Interstellar Turbulence: Theory, Implications and
Consequences
Alex Lazarian (Astronomy, Physics and CMSO)
Collaboration: H. Yan, A. Beresnyak, J. Cho, G. Kowal,
A. Chepurnov , E. Vishniac, G. Eyink, P. Desiati, G. Brunetti …
Theme 2.
Propagation and Acceleration of Cosmic
Rays in Turbulent Magnetic Fields
MHD turbulence theory induces changes on our
understanding of CRs propagation and stochastic
acceleration
Highly isotropic
Icecube measurement 2010
M. Duldig 2006
Points of Part 2:
Scattering and second order Fermi acceleration of
cosmic rays by MHD turbulence
Perpendicular diffusion of cosmic rays
Acceleation of cosmic rays by shocks in turbulent
media
Points of Part 2:
Scattering and second order Fermi acceleration of
cosmic rays by MHD turbulence
Perpendicular diffusion of cosmic rays
Acceleation of cosmic rays by shocks in turbulent
media
Cosmic rays interact with magnetic turbulence
Cosmic Rays
Magnetized medium
In case of small angle scattering, Fokker-Planck equation can be used to
describe the particles’ evolution:
S : Sources and sinks of particles
2nd term on rhs: diffusion in phase space specified by
Fokker -Planck coefficients Dxy
Correct diffusion coefficients are the key to the success of
such an approach
Turbulence induces second order Fermi process
Magnetic
“clouds”
Quic kTime™ and a
decompres sor
are needed to s ee t his pict ure.
Resonance and Transit Time Damping (TTD) are examples of
2nd order Fermi process
B
rL
QuickTime™ and a
decompressor
are needed to see this picture.
n=1
n=0
Turbulence properties determine the diffusion and acceleration
Diffusion in the fluctuating EM fields
Collisionless
Fokker-Planck equation
Boltzmann-Vlasov eq

dB, dv<<B0, V (at the scale of resonance)
Fokker-Planck coefficients: Dmm ≈ Dm2/Dt, Dpp ≈ Dp2/Dt are the
fundermental parameters we need. Those are determined by
properties of turbulence!
For TTD and gyroresonance,
tsc/ tac ≈ Dpp / p2Dmm ≈ (VA/v)2
The diffusion coefficients define characteristics of particle
propagation and acceleration
Propagation
Stochastic
Acceleration
~
~
~
•The diffusion coeffecients are determined by the statistical properties of
turbulence
Gyroresonance scattering depends on the properties of turbulence
Gyroresonance
, (n = ± 1, ± 2 …),
Which states that the MHD wave frequency (Doppler shifted)
is a multiple of gyrofrequency of particles (v|| is particle speed
parallel to B).
So,
B
rL
Alfenic turbulence injected at large scales is inefficient for
cosmic ray scattering/acceleration
1. “random walk”
B
2rL
l perp<< l|| ~ rL
l||
eddies
l
scattering efficiency is reduced
2. “steep spectrum”
steeper than Kolmogorov!
B Less energy on resonant
scale
Inefficiency of cosmic ray scattering by Alfvenic turbulence is
obvious and contradicts to what we know about cosmic rays
Scattering frequency
Alfven modes
(Kolmogorov)
Big difference!!!
Chandran 2000)
(
Total path length is ~
104 crossings at GeV
from the primary to
secondary ratio.
Kinetic energy
Alternative solution is needed for CR scattering (Yan & Lazarian 02,04
Brunetti & Lazarian 0,).
Scattering frequency
Fast modes efficiently scatter cosmic rays solving problems mentioned
earlier
fast modes
modesmode
s momodes
Depends on
damping
plot w. linear scale
Kinetic energy
Fast modes are identified as the dominate source for CR
scattering (Yan & Lazarian 2002, 2004).
Damping is for fast modes is usually defined for laminar
fluids and is not applicable to turbulent environments
Damping increases with plasma b= Pgas/Pmag and the angle q between k
and B.
Viscous damping (Braginskii 1965)
Collisionless damping (Ginzburg 1961, Foote & Kulsrud 1979)
To calculate fast mode damping one should take into account wandering of
magnetic field lines induced by Alfvenic turbulence
Magnetic field wandering induced by Alfvenic turbulence was described in
Lazarian & Vishniac 1999
dB direction changes during cascade
Randomization of local B: field line
wandering by shearing via Alfven modes:
k
dB/B ≈ (V/L)1/2 tk1/2
Q
Randomization of wave vector k:
dk/k ≈ (kL)-1/4 V/Vph
B
Yan & Lazarian 2004
Field line wandering
Lazarian, Vishniac
& Cho 2004
Mean free path (pc)
Modeling that accounts for damping of fast modes agrees with
observations
CR Transport in ISM
Palmer consensus
WIM
Text
from Bieber et al 1994
halo
Kinetic energy
Flat dependence of mean free path can occur due to
collisionless damping.
Take home message 8:
• Alfvenic turbulence is inefficient for scattering if it is
generated on large scales.
• Fast modes dominate scattering, but damping of them is
necessary to account for.
• Calculation of fast mode damping requires accounting for
field wandering by Alfvenic turbulence.
