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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)

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Perm 7.09.06

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  1. 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) Perm 7.09.06 A theoretical plasma physicist’s take onTurbulence in the ISM:popular beliefs, some observational data,some speculations about their meaning,and some rigourous approaches Reprints/references on http://www.damtp.cam.ac.uk/user/as629 or ask me for a copy

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

  3. The Great Power Law in the Sky • 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 k–5/3 Electron-density fluctuations in the interstellar medium [Armstrong et al. 1995, ApJ 443, 209]

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

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

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

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

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

  9. Strong interactions: • (critical balance) GS95 Anisotropy: It Is Really There • Simulations of MHD turbulence unambiguously demonstrate • that it is anisotropic and are consistent with GS95 • [Maron & Goldreich 2001, ApJ554, 1175; Cho et al. 2002, ApJ 564, 291]

  10. 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]. • Strong interactions: • (critical balance) GS95 Anisotropy: It Is Really There • Simulations of MHD turbulence unambiguously demonstrate • that it is anisotropic and are consistent with GS95 • [Maron & Goldreich 2001, ApJ554, 1175; Cho et al. 2002, ApJ 564, 291]

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

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

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

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

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

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

  18. Dynamo: The Movie

  19. Fluctuation Dynamo: Saturated State |u| |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]

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

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

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

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

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

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

  26. 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

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

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

  29. ISM: Two Types of Turbulence? INTERARMS: This is only a speculation: let us discuss it! Here are some points in favour: Strong guide field • Turbulence in the arms is stronger? • [Rohlfs & Kreitschmann 1987, A&A 178, 95] • Stronger urms gives stronger Brms in arms • Mean-field dynamo in the interarms • is more efficient? • [Shukurov & Sokoloff 1998, SGG 42, 391] • Stronger B0 in interarms • Mean field pushed out of arms by • turbulence diamagnetism? • Stronger B0 in interarms • Stronger B0 in interarms indeed observed? • [in other galaxies: Beck 2006, astro-ph/0603531] • Marijke’s estimates yesterday consistent with • Brms < urms in arms, Brms > urms in interarms Alfvénic turbulence ARMS: Weak guide field Saturated small-scale dynamo

  30. 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)]

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

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

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

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

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

  36. 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. Gyrokinetics: Kinetics of Larmor Rings [Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209] [Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]

  37. 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. Gyrokinetics: Kinetics of Larmor Rings ++ Maxwell’s equations

  38. Gyrokinetics: Kinetics of Larmor Rings • Averaged gyrocentre drifts: • EB0drift • B drift • motion along • perturbed fieldline Averaged wave-ring interaction ++ Maxwell’s equations

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

  40. 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]

  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. 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]

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

  43. 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]

  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] • 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]

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

  46. 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]

  47. 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]

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

  49. 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]

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