1. A theoretical plasma physicist’s take on Turbulence in the ISM: popular beliefs, some observational data, some speculations about their meaning, and some rigourous approaches Perm 7.09.06 Alexander Schekochihin (DAMTP, Cambridge) in collaborations with Steve Cowley, Alexey Iskakov & Jim McWilliams (UCLA) Bill Dorland & Tomo Tatsuno (Maryland) Greg Hammett (Princeton) Greg Howes & Eliot Quataert (Berkeley) Tarek Yousef & François Rincon (Cambridge) Torsten Enßlin & André Waelkens (MPA, Garching) Reprints/references on http://www.damtp.cam.ac.uk/user/as629 or ask me for a copy

2. Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209] The Great Power Law in the Sky k –5/3 Turbulence is stirred by supernovae at L ~ 100 pc Fluctuations of velocity and magnetic field are Alfvénic:

3. Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209] The Great Power Law in the Sky k –5/3 Turbulence is stirred by supernovae at L ~ 100 pc Fluctuations of velocity and magnetic field are Alfvénic: They have a Kolmogorov k –5/3 spectrum Density is a passive tracer: so it has the same spectrum

4. MHD Turbulence à la K41 Energy at scale l Cascade time (rate of transfer) Universality Alfvénic: Locality in scale space energy injected Kinetic energy k energy flux  energy dissipated Magnetic energy k –??

5. MHD Turbulence à la K41 energy injected Kinetic energy k energy flux  energy dissipated Magnetic energy k –?? Energy at scale l Cascade time (rate of transfer) Two time scales available: and, so MHD turbulence spectrum not fixed solely by dimensional analysis Universality Alfvénic: Locality in scale space

6. Goldreich-Sridhar Turbulence Energy at scale l Cascade time (rate of transfer) Strong interactions: (critical balance) [Goldreich & Sridhar 1995, ApJ 438, 763] energy injected Kinetic energy k energy flux  energy dissipated Magnetic energy k  –5/3 Universality Alfvénic: Locality in scale space

7. Goldreich-Sridhar Turbulence Energy at scale l Cascade time (rate of transfer) Strong interactions: (critical balance) energy injected Kinetic energy k energy flux  energy dissipated Magnetic energy k  –5/3 ANISOTROPIC! [Goldreich & Sridhar 1995, ApJ 438, 763] Universality Alfvénic: Locality in scale space

8. Goldreich-Sridhar Turbulence Energy at scale l Cascade time (rate of transfer) Strong interactions: (critical balance) GS95 energy injected Kinetic energy k energy flux  energy dissipated Magnetic energy k  –5/3 [Goldreich & Sridhar 1995, ApJ 438, 763] ANISOTROPIC! Universality Alfvénic: Locality in scale space

9. Anisotropy: It Is Really There Strong interactions: (critical balance) GS95 Simulations of MHD turbulence unambiguously demonstrate that it is anisotropic and are consistent with GS95 [Maron & Goldreich 2001, ApJ 554, 1175; Cho et al. 2002, ApJ 564, 291]

10. Anisotropy: It Is Really There Strong interactions: (critical balance) GS95 Observations of SW and ISM also show that turbulence there is anisotropic with, although it is difficult to check the GS95 scaling. In SW, it has recently been found that while as should be the case in GS95 [T. Horbury 2006, private communication]. Simulations of MHD turbulence unambiguously demonstrate that it is anisotropic and are consistent with GS95 [Maron & Goldreich 2001, ApJ 554, 1175; Cho et al. 2002, ApJ 564, 291]

11. Solar Wind: Alfvénic Turbulence Magnetic- and electric-field fluctuations in the solar wind at ~1 AU (19 Feb. 2002) [Bale et al. 2005, PRL 94, 215002] Alfvénic fluctuations k –5/3

12. ISM: Alfvénic Turbulence? Alfvénic fluctuations I have not seen a nice plot like this for the ISM… Bottle of port to anyone who can give me one!

