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Intermittency of MHD Turbulence

Intermittency of MHD Turbulence. A. Lazarian UW-Madison: Astronomy and Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas Special thanks to: A. Beresnyak (UW-Madison) A. Esquivel (UW-Madison) G. Kowal (Kracow, Poland) J. Cho (Chungnam, Korea)

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Intermittency of MHD Turbulence

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  1. Intermittency of MHD Turbulence A. Lazarian UW-Madison: Astronomy and Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas Special thanks to: A. Beresnyak (UW-Madison) A. Esquivel (UW-Madison) G. Kowal (Kracow, Poland) J. Cho (Chungnam, Korea) E. Vishniac (Johns Hopkins)

  2. Da Vinci’s view Chaotic Order! Vortices inside flow Turbulence = S eddies !

  3. Experimental insight Reynolds number Re = VL/n Eddies inside eddies Re ~ 15,000 Stochasticity depends on

  4. Astrophysical relevance Re ~VL/n ~1010 >> 1 n ~ rLvth, vth < V, rL<< L

  5. Is dissipation smooth? • Kolmogorov theory-- yes it is smooth. • Laboratory data shows intermittency. • She & Leveque 95 proposed scaling for hydro turbulence. • Politano & Pouquet 95 proposed scaling for MHD turbulence.

  6. Why do we care? • Intermittent dissipation changes interstellar heating, allows funny chemistry as discussed for years by Falgarone’s group. • Exciting effects for different astro problems. • Gives insights into the very nature of turbulent cascade and its evolution.

  7. She-Leveque and Politano-Pouquet models Scaling No intermittency Kolmogorov model Filaments She-Leveque model Above is hydro. What about MHD? General: Politano-Pouquet model for where tcas~lx, zl~l1/g, C =3- (dimension of dissipation structure) For IK theory g=4, x=1/2, C=1 for sheet-like dissipation structures But does not account for anisotropy!

  8. Scale-dependent Anisotropy B0 Cho, Lazarian & Vishniac 03 Magnetic field B0 B0

  9. Confusing results • Pioneering study by Muller & Biscamp 00 got C=3-2=1 for z in incompressible MHD • Cho, Lazarian & Vishniac 02 got C=2 for velocity in incompressible MHD accounting for anisotropy • Boldyrev 02 assumed C=1 and Padoan et al. 03 got C=1 for velocity in supersonic compressible MHD and C=2 in subsonic case

  10. Scaling in system of local B Local system of reference is related to local magnetic field Cho, Lazarian & Vishniac 02 In local system of reference Alfvenic turbulence exhibits C=1 for velocities, equivalent to She-Leveque

  11. Scalings of velocity and magnetic field Global system of reference Local system of reference MA~0.7 Incompressible Cho, Lazarian & Vishniac 03 Scaling is different for V and B!!! Scaling is different for local and global reference system. Scaling of z in global system corresponds to MB 00 Scaling of v in local system corresponds to CLV 02

  12. Compressible and incompressible MHD Compressible simulations for Mach ~ 0.7, mean B~0 Cho, Lazarian & Vishniac 03 Elsasser variables Z scale closer to MB, while velocities indeed show C=2 in accordance with Padoan et al. 03. However it is clear that MHD turbulence is more complex than hydro. Caution is needed!

  13. B B k B k MHD modes (for Pmag > Pgas) Alfven mode (v=VA cosq) incompressible; restoring force=mag. tension slow mode (v=cs cosq) restoring force = Pgas fast mode (v=VA) restoring force = Pmag + Pgas Theoretical discussion in Lithwick & Goldreich 01 Cho & Lazarian 02

  14. Basis Decomposition of MHD Turbulence into Modes Cho & Lazarian 02 • Decomposition over basis in Fourier space: xs ~ [(1-D1/2+b/2)/(1+D1/2-b/2)](k^/k||)2 k|| + k^xf ~ k|| + [(1-D1/2-b/2)/(1+D1/2+b/2)](k||/k^)2 k^xA ~ k|| x k^ *D=(1+b/2)2-2b cosq

  15. Generation of compressible components by Alfven modes is marginal. Fast decay of MHD turbulence is not due to compressibility!!! How Good is Mode Coupling? From Cho & Lazarian 02, 03

  16. Generation of Compressible Mode Normalized Compres energy • Generalize scaling of compressive mode generation from hydro (Zank &Matthaeus93). • For MHD total Mach number is appropriate. • Energy diffuses from GS95 cone predicted (M2totalVA/dV)-1 X Cho & Lazarian 02 M2totalVA/dV Predicted scaling for Mtotal<1 is

  17. Scaling relations Alfven ~k-5/3 slow ~k-5/3 fast ~k-3/2 Spectra isotropic anisotropic (GS) anisotropic (GS) Correlation functions M=2 Magnetically dominated Cho & Lazarian 02

  18. How good is our decomposition? Decomposition: dashed lines Anistoropy obtained without decomposition M=2.3 M=7 Cho & Lazarian 03 • Our decomposition into modes is statistical • Testing of it for slow modes is successful For low beta plasma velocity of slow modes are nearly parallel to the local magnetic field. Therefore correlation functions calculated in the local reference frame can be used.

