Modélisation & simulation de la génération de champs magnetiques par des écoulements de métaux liquides. Wietze Herreman
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1 Modélisation & simulation de la génération de champs magnetiques par des écoulements de métaux liquides Wietze Herreman 19ième Colloque Alain Bouyssy!!"#$%&'()*+(,#-*.#*+( )/01+"2(!!!!!!!
2 Origine des champs magnétiques des corps célestes? Terre Soleil
3 Origine des champs magnétiques des corps célestes? Terre?? Soleil
4 Effet Dynamo ( J. Larmor 1919 ) Un écoulement de fluide conducteur peut spontanément exciter et entretenir des champs magnétiques générateur simple dynamo solide dynamo fluide
5 1 Générateur simple B J énergie mécanique énergie électromagnétique phénomène de l induction magnétique
6 1 Générateur simple B J énergie mécanique énergie électromagnétique Nécessite la présence d aimants permanents!
7 1 Générateur simple B J énergie mécanique énergie électromagnétique Nécessite la présence d aimants permanents!
8 2 Dynamo solide 0. On imagine un petit champ B vertical 1. Dans le disque un courant radial est alors induit U x B 2. Le courant est collecté au bord et retourne vers l axe par des contacts glissants. Dynamo disque homopolaire (Bullard 1955) 3. En passant par la bobine le champ initial est regénéré
9 2 Dynamo solide 0. On imagine un petit champ B vertical 1. Dans le disque un courant radial est alors induit U x B 2. Le courant est collecté au bord et retourne vers l axe par des contacts glissants. Courants guidés et mouvements de solides 3. En passant par la bobine le champ initial est regénéré
10 2 Dynamo solide 0. On imagine un petit champ B vertical 1. Dans le disque un courant radial U x B Même chose est dans alors un induit écoulement de fluide conducteur 2. Le courant? est collecté au bord et retourne vers l axe par des contacts glissants. Courants guidés et mouvements de solides 3. En passant par la bobine le champ initial est regénéré
11 3 Dynamo fluide : mettre en equation le minimum Champ magnétique Diffusivité magnétique η = 1/σ e µ Equation d induction t b = (u b) + η b, b = 0 Champ de vitesse Un écoulement u est une dynamo cinématique s il permet de faire croître le champ magnétique
12 3 Dynamo fluide : quelques exemples testés en labo Ponomarenko Riga 2001 G.O. Roberts Karlsruhe 2001 Von Karman Cadarache 2006
13 3 Dynamo fluide : convection dans un corps célèste t u + (u )u = 1 1 p + αgt + ν u + ( b) b ρ µρ t b + (u )b = (b )u + η b t T + (u )T = κ T + Q u = 0 b = 0 u b p T Q champ de vitesse champ magnétique pression température sources de chaleur ν, η, κ diffusivités α dilatation thermique g gravité ρ densité µ perméabilité On comprend qu un ordinateur peut-être utile
14 3 Dynamo fluide : convection dans un corps célèste t u + (u )u = 1 1 p + αgt + ν u + ρ µρ t b + (u )b = (b )u + η b t T + (u )T = κ T + Q u = 0 b = 0 ( b) b u b p T Q champ de vitesse champ magnétique ν, η, κ α diffusivités dilatation thermique pression g gravité ρ température densité sources Glatzmaier µ de chaleur & Roberts 1995 perméabilité On comprend qu un ordinateur peut-être utile
15 Autres possibilités pour mettre en mouvement le fluide? Ω ω Ω g ε précession elliptique Dynamo précession retrouvée en simu Tilgner ( 05-07) Wu & Roberts ( 07-09) Comprendre les effets de marées & modéliser la dynamo éventuelle
16 1 écoulements induits par des marées Ω g ε
17 1 écoulements induits par des marées Lacaze 2004
18 2 Effets d un champ magnétique imposé Ω Ω ɛ g B ε ε Io Jupiter
