arXiv:astro-ph/0205286v1 17 May 2002
MHD Turbulence: Scaling Laws and
Jungyeon Cho1, A. Lazarian1, and Ethan T. Vishniac2
1Univ. of Wisconsin, Madison WI53706, USA
2Johns Hopkins Univ., Baltimore MD21218, USA
Abstract. Turbulence is the most common state of astrophysical flows. In typical as-
trophysical fluids, turbulence is accompanied by strong magnetic fields, which has a
large impact on the dynamics of the turbulent cascade. Recently, there has been a signif-
icant breakthrough on the theory of magnetohydrodynamic (MHD) turbulence. For the
first time we have a scaling model that is supported by both observations and numerical
simulations. We review recent progress in studies of both incompressible and compress-
ible turbulence. We compare Iroshnikov-Kraichnan and Goldreich-Sridhar models, and
discuss scalings of Alfv´ en, slow, and fast waves. We also discuss the completely new
regime of MHD turbulence that happens below the scale at which hydrodynamic tur-
bulent motions are damped by viscosity. In the case of the partially ionized diffuse
interstellar gas the viscosity is due to neutrals and truncates the turbulent cascade at
∼parsec scales. We show that below this scale magnetic fluctuations with a shallow
spectrum persist and discuss the possibility of a resumption of the MHD cascade after
ions and neutrals decouple. We discuss the implications of this new insight into MHD
turbulence for cosmic ray transport, grain dynamics, etc., and how to test theoretical
predictions against observations.
Most astrophysical systems, e.g. accretion disks, stellar winds, the interstellar
medium (ISM) and intercluster medium are turbulent with an embedded mag-
netic field that influences almost all of their properties. This turbulence which
spans from km to many kpc (see discussion in [2,150,86]) holds the key to many
astrophysical processes (e.g., transport of mass and angular momentum, star
formation, fragmentation of molecular clouds, heat and cosmic ray transport,
magnetic reconnection). Statistics of turbulence is also essential for the cosmic
microwave background (CMB) radiation foreground studies .
All turbulent systems have one thing in common: they have a large “Reynolds
number” (Re ≡ LV/ν; L= the characteristic scale or driving scale of the sys-
tem, V=the velocity difference over this scale, and ν=viscosity), the ratio of the
viscous drag time on the largest scales (L2/ν) to the eddy turnover time of a
parcel of gas (L/V ). A similar parameter, the “magnetic Reynolds number”, Rm
(≡ LV/η; η=magnetic diffusion), is the ratio of the magnetic field decay time
(L2/η) to the eddy turnover time (L/V ). The properties of the flows on all scales
depend on Re and Rm. Flows with Re < 100 are laminar; chaotic structures
develop gradually as Re increases, and those with Re ∼ 103are appreciably less
2Cho, Lazarian, & Vishniac
chaotic than those with Re ∼ 107. Observed features such as star forming clouds
and accretion disks are very chaotic with Re > 108and Rm > 1016.
Let us start by considering incompressible hydrodynamic turbulence, which
can be described by the Kolmogorov theory . Suppose that we excite fluid
motions at a scale L. We call this scale the energy injection scale or the largest
energy containing eddy scale. For instance, an obstacle in a flow excites motions
on scales of the order of its size. Then the energy injected at the scale L cascades
to progressively smaller and smaller scales at the eddy turnover rate, i.e. τ−1
vl/l, with negligible energy losses along the cascade1. Ultimately, the energy
reaches the molecular dissipation scale ld, i.e. the scale where the local Re ∼ 1,
and is dissipated there. The scales between L and ldare called the inertial range
and it typically covers many decades. The motions over the inertial range are
self-similar and this provides tremendous advantages for theoretical description.
The beauty of the Kolmogorov theory is that it does provide a simple scaling
for hydrodynamic motions. If the velocity at a scale l from the inertial range
is vl, the Kolmogorov theory states that the kinetic energy (ρv2
density is constant) is transferred to next scale within one eddy turnover time
(l/vl). Thus within the Kolmogorov theory the energy transfer rate (v2
and we get the famous Kolmogorov scaling
The one-dimensional2energy spectrum E(k) is the amount of energy between
the wavenumber k and k+dk divided by dk. When E(k) is a power law, kE(k) is
the energy near the wavenumber k ∼ 1/l. Since v2
E(k) ∝ k−5/3.
Kolmogorovscalings were the first major advance in the theory of incompress-
ible turbulence. They have led to numerous applications in different branches of
science (see ). However, astrophysical fluids are magnetized and the a dy-
namically important magnetic field should interfere with eddy motions.
Paradoxically, astrophysical measurements are consistent with Kolmogorov
spectra (see LPE02 ). For instance, interstellar scintillation observations in-
dicate an electron density spectrum is very close to −5/3 for 108cm - 1015cm
(see ). At larger scales LPE02 summarizes the evidence of −5/3 velocity power
spectrum over pc-scales in HI. Solar-wind observations provide in-situ measure-
ments of the power spectrum of magnetic fluctuations and Leamon et al.  also
obtained a slope of ≈ −5/3. Is this a coincidence? What properties is the mag-
netized compressible ISM expected to have? We will deal with these questions,
and some related issues, below.
l≈ kE(k), Kolmogorov scaling
1This is easy to see as the motions at the scales of large eddies have Re ≫ 1.
2Dealing with observational data, e.g. in LPE02 , we deal with three dimensional
energy spectrum P(k), which, for isotropic turbulence, is given by E(k) = 4πk2P(k).
Our approach here is complementary to that in Vazquez-Semadeni (this vol-
ume) and Mac Low (this volume). These reviews discuss attempts to simulate
the turbulent ISM in all its complexity by including many physical processes
(e.g. heating, cooling, self-gravity) simultaneously. This provides a possibility of
comparing observations and simulations (see review by Ostriker, this volume).
The disadvantage is that such simulations cannot distinguish between the con-
sequences of different processes. Note, that in studies of turbulence the adaptive
mesh does not help as the fine structures emerge through the entire computa-
Here we discuss a focused approach which aims at obtaining a clear under-
standing on the fundamental level, and considering physically relevant compli-
cations later. The creative synthesis of both approaches is the way, we think,
that studies of astrophysical turbulence should proceed3. Certainly an under-
standing of MHD turbulence in the most ideal terms is a necessary precursor to
understanding the complications posed by more realistic physics and numerical
effects. For review of general properties of MHD, see a recent book by Biskamp
In what follows, we first consider observational data that motivate our study
(§2), then discuss theoretical approaches to incompressible MHD turbulence (§3).
In §4 we discuss testing and extending of the Goldreich-Sridhar theory of turbu-
lence, then in §5 we deal with viscous damping of incompressible turbulence and
describe a new regime of MHD turbulence that is present below the viscous cut-
off scale. We move to the effects of compressibility in §6 and discuss implications
of our new understanding of MHD turbulence for the problems of dust motion,
cosmic ray dynamics, support of molecular clouds, heating of ISM etc in §7. We
propose observational testing of our results in §8 and present the summary in
2 Observational Data
Kolmogorov turbulence is the simplest possible model of turbulence. Since it
is incompressible and not magnetized, it is completely specified by its velocity
spectrum. If a passive scalar field, like “dye particles” or temperature inhomo-
geneities, is subjected to Kolmogorov turbulence, the resulting spectrum of the
passive scalar density is also Kolmogorov (see [95,171]). In compressible and
magnetized turbulence this is no longer true, and a complete characterization of
the turbulence requires not only a study of the velocity statistics but also the
statistics of density and magnetic fluctuations.
3Potentially our approach leads to an understanding of the relationship between mo-
tions at a given time at small scales (subgrid scales) and the state of the flow at a
previous time at some larger, resolved, scale. This could lead to a parametrization
of the subgrid scales and to large eddy simulations of MHD.
4Cho, Lazarian, & Vishniac
Direct studies of turbulence4have been done mostly for interstellar medium
and for the Solar wind. While for the Solar wind in-situ measurements are possi-
ble, studies of interstellar turbulence require inverse techniques to interpret the
Attempts to study interstellar turbulence with statistical tools date as far
back as the 1950s [59,64,117,172] and various directions of research achieved
various degree of success (see reviews by [65,31,2,78,79,86]).