  19. Intermittency Alfven, slow and fast modes: M<1 and M>>1 Alfven slow fast M~0.7 MA~0.7 2563 Alfven fast Alfven slow M~7 MA~0.7 Alfven is pretty much the same, Slow is affected; fast is unclear Kowal & Lazarian 05

  20. Local Frame Results M~0.7 M~7 Kowal & Lazarian 05

  21. Solenoidal & Potential M~0.7, Alfvenic solenoidal potential M~2.5, SuperAlfvenic MA~8 Decoupled only at small scales! Caution is needed! M~7, Alfvenic Kowal & Lazarian 2005

  22. Correlation contours of Density M~0.7 M~7 M=7 M=2 Flat density Lazarian & Beresnyak 04 Density anisotropy depends Mach number! Spectrum of density is flat for high M.

  23. Logarithm of density at Mach=7 before after • At high Mach number density is isotropic due to dominance of high peaks due to driving • Filtering of high peaks reveals GS pattern Beresnyak, Lazarian & Cho 05

  24. Scaling of Density Testing of predictions in Boldyrev 02 log M~10 5123 M~3 5123 Log of density scales similar to velocity M~0.7 2563 Kowal & Lazarian 05

  25. Turbulence in Partially Ionized Gas B Viscosity is important while resistivity is not. Viscous magnetized fluid Does viscous damping scale is the scale at which MHD turbulence ends? ~0.3pc in WNM

  26. Viscosity Damped Turbulence: New Regime of MHD Turbulence E(k)~k-1 intermittent Expected: k-1 for magnetic field k-4 for kinetic energy Cho, Lazarian & Vishniac 02, Numerical testing confirms that magnetic turbulence does not die!!!

  27. Scale-Dependent Intermittency Predicted in Lazarian, Vishniac & Cho 04 Large scales perp. B • filling factor of high • intensity magnetic • field Magnetic field gets more intermittent as scale gets smaller Small scales perp. B Cho, Lazarian & Vishniac 03

  28. Fraction of energy versus volume Scale-dependent intermittency Ordinary turbulence New regime Cho, Lazarian & Vishniac 03 In viscosity-damped turbulence most of magnetic energy is in a small fraction of volume

  29. High moment scaling Cho, Lazarian & Vishniac 03 The exponent is between 0.5 and 0 Using predictions for intermittent magnetic field from Lazarian, Vishniac & Cho 04

  30. Density in viscosity-dominated regime Incompressible phys. diffusion compressible Cho & Lazarian 03 intermittency magnetic magnetic density Cho, Lazarian & Vishniac 03

  31. rs = antennae temperature at frequency n (depends on both velocity and density) rs Observational testing: Can we use Velocity Centroids? Definition: n Structure function of centroids Can be obtained from observational data.

  32. Velocity High Moments? Not yet available. Problem with tools Centroids properly reflect velocity only at Mach number M<3 Modification of centroids proposed by Lazarian & Esquiel 03 may help Esquiel & Lazarian 05

  33. A 2D genus number defined as: Genus analysis For a Gaussian map the genus-threshold curve is symmetric around the mean: Work with A. Esquivel

  34. Genus analysis SMC • A shift from the mean can reveal “meatball” or “Swiss cheese” topology. • Genus curve of the HI in the SMC and from MHD simulations are different although the spectra are similar • The SMC show a evident “Swiss cheese” topology, the simulations are more or less symmetric. MHD Lazarian, Pogosyan & Esquivel 2003

  35. Summary • Turbulence intermittency is astrophysically important. • In low M local magnetic field system velocity intermittency is similar to hydro. • Intermittency of B is larger than that of V. • Intermittencies of Alfven, slow and fast modes are different (Alfven is most stable with Mach number). • Log of density intermittency is similar to velocity. • Viscosity-dominated regime demonstrates scale-dependent intermittency. • Observational testing is possible and necessary.

  36. Implications for CR Transport Big difference!!! 10-7 (Kolmogorov) Scattering efficiency Fast modes 10-10 Yan & Lazarian02 Fast modes determine scattering!

  37. Viscosity-Dominated Regime (Lazarian, Vishniac & Cho 04) • MHD turbulence does not vanish at the viscous damping scale. Magnetic energy cascades to smaller scales. • Magnetic intermittency increases with decrease of the scale. • Turbulence gets resurrected at ion decoupling scale.

  38. Density, compressible and Alfven modes (5123) Cho & Lazarian 05

  39. l^ V^ ~ l^2/3 ~ l|| l|| VA Incompressible MHD: GS model • It is easy to mix magnetic field lines: V^ ~ l^1/3 • Coupling between || and ^: (E(k^)~k^-5/3) Kolmogorov in ^ direction Basics of Goldreich & Sridhar model (1995) Anisotropy is larger at small scales

  40. What are the scattering rates for different ISM phases? (Cont.) gyroresonance TTD Solid line is analytical results Symbols are numerical results (c) scattering frequency by gyroresonance vs. pitch angle cosine; (d) near 90o transit time damping should be taken into account.

  41. Spectroscopic Observations and velocity statistics (slide composition by A. Goodman)

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