Numerical Simulation of Rhone's Glacier between 1874 and 2100sma.epfl.ch/~picasso/ottawa.pdf ·...
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Numerical Simulation of Rhone’s Glacier between1874 and 2100
G. Jouvet 1 M. Picasso 1 J. Rappaz1
H. Blatter 2 M. Funk 3 M. Huss3
1Institut d’Analyse et Calcul ScientifiqueEPF Lausanne, Switzerland
2Institute for Atmospheric and Climate ScienceETH Zurich, Switzerland
3Laboratory of Hydraulics, Hydrology and GlaciologyETH Zurich, Switzerland
Ottawa, sept. 2008
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Rhone’s river
Source : wikipedia
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Rhone’s glacier in 1850
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier in 1870
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier in 1900
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier in 1914
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier in 1925
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier in 1985
Source : http://www.unifr.ch/geosciences/geographie/glaciers
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Rhone’s glacier : comparison at 2000 m
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Rhone’s glacier in 1860 (M. Funk’s reconstruction)
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Rhone’s glacier in 1970 (M. Funk’s reconstruction)
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Rhone’s glacier in 2050 (M. Funk’s prediction)
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Rhone’s glacier from 1874 to 2100 (Numerical simulation)
1874 2008 2050 2075 2100
From 1874 to 2007 Animation.
From 2007 to 2100, median scenario Animation
Temperature trend : +3.8oCPrecipitation trend : −6m/yearData from M. Huss and D. Farinotti, ETHZ.
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Rhone’s glacier in 2100 (3 scenario)
Scenario 1 : cold-wet Scenario 2 : median Scenario 3 : warm-dry
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Outline
Starting point : 3D fluid flows with complex free surfaces.
Model.
Numerical method.
Ongoing work : crevasse formation.
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Starting Point : 3D fluid flows with complex free surfaces
Newtonian flows : V. Maronnier, M. Picasso, J. Rappaz (JCP1999, IJNMF 2003).
Broken damMould filling
Newtonian flows and compressible gas and surface tension :A. Caboussat, M. Picasso, J. Rappaz (JCP 2005).
Mould filling
Viscoelastic flows : A. Bonito, M. Picasso, M. Laso (JCP2006).
Jet buckling.Fingering instabilities : experiment, G. McKinley, MITFingering instabilities : simulation
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The model
Bedrock
Ice
ϕ = 1 ϕ = 0
Air
The computational domain is the region inside the dotted line.
The ice region is defined by the volume fraction of ice ϕ.
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Ice flows in glaciers
Trift glacier, one picture a day in 2003 Animation
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The model (Volume of Fluid)
Climatic input : b.
Unknowns :velocity u and pressure p,volume fraction of ice ϕ.
Ice accumulation
Melting
ρ∂u
∂t+ ρ(u · ∇)u − div
(2µε(u)
)+∇p = ρg ,
div u = 0,
∂ϕ
∂t+ u · ∇ϕ = bδΓ,
on the ice/air interface Γ : (2µε(u)− pI )n = 0,
on the bedrock : u = 0,
Glen’s law for the viscosity.Jouvet Picasso Rappaz Blatter Funk Huss Numerical Simulation of Rhone’s Glacier
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The model (Volume of Fluid)
Climatic input : b.
Unknowns :velocity u and pressure p,volume fraction of ice ϕ.
Ice accumulation
Melting
−div(2µε(u)
)+∇p = ρg ,
div u = 0,
∂ϕ
∂t+ u · ∇ϕ = bδΓ,
on the ice/air interface Γ : (2µε(u)− pI )n = 0,
on the bedrock : u = 0,
Glen’s law for the viscosity.Jouvet Picasso Rappaz Blatter Funk Huss Numerical Simulation of Rhone’s Glacier
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Glen’s flow law
Given ε(u) find µ such that
1
2µ= A
σn−10 +
(2µ
√1
2ε(u) : ε(u)
)n−1 .
Viscosity with respect to s =√
12ε(u) : ε(u)
The nonlinear Stokes problem has a solution in W 1,1+1/n
(minimum of a strictly convex functional).
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Climatic input : b (m of ice per year)
Cold year 1913 Warm year 2003
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Climatic input : b (m of ice per year)
Based on 150 years of measurements.
b = P −M.