13. So, It’s All Sorted Then? Magnetic- and electric-field fluctuations in the solar wind at ~1 AU (19 Feb. 2002) [Bale et al. 2005, PRL 94, 215002] k –5/3 Does all this mean we understand plasma turbulence in the sky? SEE PART II OF THIS TALK

14. What if there is no guide field? Clusters of galaxies Some parts of the ISM Strong guide field: Weak guide field: waves, random tangle,

15. Fluctuation Dynamo Stretching by random fluid motions: Exponential growth with Direction reversals at the resistive scale, k  ~ k  Field varies slowly along itself: k || ~ k flow Stretch/shear [AAS et al. 2002, PRE 65, 016305; AAS et al. 2004, ApJ 612, 276 Review: AAS & Cowley, astro-ph/0507686]

16. Fluctuation Dynamo: DNS (Pm >> 1) [AAS et al. 2004, ApJ 612, 276 Review: AAS & Cowley, astro-ph/0507686] Folded structure

17. Fluctuation Dynamo: DNS (Pm >> 1) [AAS et al. 2004, ApJ 612, 276 Review: AAS & Cowley, astro-ph/0507686] Folded structure

18. Dynamo: The Movie

20. Fluctuation Dynamo: Saturated State |u||u||B||B| Magnetic energy at resistive scales [AAS et al. 2004, ApJ 612, 276; Yousef, Rincon & AAS 2006, JFM, submitted Review: AAS & Cowley, astro-ph/0507686]

21. Folded Fields Observed in Clusters [AAS et al. 2004, ApJ 612, 276] A2256: polarised emission [Enßlin & Clarke 2005, AJ, submitted]

22. What Are the Saturated Spectra? [AAS et al. 2004, ApJ 612, 276] with prob. 1/2

23. What Are the Saturated Spectra? [Yousef, Rincon & AAS 2006, JFM submitted] with prob. 1/2

24. What Are the Saturated Spectra? with prob. 1/2 [AAS et al. 2004, ApJ 612, 276] k–1k–1

25. What Are the Saturated Spectra? with prob. 1/2 [AAS et al. 2004, ApJ 612, 276] k–?k–? This is probably too simplistic a model…

26. Saturated Spectra: DNS [AAS et al. 2004, ApJ 612, 276] NB: Velocity spectrum still has a negative exponent, possibly Kolmogorov (Alfvén waves can propagate along the folds)

27. Spectra Observed in Clusters Coma cluster: pressure fluctuations [Schuecker et al. 2004, A&A 426, 387] Core of Hydra A cluster: magnetic fields [Vogt & Enßlin 2005, A&A 434, 67] Outer scale of turbulence is roughly here Viscous scale is roughly here

28. ISM: Spiral Arms vs. Interarm Regions Structure functions of Faraday rotation measure in ISM [Haverkorn et al. 2006, ApJ 637, L33] Kolmogorov? Flat? INTERARMS: ARMS:

29. ISM: Two Types of Turbulence? Structure functions of Faraday rotation measure in ISM [Haverkorn et al. 2006, ApJ 637, L33] Strong guide field Weak guide field Alfvénic turbulence Saturated small-scale dynamo ARMS: INTERARMS: [AAS, Cowley & Dorland 2006, PPCF to be published Iskakov, Cowley & AAS 2006, in preparation]

30. ISM: Two Types of Turbulence? Strong guide field Weak guide field Alfvénic turbulence Saturated small-scale dynamo ARMS: INTERARMS: This is only a speculation: let us discuss it! Here are some points in favour: Turbulence in the arms is stronger? [Rohlfs & Kreitschmann 1987, A&A 178, 95] Stronger u rms gives stronger  B rms in arms Mean-field dynamo in the interarms is more efficient? [Shukurov & Sokoloff 1998, SGG 42, 391] Stronger B 0 in interarms Mean field pushed out of arms by turbulence diamagnetism? Stronger B 0 in interarms Stronger B 0 in interarms indeed observed? [in other galaxies: Beck 2006, astro-ph/0603531] Marijke’s estimates yesterday consistent with  B rms u rms in interarms

