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Introduction to Magnetized Turbulence in Astrophysical F luids

Introduction to Magnetized Turbulence in Astrophysical F luids. Jungyeon Cho ( Chungnam National Univ., Korea). Plan . Weak B 0 case Strong B 0 case. MHD. * Small-scale turbulence ?. ( V 2 /L) / ( n V/L 2 ) . +. = -. V 2 /L n V/L 2. What is turbulence?.

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Introduction to Magnetized Turbulence in Astrophysical F luids

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  1. Introduction toMagnetized Turbulence in Astrophysical Fluids Jungyeon Cho (Chungnam National Univ., Korea)

  2. Plan Weak B0 case Strong B0case MHD • *Small-scale turbulence?

  3. (V2/L) / (nV/L2) + = - V2/L nV/L2 What is turbulence? • Reynolds number: Re=VL/n • When Re << Recritical, flow = laminar When Re >> Recritical, flow = turbulent

  4. Example of turbulence: Turbulence created by an alligator Photo taken during the boat trip nwater ~ 0.01 (cgs) V=50 cm/s & L=100 cm  Re~5x105!

  5. Re ~ 40 Re ~ 104 Onset of turbulence Re < 1 www-pgss.mcs.cmu.edu

  6. Turbulence = Seddies! Energy cascade -Da Vinci’s view -Richardson (1920’s): concept of eddy and energy cascade Big whorls have little whorls / That feed on their velocity And little whorls have lesser whorls / And so on to viscosity ... a Flea Hath smaller Fleas that on him prey, And these have smaller Fleas to bite 'em, And so proceed ad infinitum. [1733 Swift]

  7. Vl2 Vl3 = const = const, Vl ~ l1/3 tcas l tcas= l/V Or, E(k)~k-5/3 l v Kolmogorov theory: incompressible hydrodynamic turbulence

  8. Measured spectrum (on the Earth) Energy injection E(k) ~ k-5/3 dissipation Inertial range

  9. Turbulence is everywhere! ( Re is huge!) Intracluster medium Turbulence! Interstellar medium 3C 465 -- Abell 2634 The solar wind

  10. We also observe power-law spectra: e.g.) electron density spectrum in the ISM e.g.) Magnetic spectrum in the solar wind Slope ~ -5/3 Slope = -5/3 Spacecraft-frame frequency (Hz) Leamon+ (1999) pc AU Armstrong, Rickett& Spangler (1995)

  11. Topic 1. Amplification of B fields in turbulence - How can MHD turbulence amplify B fields? Weak seed field (B0)

  12. B0 Fluid elements and field lines move together *Back reactions are negligible if Emag<Ekin Stretching of field lines t=0:

  13. Small-scale structures change faster

  14. Expectations: Stretching on the dissipation scale will occur first because eddy turnover time is shortest there E(k) B k Exponential growth (Batchelor50; Kazantsev 67; Zel’dovich+84; …)

  15. Earlier simulations confirmed this Meneguzzi et al. 1981 (Resolution = 643)

  16. Expectations: Eturb(k) E(k) What will happen when Eturb ~ Emag on the dissipation scale? k  Exponential growth stage will end!  Stretching scale gradually moves to larger scales. (see, for example, Cho & Vishniac 2000)

  17. Efficiency of stretching Magnetic spectra Dissipation scale Cho & Vishniac (2000a)

  18. Cho & Vishniac (2000a)  Saturation is reached when B2 ~ V2 Schekochihin+(2007) later showed that the growth rate of the 2nd stage is linear. B2 time

  19. Results of simulations linear exponential Cho, Vishniac, Beresnyak, Lazarian, Ryu (2009); * See also Cho & Vishniac (2000)

  20. linear growth exponential growth Cho et al. (2009)

  21. Conclusions for Topic 1 -Turbulence can amplify weak seed B fields -Two stages of amplification: exp. and linear E(k) B2 time k

  22. Plan -Weak B0 case -Strong B0case *Small-scale turbulence?

  23. Topic 2: Strong B0 case Alfven wave Suppose that we perturb magnetic field lines. We will only consider Alfvenic perturbations. (restoring force=tension) We can make the wave packet move in one direction. (We need to specify velocity)