2.1 Solar wind
Solar wind (see review ) studies allow pointwise statistics to be measured
directly using spacecrafts. These studies are the closest counterpart of laboratory
The solar wind flows nearly radially away from the Sun, at up to about 700
km/s. This is much faster than both spacecraft motions and the Alfv´ en speed.
Therefore, the turbulence is “frozen” and the fluctuations at frequency f are
directly related to fluctuations at the scale k in the direction of the wind, as
k = 2πf/v, where v is the solar wind velocity .
Usually two types of solar wind are distinguished, one being the fast wind
which originates in coronal holes, and the slower bursty wind. Both of them show,
however, f−5/3scaling on small scales. The turbulence is strongly anisotropic
(see ) with the ratio of power in motions perpendicular to the magnetic
field to those parallel to the magnetic field being around 30. The intermittency
of the solar wind turbulence is very similar to the intermittency observed in
hydrodynamic flows .
2.2Electron density statistics
Studies of turbulence statistics of ionized media (see ) have provided infor-
mation on the statistics of plasma density at scales 108-1015cm. This was based
on a clear understanding of processes of scintillations and scattering achieved by
theorists5(see [121,49]). A peculiar feature of the measured spectrum (see )
is the absence of the slope change at the scale at which the viscosity by neutrals
Scintillation measurements are the most reliable data in the “big power law”
plot in Armstrong et al. . However there are intrinsic limitations to the scintil-
lations technique due to the limited number of sampling directions, its relevance
only to ionized gas at extremely small scales, and the impossibility of getting ve-
locity (the most important!) statistics directly. Therefore with the data one faces
the problem of distinguishing actual turbulence from static density structures.
4Indirect studies include the line-velocity relationships  where the integrated ve-
locity profiles are interpreted as the consequence of turbulence. Such studies do not
provide the statistics of turbulence and their interpretation is very model dependent.
5In fact, the theory of scintillations was developed first for the atmospheric applica-
Moreover, the scintillation data does not provide the index of turbulence di-
rectly, but only shows that the data are consistent with Kolmogorov turbulence.
Whether the (3D) index can be -4 instead of -11/3 is still a subject of intense
debate [56,121]. In physical terms the former corresponds to the superposition
of random shocks rather than eddies.
Additional information on the electron density is contained in the Faraday
rotation measures of extragalactic radio sources (see [154,155]). However, there
is so far no reliable way to disentangle contributions of the magnetic field and the
density to the signal. We feel that those measurements may give us the magnetic
field statistics when we know the statistics of electron density better.
2.3 Velocity and density statistics from spectral lines
Spectral line data cubes are unique sources of information on interstellar tur-
bulence. Doppler shifts due to supersonic motions contain information on the
turbulent velocity field which is otherwise difficult to obtain. Moreover, the sta-
tistical samples are extremely rich and not limited to discrete directions. In
addition, line emission allows us to study turbulence at large scales, comparable
to the scales of star formation and energy injection.
However, the problem of separating velocity and density fluctuations within
HI data cubes is far from trivial [77,79,84,86]. The analytical description of the
emissivity statistics of channel maps (velocity slices) in Lazarian & Pogosyan
 (see also [79,86] for reviews) shows that the relative contribution of the
density and velocity fluctuations depends on the thickness of the velocity slice.
In particular, the power-law asymptote of the emissivity fluctuations changes
when the dispersion of the velocity at the scale under study becomes of the order
of the velocity slice thickness (the integrated width of the channel map). These
results are the foundation of the Velocity-Channel Analysis (VCA) technique
which provides velocity and density statistics using spectral line data cubes. The
VCA has been successfully tested using data cubes obtained via compressible
magnetohydrodynamic simulations and has been applied to Galactic and Small
Magellanic Cloud atomic hydrogen (HI) data [87,84,159,28]. Furthermore, the
inclusion of absorption effects  has increased the power of this technique.
Finally, the VCA can be applied to different species (CO, Hαetc.) which should
further increase its utility in the future.
Within the present discussion a number of results obtained with the VCA
are important. First of all, the Small Magellanic Cloud (SMC) HI data exhibit a
Kolmogorov-type spectrum for velocity and HI density from the smallest resolv-
able scale of 40 pc to the scale of the SMC itself, i.e. 4 kpc. Similar conclusions
can be inferred from the Galactic data  for scales of dozens of parsecs, al-
though the analysis has not been done systematically. Deshpande et al. 
studied absorption of HI on small scales toward Cas A and Cygnus A. Within
the VCA their results can be interpreted as implying that on scales less than
1 pc the HI velocity is suppressed by ambipolar drag and the spectrum of density
fluctuations is shallow P(k) ∼ k−2.8. Such a spectrum  can account for the
small scale structure of HI observed in absorption.
6Cho, Lazarian, & Vishniac
2.4 Magnetic field statistics
Magnetic field statistics are the most poorly constrained aspect of ISM turbu-
lence. The polarization of starlight and of the Far-Infrared Radiation (FIR) from
aligned dust grains is affected by the ambient magnetic fields. Assuming that
dust grains are always aligned with their longer axes perpendicular to magnetic
field (see the review ), one gets the 2D distribution of the magnetic field
directions in the sky. Note that the alignment is a highly non-linear process
in terms of the magnetic field and therefore the magnetic field strength is not
The statistics of starlight polarization (see ) is rather rich for the Galactic
plane and it allows to establish the spectrum7E(K) ∼ K−1.5, where K is a two
dimensional wave vector describing the fluctuations over sky patch.8
For uniformly sampled turbulence it follows from Lazarian & Shutenkov 
that E(K) ∼ Kαfor K < K0and K−1for K > K0, where K−1
angular size of fluctuations which is proportional to the ratio of the injection
energy scale to the size of the turbulent system along the line of sight. For
Kolmogorov turbulence α = −11/3.
However, the real observations do not uniformly sample turbulence. Many
more close stars are present compared to the distant ones. Thus the interme-
diate slops are expected. Indeed, Cho & Lazarian  showed through direct
simulations that the slope obtained in  is compatible with the underlying
Kolmogorov turbulence. At the moment FIR polarimetry does not provide maps
that are really suitable to study turbulence statistics. This should change soon
when polarimetry becomes possible using the airborne SOFIA observatory. A
better understanding of grain alignment (see ) is required to interpret the
molecular cloud magnetic data where some of the dust is known not to be aligned
(see  and references therein).
Another way to get magnetic field statistics is to use synchrotron emission.
Both polarization and intensity data can be used. The angular correlation of po-
larization data  shows the power-law spectrum K−1.8and we believe that the
interpretation of it is similar to that of starlight polarization. Indeed, Faraday
depolarization limits the depth of the sampled region. The intensity fluctuations
were studied in  with rather poor initial data and the results were inconclu-
sive. Cho & Lazarian  interpreted the fluctuations of synchrotron emissivity
[43,44] in terms of turbulence with Kolmogorov spectrum.
is the critical
6The exception to this may be the alignment of small grains which can be revealed
by microwave and UV polarimetry .
7Earlier papers dealt with much poorer samples (see ) and they did not reveal
This spectrum is obtained by  in terms of the expansion over the spherical
harmonic basis Ylm. For sufficiently small areas of the sky analyzed the multipole
analysis results coincide with the Fourier analysis.
3 Theoretical Approaches to MHD Turbulence
Here we consider mainly Kolmogorov-type theories. Other theories not discussed
in this section include the eddy-damped quasinormal Markovian (EDQNM) ap-
proximation , the renormalization group technique [39,174], and the direct
interaction approximation .
3.1 Iroshnikov-Kraichnan theory
Attempts to describe magnetic turbulence statistics were made by Iroshnikov
 and Kraichnan . Their model of turbulence (IK theory) is isotropic in
spite of the presence of the magnetic field.
We can understand the IK theory as follows.9For simplicity, let us suppose
that a uniform external magnetic field (B0) is present. In the incompressible
limit, any magnetic perturbation propagates along the magnetic field line. To
the first order, the speed of propagation is constant and equal to the Alfv´ en
speed VA = B0/√4πρ, where ρ is the density. Since wave packets are moving
along the magnetic field line, there are two possible directions for propagation.
