Geomagnetism, geodynamo, and computer science

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Geomagnetism, geodynamo, and computer science

Alexandre Fournier

LGIT/Universit´e Joseph-Fourier

Module Applications de l’informatique à la recherche et au développement technologique, Écoles doctorales de Lyon, Lyon, 17 avril 2007

alexandre.fournier@ujf-grenoble.fr

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1 Some definitions

2 Theoretical background

3 Observations

4 Some open questions

5 Numerical modelling

6 Experiments

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Informatique - Computer science (?)

Definition

Informatique : n.f. et adj. (1962), mot créé par Ph. Dreyfus sur le modèle de mathématique,électronique, et qui a lui-même servi de modèle pour de nombreux dérivés analogiques (bureautique,robotique, etc.). Le mot désigne la science et l’ensemble des techniques automatisées relatives aux informations (collecte, mise en mémoire, utilisation, etc.) et l’activité économique mettant en jeu cette science et ces techniques. L’informatique est en rapport avec les notions de calcul, de classement, d’ordre (cf. ordinateur), de mise en mémoire et, avec la gestion automatique des données, des informations au moyen de logiciels et du matériel approprié. (Robert historique de la langue fran¸caise, A. Rey)

Definition (Tentative translation)

Science and automated techniques related to information and its processing (gathering, storage, etc.). Computer science is related to computations, sorting issues, storage, and the associated soft- and hardware.

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Dynamo I

Definition

G´en´eratrice de courant continu. La dynamo d’une bicyclette. (Larousse) Direct current (DC) generator.

Conversion d’énergie mécanique en énergie électromagnétique.

Conversion of mechanical energy into electromagnetic energy.

A (solid) example

L R z

O

A

piste circulaire ω

i

Key ingredients:

Moving conductor in an ambient magnetic field: induction (electromotive force: electric currenti is created).

Mechanical feedback via the Lorentz force (Lenz’ law).

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Dynamo II

The energy injected into the system controls its behaviour. Possibility of a self-sustained magnetic field. (Solve set of nonlinear PDEs.)

Too weak forcing:

0 1

0 10 20

currenti(t)

time

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Dynamo III

Increase value of (externally applied) torque: Oscillatory current is sustained.

0 1

0 10 20

currenti(t)

time

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Dynamo IV

Bursts of current intensity are related to Lenz’ law

0 1

0 10 20

angular velocityω(t)

time

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Dynamo V

Such nonlinear physics are prone to non-deterministic dynamics.

L R

zO A

! i2 L R

zO A

! i2

Mutual Induction

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Dynamo VI

‘Reversals’ in a double disc dynamo:

0 1

0 10 20

currenti(t)

time

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Geo I

Athanasius Kichner,Mundus Subterraneus(ca. 1664)

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Geo II

Lamb & Sington (1998)

Seismic sounding of earth’s interior

0 1000 2000 3000 4000 5000 6000 0

2.5 5 7.5 10 12.5 15 17.5 20

0 1000 2000 3000 4000 5000 60000 500 1000 1500 2000 2500 3000 3500 4000

PSfrag replacements

pressure(kbar) density(M g/m³)

P-wave speed(km/s) S-wave speed(km/s)

depth(km) depth(km) Mantle

Outer Core

Inner Core

Dziewonski & Anderson (1981)

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Physical properties of earth’s core

Dynamical viscosityη_v – molecular transport of momentum.

I ηv ≈6 mPa.s (Poirier, 1988) .

I ηv ≈10 mPa.s (Alf`e & Gillan, 1998).

Thermal conductivityK_T – molecular transport of heat. K_T ≈50 W.m⁻¹.K⁻¹ (Stacey & Anderson, 2001). Electrical conductivityσ.

Linked to magnetic diffusivity

η= 1 µ₀σ.

η≈2 m².s⁻¹(Roberts & Glatzmaier, 2000).

