Download - 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:
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:• 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
Cascade time(rate of transfer)
• Two time scales available:
and , so
MHD turbulence spectrum not fixed solely by dimensional analysis
• Universality• Alfvénic:• Locality in scale space
Goldreich-Sridhar Turbulence
Energy at scale l
Cascade time(rate of transfer)
• Strong interactions: (critical balance)
[Goldreich & Sridhar 1995, ApJ 438, 763]
energyinjected
Kinetic energy
k
energy flux
energy
dissipated
Magnetic energy
k–5/3
• Universality• Alfvénic:• Locality in scale space
Goldreich-Sridhar Turbulence
Energy at scale l
Cascade time(rate of transfer)
• Strong interactions: (critical balance)
energyinjected
Kinetic energy
k
energy flux
energy
dissipated
Magnetic energy
k–5/3
ANISOTROPIC![Goldreich & Sridhar 1995, ApJ 438, 763]
• Universality• Alfvénic:• Locality in scale space
Goldreich-Sridhar Turbulence
Energy at scale l
Cascade time(rate of transfer)
• 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!
• Universality• Alfvénic:• Locality in scale space
Anisotropy: It Is Really There
• Strong interactions: (critical balance) GS95
• 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
• Strong interactions: (critical balance) GS95
• 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?
Magnetic- and electric-field fluctuationsin the solar wind at ~1 AU (19 Feb. 2002)
[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
What Are the Saturated Spectra?
[Yousef, Rincon & AAS 2006, JFM submitted]
with prob. 1/2
What Are the Saturated Spectra?
with prob. 1/2
[AAS et al. 2004, ApJ 612, 276]
k–1
What Are the Saturated Spectra?
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)
collisionless(kinetic)
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]
Gyrokinetics: Ordering
• Critical balance as an ordering assumption:
• Small parameter:
• Finite Larmor radius:
Low frequency
Ordering based on anisotropy + critical balanceapplied to kinetic theory gives GK
[Taylor & Hastie 1968, Plasma Phys. 10, 479; Frieman & Chen 1982, Phys. Fluids 443, 209][Howes, Cowley, Dorland, Hammett, Quataert, AAS, astro-ph/0511812]
Gyrokinetics: Ordering
• Critical balance as an ordering assumption:
• Small parameter:
• Finite Larmor radius:
Low frequencyGK ORDERING:
Ordering based on anisotropy + critical balanceapplied to kinetic theory gives GK
[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]
Gyrokinetics: Kinetics of Larmor Rings
Particle dynamics can beaveraged over the Larmororbit and everything reduces to kinetics of Larmor ringscentered at
and interacting withthe electromagnetic fluctuations.
++ Maxwell’s equations
Gyrokinetics: Kinetics of Larmor Rings
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)
collisionless(kinetic)
Alfvén waves:
GYROKINETICSFLUID THEORY
Gyrokinetics: DNS
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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
are needed to see this picture.
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)
collisionless(kinetic)
Alfvén waves:
GYROKINETICSFLUID THEORY
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)
[AAS, Cowley & Dorland 2006, PPCF to be publishedcf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]
Alfvén-Wave Cascade in the Solar Wind
• Alfvénic fluctuations
Magnetic- and electric-field fluctuationsin the solar wind at ~1 AU (19 Feb. 2002)
[Bale et al. 2005, PRL 94, 215002]
k–5/3 KRMHD
KRMHD: Density and Field Strength
• Density and field strength require kinetic description
[AAS, Cowley & Dorland 2006, PPCF to be publishedcf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]
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
[AAS, Cowley & Dorland 2006, PPCF to be publishedcf. Higdon 1984, ApJ 285, 109; Lithwick & Goldreich 2001, ApJ 562, 279]
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
Electron-density fluctuations in the interstellar medium[Armstrong et al. 1995, ApJ 443, 209]
Density and Field Strength in the ISM
Anyone knows anything?
k–5/3
Electron-density fluctuations in the interstellar medium[Armstrong et al. 1995, ApJ 443, 209]
Is this scaling correct?
Density and Field Strength in the ISM
Anyone knows anything?
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
GYROKINETICSFLUID THEORY
KRMHD
GKions
(and isothermalelectrons)
Electron Reduced MHD
k–5/3
k–7/3
energyinjected
ionheating electron
heating
GYROKINETICSFLUID THEORY
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
• Alfvénic fluctuations
Magnetic- and electric-field fluctuationsin the solar wind at ~1 AU (19 Feb. 2002)
[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?
• Alfvénic fluctuations
• Ion Heating
• KAW
Electron-density fluctuations in the interstellar medium[Armstrong et al. 1995, ApJ 443, 209]
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?