If all the wave packets are moving in one direction, then they are stable to
nonlinear order. Therefore, in order to initiate turbulence, there must be
opposite-traveling wave packets and the energy cascade occurs only when they
collide. The IK theory starts from this observation.
The IK theory assumes that, when two opposite-traveling wave packets of
size l collide, they lose the following amount of energy to smaller scales:
∆E ∼ (dv2/dt)∆t ∼ vl· ˙ vl∆t ∼ vl(v2
where the IK theory assumes that only collisions between similar size packets are
important and ∆t ∼ l/VA. The latter means that the duration of the collision
is the size of the wave packet divided by the speed of the wave packets. These
assumptions look reasonable at first. But, it is important to note that they fail
when eddies are anisotropic. That is, if eddies are elongated along the magnetic
field line, then ∆t is not l/VA, but l?/VA, where l?is the parallel size of the wave
packet (or ‘eddy’).
Equation (4) tells us that the energy change per collision is v2
is only a tiny fraction of v2
lwhen VA≫ vl. Therefore, in order for the eddy to
transfer all the energy to small eddies, the eddy must go through many collisions.
When such collisions are incoherent, we require a total (v2/∆E)2collisions to
complete the cascade. This means that the energy cascade time tcasis
l/l)∆t ∼ (v3
which means that this new cascade time is (VA/vl) times longer than the eddy
turnover time (l/vl). As in the Kolmogorov theory, the IK theory assumes the
9We follow arguments in .
8 Cho, Lazarian, & Vishniac
constancy of energy cascade (eq. 1): (v4
vl∝ l1/4, or,
l)/(lVA) = constant, which, in turn, yields
E(k) ∝ k−3/2. (6)
A uniform component to the magnetic field defines a special direction, which will
be reflected in the dynamics of turbulent fluctuations. One obvious effect is that
it is easy to mix field lines in directions perpendicular to the local mean magnetic
field and much more difficult to bend them. The IK theory assumes isotropy of
the energy cascade in Fourier space, an assumption which has attracted severe
criticism [115,153,114,158,104]. Mathematically, anisotropy manifests itself in
the resonant conditions for 3-wave interactions:
k1+ k2= k3,
ω1+ ω2= ω3,
where k’s are wavevectors and ω’s are wave frequencies. The first condition is
a statement of wave momentum conservation and the second is a statement of
energy conservation. Alfv´ en waves satisfy the dispersion relation: ω = VA|k?|,
where k?is the component of wavevector parallel to the background magnetic
field. Since only opposite-traveling wave packets interact, k1and k2must have
opposite signs. Then from equations (7) and (8), either k?,1or k?,2must be
equal to 0 and k?,3must be equal to the nonzero initial parallel wavenumber.
That is, zero frequency modes are essential for energy transfer . Therefore,
in the wavevector space, 3-wave interactions produce an energy cascade which is
strictly perpendicular to the mean magnetic field. However, in real turbulence,
equation (8) does not need to be satisfied exactly, but only to within an an error
of order δω ∼ 1/tcas . This implies that the energy cascade is not strictly
perpendicular to B0, although clearly very anisotropic.
It is noteworthy that there has been a claim that “by increasing the mag-
nitude of the mean field in 3-D simulations one finds that the transition from
the isotropic 3-D scaling properties toward those observed in 2-D” . (see also
[7,175,6] for recent development in 2-D MHD turbulence). This claim has yet
to be substantiated, however. We feel that the available numerical simulations
[116,20,101] are reasonably consistent with the Goldreich & Sridhar  model
that we review in the next section. It is also worth noting that the idea of an
anisotropic (perpendicular) cascade has been incorporated into the framework
of the reduced MHD approximation [161,146,113,182,4].
3.3 Goldreich-Sridhar theory
We assume throughout this discussion that the rms turbulent velocity at the
energy injection scale is comparable to the Alfv´ en speed of the mean field and
consider only scales below the energy injection scale. Consequently, we are not
concerned with the problem of magnetic field generation, or the magnetic dy-
namo which is considered elsewhere in this volume
(see also [111,132,72] for reviews, [107,123,12,15] for numerical calculations,
[173,53,15] for suppression of the α dynamo effect in highly conducting fluids,
and [73,11,176,100,185] for recent developments).
An ingenious model very similar in its beauty and simplicity to the Kol-
mogorov model has been proposed by Goldreich & Sridhar  (1995; hereinafter
GS95) for incompressible MHD turbulence. They pointed out that motions per-
pendicular to the magnetic field lines mix them on a hydrodynamic time scale,
i.e. at a rate t−1
cas≈ k⊥vl, where k⊥is the wavevector component perpendicular
to the local mean magnetic field and l ∼ k−1(≈ k−1
ple to the wave-like motions parallel to magnetic field giving a critical balance
where k?is the component of the wavevector parallel to the local magnetic field.
When the typical k?on a scale k⊥falls below this limit, the magnetic field tension
is too weak to affect the dynamics and the turbulence evolves hydrodynamically,
in the direction of increasing isotropy in phase space. This quickly raises the
value of k?. In the opposite limit, when k?is large, the magnetic field tension
dominates, the error δω in the matching conditions is reduced, and the nonlinear
cascade is largely in the k⊥direction, which restores the critical balance.
If conservation of energy in the turbulent cascade applies locally in phase
space then the energy cascade rate (v2
constant. Combining this with the critical balance condition we obtain an
anisotropy that increases with decreasing scale
⊥). These mixing motions cou-
l/tcas) is constant (eq. 1): (v2
and a Kolmogorov-like spectrum for perpendicular motions
vl∝ l1/3, or, E(k) ∝ k−5/3
which is not surprising since the magnetic field does not influence motions that
do not bend it. At the same time, the scale-dependent anisotropy reflects the
fact that it is more difficult for the weaker, smaller eddies to bend the magnetic
GS95 shows the duality of motions in MHD turbulence. Those perpendicular
to the mean magnetic field are essentially eddies, while those parallel to magnetic
field are waves. The critical balance condition couples these two types of motions.
3.4 Weak/Intermediate turbulence.
Let us reconsider the interaction of two wave packets moving oppositely along
the mean magnetic field line. As in equation (4), the energy loss per collision is
(v2)∆t ∼ (v3
10Cho, Lazarian, & Vishniac
where we explicitly distinguish the parallel (l?) and the perpendicular size (l⊥).
The kinetic energy of the eddy is v2
l. Therefore, the ratio of ∆E to E,
characterizes the strength of the nonlinear interaction . In the GS95 theory,
ζl∼ 1. This means that VA≈ V is required at the energy injection scale when
energy injection is isotropic (k⊥,L∼ k?,L). When this condition is satisfied, the
turbulence is called strong turbulence.
There are some astrophysical situations, e.g. the Jovian magnetosphere ,
where the parameter ζLis much smaller than the unity over a broad range of
length scales. Although as noted above the cascade will evolve in the direction of
increasing ζlfor decreasing l, and may reach the strong turbulent regime on very
small scales. In this regime, the parallel cascade is strongly suppressed so the
turbulence is qualitatively different from the strong turbulence discussed above.
This is the weak turbulence regime. We do not discuss this type of turbulence here
due to its limited astrophysical applicability and restricted inertial range. For
more information, see [129,158,122,46,42]. Note that Galtier et al.  obtained
E(k) ∼ k−2
⊥(see also ).
4 Testing and Extending Incompressible Theory
Here we focus on recent direct numerical simulations related to the anisotropic
structure of MHD turbulence. A discussion of earlier pioneering numerical sim-
ulations of MHD turbulence can be found in [140,107].
Despite its attractiveness, the Goldreich-Sridhar model is a conjecture that re-
quires testing. The first such test was done by Cho & Vishniac  who used an
incompressible pseudo-spectral MHD code with 2563grid points. In their simu-
lations, they used VA= B0/√4πρ ∼ V . Their results for eddy shapes are shown
in Figure 4a, which shows a reasonable agreement with the predicted scale-
dependent anisotropy of the turbulence (eq. (10)) in the inertial range (i.e. the
scales between the energy injection scale and the dissipation scale). They also
obtained E(k) ∼ k−5/3in the inertial range. Yet although the velocity does show
the expected scaling, the magnetic field scaling is a bit more uncertain.
