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Plasma Dynamos

Plasma Dynamos. UCLA January 5th 2009 Steve Cowley, UKAEA Culham and Imperial Thanks to Alex Schekochihin, Russell Kulsrud, Greg Hammett and Mark Rosin. After re-ionization the universe was probably a reasonably collisionless turbulent high  plasma.

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Plasma Dynamos

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  1. Plasma Dynamos UCLA January 5th 2009 Steve Cowley, UKAEA Culham and Imperial Thanks to Alex Schekochihin, Russell Kulsrud, Greg Hammett and Mark Rosin.

  2. After re-ionization the universe was probably a reasonably collisionless turbulent high  plasma. Many large scale plasmas are quite collisionless. I will argue that (random) magnetic fields grow rapidly in such a plasma. I will also argue that we need to know a lot more about the small scale dynamics of high  plasmas. We need an experiment at >> 1! Early magnetic fields -- what, when and how.

  3. Cluster Turbulence • Mergers • AGNs • Wakes L ~ 102…103 kpc U ~ 102…103 km/s (subsonic) L/U ~ 108…109yr The Coma Cluster: pressure map [Schuecker et al. 2004, A&A 426, 387]

  4. Cluster Turbulence • Mergers • AGNs • Wakes L ~ 102…103 kpc U ~ 102…103 km/s (subsonic) L/U ~ 108…109yr mfp ~ 0.1…10 kpc Re ~ 1…102 Note: it is not obvious that there is turbulence! [A. Fabian 2003, MNRAS 344, L48] The Coma Cluster [Schuecker et al. 2004, A&A 426, 387]

  5. Cluster Magnetic Fields 900 kpc Abell 400 cluster [Eilek & Owen 2002, ApJ567, 202]

  6. Cluster MHD Turbulence TURBULENCE Coma cluster [Schuecker et al. 2004, A&A 426, 387] MAGNETIC FIELDS Hydra A Cluster [Vogt & Enßlin 2005, A&A 434, 67] Turbulence scale is around here • Magnetic Reynolds #, Rm ~ 1029.

  7. 8 t = 10 years  The Large Prandtl Number Case: Galaxies, Clusters etc. • Magnetic Prandtl number = Pr =  /. • On the turnover time of the viscous eddies the “seed field” grows. The field develops structure below the viscous scale down to the resistive scale l= Pr -1/2 l l 10 -30kpc =  Viscous scale Viscous eddy Turnover. l 

  8. Isotropic Homogeneous Dynamo Folded Structure at Resistive Scale Grayscale is |B|. Scalar Viscosity

  9. Plasma not Fluid

  10. Magnetized Viscosity --Anisotropic Pressure DEFINITION OF PRESSURE TENSOR. Anisotropic pressure tensor in magnetized plasma. Because of fast motion around the field the tensor must be of the form:

  11. Magnetized Viscosity. B Collisionless particle motion restricted to being close to field line and conserving . Collisionless. Relaxed by Collisions. P Compressing Field

  12. Incompressible Braginskii MHD. Collisional limit Unit vector along B Coefficients worked out by Braginskii Reviews of Plasma Physics Vol. 2.

  13. Equilibrium -- Decreasing B. B0 B0 V0 V0 Stretching rate

  14. Firehose Instability. Look at instabilities that are smaller scale than the field and growing faster than the stretching rate. Treat B0 as quasi-constant during the growth. We take perturbed velocity to go as: The condition that the growth rate is faster than stretching rate is:

  15. Firehose Instability. Linearized: The x component becomes: Perturbed field line Curvature.

  16. Firehose Instability. Putting this into force equation we get. Alfven wave when no anisotropy

  17. More Firehose. Unstable when Growth rate at negligible B Parallel pressure forces squeeze tube out. Tighter bend grows faster. Rosenbluth 1956 Southwood and Kivelson 1993 P|| P||

  18. So What!? -- Nonlinear Firehose. Schekochihin et. al. Phys. Rev. Lett. Nonlinear kinetic theory gives: Rate of change of B2averaged along B. Instability tries to keep average B constant by bending the field. Diverging Flow. Makes finite wiggles

  19. Nonlinear Mirror Mode. When the field increasing the plasma is unstable to the mirror Mode which creates little traps in the plasma. Converging

  20. Stretching and compressing Stretched at the turnover rate of the viscous eddies. Using Braginskii’s Expression we get P||-P ~ Re-1/2P ~ P/6 Field increasing P||< P Mirror mode Unstable. Field decreasing P||>P Firehose Unstable.

  21. Scales  EV EB ? k Ion Larmor Radius Scale @ B = 1G ~105km. Mean-Free Path. mfp ~ l0Re-1~1-10kpc Viscous Scale l ~ l0Re-3/4~ 10 - 30 kpc l0~ 1-3Mpc Resistive Scale ~ l0Rm-1/2 ~ 104km l0 /u0~109 years l/ u~108 years Maximum growth rate

  22. Scales EV EB ? ? ? k Ion Larmor Radius Scale @ B = 1G ~105km. Mean-Free Path. mfp ~ l0Re-1~1-10kpc Viscous Scale l ~ l0Re-3/4~ 10 - 30 kpc l0~ 1-3Mpc Resistive Scale ~ l0Rm-1/2 ~ 104km l0 /u0~109 years l/ u~108 years

  23. What does small scale field do? Sharma, Hammett, Schekochihin, Kulsrud etc. Enhanced particle scattering? Effective collisions increase --i? If so viscosity decreases -- Re gets large and turbulence has faster motions. Dynamo Growth Time:  ~ 0( i/L)(1/2) ~ 1000 years! MAGNETIC FIELD CAN GROW ON TRIVIAL TIMESCALES.

  24. Small scale fast growing instabilities to be expected in weak field magnetized fully ionized plasmas. Make finite wiggles on the scale almost of the ion larmor radius. May enhance collisions, dissipation and change the transport properties. Conclusions.

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