Physics of Alfvenic MHD Turbulence

1. Physics of Alfvenic MHD Turbulence Jungyeon Cho Chungnam National Univ., Korea

2. 들어 가기에 앞서… 1.Heisenberg가 죽기 전, 신을 만나면 다음 두가지 질문을 하겠다고 했다 한다: Why relativity and why turbulence? 2. Feynman said … “turbulence is the last great unsolved problem of classical physics.”

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

4. Example: wake behind a sphere critical Re = 40~50 Re~15,000

5. Example: cylinder in water nwater ~ 0.01 (cgs) If v=10cm/sec & D=1 cm, Re~1000. ==> turbulence!

6. Measured spectrum E(k) ~ k-5/3

7. Interstellar gas Orion nebula Astrophysical fluids are turbulent and magnetized (Re > 1010)

8. Solar Surface Doppler image SOHO/MDI

9. Magnetic fluctuations in the solar wind

10. Plan Ordinary MHD turbulence Electron MHD turbulence = small scale turbulence 3. Relativistic force-free MHD turbulence <= extremely strong B field

11. Topic 1: ordinary MHD turbulence To study magnetized turbulence, we use the ordinary MHD equations: Orion nebula

12. Topic 1. Ordinary MHD turbulence magnetic field Suppose that we perturb magnetic field lines. We will only consider Alfvenic perturbations. (restoring force=tension) We can make the wave packet move to one direction. (We need to specify velocity)

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

14. One wave packet FFMHD 643 Nothing happens.

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

16. Two wave packets This is something we call turbulence

17. What happens? What happens when two Alfvenic wave packets collide? l|| l^ B0 VA VA B0

18. What happens? What happens when two Alfvenic wave packets collide? energy~b2/2 l|| l^ B0 VA VA =B0 *From now on, B = actually B/(4pr)1/2

19. NOTE: => db/dt ~ b2/l^ => dE/dt ~ b3/l^ • When they collide, a packet loses energy of • DE~(dE/dt)Dt~ (b3/ l^ )tcoll ~ (b3/ l^ )(l||/VA). • Therefore DE /E ~ (b3/ l^ )(l||/VA) / b2 • = (b l|| / l^B0) • = (l||/B0)/( l^ /b ) • = tw/teddy = c

20. c ~tw/teddy ~ (b l|| / l^B0) ~ DE /E • Suppose that c ~1 . e.g.) When B0~bl and l|| ~ l^ , we have c ~1. =>1 collision is enough to complete cascade!

21. c~1 E(k) c~1 k • c ~tw/teddy ~ (b l|| / l^B0) ~ DE /E • Goldreich & Sridhar (1995) found that, when c ~1 on a scale, c ~1 on all smaller scales. * c ~1 is called critical balance *This regime is called strong turbulence regime

22. l b Energy Cascade b2/tcas = constant

23. b^l2 = const (l^/b^l) b^l2 = const l^l|| tcas = b^lB0 Goldreich-Sridhar model (1995) • Critical balance • Constancy of energy cascade rate b^~l^1/3 Or, E(k)~k-5/3 l|| ~l^2/3 back

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

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

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

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

28. Summary for ordinary MHD • Spectrum: E(k)~k-5/3 • Anisotropy: l|| ~l^2/3 • Theory: Goldreich-Sridhar (1995) Numerical test: Cho-Vishniac (2000)

29. So far, we have considered ordinary MHD B0 Example of incompressible MHD simulations What about small scales? (earth magnetosphere, crust of neutron stars, ADAFs, or any small scales)

30. B What do I mean by small scales? Scales smaller than rg crust of neutron star Protons=background; only electrons move

31. Topic 2: EMHD - Introduction B B Protons => smooth background Electrons carry current => J  v

32. J  v + 0 Electron MHD eq v B

33. Ordinary MHD vs. EMHD turbulence incompressible -Studied since 1960’s -spectrum: Kolmogorov -scale-dependent anisotropy • Studied since 1990’s • Energy spectrum: known • Biskamp-Drake group: • E(k)  k-7/3 • -Anisotropy: not known

34. Scaling of EMHD turbulence Consider two EMHD wave packets: l|| l^ B0 Vw Vw  kB0

35. b^l2 = const (l^2/b^l) b^l2 = const l^2l^l|| tcas = b^lB0 Cho & Lazarian (2004) • Critical balance (teddy =tW) • Constancy of energy cascade rate b^~l^2/3 Or, E(k)~k-7/3 l|| ~l^1/3 Cf. Ordinary MHD

36. Numerical Results: spectrum 2883 Biskamp & Drake’s group obtained a k-7/3 spectrum in late 90’s.

37. Illustration of anisotropy This is only for illustration.

38. Numerical Results: anisotropy

39. Numerical Results: critical balance

40. Summary for EMHD • Spectrum: E(k)~k-7/3 • Anisotropy: l|| ~l^1/3 • Critical balance: c ~ 1 • Theory & test: Cho & Lazarian (2004)

41. Topic3: Relativistic Force Free MHD • Force-free ( B2 >> rc2 => reE+B x J=0 ) e.g) magnetospheres of NS, BH, … • Theory: Thompson & Blaes (1998) * c=1, flat space-time Conserved form!

42. Scaling of Relativistic FF-MHD turbulence Consider two wave packets: l|| l^ B0 Vw Vw =c =1

43. Simulation -5123 -MUSCL type scheme with HLL flux -Constrained transport scheme for div B=0 (Toth 2000)

44. E(k) k 4 6 t=0 t > 0

45. spectrum anisotropy Results: Relativistic MHD ~ classical MHD ! Cho (2005)

46. Results: eddy shapes Scale-dependent anisotropy

47. Results: critical balance • ~ DE /E ~ tw/teddy

48. Summary for Relativistic FFMHD • Kolmogorov spectrum: E(k) ~ k-5/3 • Scale-dependent anisotropy: l|| ~ l^2/3 • Theory: Thompson & Blaes (1998) Numerical test: Cho (2005)

49. Summary • We have considered 3 types of Alfvenic turbulence: • - ordinary MHD turbulence • - electron MHD turbulence • - relativistic force-free MHD turbulence • They all show anisotropy and critical balance