Perm 7.09.06

A theoretical plasma physicist’s take on

Turbulence in the ISM:popular beliefs, some observational data,some speculations about their meaning,

and some rigourous approaches

Perm 7.09.06

Alexander Schekochihin (DAMTP, Cambridge)

in collaborations withSteve 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) Reprints/references on http://www.damtp.cam.ac.uk/user/as629 or ask me for a copy

Electron-density fluctuations in the interstellar medium[Armstrong et al. 1995, ApJ 443, 209]

The Great Power Law in the Sky

k–5/3

• Turbulence is stirred by supernovae at L ~ 100 pc• Fluctuations of velocity and magnetic field are Alfvénic:


The Great Power Law in the Sky

k–5/3

• 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

MHD Turbulence à la K41

Energy at scale l

Cascade time(rate of transfer)

• Universality• Alfvénic:• Locality in scale space

energyinjected

Kinetic energy

k

energy flux

energy

dissipated

Magnetic energy

k–??

MHD Turbulence à la K41

energyinjected

Kinetic energy

k

energy flux

energy

dissipated

Magnetic energy

k–??

Energy at scale l


• Two time scales available:

and , so

MHD turbulence spectrum not fixed solely by dimensional analysis


Goldreich-Sridhar Turbulence

Energy at scale l


• Strong interactions: (critical balance)

[Goldreich & Sridhar 1995, ApJ 438, 763]

energyinjected

Kinetic energy

k

energy flux

energy

dissipated

Magnetic energy

k–5/3



Energy at scale l


• Strong interactions: (critical balance)

energyinjected

Kinetic energy

k

energy flux

energy

dissipated

Magnetic energy

k–5/3

ANISOTROPIC![Goldreich & Sridhar 1995, ApJ 438, 763]



Energy at scale l


• Strong interactions: (critical balance) GS95

energyinjected

Kinetic energy

k

energy flux

energy

dissipated

Magnetic energy

k–5/3

[Goldreich & Sridhar 1995, ApJ 438, 763] ANISOTROPIC!


Anisotropy: It Is Really There


• Simulations of MHD turbulence unambiguously demonstrate that it is anisotropic and are consistent with GS95

[Maron & Goldreich 2001, ApJ 554, 1175; Cho et al. 2002, ApJ 564, 291]

Anisotropy: It Is Really There


• Observations of SW and ISM also show that turbulencethere is anisotropic with , although it is difficult

to check the GS95 scaling. In SW, it has recently beenfound that while

as should be the case in GS95[T. Horbury 2006, private communication].

• Simulations of MHD turbulence unambiguously demonstrate that it is anisotropic and are consistent with GS95

[Maron & Goldreich 2001, ApJ 554, 1175; Cho et al. 2002, ApJ 564, 291]

Solar Wind: Alfvénic Turbulence

Magnetic- and electric-field fluctuationsin the solar wind at ~1 AU (19 Feb. 2002)

[Bale et al. 2005, PRL 94, 215002]

Alfvénicfluctuations

k–5/3k–5/3k–5/3

ISM: Alfvénic Turbulence?

Alfvénicfluctuations

I have not seen a nice plot likethis for the ISM…

Bottle of port to anyone whocan give me one!

So, It’s All Sorted Then?


[Bale et al. 2005, PRL 94, 215002]

k–5/3k–5/3k–5/3

Does all this meanwe understand plasmaturbulence in the sky?

SEE PART II OFTHIS TALK

What if there is no guide field?

• Clusters of galaxies• Some parts of the ISM

Strong guide field: Weak guide field:

waves, random tangle,

Fluctuation Dynamo

Stretching by random fluid motions:

• Exponential growth with • Direction reversals at the resistive scale, k ~ k

• Field varies slowly along itself: k|| ~ kflow

Stretch/shear

[AAS et al. 2002, PRE 65, 016305; AAS et al. 2004, ApJ 612, 276Review: AAS & Cowley, astro-ph/0507686]

Fluctuation Dynamo: DNS (Pm >> 1)

[AAS et al. 2004, ApJ 612, 276Review: AAS & Cowley, astro-ph/0507686]

Folded structure

Fluctuation Dynamo: DNS (Pm >> 1)

[AAS et al. 2004, ApJ 612, 276Review: AAS & Cowley, astro-ph/0507686]

Folded structure

Dynamo: The Movie

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

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]

Folded Fields Observed in Clusters

[AAS et al. 2004, ApJ 612, 276]A2256: polarised emission[Enßlin & Clarke 2005, AJ, submitted]

What Are the Saturated Spectra?

[AAS et al. 2004, ApJ 612, 276]

with prob. 1/2


[Yousef, Rincon & AAS 2006, JFM submitted]

with prob. 1/2


with prob. 1/2

[AAS et al. 2004, ApJ 612, 276]

k–1


with prob. 1/2

[AAS et al. 2004, ApJ 612, 276]

k–?

This is probably too simplistica model…

Saturated Spectra: DNS

[AAS et al. 2004, ApJ 612, 276]

NB:Velocity spectrumstill has a negative

exponent,possibly Kolmogorov

(Alfvén waves can propagatealong the folds)

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 turbulenceis roughly here

Viscous scale isroughly here

ISM: Spiral Arms vs. Interarm Regions

Structure functions of Faraday rotation measure in ISM [Haverkorn et al. 2006, ApJ 637, L33]

Kolmogorov?

Flat?

INTERARMS:

ARMS:

ISM: Two Types of Turbulence?

Structure functions of Faraday rotation measure in ISM [Haverkorn et al. 2006, ApJ 637, L33]

Strong guide field

Weak guide field

Alfvénic turbulence

Saturated small-scaledynamo

ARMS:

INTERARMS:

[AAS, Cowley & Dorland 2006, PPCF to be published Iskakov, Cowley & AAS 2006, in preparation]

ISM: Two Types of Turbulence?

