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. Interstellar gas Orion nebula Astrophysical fluids are turbulent and magnetized

4. Magnetic fluctuations in the solar wind

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

6. We can do many things with the equations! B0 Example of incompressible MHD simulations But the equations should be modified when we deal with small scalesor relativistic cases… ==> Electron MHD (Topic 2) or Relativistic Force-Free MHD (Topic 3)

7. What do we want to understand?==> dynamics of wave packets 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)

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

9. One wave packet FFMHD 643 Nothing happens.

10. There is a good Chinese expression for this:

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

12. Two wave packets This is something we call turbulence

13. Energy spectrum is useful! Energy injection E(k) E(k) ~ k-5/3 for hydrodynamic turbulence dissipation Inertial range k

14. Energy spectrum is useful! Energy injection E(k) E(k) ~ k-5/3 for hydrodynamic turbulence dissipation Inertial range k What is spectrum for MHD turbulence?

15. Let’s consider ordinary incompressible Alfvenic MHD first. Goldreich & Sridhar (1995) considered dynamics of Alfvenic wave packets. l|| l^ B0 VA VA B0

16. Let’s consider non-relativistic incompressible Alfvenic MHD first. Goldreich & Sridhar (1995) considered dynamics of Alfvenic wave packets. energy~b2/2 l|| l^ B0 VA VA =B0 *From now on, B = actually B/(4pr)1/2

17. 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

18. 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!

19. 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

20. l b Energy Cascade b2/tcas = constant

21. 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

22. Numerical test: Cho & Vishniac (2000) B

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

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

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

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

27. 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)

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

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

30. J  v + 0 Electron MHD eq v B

31. 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

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

33. 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

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

35. Illustration of anisotropy This is only for illustration.

36. Numerical Results: anisotropy

37. Numerical Results: critical balance

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

39. 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!

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

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

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

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

44. Results: eddy shapes Scale-dependent anisotropy

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

46. 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)

47. 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

48. W=B0z +v-b W=B0z +v+b tW+B0zW=-W W - P tW -B0zW=-W W - P Why? tv=-vv + Bb - P tb=-vb + bv