A subsequent numerical study by Maron & Goldreich  performed with
a different code and in a different physical regime, namely, for VA ≫ V , also
supported the GS95 model and clarified the role of pseudo-Alfv´ en and Alfv´ en
modes. In particular, they confirmed that the pseudo-Alfv´ en modes are passively
carried down the cascade through interactions with the Alfv´ en modes. They also
showed that passive scalars adopt the same power spectrum as the velocity and
magnetic field fluctuations. In addition, they addressed several issues about the
L ~ 1/kL
averaged magnetic field near eddies under consideration. Eddies at different locations
(e.g. eddies 1 & 1′) can have different local mean fields. Eddies of different sizes (e.g. ed-
dies 1 & 2) can also have different local mean fields. When we do not take into account
the local mean field, calculations hardly reveal true eddy structures. From . (b) An
example of calculating the local mean field. The direction of the local mean field Bl is
obtained by the pair-wise average: Bl= (B(r1) + B(r2))/2.
(a) Eddies and local mean magnetic field. Local mean field is the properly
imbalanced cascade. Overall, they obtained results reasonably consistent with
the GS95 model, but their energy spectra scale slightly differently: E(k) ∝ k−3/2.
They attributed this result to intermittency.
M¨ uller & Biskamp  studied MHD turbulence numerically in the regime of
V ≫ VAand obtained a Kolmogorov spectrum: E(k) ∼ k−5/3. They numerically
studied the scaling exponents and obtained ζ2≈ 0.7 and ζ3≈ 1 (see the definition
of the scaling exponents in the next subsection). The value of ζ2 is consistent
with the energy spectrum.
Other related recent numerical simulations include Matthaeus et al. 
and Milano et al. . Matthaeus et al.  showed that the anisotropy of low
frequency MHD turbulence scales linearly with the ratio of perturbed and total
magnetic field strength b/B (= b/(b2+ B2
In fact we can derive the GS95 scaling using this result. Although their analysis
was based on comparing the strength of a uniform background field and the mag-
netic perturbations on all scales, we can reinterpret this result by assuming that
the strength of random magnetic field at a scale l is bl, and that the background
field is the sum of all contributions from larger scales. Then Matthaeus et al.’s re-
sult becomes a prediction that the anisotropy (k?/k⊥) is proportional to (bl/B). We
can take the total magnetic field strength B ∼ constant as long as the background
field is stronger than the perturbations on all scales. Since bl∼ (kE(k))1/2∼ k−1/3
we obtain an anisotropy (k?/k⊥) proportional to k−1/3
terpretation, smaller eddies are more elongated because they have a smaller bl/B
, and k?∝ k2/3
⊥. In this in-
12Cho, Lazarian, & Vishniac
Fig.2. Cross-sections of the data cube. (Left panels) |bl| in a plane ? to B0. (Right
panels) |bl| in a plane ⊥ to B0. In the left panels, B0 is along the horizontal axis. Large
scale eddies are obtained from the Fourier components with 1 ≤ k < 4. Medium scale
eddies are obtained from the Fourier components with 4 ≤ k < 16. Small scale eddies
are obtained from the Fourier components with 16 ≤ k < 64. The small scale eddies
show a high degree of elongation in the parallel plane. However, they do not show a
systematic behavior in the perpendicular plane.
All in all, numerical simulations so far have been largely, but not perfectly,
consistent with the GS95 theory, e.g. the Kolmogorov-type scaling and the scale-
dependent anisotropy (k?∝ k2/3
which was not included in some of the earlier work, is that the scale dependent
anisotropy can be measured only in a local coordinate frame which is aligned
with the locally averaged magnetic field direction . The necessity of using
a local frame is due to the fact that eddies are aligned along the local mean
magnetic fields, rather than the global mean field B0. Since smaller scale eddies
⊥), and helped to extend it. An important point,
Parallel Distance (grid units)
0246810 121416 1820 22 24 2628 3032
Parallel Distance (grid units)
02468 1012 14 16182022 2426 283032
Fig.3. (a) Velocity correlation function (VCF) from a simulation. The contours repre-
sent shape of different size eddies. The smaller contours (or, eddies) are more elongated.
(b) VCF generated from equation (14). From .
Semi-major Axis (~1/k||)
Semi-minor Axis (~1/k⊥)
02468 10 1214
Figure 3a. The results support k?∝ k2/3
(a) Semi-major axis and semi-minor axis of contours of the VCF shown in
⊥. From . (b) Scaling exponents. From .
are weaker, and more anisotropic, measurements of eddy shape based on a global
coordinate system are always dominated by the largest eddies in the simulation.
Figure 1 illustrates the concept of a local frame and one way to identify it.
Further research in Cho, Lazarian, & Vishniac  (2002a: hereinafter CLV02a)
showed that in the local system of reference the mixing motions perpendicular
to the magnetic field have statistics identical to hydrodynamic turbulence (cf.
M¨ uller & Biskamp ).
Fig. 2 shows the shapes of eddies of different sizes. Left panels show an
increased anisotropy as we move from the top (large eddies) to the bottom (small
eddies). The horizontal axes of the left panels are parallel to B0. Structures in
the perpendicular plane do not show a systematic elongation.
Fig. 3a and Fig. 4a quantify some of these results. The contours of the corre-
lation function obtained in  are shown in Fig. 3a and are consistent with the
predictions of the GS95 model. Fig. 4a shows that the semi-major axis (1/k?)
is proportional to the 2/3 power of the semi-minor axis (1/k⊥), implying that
⊥. While the one dimensional energy spectrum follows Kolmogorov spec-
14Cho, Lazarian, & Vishniac
trum, E(k) ∝ k−5/3, CLV02a showed that the 3D energy spectrum can be fit
P(k⊥,k?) = (B0/L1/3)k−10/3
where B0is the strength of the mean field and L is the scale of energy injection.
The velocity correlation from the 3D spectrum provides an excellent fit to the
numerical data (Fig. 3b). This allows practical applications illustrated in §5.
Intermittency refers to the non-uniform distribution of dissipative structures.
Intermittency has an important dynamical consequence: it affects the energy
spectrum. Highly intermittent turbulent structures were invoked by Falgarone et
al.  and Joulain et al.  as the primary location of endothermic interstellar
Maron & Goldreich  studied the intermittency of dissipation structures
in MHD turbulence using the fourth order moments of the Elsasser fields and
the gradients of the fields. Their simulations show strong intermittent structures.
CLV02a used a different, but complementary, method to study intermittency,
based on the higher order longitudinal structure functions. They found that
by this measure the intermittency of velocity field in MHD turbulence across
local magnetic field lines is as strong as, but not stronger than, hydrodynamic
In fully developed hydrodynamic turbulence, the (longitudinal) velocity struc-
ture functions Sp =< ([v(x + r) − v(x)] · ˆ r)p>≡< δvp
to scale as rζp. For example, the classical Kolmogorov phenomenology (K41)
predicts ζp = p/3. The (exact) result for p=3 is the well-known 4/5-relation:
L(r) >= −(4/5)ǫr, where ǫ is the energy injection rate (or, energy dis-
sipation rate) (see e.g. ). On the other hand, considering intermittency,
She & Leveque (; hereinafter S-L) proposed a different scaling relation:
= p/9 + 2[1 − (2/3)p/3]. Note that the She-Leveque model also implies
So far in MHD turbulence, to the best of our knowledge, there is no rigor-
ous intermittency theory which takes into account scale-dependent anisotropy.