• Scattering depends on the environment and plasma beta.
• Actual turbulence and acceleration in collisionless
environments may be more complex
Points of Part 2:
Scattering and second order Fermi acceleration of
cosmic rays by MHD turbulence
Perpendicular diffusion of cosmic rays
Acceleation of cosmic rays by shocks in turbulent
media
Perpendicular transport is due to
turbulent B field
Dominated by field line
wandering.
B0
Intensive studies:
e.g., Jokipii & Parker 1969, Forman 74,
Urch 77, Bieber & Matthaeus 97, Giacolone
& Jokipii 99, Matthaeus et al 03, Shalchi et
al. 04
– Particle trajectory
— Magnetic field
What if we use the tested model of turbulence?
Perpendicular
transport
Whether and to what degree CRs diffusion is
suppressed depends on Alfven Mach number,
i.e MA= Vinj/VA.
MA< 1, CRs free stream over distance L, thus
D⊥ =R2 /∆t= Lv|| MA4
Lazarian & Vishniac 1999, Lazarian 2006, Yan & Lazarian 2008
Earlier works suggested MA2 dependence
Predicted MA4 suppression is
observed in simulations!
Xu & Yan 2013
Differs from M2 dependence in classical works, e.g. in Jokipii & Parker
69, Matthaeus et al 03.
Is Subdiffusion (∆x ~ ∝ta, a<1) typical?
Subdiffusion (or compound diffusion,
Getmantsev 62, Lingenfelter et al 71, Fisk et al.
73, Webb et al 06) was observed in near-slab
turbulence, which can occur on small scales
due to instability.
What about large scale turbulence?
Example: diffusion of a dye on a rope
a) A rope allowing retracing, ∆t =lrope2 /D
b) A rope limiting retracing within pieces
lrope /n, ∆t =lrope2 /nD
Diffusion is slow if particles retrace their trajectories.
Is there subdiffusion (∆x2∝∆ta, a<1) ?
Subdiffusion (or compound
diffusion, Getmantsev 62, Lingenfelter et
al 71, Fisk et al. 73, Webb et al 06) was
observed in near-slab turbulence,
which can occur on small scales
due to instability.
Diffusion is slow only if particles retrace their trajectories.
Subdiffusion does not happen in realistic
astrophysical turbulence
In turbulence, CRs’ trajactory
become independent when field
lines are seperated by the
smallest eddy size , l⊥,min. The
separation between field lines
grows exponentially, provides
LRR =|||,min log(l⊥,min /rL)
Subdiffusion only occurs below
LRR. Beyond LRR, normal diffusion
applies.
Lazarian 06, Yan & Lazarian 08
l||,min
– Particle trajectory
— Magnetic field
General Normal Diffusion is observed in
simulations!
compressible turbulence (Xu & Yan 2013
incompressible turbulence
)
∝t
0.01
∝t
0.01
rL /L
rL /L
0.001
0.001
Beresnyak et al. (2011)
Cross field transport in 3D turbulence is in general a normal diffusion!
Perpendicular propagation is superdiffusive on
scales less than the injection scale
Xu & Yan 2013
x
Lazarian, Vishniac & Cho 2004
agnetic field separation follows the law y2_x3 (Richarson law), x<Linj
Take home message 9:
• Alfvenic Perpendicular diffusion scales as MA4, not MA2
• Subdiffusion does not happen
• Superdiffusion takes place on scales smaller than the
injection scale
Points of Part 2:
Scattering and second order Fermi acceleration of
cosmic rays by MHD turbulence
Perpendicular diffusion of cosmic rays
Acceleation of cosmic rays by shocks in turbulent
media
Point 5. Turbulence alters processes of Cosmic Ray acceleration
in shocks
Acceleration in shocks requires scattering of particles back
from the upstream region.
Downstream
Magnetic turbulence
generated by shock
Upstream
Magnetic fluctuations
generated by streaming
In postshock region damping of magnetic turbulence
explains X-ray observations of young SNRs
Chandra
Alfvenic turbulence decays in one eddy turnover time (Cho & Lazarian 02),
which results in magnetic structures behind the shock being transient
and generating filaments of a thickness of 1016-1017cm (Pohl, Yan &
Lazarian 05).
Streaming instability in the preshock region is a textbook
solution for returning the particles to shock region
vA
B
shock
Streaming instability is inefficient for producing large field in the
preshock region
B
shock
1.
2.
Beresnyak & Lazarian 08
Streaming instability is suppressed in the presence of external turbulence (Yan & Lazarian
02, Farmer & Goldreich 04, Beresnyak & Lazarian 08).
Non-linear stage of streaming instability is inefficient (Diamond & Malkov 07).
Bell (2004) proposed a solution based on the current instability
jCR
shock
B
Precursor forms in front of the shock and it gets turbulent as
precursor interacts with gas density fluctuation
Turbulence efficiently generates magnetic fields as shown by
Cho et al. 2010
hydrodynamic
cascade
MHD scale
The model allows to calculate the parameters of magnetic field
Beresnyak, Jones & Lazarian 2010
Take home message 9:
Magnetic field generated by precursor -- density fluctuations interaction
might be larger than the arising from Bell’s instability
current instability
jCR
B