Diffusive time scales for the core:

τ mag( ≈ 10000 ans) τ vis et τ tem( > 1 Ma).

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Physical properties of earth’s core

Thermal conductivityK_T – molecular transport of heat.

K_T ≈50 W.m⁻¹.K⁻¹ (Stacey & Anderson, 2001).

Electrical conductivityσ. Linked to magnetic diffusivity

η= 1 µ₀σ.

Diffusive time scales for the core:

τ mag( ≈ 10000 ans) τ vis et τ tem( > 1 Ma).

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Physical properties of earth’s core

Electrical conductivityσ.

η= 1 µ₀σ.

Diffusive time scales for the core:

τ mag( ≈ 10000 ans) τ vis et τ tem( > 1 Ma).

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Physical properties of earth’s core

Electrical conductivityσ.

η= 1 µ₀σ.

Diffusive time scales for the core:

τ mag( ≈ 10000 ans) τ vis et τ tem( > 1 Ma).

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The geodynamo

The earthLbuilds its magnetic fieldBinside its fluid outer core.

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Dynamo theory: a partial timeline

Cowling ’s antidynamo theorem

1600 1839 1919 1934 1946 1958 1994

William Gilbert’s Terrella

Carl Friedrich Gauss

Sir Joseph Larmor’s hypothesis

1949 Bullard

W.Elsasser G. Backus A.Herzenberg

Kinematic dynamos

Direct numerical integration G.Glatzmaier P. Roberts

Larmor (1919), Cowling (1933), Bullard & Gellman (1954), Backus (1958), Glatzmaier & Roberts (1995).

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Kinematic dynamos I

Imposed velocity fieldu.

The induction equation is written

∂B

∂t =Rm∇∧(u∧B) +∇²B.

Rmis the magnetic Reynolds number

Rm=U L η ,

with

U : velocity scale,

L: length scale (e.g. size of the system), η : magnetic diffusivity ([η] =L²T⁻¹)).

Interpretation ofRm:

Rm= L²/η L/U = τvis

τadv .

Rm>1 appears as a necessary condition.

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L:

uis probably due to thermo-chemical convection (cooling ofL) (e.g.

Gubbins & Roberts, 1987).

Another possibility: fluid instabilities due to the precession of the axis of rotation ofL

(Malkus, 1968).

ωd

23.5^o

β

Torque

♁

Equator

Ecliptic Plane Pole of Ecliptic

or

$

ω_p

NOT TO SCALE

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Ingredients to make a dynamo: α and ω effects

α ω

Hollerbach (1996)

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Dudley-James s2t2

umade of an antisymmetric zonal wind + two counter-rotating meridional rolls (Dudley & James, 1989)

uφ ψ_m

N

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0

1 2 3 4 5 6 7 8 9 10

−1

−2

−3

−4

−5

−6

−7

−8

−9

−10

−11

Rm

non-dimensionalgrowthrate

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Limits of the kinematic approach

No magnetic feedback on the flow (no dynamics, no Lorentz force).

Hollerbach (1996) No scale disparity betweenuandB.

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Magnetohydrodynamics

Induction equation + conservation laws (mass, momentum, energy).

∇·u = 0,

Ro(∂tu+u·∇u) + 2ˆez∧u = −∇P+E∇²u

+ qRaΘˆer+ (∇∧B)∧B,

∂tΘ +u·∇Θ = q∇²Θ,

∂_tB = ∇∧(u∧B) +∇²B,

∇·B = 0.

Dimensionless numbers: E = ν

ωd² ≤10⁻¹², q=κ

η ≤10⁻⁵, Ra= g₀αβd²

ωκ , Ro= u

ωL ≤10⁻⁶.

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Magnetohydrodynamics

∇·u = 0,

Ro(∂_tu+u·∇u) + 2ˆe_z∧u = −∇P+E∇²u

∂tΘ +u·∇Θ = q∇²Θ,

∂_tB = ∇∧(u∧B) +∇²B,

∇·B = 0.

