlatter/Pattyn modes.
9/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation
setflowequation is used to generate the approximation used to compute thevelocity
• Arguments:1 model2 approximation names3 approximation domains
• Domains can be Argus files or array of element flags• Approximation available
• stokes (Full-Stokes model)• pattyn (Higher-order model)• macayeal (Shallow Shelf Approximation)• hutter (Shallow Ice Approximation)
• Possibility of coupling models
10/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation
setflowequation is used to generate the approximation used to compute thevelocity
• Examples
1 md=setflowequation(md,'hutter','all')2 md=setflowequation(md,'stokes','all')3 md=setflowequation(md,'macayeal','all')4 md=setflowequation(md,'pattyn','all')
• To diplay the type of approximation:
1 >> plotmodel(md,'data','elements_type')
11/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation• To diplay the type of approximation:
1 >> plotmodel(md,'data','elements_type')
0 2 4 6 8 10
x 105
0
1
2
3
4
5
6
7
8
9
10x 10
5
Elements type
Hutter’s elements
MacAyeal’s elements
Pattyn’s elements
MacAyealPattyn’s elements
Stokes’s elements
MacAyealStokes’s elements
PattynStokes’s elements
None element
12/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation class
1 >> md.flowequation23 ans =45 flow equation parameters:6 ismacayealpattyn : 0 -- is the macayeal or pattyn approximation used ?7 ishutter : 0 -- is the shallow ice approximation used ?8 isstokes : 0 -- are the Full-Stokes equations used ?9 vertex_equation : N/A -- flow equation for each vertex
10 element_equation : N/A -- flow equation for each element11 bordermacayeal : N/A -- vertices on MacAyeal's border (for tiling)12 borderpattyn : N/A -- vertices on Pattyn's border (for tiling)13 borderstokes : N/A -- vertices on Stokes' border (for tiling)
13/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Diagnostic class
1 >> md.diagnostic23 ans =45 Diagnostic solution parameters:67 Convergence criteria:8 restol : 0.0001 -- mechanical equilibrium residue convergence criterion9 reltol : 0.01 -- velocity relative convergence criterion, NaN -> not applied
10 abstol : 10 -- velocity absolute convergence criterion, NaN -> not applied11 maxiter : 100 -- maximum number of nonlinear iterations12 viscosity_overshoot : 0 -- over-shooting constant new=new+C*(new-old)1314 boundary conditions:15 spcvx : N/A -- x-axis velocity constraint (NaN means no constraint)16 spcvy : N/A -- y-axis velocity constraint (NaN means no constraint)17 spcvz : N/A -- z-axis velocity constraint (NaN means no constraint)18 icefront : N/A -- segments on ice front list (last column 0-> Air, 1-> Water, ...
2->Ice1920 Rift options:21 rift_penalty_threshold : 0 -- threshold for instability of mechanical constraints22 rift_penalty_lock : 10 -- number of iterations before rift penalties are locked2324 Penalty options:25 penalty_factor : 3 -- offset used by penalties: penalty = Kmax*10^offset26 vertex_pairing : N/A -- pairs of vertices that are penalized2728 Other:29 shelf_dampening : 0 -- use dampening for floating ice ? Only for Stokes model30 stokesreconditioning : 10000000000000 -- multiplier for incompressibility equation. Only for Stokes model31 referential : N/A -- local referential32 requested_outputs : N/A -- additional outputs requested
14/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Boundary conditions
Boundary conditions created automatically or manually• Automatically:
1 >> md=SetIceSheetBC(md)2 >> md=SetIceShelfBC(md,'Front.exp')3 >> md=SetMarineIceSheefBC(md,'Front.exp')
• Manually: fields to change• md.diagnostic.spcvx• md.diagnostic.spcvy• md.diagnostic.spcvz• md.diagnostic.icefront
• To diplay the boundary conditions
1 >> plotmodel(md,'data','BC')