Strong guide field

Weak guide field

Alfvénic turbulence

Saturated small-scaledynamo

ARMS:

INTERARMS:This is only a speculation: let us discuss it!Here are some points in favour:• 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

Now the Rigourous Bit…

PART II

THE PLASMA PHYSICS OFINTERSTELLAR TURBULENCE

[Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812AAS, Cowley & Dorland 2006, PPCF to be published

(preprint on www.damtp.cam.ac.uk/user/as629)]

Turbulence in Weakly Collisional Plasma

KAW

k–5/3

k–7/3

energyinjected

ionheating electron

heating

Observed spectra

collisional(fluid)

collisionless(kinetic)

Alfvén waves:

SWISMIGM

Turbulence in Weakly Collisional Plasma

KAW

k–5/3

k–7/3

energyinjected

ionheating electron

heating

Observed spectra

collisional(fluid)


Alfvén waves:

MUST USE KINETICS, NOT MHD!

SWISMIGM

Gyrokinetics: Ordering

[Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209]

Ordering based on anisotropy + critical balanceapplied to kinetic theory gives GK

• Critical balance as an ordering assumption:

• Small parameter:

• Finite Larmor radius:

[Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]





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]





Low frequencyGK ORDERING:


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

Gyrokinetics: Kinetics of Larmor Rings

[Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]

Particle dynamics can beaveraged over the Larmororbit and everything reduces to kinetics of Larmor ringscentered at

and interacting withthe electromagnetic fluctuations.

[Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209]


Particle dynamics can beaveraged over the Larmororbit and everything reduces to kinetics of Larmor ringscentered at

and interacting withthe electromagnetic fluctuations.

++ Maxwell’s equations


Averaged gyrocentre drifts:• EB0 drift• B drift• motion along perturbed fieldline

Averagedwave-ringinteraction

++ Maxwell’s equations

Gyrokinetics Covers Everything

KAW

k–5/3

k–7/3

energyinjected

ionheating electron

heating

Observed spectra

collisional(fluid)


Alfvén waves:

GYROKINETICSFLUID THEORY

Gyrokinetics: DNS

QuickTime™ and aTIFF (LZW) decompressor


Numerical simulations(gyrokinetics in 3+2D)are possible (piece-wise!)at the limit of currentlyavailable computing powerusing codes developed forfusion problems.

Transatlantic projectunderway 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]

Gyrokinetics: DNS

QuickTime™ and aTIFF (LZW) decompressor


Numerical simulations(gyrokinetics in 3+2D)are possible (piece-wise!)at the limit of currentlyavailable computing powerusing codes developed forfusion problems.

Reduced fluid/kinetic/hybridmodels necessary to understand and to simulatewhat happens in variousparameter regimes.

Simulations using GS2 [picture courtesy Bill Dorland 2005]

Kinetic Reduced MHD

k–5/3

k–7/3

energyinjected

ionheating electron

heatingcollisional(fluid)


Alfvén waves:


magnetised ions

isothermal electrons

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 publishedcf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]

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)


Alfvén-Wave Cascade in the Solar Wind

• Alfvénic fluctuations


[Bale et al. 2005, PRL 94, 215002]

k–5/3 KRMHD

KRMHD: Density and Field Strength

• Density and field strength require kinetic description


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


Density and Field Strength in the Solar Wind

[Bershadskii & Sreenivasan 2004,PRL 93, 064501]

Spectrum of magnetic-field strengthin the solar wind at ~1 AU (1998)

Density fluctuations in the solar windat ~1 AU (31 Aug. 1981)

[Celnikier, Muschietti & Goldman1987,A&A 181, 138]

k–5/3

FLR: density modemixing with

Alfvén waves

Density and Field Strength in the ISM

Anyone knows anything?

k–5/3




k–5/3


Is this scaling correct?



k–1.46±0.20

Structure function from scintillation measurements[Smirnova, Gwinn & Shishov 2006, astro-ph/0603490]

Is this scaling correct?

Ion Heating

k–5/3

k–7/3

energyinjected

ionheating electron

heating


KRMHD

GKions

(and isothermalelectrons)

Electron Reduced MHD

k–5/3

k–7/3

energyinjected

ionheating electron

heating


KRMHD

GKions

Boltzmann ions

magnetised electrons

ERMHD

ERMHD: Kinetic Alfvén Waves

• KAW fluctuations and

• Critical balance + constant flux argument à la K41/GS95 give spectrum of magnetic field with anisotropy

[Biskamp et al. 1999, Phys. Plasmas 6, 751; Cho & Lazarian 2004, ApJ 615, L41]

• This is a cascade of KAW,

• Electric field has spectrum:

[AAS, Cowley & Dorland 2006,PPCF to be published; this is theanisotropic version of EMHD,see Kingsep et al. 1990,Rev. Plasma Phys. 16, 243]

KAW Cascade in the Solar Wind



[Bale et al. 2005, PRL 94, 215002]

• Ion Heating

k–5/3 KRMHD

ERMHD

k–7/3

k–1/3

GKions

• KAW

KAW Cascade in the ISM?


• Ion Heating

• KAW


Again, I have not seen any data.Should be there.

Conclusions

It is too early for conclusions!

Observational Desiderata• Scalings (spectra):

Arms:

Interarms:

(Alfvénic) (passive) • Measures of anisotropy:

Compare

(Alfvénic)

Is the same true for (passive)

• Turbulence below ion gyroscale: Kinetic Alfvén waves

• What is the reversal scale of the folded fields?

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Perm 7.09.06