ωd² ≤10⁻¹², q=κ

η ≤10⁻⁵, Ra= g₀αβd²

ωκ , Ro= u

ωL ≤10⁻⁶.

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Magnetohydrodynamics

∇·u = 0,

Ro(∂_tu+u·∇u) + 2ˆe_z∧u = −∇P+E∇²u

∂tΘ +u·∇Θ = q∇²Θ,

∂_tB = ∇∧(u∧B) +∇²B,

∇·B = 0.

ωd² ≤10⁻¹², q=κ

η ≤10⁻⁵, Ra= g₀αβd²

ωκ , Ro= u

ωL ≤10⁻⁶.

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Magnetohydrodynamics

∇·u = 0,

Ro(∂_tu+u·∇u) + 2ˆe_z∧u = −∇P+E∇²u

∂tΘ +u·∇Θ = q∇²Θ,

∂_tB = ∇∧(u∧B) +∇²B,

∇·B = 0.

Dimensionless numbers:

E = ν

ωd² ≤10⁻¹², q=κ

η ≤10⁻⁵, Ra=g0αβd²

ωκ , Ro= u

ωL ≤10⁻⁶.

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Magnetic elements

M•

Nord

Est

bas

X

Y

Z

B D H

I Three componentsX,Y,Z.

IntensityF (kBk),

inclinationI, declinationD.

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Terrestrial magnetic environment

(NASA)

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(Mathematical) description of the field I

Spherical harmonics

C.F. Gauss

One can write ~B =−∇V~ mag at earth’s surface. (Be- ware! This is not true in a region containing magnetic sources.)

B~ is a potential field. V_mag must obey Laplace’s equation

∇²V_mag = 0.

Spherical harmonics Denoted byY_l^m(θ, φ).

Y_l^m(θ, φ) =P_l^m(cosθ) expimφ.

l : harmonic degree.

m: harmonic order. (m<l)

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An example

Take:

champF

=

1 ×Y₁⁰ +

1 ×Y₂² +

1×Y₇⁶ +

1×Y₁₄² In this case, one has

F(θ, φ) =Y₁⁰(θ, φ) +Y₂²(θ, φ) +Y₇⁶(θ, φ) +Y₁₄²(θ, φ).

Keypoints:

Iflincreases, the associated lengthscale decreases.

In principle,l (andm) can go to∞. In practice, the expansion is truncated atl =lmax.

l= 1: dipolar terms,

l= 2: quadrupolar terms, etc.

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An example

Take:

champF

=

1 ×Y₁⁰ +

1 ×Y₂² +

1×Y₇⁶ +

Keypoints:

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An example

Take:

champF

=

1 ×Y₁⁰ +

1×Y₂² +

1×Y₇⁶ +

Keypoints:

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An example

Take:

champF

=

1 ×Y₁⁰ +

1×Y₂² +

1×Y₇⁶ +

Keypoints:

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An example

Take:

champF

=

1 ×Y₁⁰ +

1×Y₂² +

1×Y₇⁶ +

Keypoints:

In principle,l(and m) can go to∞. In practice, the expansion is truncated atl=lmax.

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Sources of the earth’s magnetic field

Thanks to the analysis ofV_mag at earth’s surface by means of theY_l^m : Internal and external sources can be separated. NB:

I External sources: sources located atr>R^L.

I Internal sources: sources located atr<R^L.

99% of the field is of internal origin (created somewhere beneath our feet).

This isthe main magnetic field.

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Spectrum of the main component of B.

1 2 4 6 8 10 121314 16 18 20 22 0

1 2 3 4 5 6 7 8 9 10

PSfragreplacements

nT

Champ du noyau

Champ crustal

km

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Recent magnetic data: observatories + satellites I

http://www.intermagnet.org 2Hz

Contribution of CS (// sismology): management of (freely available) databases.

International Geomagnetic Reference Field (IGRF), World Magnetic Model (WMM).