P : solid precipitations (snow)
P(x , y , z , t) = Pws(t)
(1 +
dP
dz(z − zws)
)CprecDIST (x , y , z).
M : ice melting
M(x , y , z , t) =
{(FM + rice/snow I (x , y , z)T (z , t) if T (z , t) > 0,
0 else.
T (z , t) = Tws(t)− 0.006(z − zws).
The coefficients FM , rice , rsnow , Cprec , dP/dz are tuned sothat ∫ 2007
1874
∫ice
(b − bmeas)2dVdt
is minimum.
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Time discretization : a decoupling scheme
Time tn−1 Time tn
ϕn−1(x) = 1
un−1(x) in the ice
ϕn−1(x) = 0
ϕn(x) = 1
ϕn(x) = 0
un(x) in the ice
Shape computation : solve between t = tn−1 and t = tn
∂ϕ
∂t+ u · ∇ϕ = bδΓ.
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Time discretization : a decoupling scheme
Time tn−1 Time tn
ϕn−1(x) = 1
un−1(x) in the ice
ϕn−1(x) = 0
ϕn(x) = 1
ϕn(x) = 0
un(x) in the ice
Velocity computation : solve
−div(2µε(u)
)+∇p = ρg ,
div u = 0,
on the ice/air interface Γ : (2µε(u)− pI )n = 0,
on the bedrock : u = 0.
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Space discretization : structured cells and finite elements
Bedrock
Ice
Air
Shape computation (transport + ice accumulation/melting) :small structured cells
Velocity computation (nonlinear Stokes) : unstructured finiteelements
To avoid numerical diffusion : FE spacingcells spacing ' 5.
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The 3D finite element mesh
1874 2008 2050 2075 2100
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The 3D finite element mesh
1874 2008 2050 2075 2100
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Shape computation : transport + iceaccumulation/melting (1/3)
Solve between t = tn−1 and t = tn
∂ϕ
∂t+ u · ∇ϕ = bδΓ.
Forward characteristics method :
ϕn(x + ∆t un−1(x)) = ϕn−1(x).
index j
index i
ϕn−1ij = 1
∆t un−1ij
116
316
916
316
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Shape computation : transport + iceaccumulation/melting (2/3)
SLIC Postprocessing (Simple Line Interface Calculation), seefor instance Scardovelli Zaleski Ann. Rev. Fluid Mech. 1999.
Without SLIC.
With SLIC.
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Shape computation : transport + iceaccumulation/melting (3/3)
Solve between t = tn−1 and t = tn
∂ϕ
∂t+ u · ∇ϕ = bδΓ.
Before
After
ELA
Filling
Emptying
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Conclusions - Perspectives - Ongoing work
Given the climatic input b, we can simulate the motion of aglacier.
Optimization problem : observed moraines → climatic input b.
Sliding condition on the bedrock.
Crevasse formation : rupture of hanging glaciers, calvingglaciers → mechanics of damage.
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Sliding condition on the bedrock 1874
u · n = 0 and (2µε(u)n) · ti = −αu · ti i = 1, 2,
α =1
c13
1(√u2 + v2 + w2 + s0
) 23
,
Source http://www.unifr.ch/geosciences/geographie/glaciers.
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Sliding condition on the bedrock 1900
Source http://www.unifr.ch/geosciences/geographie/glaciers.
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Sliding condition on the bedrock 1932
Source http://www.unifr.ch/geosciences/geographie/glaciers.
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Sliding condition on the bedrock 1960
Source http://www.unifr.ch/geosciences/geographie/glaciers.
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Sliding condition on the bedrock 1985
Source http://www.unifr.ch/geosciences/geographie/glaciers.
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Rupture of hanging glaciers : Argentiere
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Calving glaciers : Rhone Glacier
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Damage modelling
Microcracks can be modeled at macroscopic level byintroducing a state variable D(x , t) called damage : A.Pralong and M. Funk, Annals of Glaciology, 2003.
Isotropic damage : D is a scalar field, D = 0 no damage,D = 1 full damage.
Momentum equation :
ρ∂u
∂t+ ρ(u · ∇)u − div (2µ(1− D)ε(u)) +∇p = ρg ,
Damage nonlinear transport equation :
∂D
∂t+ u · ∇D = f (D, ε(u), p).
Simulation of crevasse formation : Eiger Gruben.
Jouvet Picasso Rappaz Blatter Funk Huss Numerical Simulation of Rhone’s Glacier