31. Now the Rigourous Bit… PART II THE PLASMA PHYSICS OF INTERSTELLAR TURBULENCE [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812 AAS, Cowley & Dorland 2006, PPCF to be published (preprint on www.damtp.cam.ac.uk/user/as629)]

32. Turbulence in Weakly Collisional Plasma KAW k –5/3 k –7/3 energy injected ion heating electron heating Observed spectra collisional (fluid) collisionless (kinetic) Alfvén waves: SW ISM IGM

33. Turbulence in Weakly Collisional Plasma KAW k –5/3 k –7/3 energy injected ion heating electron heating Observed spectra collisional (fluid) collisionless (kinetic) Alfvén waves: MUST USE KINETICS, NOT MHD! SW ISM IGM

34. Gyrokinetics: Ordering [Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209] Ordering based on anisotropy + critical balance applied to kinetic theory gives GK Critical balance as an ordering assumption: Small parameter: Finite Larmor radius: [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]

35. Gyrokinetics: Ordering Critical balance as an ordering assumption: Small parameter: Finite Larmor radius: Low frequency Ordering based on anisotropy + critical balance applied to kinetic theory gives GK [Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209] [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]

36. Gyrokinetics: Ordering Critical balance as an ordering assumption: Small parameter: Finite Larmor radius: Low frequency GK ORDERING: Ordering based on anisotropy + critical balance applied to kinetic theory gives GK [Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209] [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]

37. Gyrokinetics: Kinetics of Larmor Rings [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812] Particle dynamics can be averaged over the Larmor orbit and everything reduces to kinetics of Larmor rings centered at and interacting with the electromagnetic fluctuations. [Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209]

38. Gyrokinetics: Kinetics of Larmor Rings Particle dynamics can be averaged over the Larmor orbit and everything reduces to kinetics of Larmor rings centered at and interacting with the electromagnetic fluctuations. ++ Maxwell’s equations

39. Gyrokinetics: Kinetics of Larmor Rings Averaged gyrocentre drifts: E  B 0 drift  B drift motion along perturbed fieldline Averaged wave-ring interaction ++ Maxwell’s equations

40. Gyrokinetics Covers Everything KAW k –5/3 k –7/3 energy injected ion heating electron heating Observed spectra collisional (fluid) collisionless (kinetic) Alfvén waves: GYROKINETICS FLUID THEORY

41. Gyrokinetics: DNS Numerical simulations (gyrokinetics in 3+2D) are possible (piece-wise!) at the limit of currently available computing power using codes developed for fusion problems. Transatlantic project underway with Bill Dorland (Maryland) Greg Howes (Berkeley) Steve Cowley (UCLA) Tarek Yousef (Cambridge) Eliot Quataert (Berkeley) Greg Hammett (Princeton) … et al. Simulations using GS2 [picture courtesy Bill Dorland 2005]

42. Gyrokinetics: DNS Numerical simulations (gyrokinetics in 3+2D) are possible (piece-wise!) at the limit of currently available computing power using codes developed for fusion problems. Reduced fluid/kinetic/hybrid models necessary to understand and to simulate what happens in various parameter regimes. Simulations using GS2 [picture courtesy Bill Dorland 2005]

43. Kinetic Reduced MHD k –5/3 k –7/3 energy injected ion heating electron heating collisional (fluid) collisionless (kinetic) Alfvén waves: GYROKINETICS FLUID THEORY magnetised ions isothermal electrons

44. KRMHD: Alfvén Waves Alfvénic fluctuations and rigourously satisfy Reduced MHD Equations: [cf. Kadomtsev & Pogutse 1974, Sov. Phys. JETP 38, 283 Strauss 1976, Phys. Fluids 19, 134] [AAS, Cowley & Dorland 2006, PPCF to be published cf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]