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US/UK World Magnetic Model -- Epoch 2005.0 Main Field Total Intensity (F)

Map Date : 2005.0 Units : nanoTesla Contour Interval : 1000 nanoTesla Map Projection : Mercator

180°

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US/UK World Magnetic Model -- Epoch 2005.0 Main Field Declination (D)

Map Date : 2005.0

Units (Declination) : degrees (Red contours positive (east), blue negative (west)) Contour Interval : 2 degrees

Map Projection : Mercator 180°

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90 100110 120130 150140 160 0

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00

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00

0 0

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000

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US/UK World Magnetic Model -- Epoch 2005.0 Main Field Inclination (I)

Map Date : 2005.0

Units (Inclination) : degrees (Red contours positive (down), blue negative (up)) Contour Interval : 2 degrees

Map Projection : Mercator 180°

180°

210°

240°

270°

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330°

0°

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Demo

Software that can be used ‘out of the box’: a demo?

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Satellites I

Recent satellite missions: Magsat (1980) / Ørsted (1999-2004) / Champ (2000-) / SAC-C + Swarm around 2010 (ESA).

More accurate description of thesecular variationof the magnetic field.

CHAOS (Olsen et al., 2006a): ∂_tBup to harmonic degree 16 with good accuracy.

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Satellites II

Birth of a global geomagnetic community

http://www.sciences.univ-nantes.fr/geol/Swarm/1stmeeting.html

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Temporal variability of the main field of the earth

PSfrag replacements

10⁻⁵ 10⁻² 10⁻¹ 1 10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

Filtrepasse-bas:lemanteau

Secousses

variation s´eculaire

Dur´ee d’une inversion

Cretaceous superchron

ondes sonores ondes inertielles

1 jour τad

τ_M

Ondes de torsion

τ (ans)

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Reversals of the earth’s magnetic field

Inclination can vary very rapidly inside a sediment core. This reflects a reversal of the earth’s magnetic field. This phenomenon is observed at aGLOBALscale in the sedimentary records.

Glatzmaier & Roberts (1995) Duration of an inversionO(10) kyr.

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Palaeomagnetic dataset I

OldestBrecorded : >3 billion years ago (south African rocks) - see recent letter in Nature by Tarduno et al. (2007).

Working of the geodynamo on geological timescales, Virtual axial dipole hypothesis .

Fondation of plate tectonics revolution in the 1960ies.

Heirtzler (1968) Backus et al. (1996)

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Palaeomagnetic dataset II

Frequency of reversals in the past:

After Merrill et al. (1996).

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Secular variations: from years to centuries

Location of the geomagnetic pole

-80˚

-60˚

-40˚

-20˚

75˚ 75˚

80˚ 80˚

85˚ ₁₆₀₀ 85˚

1630

1660

1690

1720 1750

1780

1840 1810

1870

1990

G. L´egaut (LGIT)

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Variation of dipole strength:

Backus et al. (1996)

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Secular variation and motions at the top of the core

Frozen-flux approximation + geostrophic balance assumption.

Eymin (2004) ;→: 37km / yr.

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Secousses g´ eomagn´ etiques (geomagnetic jerks)

dD/dt

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?

PSfrag replacements

10⁻⁵ 10⁻² 10⁻¹ 1 10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

Filtrepasse-bas:lemanteau

Secousses

variation s´eculaire

Dur´ee d’une inversion

Cretaceous superchron

ondes sonores ondes inertielles

1 jour τad

τ_M

Ondes de torsion

τ(ans)

Mechanism(s) responsible for geomagnetic secular variation, Core-mantle coupling,

Reversals: mechanism, frequency.

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Numerical simulations I

Principle: space- and time- discretization of conservation laws (mass, momentum, energy) and Maxwell’s equations.

A model can be represented by a state vectorX.

X˙ =f(X,t).

Because of scaling,f depends upon a bunch of non-dimensional numbers.

