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)

E. Vishniac (Johns Hopkins)

Da Vinci’s viewDa Vinci’s view

Turbulence = eddies !

Chaotic Order!Vortices inside flow

Experimental insight

Reynolds number Re = VL/

Re ~ 15,000

Eddies insideeddies

Stochasticity depends on

Astrophysical relevanceAstrophysical relevance

Re ~VL/ ~1010 >> 1

~ rLvth, vth < V, rL<< L

Is dissipation smooth?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?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.

She-Leveque and Politano-Pouquet models

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!

Cho, Lazarian & Vishniac 03Magnetic

field B0 B0

B0

Scale-dependent Anisotropy

Confusing resultsConfusing 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 BScaling 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

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

MA~0.7

Incompressible

Compressible and incompressible MHD

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 withPadoan 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 Pmag > Pgas)

Alfven mode (v=VA cos)

incompressible;restoring force=mag. tension

k

B

slow mode (v=cs cos)

fast mode (v=VA)restoring force = Pmag + Pgas

Bk

B

restoring force = Pgas

Theoretical discussion in Lithwick & Goldreich 01 Cho & Lazarian 02

Basis

s ~ [(1-D1/2+/2)/(1+D1/2-/2)](k/k||)2 k|| + k

f ~ k|| + [(1-D1/2-/2)/(1+D1/2+/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

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

Xpredicted

totalVA/VPredicted scaling for Mtotal<1 is

totalVA/V)-1

NormalizedCompres energy

Cho & Lazarian 02

Alfven slow fast~k-5/3 ~k-5/3 ~k-3/2

anisotropic (GS) anisotropic (GS) isotropic

Spectr

a

Correlation

functions

M=2 Magnetically dominated

How good is our decomposition?How good is our decomposition?

• Our decomposition into modes is statistical

• Testing of it for slow modes is successfulFor 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 and M>>1

Intermittency Alfven, slow and fast modes: M<1 and M>>1

M~0.7

M~7

Alfven is pretty much the same, Slow is affected; fast is unclear

2563

Kowal & Lazarian 05

Alfven

Alfven

slow

slow

fast

fast

MA~0.7

MA~0.7

Alfven

Local Frame ResultsLocal Frame Results

Kowal & Lazarian 05

M~0.7

M~7

Solenoidal & PotentialSolenoidal & Potential

M~0.7, Alfvenic

M~2.5, SuperAlfvenic

M~7, Alfvenic

solenoidal potential

Kowal & Lazarian 2005

Decoupled only at small scales!Caution is needed!

MA~8

Correlation contours of Density Correlation contours of Density

Lazarian & Beresnyak 04

Density anisotropy depends Mach number!Spectrum of density is flat for high M.

M=2M=7M~7M~0.7

Flat density

Logarithm of density at Mach=7Logarithm 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

before after

Beresnyak, Lazarian & Cho 05

Scaling of DensityScaling of Density

M~10

M~3

M~0.7

Kowal & Lazarian 05

log

5123

5123

2563

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 IntermittencyScale-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

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 scalingHigh 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

Density in viscosity-dominated regime

Cho & Lazarian 03

Incompressible phys. diffusion

compressible

intermittency

magnetic magnetic density

Cho, Lazarian & Vishniac 03

Observational testing: Can we use Velocity Centroids?

Observational testing: Can we use Velocity Centroids?

zzsz dvvyxvyxS ),,(),( ∫= ρ

Structure function of centroids

Definition:

Can be obtained from observational data.

[ ] [ ]2222 )()()()()()( ∫ ∫−=− dzvdzvSS xxxxXX 1121 ραρα

ρs

ρs= antennae temperature at frequency depends on both velocity and density)

Velocity High Moments?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 analysisGenus analysis• A 2D genus number

defined as: )(

)()(2

regionsdensitylowisolatedofnumber

regionsdensityhighisolatedofnumberG

−=

For a Gaussian map the genus-threshold curve is symmetric around the mean:

Work with A. Esquivel

Genus analysisGenus 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

SummarySummary

• 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 TransportS

catt

erin

g ef

fici

e ncy

(Kolmogorov)

Fast modes

Big difference!!!

Yan & Lazarian 02

10-7

10-10

Fast modes determine scattering!

Viscosity-Dominated Regime (Lazarian, Vishniac & Cho 04)

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 (5123)

Density, compressible and Alfven modes (5123)

Cho & Lazarian 05

• It is easy to mix magnetic field lines: V ~ l1/3

• Coupling between || and :

l

l~

V

VA

l2/3 ~ 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 90o transit time damping should be taken into account.

Solid line is analytical resultsSymbols are numerical results

gyroresonance TTD

Spectroscopic Observations and velocity statistics

Spectroscopic Observations and velocity statistics

Spectral Line Observations

(slide composition by A. Goodman)