45. KRMHD: Alfvén Waves Alfvénic fluctuations and rigourously satisfy Reduced MHD Equations: [cf. Kadomtsev & Pogutse 1974, Sov. Phys. JETP 38, 283 Strauss 1976, Phys. Fluids 19, 134] Alfvén-wave cascade is indifferent to collisions and damped only at the ion gyroscale The GS95 theory describes this part of the turbulence Alfvén waves are decoupled from density and magnetic-field-strength fluctuations (slow waves and entropy mode in the fluid limit) [AAS, Cowley & Dorland 2006, PPCF to be published cf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]

46. Alfvén-Wave Cascade in the Solar Wind Alfvénic fluctuations Magnetic- and electric-field fluctuations in the solar wind at ~1 AU (19 Feb. 2002) [Bale et al. 2005, PRL 94, 215002] k –5/3 KRMHD

47. KRMHD: Density and Field Strength Density and field strength require kinetic description [AAS, Cowley & Dorland 2006, PPCF to be published cf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]

48. KRMHD: Density and Field Strength Density and field strength require kinetic description They are passively mixed by Alfvén waves Equations are linear in the Lagrangian frame, so there is no refinement of fluctuation scale along the field by nonlinear interactions Therefore, despite collisional and collisionless (Landau) damping, this cascade is also undamped above the ion gyroscale [AAS, Cowley & Dorland 2006, PPCF to be published cf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]

49. Density and Field Strength in the Solar Wind [Bershadskii & Sreenivasan 2004, PRL 93, 064501] Spectrum of magnetic-field strength in the solar wind at ~1 AU (1998) Density fluctuations in the solar wind at ~1 AU (31 Aug. 1981) [Celnikier, Muschietti & Goldman1987, A&A 181, 138] k –5/3 FLR: density mode mixing with Alfvén waves

50. Density and Field Strength in the ISM Anyone knows anything? k –5/3 Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209]

51. Density and Field Strength in the ISM Anyone knows anything? k –5/3 Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209] Is this scaling correct?

52. Density and Field Strength in the ISM Anyone knows anything? k –1.46±0.20 Structure function from scintillation measurements [Smirnova, Gwinn & Shishov 2006, astro-ph/0603490] Is this scaling correct?

53. Ion Heating k –5/3 k –7/3 energy injected ion heating electron heating GYROKINETICS FLUID THEORY KRMHD GK ions (and isothermal electrons)

54. Electron Reduced MHD k –5/3 k –7/3 energy injected ion heating electron heating GYROKINETICS FLUID THEORY KRMHD GK ions Boltzmann ions magnetised electrons ERMHD

55. ERMHD: Kinetic Alfvén Waves KAW fluctuations and Critical balance + constant flux argument à la K41/GS95 give spectrum of magnetic field with anisotropy [Biskamp et al. 1999, Phys. Plasmas 6, 751; Cho & Lazarian 2004, ApJ 615, L41] This is a cascade of KAW, Electric field has spectrum: [AAS, Cowley & Dorland 2006, PPCF to be published; this is the anisotropic version of EMHD, see Kingsep et al. 1990, Rev. Plasma Phys. 16, 243]

56. KAW Cascade in the Solar Wind Alfvénic fluctuations Magnetic- and electric-field fluctuations in the solar wind at ~1 AU (19 Feb. 2002) [Bale et al. 2005, PRL 94, 215002] Ion Heating k –5/3 KRMHD ERMHD k –7/3 k –1/3 GK ions KAW

57. KAW Cascade in the ISM? Alfvénic fluctuations Ion Heating KAW Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209] Again, I have not seen any data. Should be there.

58. Conclusions It is too early for conclusions!

59. Observational Desiderata Scalings (spectra): Arms: Interarms: (Alfvénic) (passive) Measures of anisotropy: Compare (Alfvénic) Is the same true for (passive) Turbulence below ion gyroscale: Kinetic Alfvén waves What is the reversal scale of the folded fields?