Politano & Pouquet  have developed an MHD version of the She-Leveque
g(1 − x) + C
L(r) > are expected
1 − (1 − x/C)p/g?
where C is the co-dimension of the dissipative structure, g is related to the
scaling vl∼ l1/g, and x can be interpreted as the exponent of the cascade time
tcas ∝ lx. (In fact, g is related to the scaling of Elsasser variable z=v ± b:
zl ∼ l1/g.) In the framework of the IK theory, where g = 4, x = 1/2, and
C = 1 when the dissipation structures are sheet-like, their model of intermittency
= p/8+ 1 − (1/2)p/4. On the other hand, M¨ uller & Biskamp 
MHD Turbulence 15
performed numerical simulations on decaying isotropic MHD turbulence and
obtained Kolmogorov-like scaling (E(k) ∼ k−5/3and t ∼ l2/3) and sheet-like
dissipation structures, which implies g = 3, C = 1, and x = 2/3. From equation
(15), they proposed that
= p/9 + 1 − (1/3)p/3.(16)
The intermittency results from  are shown in Fig. 4b. The filled circles
represent the scaling exponents of longitudinal velocity structure functions in
directions perpendicular to the local mean magnetic field. It is surprising that
the scaling exponents are so close the original (i.e. hydrodynamic) S-L model.
This raises an interesting question. In the simulations of CLV02a, tcas ∝ l2/3
and E(k) ∝ k−5/3scaling is observed. It is observed that MHD turbulence has
sheet-like dissipation structures . Therefore, the parameters for CLV02a
simulations should be the same as those of M¨ uller & Biskamp’s (i.e. g = 3, C = 1,
and x = 2/3) rather than suggesting C = 2. We believe this difference stems from
the different simulation settings (M¨ uller & Biskamp’s turbulence is isotropic and
CLV02a’s is anisotropic) and the way the turbulence is analyzed (global versus
local frame). In fact, we expect the the small scale behavior of MHD turbulence
should not depend on whether the largest scale fields are uniform or have the
same scale of organization as the largest turbulent eddies. Nevertheless, given the
limited dynamical range available in these simulations, it would not be surprising
if the scale of the magnetic field has an impact on the intermittency statistics. It
is not clear how scale-dependent anisotropy changes the intermittency model in
equation (15) and we will not discuss this issue further. Instead, we simply stress
that a striking similarity exists between ordinary hydrodynamic turbulence and
MHD turbulence in perpendicular directions, which further supports the picture
of the GS95 turbulence.11
In Figure 5a, we also plot the scaling exponents (represented by filled squares)
of longitudinal velocity structure functions along directions of the local mean
magnetic field. Although we show only the exponents of longitudinal struc-
ture functions, those of transverse structure functions follow a similar scaling
law. Evidently intermittency along the local mean magnetic field directions is
completely different from the scaling predicted by previous (isotropic) models.
Roughly speaking, the scaling exponents along the directions of local magnetic
field are 1.5 times larger than those of perpendicular directions. This has an
obvious similarity to the scaling of eddy shapes.
The second order exponent ζ2 is related to the the 1-D energy spectra:
E(k⊥) ∝ k−(1+ζ2)
tano, Pouquet, & Carbone  also found ζ2 ∼ 0.7. However, Biskamp, &
Schwarz  obtained ζ2∼ 0.5 from decaying 2-D MHD calculations with B0= 0.
The result of CLV02a suggests that ζ2is closer to 2/3, rather than to 1/2. (It
is not clear whether or not the scaling exponents follow the original S-L model
exactly. At the same time, our calculation shows that the original S-L model can
. Previous 2-D driven MHD calculations for B0= 0 by Poli-
11MG01 attributed the deviation of their spectrum from the Kolmogorov-type to the
intermittency present in their simulation.
16Cho, Lazarian, & Vishniac
05 10 152025
V2 + Brandom
Ev(k) : Kinetic Spectrum
Eb(k) : Magnetic Spectrum
Fig.5. (a) (Left) Imbalanced Decay. When imbalance is large, turbulence decays slow.
From CLV02a. (b) (Right) Viscous damped regime. A new inertial range emerges below
the viscous cut-off at k ∼ 7. From Cho, Lazarian, & Vishniac (2002b).
be a good approximation for our scaling exponents. The S-L model predicts that
ζ2∼ 0.696.) This is equivalent to our earlier claim that our result supports the
scaling law E(k⊥) ∝ k−5/3
results support E(k?) ∝ k−2
at least for velocity. For the parallel directions, the
although the uncertainty is large.
5Damping of Turbulence
5.1 Imbalanced cascade
Turbulence plays a critical role in molecular cloud support and star formation
and the issue of the time scale of turbulent decay is vital for understanding these
processes. If MHD turbulence decays quickly, then serious problems face the re-
searchers attempting to explain important observational facts, e.g. turbulent mo-
tions seen within molecular clouds without star formation  and rates of star
formation . Earlier studies attributed the rapid decay of turbulence to com-
pressibility effects . GS95 predicts and numerical simulations, e.g. CLV02a,
confirm that MHD turbulence decays rapidly even in the incompressible limit.
This can be understood if mixing motions perpendicular to magnetic field lines
are considered. As we discussed earlier, such eddies, as in hydrodynamic turbu-
lence, decay in one eddy turnover time.
Below we consider the effect of imbalance [103,165,50,60,101] on the turbu-
lence decay time scale. Duality of the MHD turbulence means that the turbulence
can be described by colliding wave packets. ‘Imbalance’ means that the flux of
wave packets traveling in one direction is significantly larger than those travel-
ing in the other direction. In the ISM, many energy sources are localized both
in space and time. For example, in terms of energy injection, stellar outflows
are essentially point energy sources. With these localized energy sources, it is
natural that interstellar turbulence be typically imbalanced.
Here we show results of the CLV02a study that demonstrate that imbalance
does extend the lifetime of MHD turbulence (Fig. 5a). We used a run on a grid
of 1443to investigate the decay time scale. For initial conditions we took a data
cube from a driven turbulence simulation. The initial data cube contains both
upward (denoted as +) and downward moving waves (denoted as −). To adjust
the degree of initial imbalance, we either increased or decreased the energy of the
upward moving components components and, by turning off the forcing terms,
let the turbulence decay. Note that the initial energy is normalized to 1. The
y-axis is the normalized total (=up + down) energy.
The dependence of the turbulence decay time on the degree of imbalance
is an important finding. To what degree the results persist in the presence of
compressibility is the subject of our current research. It is obvious that results of
CLV02a are applicable to incompressible, namely, Alfv´ en motions.12We show in
§6.3 that the Alfv´ en motions are essentially decoupled from compressible modes.
As the result we expect that the turbulence decay time may be substantially
longer than one eddy turnover time provided that the turbulence is imbalanced.
5.2Ion-neutral damping: a new regime of turbulence
In hydrodynamic turbulence viscosity sets a minimal scale for motion, with an
exponential suppression of motion on smaller scales. Below the viscous cutoff
the kinetic energy contained in a wavenumber band is dissipated at that scale,
instead of being transferred to smaller scales. This means the end of the hydrody-
namic cascade, but in MHD turbulence this is not the end of magnetic structure
evolution. For viscosity much larger than resistivity, ν ≫ η, there will be a broad
range of scales where viscosity is important but resistivity is not. On these scales
magnetic field structures will be created by the shear from non-damped turbu-
lent motions, which amounts essentially to the shear from the smallest undamped
scales. The created magnetic structures would evolve through generating small
scale motions. As a result, we expect a power-law tail in the energy distribution,
rather than an exponential cutoff. To our best knowledge, this is a completely
new regime for MHD turbulence.
In partially ionized gas neutrals produce viscous damping of turbulent mo-
tions. In the Cold Neutral Medium (see Draine & Lazarian  for a list of the
idealized phases) this produces damping on the scale of a fraction of a parsec.
The magnetic diffusion in those circumstances is still negligible and exerts an
influence only at the much smaller scales, ∼ 100km. Therefore, there is a large
range of scales where the physics of the turbulent cascade is very different from
the GS95 picture.
12In the long run, the imbalance will be defeated by the parametric instability, which
develops through formation of density inhomogeneities within the beam of Alfv´ en
waves [26,47,62,25]. However, this instability takes many wave periods to be estab-
lished. A similar argument can be applied when we consider completely imbalanced
cascade. That is, even in the completely imbalanced cascade decay of energy can
occur due to non-linear steepening if waves. But, this will be very slow for Alfv´ en
18 Cho, Lazarian, & Vishniac
Cho, Lazarian, & Vishniac  have explored this regime numerically. Here
we used a grid of 3843and a physical viscosity for velocity damping. The kinetic
Reynolds number is around 100. With this Reynolds number, viscous damping
occurs around k ∼ 7, or about ∼ 1/7 of the width of the computational box.