19 2 Effets d un champ magnétique imposé 6.3 Analytical study of the spin-over mode (a) Ω SO Λ = x 10 2 (b) b r Λ = t /2π t /2π Fig. 6.2 (a) Theoretical nonlinear temporal evolution of the horizontal projection of the spin-over mode amplitude ΩH = (Ω21 + Ω2 )1/2, for various values of the Elsasser number Λ. Ekmann number and eccentricity are fixed, E = ,! = Calculations started from the initial state Ω1 = 10 3, Ω2 = 10 3, Ω3 = 0, which is the linearly unstable spin-over mode with small amplitude. The arrow on the right side, indicate the saturation level of the slowly growing spin-over mode horizontal amplitude at Λ = The critical Elsasser number is Λc = for this parameter set. (b) Typical recorded magnetic field-signals for varying Elsasser number. Ekmann number and eccentricity are fixed E = ,! = 0.10 ± The experiments agree with the theoretical profiles of figure 6.2(a). 6.3 Herreman, Le Bars & Le Gal : On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids 2009 Phys. Fluids Analytical study of the spin-over mode
20 3 Est-ce que les écoulements excités par les marées ( ondes inertielles ) sont-ils des dynamos? t b = S [ (u e it + u e it) b ] + Q b U ωr 1 petit paramètres η ωr 2 1 µ Ω µ µ couche plane cylindre RESULTAT Pas de dynamos
21 4 Comprendre pourquoi ça ne marche pas? t b = S ( U b ) + Q b limite S, Q 1 U = u(r)e it + u (r)e it Modélisation champ moyen f = 1 T T 0 f dt ( ) ( b = [ b(r) + b (r, t) ] e σt σb = S 2 (U b) + Q b Le champ moyen respecte aussi une équation d induction avec un écoulement apparent U = 2 Im (u u ) ent that contributes to Circulation pure ( vent zonal ) pour ondes inertielles = pas une dynamo
22 5 Pour trouver un résultat plus général t b = S ( U b ) + Q b limite S, Q 1 U = u(r)e it + u (r)e it Modélisation champ moyen f = 1 T T 0 f dt ( ) ( b = [ b(r) + b (r, t) ] e σt σb = S 2 (U b) + Q b Le champ moyen respecte aussi une équation d induction avec un écoulement apparent U = 2 Im (u u ) ent that contributes to = Stokes drift
23 y x G.O.Roberts drift dynamo Herreman, Lesaffre : Stokes drift dynamos accepté J. Fluid Mech.
24
25
26 3 Dynamo fluide : simulation numérique t u + (u )u = 1 1 p + αgt + ν u + ( b) b ρ µρ t b + (u )b = (b )u + η b t T + (u )T = κ T + Q u = 0 b = 0 u b p T Q champ de vitesse champ magnétique pression température sources de chaleur ν, η, κ diffusivités α dilatation thermique g gravité ρ densité µ perméabilité On comprend qu un ordinateur peut-être utile
27 3 Dynamo fluide : simulation numérique t u + (u )u = 1 1 p + αgt + ν u + ( b) b ρ µρ t b + (u )b = (b )u + η b t T + (u )T = κ T + Q u = 0 b = 0 u b p T Q champ de vitesse champ magnétique pression température sources de chaleur ν, η, κ diffusivités α dilatation thermique g gravité ρ densité µ perméabilité On comprend qu un ordinateur peut-être utile
28 1 Générateur complexe B J Remplacer l aimant par circuit bien choisi!
29 1 Générateur complexe B Choisissant les bonnes connections & tournant suffisamment rapidement J Bobine & quelques boîtes noires on peut arriver à créer un courant et donc un champ magnétique
30 1 Générateur complexe 2 Dynamo solide B J Cette idée peut être mise en oeuvre Dynamo disque homopolaire (Bullard 1955)
31 3 Dynamo fluide : fluides conducteurs Métal liquide Plasma Terre Soleil Fer liquide
32 Maintenant : comprendre si les écoulements excités par les marées ( ondes inertielles ) sont des dynamos? t b = S [ (u e it + u e it) b ] + Q b U ωr 1 petit paramètres η ωr 2 1 µ Ω µ µ couche plane cylindre RESULTAT Pas de dynamos
33 Ω 55 mm Steady structure (high) g ε Silicone Roll 100 mm Lune Terre Steady structure (low) Rotating axis Motor dispositif expérimental spinover
34 3 Dynamo fluide : champ moyen si faible t b = (u b) + η b, η b 0= 0 Equivalence avec l équation de l élément de fluide t dl = (u dl) u Peu de raisons pour penser que échelles u et b si différent b
35 3 Dynamo fluide : analyse champ moyen repris Diffusivité magnétique η = 1/σ e µ si faible t b = (u b) + η b, η b 0= 0 Equivalence avec l équation de l élément de fluide t dl = (u dl) u Peu de raisons pour penser que échelles u et b si différent b
Φ B. , into the page. 2π ln(b/a).
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