15/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Models description”Everything should be made as simple as possible, but no simpler.” Albert Einstein
Model Dim. Unknowns Reference
Full-Stokes (FS) 3d 4 [Stokes, 1845]
Blatter-Pattyn (BP) 3d 2 + 1 [Blatter, 1995, Pattyn, 2003]
Shallow shelf (SSA) 2d 2 + 1 [MacAyeal, 1989]
Shallow ice (SIA) 2d 2 + 1 [Hutter, 1983]
Hutter (SIA)
(vx, vy)
MacAyeal-Morland (SSA)
(vx, vy)
Blatter-Pattyn (BP)
(vx, vy)
Full-Stokes (FS)
(vx, vy, vz, P )
16/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Penalty method
• Only to couple SSA and HO• Very stiff spring to penalize differences between degrees of freedom
Using penalties to couple models:
1 md=setflowequation(md,'macayeal','FloatingIce.exp','fill','pattyn','coupling','penalties')
17/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Domain Decomposition
• Ω = Ω1 ∪ Ω2
• ΩS = Ω1 ∩ Ω2 6= ∅• u = u1 |Ω1
+ u2 |Ω2∈ V (Ω) = (V1 (Ω1) + V2 (Ω2))
Ω1
Ω2
Ωs
Γ1Γ2
Ω1
Ω2
Ωs
Γ1 Γ2
Find u = u1 |Ω1+ u2 |Ω2
∈ V ,
∀ (v1, v2) ∈ V a (u1 + u2, v1 + v2) = l (v1 + v2)
→ Infinite number of solutions for the continuous problem
18/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Discretization
ΩS
u2 |Ω2
u1 |Ω1
u
Ω2Ω1 Γ2 Γ1
We take advantage of the discretization to avoid the redundancy:
→ Create one layer of elements in the superposition zone
19/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Multi-model formulation
Two different models: a1, a2 and l1, l2
Find u = u1 |Ω1+ u2 |Ω2
∈ (V1 + V2) , such that:
∀v = v1 |Ω1+ v2 |Ω2
∈ (V1 + V2)
a1
(u1 |Ω1
, v1 |Ω1
)︸ ︷︷ ︸
model 1
+ a2
(u2 |Ω2
, v2 |Ω2
)︸ ︷︷ ︸
model 2
+
a2
(u1 |Ω1
, v2 |Ω2
)+ a1
(u2 |Ω2
, v1 |Ω1
)︸ ︷︷ ︸
model coupling
= l1(
v1 |Ω1
)︸ ︷︷ ︸
model 1
+ l2(
v2 |Ω2
)︸ ︷︷ ︸
model 2
• Coupling different mechanical models
• Easy to implement (local modification of stiffness matrices)
20/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation
setflowequation is used to generate the approximation used to compute thevelocity
• Examples
1 md=setflowequation(md,'pattyn',md.elementongroundedice,'fill','macayeal','coupling','penalties')2 md=setflowequation(md,'pattyn',md.elementongroundedice,'fill','macayeal','coupling','tiling')3 md=setflowequation(md,'stokes','Contour.exp','fill','pattyn')
• Use exptool to create EXP contours
1 >> exptool('Contour.exp')
21/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Flow equation• To diplay the type of approximation:
1 >> ...plotmodel(md,'data','elements_type','edgecolor','k','expdisp','Contour.exp')
0 2 4 6 8 10
x 105
0
1
2
3
4
5
6
7
8
9
10x 10
5
Elements type
Hutter’s elements
MacAyeal’s elements
Pattyn’s elements
MacAyealPattyn’s elements
Stokes’s elements
MacAyealStokes’s elements
PattynStokes’s elements
None element
22/23
Ice flow models
Larour et al.
Ice flow equationsApproximationsimplemented
Ice flow equation
Diagnostic parameters
Boundary conditions
Combining modelsMethods implemented inISSM
Penalties
Tiling method
Utilization
I S S M W O R K S H O P 2 0 1 1 J E T P R O P U L S I O N L A B O R A T O R Y
Bibliography I
Blatter, H. (1995).Velocity and stress-fields in grounded glaciers: A simple algorithm for includingdeviatoric stress gradients.J. Glaciol., 41(138):333–344.
Hutter, K. (1983).Theoretical glaciology: material science of ice and the mechanics of glaciersand ice sheets.D. Reidel Publishing Co, Dordrecht, The Netherlands.
MacAyeal, D. (1989).Large-scale ice flow over a viscous basal sediment - Theory and application toice stream-B, Antarctica.J. Geophys. Res., 94(B4):4071–4087.
Pattyn, F. (2003).A new three-dimensional higher-order thermomechanical ice sheet model:Basic sensitivity, ice stream development, and ice flow across subglacial lakes.J. Geophys. Res., 108(B8):1–15.
Stokes, G. (1845).On the theories of internal friction of fluids in motion.Trans. Cambridge Philos. Soc., 8:287–305.
23/23
Thanks!