Large-scale field and small scale dynamo

Transcription

1 Large-scale field and small scale dynamo Franck Plunian & Yannick Ponty Université de Grenoble, LGIT Observatoire de la Côte d'azur

2 Large scale magnetic fields are ubiquitous in planetary and stellar objects Radial magnetic field at the Earth s surface

3 Large scale magnetic fields are ubiquitous in planetary and stellar objects Radial magnetic field at the Earth s surface Sunspots at the photosphere

4 Large scale magnetic fields are ubiquitous in planetary and stellar objects Though, strong small-scale fields are also there Radial magnetic field at the Earth s surface At the core-mantle boundary (E. Canet 2009) Sunspots at the photosphere

5 Large scale magnetic fields are ubiquitous in planetary and stellar objects Though, strong small-scale fields are also there Radial magnetic field at the Earth s surface At the core-mantle boundary (E. Canet 2009) Sunspots at the photosphere Solar granulation

6 Large scale magnetic fields are ubiquitous in planetary and stellar objects Though, strong small-scale fields are also there Radial magnetic field at the Earth s surface At the core-mantle boundary (E. Canet 2009) Sunspots at the photosphere Solar granulation - How compatible? - How does the mean-field approach cope with it?

7 DNS A numerical approach

8 A numerical approach DNS Parallelized pseudo-spectral code Periodic box of aspect ratio N=2

9 A numerical approach DNS Parallelized pseudo-spectral code Periodic box of aspect ratio N=2 Two control parameters: viscosityν and diffusivity η =>

10 A numerical approach DNS Parallelized pseudo-spectral code Periodic box of aspect ratio N=2 Two control parameters: viscosityν and diffusivity η => Roberts forcing: - A good candidate for large-scale dynamo mechanism (strong helicity) - Anisotropic => easy to understand in terms of mean fields

11 The Roberts forcing If ν is large, without Lorentz forces => stationary flow U=F/2ν Periodic array of vorticies strongly helical

12 The Roberts forcing If ν is large, without Lorentz forces => stationary flow U=F/2ν Periodic array of vorticies strongly helical => Roberts dynamo (Roberts 1972) Section AA Vertical advection Expulsion Stretching

13 The Roberts forcing If viscosity ν is large, without Lorentz forces => stationary flow U=F/2ν Periodic array of vorticies strongly helical => Roberts dynamo (Roberts 1972) The field is large scale in x and y helical along z

14 The Roberts forcing has led to a number of investigations A non exhaustive list: Roberts GO Phil. Trans. R. Soc. Lond (1972) Soward AM J. Fluid Mech. (1987) Soward AM GAFD (1989) Tilgner A & Busse FH Proc. R. Soc.Lond. A (1995) Tilgner A Phys Lett (1997) Rüdiger S et al (1998) Rädler KH et al Stud. Geophys. Geod. (1998) Stieglitz R & Müller U PoF (2001) Plunian F & Rädler KH GAFD (2002) Plunian F & Rädler KH Magnetohydrodynamics (2002) Rädler KH et al Nonlin. Process. Geophys. (2002) Müller U & Stieglitz R Nonlin. Process. Geophys. (2002) Feudel et al PRE (2003) Rädler KH & Brandenburg A PRE (2003) Avalos etal PRE (2003) Müller etal PoF (2004) Müller etal JFM (2004) Tilgner A GAFD (2004) Minini PD & Montgomery DC PRE (2005) Plunian F PoF (2005) Sarkar A & Tilgner A Astr. Nach. (2005) Müller U et al JFM (2006) Pétrélis F & Fauve S (2006) Europhys Lett Tilgner A NJP (2007) Rädler KH & Brandenburg A PRE (2008) Tilgner A & Brandenburg A MNRAS (2008) Hori K & Yoshida S GAFD (2008) Müller U & Stieglitz R PoF (2009) Kleeorin N etal PRE (2009) Courvoisier A etal Proc. Roy S. Lond. A (2009)

15 Rm < Rmc E U Exponential decrease E B

16 Rm > Rmc E U Saturation E B Exponential growth

17 Roberts dynamo Dynamo threshold

18 Dynamo threshold Flow = rolls Roberts dynamo

19 Dynamo threshold Large scale hydrodynamic instability Flow = rolls Roberts dynamo

20 Dynamo threshold Large scale hydrodynamic instability Flow = rolls Averaged flow = rolls Roberts dynamo

21 Explore the dynamo range far above threshold New simulations Fixed viscosity ν = 0.02 Varying diffusivity 0.01 < η < 0.85 ( < Pm < 2)