Upon discretization,tⁿ=n∆t,

Xⁿ⁺¹=Xⁿ+ ∆tf Xⁿ,Xⁿ⁺¹,tⁿ,tⁿ⁺¹ .

Size: 8 scalar fields to evaluate at 50 grid points in each direction of space: 10⁶ values.

Number of time iterations: if ∆t = 2 weeks, for 10⁵yr: 10⁵×25 = 2.5 millions of iterations.

→high performance computing.

Implementation:

Space: spherical harmonics and/or grid-based methods (finite differences, finite element, finite volume, spectral element).

Time: finite differences (explicit, implicit, mixt).

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A quick tour of today’s models I

Horizontal expansion in spherical harmonics, In radius

• Chebyshev polynomials(Glatzmaier, 1984).

• Finite differences (Dormy et al., 1998; Kuang & Bloxham, 1999).

Pros ;−) : a lot.

1 Weak numerical dispersion.

2 No pole problem.

3 B: natural connection with an exterior potential field.

Cons :−( :

1 Pseudo-spectral calculation of nonlinear terms. Cost of Legendre transform (M²vs.

MlogM for FFT).

2 Global basis: parallel implementation is not straightforward.

3 Restricted by essence to spherical geometry (precession).

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Characteristics of current models (Dormy et al., 2000):

Reversals

First order morphology ofB.

Westward drift.

model

data Kuang & Bloxham (1997)

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Transition to local, grid-based methods

Over the past 10 years, a lot has been learned on the working of the geodynamo and core dynamics thanks toY_l^m-based numerical models.

Now, several groups are involved in the building of codes that should allow longer integration times,

to decrease the viscosity of the modelled fluid,

to compute in non-spherical geometries (ellipticity, CMB topography).

Methods:

finite element (Matsui & Okuda, 2003),

spectral element (Fournier et al., 2004, 2005; Fournier, 2006).

finite volume(Harder & Hansen, 2005).

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The spectral element method

Idea: combine the geometrical flexibility of the finite element method with the accuracy of spectral methods (Maday & Patera, 1989).

Local bases of high-order polynomials (7−14).

Fournier et al. (2004)

Properties:

Spectral convergence.

Weak numerical dispersion.

Tensorized geometry.

Domain decomposition is natural.

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Spectral element and geophysics

Atmosphere and ocean dynamics - shallow water equations. Taylor et al. (1997);

Levin et al. (2000); Giraldo (2001)

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Seismology at the local and global scale - the wave equation in elastodynamics.

Komatitsch & Vilotte (1998); Komatitsch & Tromp (1999); Chaljub (2000);

Komatitsch et al. (2002); Chaljub et al. (2003)

Vector function = Uvec

Cubed Sphere U_X from -0.0081 to 0.0040

[Merci `a Emmanuel Chaljub]

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Practical considerations

http://bladerunner.princeton.edu With this type of approach, calculations are

mostly made on Linux-operated PC-clusters, using the message-passing interface (MPI) (Gropp et al., 1999).

Given their unbeatable performance-price ra- tio, these computers are the democratic fu- ture of High-Performance Computing (Bunge

& Tromp, 2003).

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Application of the SEM to the earth’s core

Mesh of a spherical shell.

Fourier–spectral element approach

Hyp: computational domain is axisymmetric.

Cylindrical coordinates (s, φ,z).

Expansion in Fourier series inφ.

Parallel SEM applied to collection of meridional sub-problems.

Consistent algebraic splitting to update pressure fields.

Multi-level elliptic solvers (≈multigrid) .

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Tools

Languages: f77,f90, c, c++.

Libraries (http://www.netlib.org)

BLAS, LAPACK (Matrix-vector, matrix-matrix products, factorization, inversion).

FFTW (Fastest Fourier Transform in the West) (MIT) http://www.fftw.org.

OPENMP - shared memory.

