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)

Da Vincis view Turbulence = eddies ! Chaotic Order! Vortices inside flow

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

Astrophysical relevance Re ~VL/ ~10 10 >> 1 ~ r L v th, v th < V, r L << L

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.

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

She-Leveque and Politano- Pouquet models Scaling No intermittencyKolmogorov model FilamentsShe-Leveque model Above is hydro. What about MHD? General: Politano-Pouquet model for where t cas ~l x, z l ~l 1/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!

Cho, Lazarian & Vishniac 03 Magnetic field B0B0 B0B0 B0B0 Scale-dependent Anisotropy

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

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

Scalings of velocity and magnetic field Local system of reference Global system of reference 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 Cho, Lazarian & Vishniac 03 M A ~0.7 Incompressible

Compressible and incompressible MHD Compressible simulations for Mach ~ 0.7, mean B~0 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! Cho, Lazarian & Vishniac 03

MHD modes (for P mag > P gas ) Alfven mode (v=V A cos ) incompressible; restoring force=mag. tension k B slow mode (v=c s cos ) fast mode (v=V A ) restoring force = P mag + P gas B k B restoring force = P gas Theoretical discussion in Lithwick & Goldreich 01 Cho & Lazarian 02

Basis s ~ [(1-D 1/2 + /2)/(1+D 1/2 - /2)](k /k || ) 2 k || + k f ~ k || + [(1-D 1/2 - /2)/(1+D 1/2 + /2)](k || /k ) 2 k A ~ k || x k *D=(1+ /2) 2 -2 cos Decomposition over basis in Fourier space: Cho & Lazarian 02

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

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

Cho & Lazarian 02 Alfvenslowfast~k -5/3 ~k -3/2 anisotropic (GS) isotropic Spectr a Correlation functions M=2 Magnetically dominated

How good is our decomposition? 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. Cho & Lazarian 03 Decomposition: dashed lines M=7 M=2.3 Anistoropy obtained without decomposition

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

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

Solenoidal & Potential M~0.7, Alfvenic M~2.5, SuperAlfvenic M~7, Alfvenic solenoidalpotential Kowal & Lazarian 2005 Decoupled only at small scales! Caution is needed! M A ~8

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

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

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

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

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

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

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

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

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

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

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

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

Genus analysis 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. SMC MHD Lazarian, Pogosyan & Esquivel 2003

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.

Implications for CR Transport Scattering efficiency (Kolmogorov) Fast modes Big difference!!! Yan & Lazarian Fast modes determine scattering!

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.

Density, compressible and Alfven modes (512 3 ) Cho & Lazarian 05

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

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

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