22 Rm ~ Rmc E U E B E B Snapshot of magnetic energy

23 Rm ~ Rmc Rm >> Rmc E U E U E B E B E B E B Snapshot of magnetic energy

24 O(Rm -1 ) At threshold Far from equipartition Most magnetic Far above threshold Equipartition Most magnetic energy is small

25 The mean field: Mean field symmetries Introduce a symmetry factor : «Roberts dynamo» symmetry

26 The mean field: Mean field symmetries Introduce a symmetry factor : «Roberts dynamo» symmetry s Rm ~ Rmc s Rm >> Rmc t t «Roberts dynamo» symmetry during saturation, => Mean field approach can be applied

27 Alpha tensor Calculating: the mean emf the mean magnetic field we find the alpha tensor:

28 Calculating: Alpha tensor Rm ~ Rmc Rm >> Rmc the mean emf the mean magnetic field we find the alpha tensor:

29 Alpha tensor Calculating: the mean emf Alpha coefficients in the saturated regime the mean magnetic field we find the alpha tensor:

30 Interpretation The flow contains two parts: a long-time average => Roberts like a short-time variation => chaotic

31 Interpretation The flow contains two parts: a long-time average => Roberts like a short-time variation => chaotic Enstrophy at long timescale

32 Interpretation The flow contains two parts: a long-time average => Roberts like a short-time variation => chaotic Enstrophy at long timescale Enstrophy at short timescale

33 Flow spectrum Eu Roberts like Chaoti c k F k ν k

34 Rm ~Rmc B spectral window Eu Flow spectrum Large-scale dynamo mechanism: E B / E B ~1 α ii 1 E B k F k ν k

35 Rm >> Rmc B spectral window Flow spectrum Competition between the large and small scales flow for dynamo action Eu E B E B / E B ~ O(Rm -1 ) α O(Rm -1 ) Equipartition Fast The small-scale dynamo seems to be the most efficient (Vainshtein & Cattaneo 1992, Cattaneo & Hughes 1996) k F k ν k

36 Spectra 1 Rm ~ Rmc E U E B 1 Rm >> Rmc k k

37 Rm >> Rmc B spectral window Eu Flow spectrum E B Additional diagnostic: During saturation if the small scale flow is sufficiently chaotic it should still act as a fast dynamo. k F k ν k

38 Rm >> Rmc B spectral window Eu Flow spectrum E B Additional diagnostic: During saturation if the small scale flow is sufficiently chaotic it should still act as a fast dynamo. Growth of a passive vector? (Cattaneo & Tobias 2009) k F k ν k

39 Passive vector Rm ~ Rmc Rm >> Rmc E U E U E B E B E B E B t t E C E C E C E C t t

40 Passive vector Rm ~ Rmc Rm >> Rmc E U E U E B E B E B E B t t E C E C E C E C t t

41 Passive vector Growthrates γ B γ C B threshold C threshold

42 Passive vector Growthrates γ B γ C O(Rm -1 ) B threshold C threshold

43 Passive vector Growthrates γ B γ C O(Rm -1 ) B threshold C threshold At threshold C saturates (as B) Far above threshold C grows exponentially

44 Mean-field interpretation Splitting U and B in large and small scales: and assume

45 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation

46 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation

47 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation

48 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation

49 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation Large scale dynamo mechanism

50 Mean-field interpretation Splitting U and B in large and small scales: and assume Induction equation Large scale dynamo mechanism Small scale dynamo mechanism

51 Mean-field interpretation To test this idea we solve the MHD problem without mean emf: Solve if the mechanism is large scale => dynamo is impossible if the mechanism is small scale => dynamo should be possible

52 Mean-field interpretation Rm ~ Rmc Rm >> Rmc E U E U E B E B

53 Mean-field interpretation Rm ~ Rmc Rm >> Rmc E U E U E B E B

54 Three dynamo thresholds: Without mean emf Passive vector Kinematic

55 Large scale magnetic fields do not require large scale dynamo mechanisms. They can appear from small scale dynamos, as a by-product.

56 Large scale magnetic fields do not require large scale dynamo mechanisms. They can appear from small scale dynamos, as a by-product. Thank you!