We minimized magnetic diffusion through the use of a hyper-diffusion term of
order 3. To test for possible “bottle neck” effects we also did simulations with
normal magnetic diffusion and reproduced similar results but only over a re-
duced dynamical range. The bottleneck effect is a common feature in numerical
hydrodynamic simulations with hyperviscosity. See  for MHD simulations.
In Fig. 5b, we plot energy spectra. The spectra consist of several parts. First,
the peak of the spectra corresponds to the energy injection scale. Second, for
2 < k < 7, kinetic and magnetic spectra follow a similar slope. This part is
more or less a severely truncated inertial range for undamped MHD turbulence.
Third, the magnetic and kinetic spectra begin to decouple at k ∼ 7. Fourth, after
k ∼ 20, a new damped-scale inertial range emerges. In the new inertial range,
magnetic energy spectrum follows
Eb(k) ∝ k−1,(17)
implying considerable magnetic structures below the viscous damping scale. The
velocity power spectrum steepens in this regime, but does not fall exponentially.
Figure 6a shows that the small magnetic structures are highly intermittent
in the viscous-damped regime. Here we obtained the small scale magnetic field
by eliminating Fourier modes with k < 20. We can see that the typical radius of
curvature of field lines in the plane is much larger than the typical perpendicular
scale for field reversal. The typical radius of curvature of field lines corresponds to
the viscous damping scale, indicating that stretched structures are results of the
shearing motions at the viscous damping scale. There is no preferred direction for
these elongated structures. A similar plot for ordinary (not viscously damped)
MHD turbulence (Fig. 6b) shows much less intermittent structures. We remove
Fourier modes with k < 20 also in Fig. 6b.
A theoretical model for this new regime and its consequences for stochastic
reconnection  will be found in an upcoming paper (Lazarian, Vishniac, & Cho
2002). Here we summarize the main points of the model. We begin by noting
that the strong intermittency seen in this regime suggests a new parameter, fl,
which is the volume filling fraction of structures with scales comparable to l. This
in turn implies that we need to distinguish between volume averaged means and
typical values of velocity and magnetic field perturbations in fraction of space
where they are concentrated. We denote the latter with a ‘ˆ’, so that v2
l. This model does not include information about the range of field
strengths or velocities with structure on a scale l in the bulk of the volume, aside
from assuming that they are sufficiently weak that they do not contribute to any
The second new fundamental parameter in this model is the eddy turn over
rate at the damping scale, kd, i.e.
l= flˆ v2
∼ kdvd∼ k2
Fig.6. (Left) Viscous-damped turbulence. Strength of magnetic field in a plane per-
pendicular to B0. Arrows are magnetic fields in the plane. Only a part of the plane is
shown. Note highly intermittent structures. From . (Right) Same as Left, but for
ordinary (not viscous-damped) MHD turbulence. Structures are less intermittent.
where vdis the velocity at ld∼ 1/kdand νnis the viscosity of the plasma due
to neutral particles, which differs from the viscosity of neutral fluid by the ratio
of atomic to total densities.
Since motions on smaller scales are strongly damped, the cascade of magnetic
energy to smaller scales is due to motions on the damping scale. This implies
so that bl is maintained approximately at the level of magnetic field at the
damping scale, bd.
This folding and refolding, perpendicular to the mean field direction, has a
weak effect on the field line curvature. (This result is also seen in the simulations.)
The decrease in the structure length l is due to an increase in the magnetic
field gradient perpendicular to the mean field direction. The resulting magnetic
pressure gradients are balanced by plasma pressure gradients. Thus on the scales
below the viscous cutoff, the tension forces are balanced by viscous drag, i.e.
l2ˆ vl∼ max[kdˆbl,k?B0]ˆbl.(20)
Finally, this dynamic equilibrium can be maintained only if the small scale
motions are strong enough to counteract the shear, τ−1
Combining these results we see that fl ∼ kdl,ˆbl ∼ bd(kdl)−1/2, and vl ∝
l3/2. These scaling laws are at least qualitatively consistent with the simulation
results, although the velocity power spectrum may be slightly steeper than the
s . In other words, ˆ vl/l ∼
20 Cho, Lazarian, & Vishniac
Unfortunately, a realistic treatment of the ISM requires an explicit recogni-
tion of the two fluid nature of the partially ionized plasma, rather than simply
representing neutral drag with an effective viscosity. Here we are beyond the
reach of available simulations, and need to rely on an extension of the scaling
arguments given above.
First, at sufficiently small values of l, the ambipolar diffusion time
than τs, and the magnetic pressure gradients will be supported entirely by the
ionized particle pressure. This means that only a fraction, ∼ ρi/ρtot, of the
energy will continue the cascade to smaller scales.
Next, for some ion-neutral collision time, tin, there will be some decoupling
scale, lc, where
l/ρtot)(ρn/ρi),where tin = ion-neutral collision time) will become less
At smaller scales the ions will be dragged through a more or less uniform neutral
background. The argument above needs to be modified by replacing the viscous
drag coefficient in equation (20) with the right hand side of equation (21). This
Finally, at some sufficiently small scale the filling factor will rise to unity,
and the gradients in the magnetic field will become strong enough that neutral
drag can be ignored. These conditions are satisfied simultaneously when
kdl ∼ tin/τs.(24)
Below this scale we expect to see a resumption of the turbulent cascade, now
involving only the ionized component of the plasma, down to scales where plasma
resistivity and viscosity finally dissipate the remaining energy. Since the longest
cascade time for this regime is tin≪ τs, we expect this small scale turbulence
to be intermittent, with a duty cycle ∼ tin/τs.
All the consequences of the new regime of the MHD turbulence have not yet
been appreciated, but we expect that it will have a substantial impact on our
understanding of the interstellar physics. Moreover, the treatment given above
actually applies only when ρi/ρ is not very small. Otherwise the decoupling scale
can be larger than the viscous damping scale.
For the rest of the review, we consider MHD turbulence of a single conducting
fluid. While the GS95 model describes incompressible MHD turbulence well,
no universally accepted theory exists for compressible MHD turbulence despite
various attempts (e.g., ). Earlier numerical simulations of compressible MHD
turbulence covered a broad range of astrophysical problems, such as the decay
of turbulence (e.g. [98,160]) or turbulent modeling of the ISM (see recent review
Fig.7. (a) Directions of fast and slow basis vectors.ˆξf andˆξs represent the directions
of displacement of fast and slow modes, respectively. In the fast basis (ˆξf) is always
betweenˆk andˆk⊥. In the slow basis (ˆξs) lies betweenˆθ andˆB0. Here,ˆθ is perpendicular
toˆk and parallel to the wave front. All vectors lie in the same plane formed by B0
and k. On the other hand, the displacement vector for Alfv´ en waves (not shown) is
perpendicular to the plane. (b) Directions of basis vectors for a very small β drawn
in the same plane as in (a). The fast bases are almost parallel toˆk⊥. (c) Directions
of basis vectors for a very high β. The fast basis vectors are almost parallel to k. The
slow waves become pseudo-Alfv´ en waves.
; see also [133,168,134,169] for earlier 2D simulations and [125,126,131,67,9]
for recent 3D simulations). In what follows, we concentrate on the fundamental
properties of compressible MHD.
6.1Alfv´ en, slow, and fast modes
Let us start by reviewing different MHD waves. In particular, we describe the
Fourier space representation of these waves. The real space representation can
be found in papers on modern shock-capturing MHD codes (e.g. [13,147]). For
the sake of simplicity, we consider an isothermal plasma. Figure 7 and Figure 8
Table 1. Notations for compressible turbulence
a, cs, cf, VA
δV , (δV )s, (δV )f, (δV )A
sound, slow, fast, and Alfv´ en speed
random (rms) velocity
Previously we used V for the rms velocity
velocity at scale l
velocity vector at wavevector k
vl, (vl)s, (vl)f, (vl)A
vk, (vk)s, (vk)f, (vk)A
ˆB0 (=ˆk?),ˆk⊥,ˆk,ˆθ, ...
