intermittency of mhd turbulence a. lazarian uw-madison: astronomy and center for magnetic...
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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)
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)