Emmanuel DORMY (CNRS / ENS)

Transcription

1 Emmanuel DORMY (CNRS / ENS) dormy@phys.ens.fr

2

3 The Earth s internal structure Mantle Fluid outer core ICB 3480 km 6366 km 1221 km Inner core CMB Roberts & King 2013

4 Governing Equations

5 Governing Equations

6 Governing Equations

7 Governing Equations

8 Governing Equations

9 Governing Equations

10 Governing Equations

11 E η E/Pm=E η

12 E η E/Pm=E η

13 E η E/Pm=E η

14 Weak and Strong field branches Chandrasekhar, 1961 Eltayeb & Roberts, 1970 Eltayeb & Kumar, 1977 Roberts & Soward, 1978 Soward, 1979 Fearn, 1979 Fautrelle & Childress, 1982 Proctor & Weiss, 1982

15 Weak and Strong field branches Rac Λ = B2/ρµηΩ Fearn (1979)

16 Weak and Strong field branches Λ = B2/ρµηΩ Ο(1) (Roberts, GAFD, 1988)

17 E η E/Pm=E η

18 Glatzmaier & Roberts, 1995

19 Ema Roberts number q = Pm / Pr Carsten Kutzner, Uli Christensen 2003 g No dynamo

20 Christensen-Aubert 2006

21 Power based scaling laws for the magnetic field (Christensen & Aubert, 2006) : Lorentz number : fraction of ohmic dissipation : flux-based Rayleigh number (directly related to the injected power) Parameter range

22 Statistical energy balance between production and dissipation where : magnetic dissipation length scale Oruba & Dormy, GJI, 198, (2014).

23 Statistical energy balance between production and dissipation Verified by any statistically steady dynamo Oruba & Dormy, GJI, 198, (2014).

24 Statistical energy balance between production and dissipation The simplest approximation: constant Oruba & Dormy, GJI, 198, (2014).

25 Power based scaling laws for the magnetic field Energy balance + constant Energy balance + more precise description of : (Christensen & Aubert, 2006) Oruba & Dormy, GJI, 198, (2014).

26 Power based scaling laws for the magnetic field Energy balance + constant Energy balance + more precise description of : (Christensen & Aubert, 2006) Oruba & Dormy, GJI, 198, (2014).

27 =Verified by any statistically steady dynamo Power based magnetic field scalings = TOO GENERAL: applicable to any model irrespectively of the magnetic field generation mechanism Oruba & Dormy, GJI, 198, (2014).

28 Viscous length scale of the flow (see King & Buffett, 2013) In numerical models: non negligible viscous effects in the bulk of the flow Oruba & Dormy, GJI, 198, (2014).

29 Magnetic field strength Bifurcation from a laminar flow (Fauve & Pétrélis, 2007): Lorentz force modified viscous force with : Elsasser (Reduced database) Oruba & Dormy, GJI, 198, (2014).

30 Magnetic field strength Bifurcation from a laminar flow (Fauve & Pétrélis, 2007): Lorentz force modified viscous force with : Elsasser Application to the Earth s core? (Reduced database) much smaller than Oruba & Dormy, GJI, 198, (2014).

31 Intermediate result Dipolar Dynamos at moderate forcing are dominated by a balance between viscous and Coriolis forces

32 Ema Roberts number q = Pm / Pr Carsten Kutzner, Uli Christensen 2003 g No dynamo

33 Transition from dipolar to multipolar regime Relative dipole field strength vs the local Rossby number: Christensen & Aubert 2006

34 Transition from dipolar to multipolar regime Dominance of the Coriolis force in both regimes: Transition controlled by the relative strength of inertial to viscous forces: Soderlund et al 2012

35 Inertial forces versus Coriolis Typical length scales the kinematic dissipation length scale the length scale the parallel length scale : : (Oruba & Dormy 2014) (quasi-geostrophy) with Oruba & Dormy, GRL in press.

36 Inertial forces versus Coriolis Test with the numerical database (U. Christensen): Oruba & Dormy, GRL in press.

37 Viscous forces versus Coriolis Well verified by dipolar dynamos: (King & Buffet 2013 and Oruba & Dormy 2014) Oruba & Dormy, GRL in press.

38 Inertial versus viscous forces Oruba & Dormy, GRL in press.

39 Inertial versus viscous forces Test against the numerical database: Oruba & Dormy, GRL in press.

40 A dominant three forces balance at the transition C I V Oruba & Dormy, GRL in press.

41 Unified description of the transition Oruba & Dormy, GRL in press.

42 Bistability between both branches Simitev & Busse, Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells, Europhysics Letters (EPL), 85 (2009) 19001

43 Bistability between both branches Schrinner, Petitdemange, Dormy, ApJ, 2012

44 Intermediate results - Dipolar Dynamos at moderate forcing are dominated by a balance between viscous and Coriolis forces - Loss of dipolarity, and multipolar Dynamos at larger forcing are associated with the increasing strength of inertia

45 Key question: Is the Magnetostrophic balance achievable with todays computer?

46 Ema Roberts number q = Pm / Pr Carsten Kutzner, Uli Christensen 2003 g No dynamo

47 q Ra / Rac Morin & Dormy (2005, 2009)

48 Magnetic energy -4 E=3.10, q=6 Morin & Dormy (2005, 2009) Magnetic energy q

49 -4 E=3.10, q=3 Morin & Dormy (2005, 2009) Magnetic energy q

50 -4 E=3.10, q=1.5 Morin & Dormy (2005, 2009) Magnetic energy q

51

52 q Morin & Dormy (2005, 2009) E

53 q -4 E=3.10, q=6

54 Weak and Strong field branches q=pm=12 (Dormy, in prep)

55 Weak and Strong field branches q=pm=18 (Dormy, in prep)

56 Weak and Strong field branches Vφ, E = , Pm = 18, Ra/Rac = 1.72 (Dormy, in prep)

57 E η E/Pm=E η

58 Weak and Strong field branches (Dormy, in prep)

59 Weak and Strong field branches Λ = B2/ρµηΩ Ο(1) (Roberts, GAFD, 1988)

60 Weak and Strong field branches Ra/Rac = > 1.83 (Dormy, in prep)

61 Weak and Strong field branches Ra/Rac = 1.73-> 1.68 (Dormy, in prep)

62 3D bifurcation diagram E= (Dormy, in prep)

63 Weak and Strong field branches Λ = B2/ρµηΩ Ο(1) (Roberts, GAFD, 1988)

64 Relevant parameter space

65 Toward a distinguished limit Ema Roberts number q = Pm / Pr Carsten Kutzner, 2003 g No dynamo

66 Toward a distinguished limit Dormy & Le Mouël (2008)

67 Toward a distinguished limit (Dormy, in prep)

68 The Elsasser number Christensen-Aubert 2006

69 The Elsasser number

70 The Elsasser number (Dormy, in prep)

71 CLAIMS - None of the published spherical dynamo models correspond to a force balance relevant to the geodynamo! - This force balance can be approached in numerical models.

72 OPENED ISSUES - All terms are of similar amplitude, Yet well identified balances seem to emerge. Why? - How do these models evolve at lower E, and lower Pm? - Is it possible to sustain dynamo action for Ra<Ra? c

73 Glatzmaier & Roberts, 1995