22Cho, Lazarian, & Vishniac
Fig.8. Waves in real space. We show the directions of displacement vectors for a slow
wave (left) and a fast wave (right). Note thatˆξs lies betweenˆθ andˆB0 (=ˆk?) and
ˆξf betweenˆk andˆk⊥. Again,ˆθ is perpendicular toˆk and parallel to the wave front.
Note also that, for the fast wave, for example, density (inferred by the directions of the
displacement vectors) becomes higher where field lines are closer, resulting in a strong
restoring force, which is why fast waves are faster than slow waves.
give schematics of slow and fast waves. For slow and fast waves, B0, vk(∝ ξ),
and k are in the same plane. On the other hand, for Alfv´ en waves, the velocity
of the fluid element (vk)Ais orthogonal to the B0− k plane.
As before, the Alfv´ en speed is VA = B0/√4πρ0, where ρ0 is the average
density. Fast and slow speeds are
where θ is the angle between B0and k. See Table 1 for the definition of other
variables. When β (β = Pg/PB=2a2/V2
sure; hereinafter β = average β ≡¯Pg/¯PB) goes to zero, we have
cs ≈ acosθ.
Figure 7 shows directions of displacement (or, directions of velocity) vectors
for these three modes. We will call them the basis vectors for these modes.
The Alfv´ en basis is perpendicular to bothˆk andˆB0, and coincides with the
azimuthal vectorˆφ in a spherical-polar coordinate system. Here hatted vectors
A; Pg= gas pressure, PB= magnetic pres-
are unit vectors. The fast basisˆξf lies betweenˆk andˆk⊥:
ˆξf∝1 −√D + β/2
1 +√D − β/2
where D = (1+β/2)2−2β cos2θ, and β is the averaged β (=¯Pg/¯PB). The slow
basisˆξslies betweenˆθ andˆB0(=ˆk?):
ˆξs∝ k?ˆk?+1 −√D − β/2
1 +√D + β/2
The two vectorsˆξf andˆξsare mutually orthogonal. Proper normalizations are
required for both bases to make them unit-length.
When β goes to zero (i.e. the magnetically dominated regime),ˆξf becomes
parallel toˆk⊥ andˆξs becomes parallel toˆB0 (Fig. 7b). The sine of the angle
betweenˆB0 andˆξs is (β/2)sinθ cosθ. When β goes to infinity (i.e. gas pres-
sure dominated regime)13,ˆξf becomes parallel toˆk andˆξsbecomes parallel to
ˆθ (Fig. 7c). This is the incompressible limit. In this limit, the slow mode is
sometimes called the pseudo-Alfv´ en mode .
Here we address the issue of mode coupling in a low β plasma. It is reasonable
to suppose that in the limit where β ≫ 1 turbulence for Mach numbers (Ms=
δV/a) less than unity should be largely similar to the exactly incompressible
regime. Thus, Lithwick & Goldreich  conjectured that the GS95 relations
are applicable to slow and Alfv´ en modes with the fast modes decoupled. They
also mentioned that this relation can carry on for low β plasmas. For β ≫ 1
and Ms> 1, we are in the regime of hydrodynamic compressible turbulence for
which no theory exists, as far as we know.
In the diffuse interstellar medium β is typically less than unity. In addition, it
is ∼ 0.1 or less for molecular clouds. There are a few simple arguments suggest-
ing that MHD theory can be formulated in the regime where the Alfv´ en Mach
number (≡ δV/VA) is less than unity, although this is not a universally accepted
assumption. Alfv´ en modes describe incompressible motions. Arguments in GS95
are suggestive that the coupling of Alfv´ en to fast and slow modes will be weak.
Consequently, we expect that in this regime the Alfv´ en cascade should follow
the GS95 scaling. Moreover the slow waves are likely to evolve passively . For
a ≪ VAtheir nonlinear evolution should be governed by Alfv´ en modes so that
we expect the GS95 scaling for them as well. The phase velocity of Alfv´ en waves
and slow waves depend on a factor of cosθ and this enables modulation of the
slow waves by the Alfv´ en ones. However, fast waves do not have this factor and
therefore cannot be modulated by the changes of the magnetic field direction
13In this section, we assume that external mean field is strong (i.e. VA > (δV )) but
finite, so that β → ∞ means the gas pressure¯Pg → ∞.
24Cho, Lazarian, & Vishniac
Alfv´ enic regime (shaded region). In this figure, Vtot
ible MHD regime. Similarly, for compressible case, the small scales of super-Alfv´ enic
compressible turbulence are expected to fall in the sub-Alfv´ enic compressible regime.
Moreover, winding of magnetic field by turbulence increases the magnetic field energy
and the super-Alfv´ enic turbulence becomes more and more magnetically dominated
with V/VA → 1. Region B is the region where, within the density fluctuations, the
velocities can get super-Alfv´ enic. In the figure, we used equation (33) to determine the
borderline between region A and B.
Different regimes of MHD turbulence. We consider the compressible sub-
represents the total Alfv´ en speed
0+ δB2/√4πρ). V ≡ Vflow= δV . Cho & Vishniac  argued that, even when
the external field is weak, small scales can follow the GS95-like scaling in incompress-
associated with Alfv´ en waves. The coupling between the modes is through the
modulation of the local Alfv´ en velocity and therefore is weak.
For Alfv´ en Mach number (MA) larger than unity a shock-type regime is
expected. However, generation of magnetic field by turbulence  is expected
for such a regime. It will make the steady state turbulence approach MA∼ 1.14
Therefore in Cho & Lazarian  we consider turbulence in the limit Ms> 1,
MA< 1, and β < 1 (Fig. 9). For these simulations, we mostly used Ms∼ 2.2,
MA∼ 0.7, and β ∼ 0.2. The Alfv´ en speed of the mean external field is similar
to the rms velocity (VA = 1,δV ∼ 0.7,a =
equation of state.
√0.1), and we used an isothermal
14We suspect that simulations that show super-Alfv´ enic turbulence is widely spread
in the ISM might not evolve for a long enough time to reach the steady state.
Fig.10. (Left) Decay of Alfv´ enic turbulence. The generation of fast and slow waves is
not efficient. β ∼ 0.2, Ms ∼ 3. (Right) The ratio of (δV )2
external field (B0) is, the more suppressed the coupling is. The ratio is not sensitive to
β. From 
fto (δV )2
A. The stronger the
Although the scaling relations presented below are applicable to sub-Alfv´ enic
turbulence, we cautiously speculate that small scales of super-Alfv´ enic turbu-
lence might follow similar scalings. Boldyrev, Nordlund, & Padoan  obtained
energy spectra close to E(k) ∼ k−1.74in solenoidally driven super-Alfv´ enic su-
personic turbulence simulations. The spectra are close to the Kolmogorov spec-
trum (∼ k−5/3), rather than shock-dominated spectrum (∼ k−2). This result
might imply that small scales of super-Alfv´ enic MHD turbulence can be de-
scribed by our sub-Alfv´ enic model presented below, which predicts Kolmogorov-
type spectra for Alfv´ en and slow modes.
6.3Coupling of MHD modes and Scaling of Alfv´ en modes
Alfv´ en modes are not susceptible to collisionless damping (see [156,110] and ref-
erences therein) that damps slow and fast modes. Therefore, we mainly consider
the transfer of energy from Alfv´ en waves to compressible MHD waves (i.e. to
the slow and fast modes).
In Cho & Lazarian , we carry out simulations to check the strength of the
mode-mode coupling. We first obtain a data cube from a driven compressible
numerical simulation with B0/√4πρ0 = 1. Then, after turning off the driving
force, we let the turbulence decay. We go through the following procedures be-
fore we let the turbulence decay. We first remove slow and fast modes in Fourier
space and retain only Alfv´ en modes. We also change the value of B0preserving
its original direction. We use the same constant initial density ρ0for all simula-
tions. We assign a new constant initial gas pressure Pg
15. After doing all these
The changes of both B0 and Pg preserve the Alfv´ en character of perturbations.