  24. Dynamics of one wave packet Suppose that this packet is moving to the right. What will happen? VA: Alfven speed =

  25. One wave packet 643 Nothing happens.

  26. Dynamics of two opposite-traveling wave packets Now we have two colliding wave packets. What will happen?

  27. Two wave packets This is something we call turbulence

  28. What happens? What happens when two Alfvenic wave packets collide? l|| l^ B0 VA VA =B0

  29. Goldreich & Sridhar (1995): In strong turbulence, 1 collision is enough to complete cascade!

  30. 1 collision is enough to complete cascade! -Distortion time scale ~ l^/vl • -Duration of collision ~ l|| /B0 tw/teddy~ (l|| /B0) /(l^/v)~(b l|| / l^B0) ~1

  31. l bl Energy Cascade bl2/tcas= constant

  32. bl2 = const (l^/bl) bl2 bl~l^1/3 Or, E(k)~k-5/3 = const l^l|| tcas = blB0 Goldreich-Sridhar model (1995) • Critical balance • Constancy of energy cascade rate l|| ~l^2/3

  33. Numerical test: Cho & Vishniac (2000b) B -pseudo-spectral method -2563

  34. |B| B0 Spectra: See also Muller & Biskamp (2000); Maron & Goldreich (2001)

  35. B Anisotropy Smaller eddies are more elongated => Relation between parallel size and perp size?

  36. Anisotropy: Cho & Vishniac (2000) * Maron & Goldreich (2001) also obtained a similar result

  37. Summary for strong B0 case (i.e. Scaling relations for Alfvenic MHD turbulence) • Theory: Goldreich & Sridhar (1995) • Numerical test: Cho & Vishniac (2000) Maron & Goldreich (2001) • Spectrum = Kolmogorov • But, structures are anisotropic. *Recent issues: 1.Spectrum: Mueller+03; Boldyrev 05; Beresnyak& Lazarian 06; Mason+ 06; Gogoberidze07; Matthaeus+08; Cho 10, … 2.Imbalance: Lithwick+ 07; Beresnyak& Lazarian08; Chandran 08; Perez & Boldyrev09; Podesta& Bhattacharjee 09, …

  38. Actually strong turbulence is very common…(For simplicity, let’s suppose that driving is isotropic.) b>>B0 (b l|| / l^B0) =(bk^/k||B0)~1 b<<B0

  39. Critical balance may be a very common state in strongly magnetized plasmas… -Relativistic force-free MHD turbulence (in magnetospheres of BHs or NSs) * Thompson & Bleas (1999): theory Cho (2005): numerical test -EMHD model for small-scale MHD turbulence * Cho & Lazarian (2004, 2009) crust Neutron star

  40. Conclusion for MHD turbulence (i.e. large-scale magnetized turbulence) -Turbulence can efficiently amplify weak seed fields -Alfvenic MHD turbulence : Kolmogorov spectrum + anisotropy

  41. Small-scale turbulence: spectrum=? Spectrum of magnetic fluctuations in the solar wind Leamonet al (1999)

  42. How can we describe small-scale physics?EMHD B B Protons  smooth background Electrons carry current  J  v

  43. J  v + 0 Electron MHD eq v B

  44. Ordinary MHD vs. EMHD turbulence incompressible • Studied since 1990’s • Energy spectrum: • E(k)  k-7/3 • (Vainshtein 1973; • Biskamp-Drake 1990’s) • -Anisotropy: • k||  k^1/3 • (Cho & Lazarian2004) • -Studied since 1960’s • Goldreich & Sridhar 1995 • E(k)  k-5/3 • k||  k^2/3 • Numerical test: • Cho & Vishniac 2000

  45. Conclusion for small-scale turbulence -Small scale turbulence k-7/3 spectrum + stronger anisotropy

  46. Small-scale turbulence: spectrum? k-5/3 E(k) k-7/3 Alfvenic turbulence No more turbulence ~ lmfp ~re k Biskamp’sgroup (1990’s), Cho & Lazarian (04, 09) ~ri

  47. Small-scale turbulence: spectrum=? 2883 Biskamp & Drake’s group obtained this in late 90’s. Cho & Lazarian (2004, 2009)

  48. Small-scale turbulence: anisotropy=? Cho & Lazarian (2009; see also 2004)

  49. Gyro-kinetic Simulations (Howes et al 2008) ri

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