numerical simulation of rhône's glacier from 1874 to 2100sma.epfl.ch/~picasso/grenoble.pdf ·...

Numerical simulation of Rhone’s glacier from1874 to 2100

G. Jouvet 1 M. Picasso 1 J. Rappaz1

H. Blatter 2 M. Funk 3 M. Huss3

1Mathematics Intitute of Computational Science and EngineeringEPF Lausanne, Switzerland

2Institute for Atmospheric and Climate ScienceETH Zurich, Switzerland

3Laboratory of Hydraulics, Hydrology and GlaciologyETH Zurich, Switzerland

Seminaire MODANT, LJK, Grenoble, 10 nov. 2010

Jouvet Picasso Rappaz Blatter Funk Huss Numerical simulation of Rhone’s glacier

Rhone’s glacier 20 000 years ago (Wurm ice age)

Source: geologie-montblanc.fr


Rhone’s glacier in 1850

Source: unifr.ch/geosciences/geographie/glaciers


Rhone’s glacier: comparison at 2000 m


Rhone’s glacier in 1860 (M. Funk’s reconstruction)


Rhone’s glacier in 1970 (M. Funk’s reconstruction)


Rhone’s glacier in 2050 (M. Funk’s prediction)


Mathematical model: ice flows in glaciers

For long time scales, ice behaves as a fluid: Trift glacier, onepicture a day in 2003, Animation. Free surface flow.Climatic input: meters of ice per year, model based on 150years of measurements.

Cold year 1913 Warm year 2003

Eq. line Acc.

Accumulation

Ablation

Eq. line Acc.


Numerical validation: 1900

Only one parameter to tune: sliding coefficient along the bedrock.


Numerical simulation from 2008 to 2100

1874 2008 2050 2075 2100

Median climatic scenario (occh.ch, Organe Consultatif pourles Changements Climatiques), temperature trend +3.8oC ,precipitation trend −6%: Animation.

Cold-wet scenario: Animation.

Warm-dry scenario: Animation.


Numerical simulation of Aletsch’s glacier

Largest alpine glacier, 25% of Swiss ice, 23 km long, max.depth 900 m:

Three scenario: Animation.


The mathematical model: 3D fluid flows with complex freesurfaces

Newtonian flows: Maronnier Picasso Rappaz (JCP 1999,IJNMF 2003).

Broken damMould filling

Newtonian flows and compressible gas and surface tension:Caboussat Picasso Rappaz (JCP 2005).

Mould filling

Viscoelastic flows: Bonito Picasso Laso (JCP 2006).

Jet buckling.Fingering instabilities: experiment, G. McKinley, MITFingering instabilities: simulation

Dynamics of glaciers: Jouvet Huss Blatter Picasso Rappaz(JCP 2009).


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,

along the bedrock : slip or no-slip.


Ice rheology

Glen’s law: viscosity µ(|ε(u)|) = O

(1

(1 + |ε(u)|)1− 1m

).

Sliding law u · n = 0 and (2µε(u)n) · ti = −αu · ti , i = 1, 2with

α(|u|) = O

(1

(1 + |u|)1− 1m

)Following Barrett Liu 1994, the nonlinear Stokes problem withsliding in a given domain has a solution in W 1,1+1/m

(minimum of a strictly convex functional), Jouvet Rappaz2010.


Climatic input : b (m of ice per year)

Cold year 1913 Warm year 2003


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.


Time discretization : a splitting 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

∂u

∂t+ (u · ∇)u = 0,

∂ϕ

∂t+ u · ∇ϕ = bδΓ.


Time discretization : a splitting 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

ρ∂u

∂t− div

(2µε(u)

)+∇p = ρg ,

div u = 0,

on the ice/air interface Γ : (2µε(u)− pI )n = 0,

on the bedrock : no-slip or sliding.Jouvet Picasso Rappaz Blatter Funk Huss Numerical simulation of Rhone’s glacier

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.


Diffusion step: the 3D finite element mesh

1874 2008 2050 2075 2100


Shape computation : transport + iceaccumulation/melting (1/3)

Solve between t = tn−1 and t = tn

∂ϕ

∂t+ u · ∇ϕ = 0.

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



SLIC Postprocessing (Simple Line Interface Calculation), seefor instance Scardovelli Zaleski Ann. Rev. Fluid Mech. 1999.

Without SLIC.

With SLIC.



Solve between t = tn−1 and t = tn

∂ϕ

∂t= bδΓ.

Before

After

ELA

Filling

Emptying


Conclusions and perspectives

Accurate model to reproduce the dynamics of alpine glaciers.

Data: bedrock, accumulation or ablation (meters of ice peryear).

Tuning: sliding coefficient along the bedrock.

Can be used to predict the glacier shape during the next 100years, given several climatic scenario.

Useful for hydro-electric companies (water captation fordams).

Inverse modelling: bedrock optimization.

Free surface problems with anisotropic finite elements: T.Coupez (Mines Sophia), F. Alauzet (INRIA Rocquencourt).


Free surfaces with anisotropic finite elements

Jet buckling of a viscoelastic fluid.

Navier-Stokes with level set, finite elements.

Anisotropic adaptive remeshing, criterion: distance tointerface.

Coupez 2010.


Free surfaces with anisotropic finite elements

Free surface Mesh

Navier-Stokes with level set, finite volumes.Anisotropic adaptive remeshing, criterion: distance tointerface.Alain Guegan Alauzet 2009.


numerical simulation of rhône's glacier from 1874 to 2100sma.epfl.ch/~picasso/grenoble.pdf ·...

Documents