MPI (Message Passing Interface) - distributed memory.

I MPICH (Los Alamos)

I LAMMPI

I Now: OPENMPI (routines globales optimis´ees)http://www.open-mpi.org.

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Simulations: where we stand

Simulation closest toLup to this day: Takahashi et al. (2005). E = 4.10⁻⁶. Several weeks on 512 nodes of the Earth simulator (1 node: 8 processors).

www.es.jamstec.go.jp/esc/eng/

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Post-processing of simulations I

R´educe dimensionality of synthetic dataset: integral quantities, spectra.

3D visualization. http://www.paraview.org

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Post-processing of simulations II

Rotating convection:

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DTS

Scientific goals

DTS

L

Leff

ES

PARODY (FD- Yl^m)

0. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0. 1 2 3 4 5 6

−logE

−logPm

QG : Schaeffer & Cardin (2006)

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DTS I

Scientific goals

Feasibility of a geophysical, experimental fluid dynamo.

Rm≤ O(30)

Experimental fluid dynamos: Riga, Karlsruhe, and Cadarache.

Stieglitz & M¨uller (2001). See also Gailitis et al. (2001).

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DTS II

Scientific goals

Cadarache : Von-Karman sodium (VKS), turbulent flow!!! MAJOR BREAKTROUGH (Monchaux & 15 coauthors, 2007; Berhanu & 13 coauthors, 2007)

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DTS III

Scientific goals

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DTS IV

Scientific goals

(see also the VKS website -google: VKS experiment.)

Experiments are an ideal complement for the study of the geomagnetic SV. .

Alexandre Fournier (LGIT/Universit´e Joseph-Fourier) Geomagnetism, geodynamo, and computer science 17/04/07 60 / 81

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The Derviche Tourneur Sodium experiment

Principle :

Spherical Taylor–Couette MHD flow

ω + ∆ ω ω

b a

Γ

R´ealisation O(300) ke(design, etc.).

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YMCA

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The sphere

Nataf et al. (2006)

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Input from CS I

2 computers: 1 monitor / 1 command. Interfacing with Labview.

CS Budget'5% of total.

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Jumping from one branch to the other:

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Stop: spin down.

Alexandre Fournier (LGIT/Universit´e Joseph-Fourier) Geomagnetism, geodynamo, and computer science 17/04/07 66 / 81

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Data acquisition and control in real time

Data processing

P, Γ,ω,B, ∆V.

Sampling rate: 1−2 kHz. Low-passRC filter,fc = 500 Hz

+ Doppler velocimetry (radial or azimutal velocity depending on the shoot angle).

A ‘run’ : a few hours of recording. →several Gigabytes of data.

Processing and analysis: matlab/scilab. Large amount of data: needs meso-scale computing power (16 Go of RAM on some nodes).

http://www.obs.ujf-grenoble.fr/SCCI/

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Electric potential measurements (Denys Schmitt) I

1 Raw signal. 3 points median.

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Electric potential measurements (Denys Schmitt) II

2 PSD (spectral density): amplitude of the Fourier transform

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Electric potential measurements (Denys Schmitt) III

3 Same after 2 Hz sliding window averaging.

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V´ elocim´ etrie Doppler (Daniel Brito) I

Principle : Brito et al. (2001).

file2 BRUT

Time in seconds

Distance to the probe in mm

0 50 100 150 200 250 300

50 100 150 200 250 300 350

0 100 200 300 400 500 600 700 800 900

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V´ elocim´ etrie Doppler (Daniel Brito) II

file2 PDF BRUT

Velocity (mm/s)

Distance to the probe in mm

0 100 200 300 400 500 600 700 800 900

50 100 150 200 250 300 350

0 0.02 0.04 0.06 0.08 0.1

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Basic state and instabilities / Waves I

Ha= 10,E = 1 Ha= 10,E = 10⁻² Ha= 10,E = 10⁻⁴

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Basic state and instabilities / Waves II

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