Alfvén wave turbulence: new results with applications to astrophysics Sébastien GALTIER Université Paris-Sud & Institut Universitaire de France 1
Recent co-workers : - Barbara Bigot (France/USA) - Ben Chandran (USA) - Sergey Nazarenko (UK) - Hélène Politano (France) 2
The Sun as a turbulent laboratory 60 000 K EIT/SoHO (30.4nm) TRACE (17.1nm) δx = 350 km EIT/SoHO (17.1nm) 106 K 3 δx = 1850 km
The solar wind as a turbulent laboratory - Continual and variable outflow from the Sun (heliosphere ~ 100AU) - Fast and slow winds (> 20R SUN ) Inner solar wind 4
The solar wind as a turbulent laboratory Taylor hypothesis : k 2π f / V SW E b (k) ~ k α Pdf(α) E b (f) ~ 1AU α [Smith aet al., 2006] b Pdf(α) WIND, Helios [Salem, 2000] 5
MHD Models for Astrophysical Turbulence 6
Magnetohydrodynamics Basic equations Equations for energy Standard Ohm s law 7
Regimes of turbulence B = Bo + εb Wave turbulence B = b Isotropic turbulence Real world : Anisotropic turbulence 8
Concept in MHD turbulence The main difference between neutral fluids and MHD is the presence of Alfvén waves. Wavepackets interact nonlinearly on a typical time: z z + B 0 Sporadic collisions between wavepackets : Nonlinear transfers slower in MHD than in HD! 9
Which energy spectrum? Homogeneous, isotropic medium [Iroshnikov, 1964; Kraichnan, 1965] 1) Very close to the Kolmogorov prediction 2) Heuristic result not compatible with the 4/3 s exact law! Homogeneous, isotropic medium, at Re + r [Kolmogorov, 1941] => E (k) ~ k -5/3 10
Which energy spectrum? Homogeneous, isotropic medium [Iroshnikov, 1964; Kraichnan, 1965] 1) Very close to the Kolmogorov prediction 2) Heuristic result not compatible with the 4/3 s exact law! Homogeneous, isotropic medium, at Re, Rm + r [Politano & Pouquet, 1998] => E (k) ~ k -5/3 11
Solar wind energy spectrum Taylor hypothesis : k 2π f / V SW => E (k) ~ k α Pdf(α) ACE f < 1Hz [Smith et al., 2006] Iroshnikov Kraichnan or Kolmogorov? or something else? α 12
4/3 s exact law and the solar wind [Sorriso-Valvo et al., 2007] Only one decade We NEED an anisotropic 4/3s law! 13
4/3 s exact law and the solar wind [Sorriso-Valvo et al., 2007]? Extension to Hall MHD : It gives a theoretical background to the heuristic prediction: E(k) ~ k -7/3 [Biskamp et al., 1996] [Galtier, 2008] 14
Alfvén wave turbulence Wave turbulence in MHD : E(k,k // ) ~ k 2 g(k // ) [Galtier et al., 2000] only valid for k >> k // no transfer along B 0 Wave turbulence Strong turbulence (k // =0) Wave turbulence 15
Alfvén wave turbulence Other exact solutions are possible with a power law forcing: [Galtier & Nazarenko, 2008] Wave turbulence predictions may be recovered with filtered MHD equations [Galtier & Chandran, 2006] First evidence of k 2 spectrum [Perez & Boldyrev, 2008] Wave turbulence in electron and Hall MHD [Galtier, 2006] Spectral steepening at small scales like in the solar wind 16
Recent results Bigot, Galtier & Politano (2008ab) : - DNS in 3D, 512 2 x64, ν=η, B 0 30, <v 2 +b 2 >~1 - Pseudo-spectral code - Freely decaying turbulence - Isotropic initial condition at large-scales 17
Energy decay laws Wave turbulence code AA -2/3 18
Alfvén ratio B 0 =0 : E u /E b ~0.5 (like in the solar wind at 1AU) B 0 >0 : E u /E b ~0.8 19
Spectral Alfvén ratio E u (k // ) / E b (k // ) We recover the equipartition between kinetic and magnetic energies at k // >0 Coexistence of wave and strong turbulence 20
Critical balance: χ = τ A / τ nl = constant? τ A = L // / B 0 ; τ nl = L / z rms χ ~ 1 : k // ~ k 2/3 E(k ) ~ k 5/3 [Goldreich & Sridhar, 1995] χ χ <1 : k // ~ k 2/3 / B 0 E(k,k // ) ~ k α k // β, 3α+2β=7 [Galtier, Mangeney & Pouquet, 2005] We NEED an anisotropic 4/5s law! 21
Anisotropic MHD turbulence vorticity w and current j filaments! Δλ>δx A ~ 35% of wmax and jmax ~ 25% of wmax and jmax 22
Implication for astrophysics First evidence of a condensation into filaments of (current) sheets Does it explain coronal loops on the Sun? Classical turbulent MHD models : Real world? : 23
2D energy spectra Isotropic energy transfer B 0 Anisotropic energy transfer 24
Energy spectra interpretation Wave turbulence Strong turbulence Wave turbulence Total decoupling of planes in the wave turbulence dynamics In real life, only partial decoupling of planes We must plot E(k,k // ) for different k // ; and not E(k ) or E(k // ) 25
Energy spectra (B 0 =15) E(k,0) ~ k 3/2 E(k, k // =1) ~ k 7/3 or k 2 Shear-Alfvén wave Shear-Alfvén wave Pseudo-Alfvén wave Shear-Alfvén wave 26
A unified vision (z + ~ z - ) ENERGY SPECTRA COMMENTS STRUCTURES 3D B 0 = 0 E(k) ~ k 5/3 In agreement with Current and vorticity sheets with rolling up [Mininni et al., 2006] 3D B 0 ~ δb 3D B 0 >> δb 5/3 E(k ) ~ k 3/2 E(k,0) ~ k 2 E(k,k // >0) ~ k Critical balance : τ A ~ τ nl Exact solutions of axisymmetric MHD turbulence! [Galtier, 2009] Current and vorticity sheets parallel to B 0 Sheets may condensate into filaments parallel to B 0 2D B 0 = 0 E(k) ~ k 3/2 Iroshnikov and Kraichnan may still be right Current and vorticity «sheets» 27