In Fourier space, the mean magnetic field (B0) is the amplitude of k = 0 compo-
nent. Alfv´ en components in Fourier space are for k ?= 0 and their directions are
parallel/anti-parallel toˆξA (=ˆB0×ˆk⊥). The direction ofˆξA does not depend on the
magnitude of B0 or Pg.
26 Cho, Lazarian, & Vishniac
Fig.11. (a) Alfv´ en spectra follow a Kolmogorov-like power law. (b) The second-order
structure function (SF2 =< v(x + r) − v(x) >) for Alf´ ven velocity shows anisotropy
similar to the GS95. Conturs represent eddy shapes. From .
procedures, we let the turbulence decay. We repeat the above procedures for
different values of B0and Pg. Fig. 10a shows the evolution of the kinetic energy
of a simulation. The solid line represents the kinetic energy of Alfv´ en modes. It
is clear that Alfv´ en waves are poorly coupled to the compressible modes, and
do not generate them efficiently16Therefore, we expect that Alfv´ en modes will
follow the same scaling relation as in the incompressible case. Fig. 10b shows
that the coupling gets weaker as B0increases:
The ratio of (δV )2
This marginal coupling is in good agreement with a claim in GS95, as well as
earlier numerical studies where the velocity was decomposed into a compressible
component vCand a solenoidal component vS. The compressible component is
curl-free and parallel to the wave vector k in Fourier space. The solenoidal com-
ponent is divergence-free and perpendicular to k. The ratio χ = (δV )C/(δV )S
is an important parameter that determines the strength of any shock [133,141].
Porter, Woodward, & Pouquet  performed a hydrodynamic simulation of
decaying turbulence with an initial sonic Mach number of unity and found that
χ2evolves toward ∼ 0.11. Matthaeus et al.  carried out simulations of de-
caying weakly compressible MHD turbulence  and found that χ2∼ O(M2
where Msis the sonic Mach number. In  a weak generation of compressible
components in solenoidally driven super-Alfv´ enic supersonic turbulence simula-
tions was obtained.
Fig. 11 shows that the spectrum and the anisotropy of Alfv´ en waves in this
limit are compatible with the GS95 model:
sto (δV )2
Ais proportional to (δV )2
Spectrum of Alfv´ en Modes:
E(k) ∝ k−5/3
16As correctly pointed out by Zweibel (this volume) there is always residual coupling
between Alfv´ en and compressible modes due to steepening of Alfv´ en modes. However,
this steepening happens on time-scales much longer than the cascading time-scale.
Fig.12. (a) Slow spectra also follow a Kolmogorov-like power law. (b) Slow modes
show anisotropy similar to the GS95 theory. From .
and scale-dependent anisotropy k?∝ k2/3
that is compatible with the GS95
6.4 Scaling of the slow modes
Slow waves are somewhat similar to pseudo-Alfv´ en waves (in the incompressible
limit). First, the directions of displacement (i.e. ξs) of both waves are similar
when anisotropy is present. The vector ξsis always betweenˆθ andˆk?. In Figure
7, we can see that the angle betweenˆθ andˆk?gets smaller when k?≪ k⊥.
Therefore, when there is anisotropy (i.e. k?≪ k⊥),ˆξs of a low β plasma be-
comes similar to that of a high β plasma. Second, the angular dependence in the
dispersion relation cs≈ acosθ is identical to that of pseudo-Alfv´ en waves (the
only difference is that, in slow waves, the sound speed a is present instead of the
Alfv´ en speed VA).
Goldreich & Sridhar  argued that the pseudo-Alfv´ en waves are slaved to
the shear-Alfv´ en (i.e. ordinary Alfv´ en) waves in the presence of a strong B0,
meaning that the energy cascade of pseudo-Alfv´ en modes is primarily mediated
by the shear-Alfve´ en waves. This is because pseudo-Alfv´ en waves do not provide
efficient shearing motions. Similar arguments are applicable to slow waves in a
low β plasma  (see also  for high-β plasmas). As a result, we conjecture
that slow modes follow a scaling similar to the GS95 model :
Spectrum of Slow Modes:
Es(k) ∝ k−5/3
Fig. 12a shows the spectra of slow modes. For velocity, the slope is close
to −5/3. Fig. 12b shows the contours of equal second-order structure function
(SF2) of slow velocity, which are compatible with k?∝ k2/3
In low β plasmas, density fluctuations are dominated by slow waves .
From the continuity equation ˙ ρ = ρ∇ · v
ωρk= ρ0k · vk,
28 Cho, Lazarian, & Vishniac
we have, for slow modes, (ρk)s∼ ρ0(vk)s/a. Hence, this simple argument implies
where we assume that (δV )s ∼ (δV )A and Ms is the Mach number. On the
other hand, only a small amount of magnetic field is produced by the slow
waves. Similarly, using the induction equation (ωbk= k × (B0× vk)), we have
which means that equipartition between kinetic and magnetic energy is not guar-
anteed in low β plasmas. In fact, in Fig. 12a, the power spectrum for density
fluctuations has a much larger amplitude than the magnetic field power spec-
trum. Since density fluctuations are caused mostly by the slow waves and mag-
netic fluctuation is caused mostly by Alfv´ en and fast modes, we do not expect
a strong correlation between density and magnetic field, which agrees with the
ISM simulations [130,126,170].
6.5Scaling of the fast modes
Figure 13 shows fast modes are isotropic. The resonance conditions for interact-
ing fast waves are:
ω1+ ω2= ω3,
k1+ k2= k3.
Since ω ∝ k for the fast modes, the resonance conditions can be met only when
all three k vectors are collinear. This means that the direction of energy cascade
is radial in Fourier space, and we expect an isotropic distribution of energy in
Using the constancy of energy cascade and uncertainty principle, we can
derive an IK-like energy spectrum for fast waves. The constancy of cascade rate
On the other hand, tcascan be estimated as
= constant. (37)
(v · ∇v)k
If contributions are random, the denominator can be written by the square root
of the number of interactions (√N) times strength of individual interactions
To be exact, the strength of individual interactions is ∼ kv2
angle between k and B0. Thus marginal anisotropy is expected. It will be investigated
k)17. Here we assume locality of interactions: p ∼ q ∼ k. Due to the
ksinθ, where θ is the
Fig.13. (a) The power spectrum of fast waves is compatible with the IK spectrum.
(b) The magnetic second-order structure function of fast modes shows isotropy.From
uncertainly principle, the number of interactions becomes N ∼ k(∆k)2, where
∆k is the typical transversal (i.e. not radial) separation between two wave vec-
tors p and q (with p + q = k). Therefore, the denominator of equation (38) is
k. We obtain an independent expression for tcasfrom the uncer-
tainty principle (tcas∆ω ∼ 1 with ∆ω ∼ ∆k(∆k/k)). From this and equation
(38), we get tcas∼ t1/2
Combining equations (37) and (39), we obtain v2
k−3/2. This is very similar to acoustic turbulence, turbulence caused by inter-
acting sound waves [180,181,97]. Zakharov & Sagdeev  found E(k) ∝ k−3/2.
However, there is debate about the exact scaling of acoustic turbulence. Here we
cautiously claim that our numerical results are compatible with the Zakharov &
cas/(k2vk), which yields
k∼ k−7/2, or Ef(k) ∼ k2v2
Spectrum of Fast Modes:
Ef(k) ∼ k−3/2.(40)
Magnetic field perturbations are mostly affected by fast modes  when β
if (δV )A∼ (δV )s.
The turbulent cascade of fast modes is expected to be slow and in the absence
of collisionless damping they are expected to propagate in turbulent media over
distances considerably larger than Alfv´ en or slow modes. This effect is difficult to
observe in numerical simulations with ∆B ∼ B0. A modification of the spectrum
in the presence of the collisionless damping is presented in .
Many astrophysical problems require some knowledge of the scaling properties of
turbulence. Therefore we expect a wide range of applications of the established
30Cho, Lazarian, & Vishniac
scaling relations. Here we show how recent breakthroughs in understanding MHD
turbulence affect a few selected issues.
7.1Cosmic ray propagation
The propagation of cosmic rays is mainly determined by their interactions with
electromagnetic fluctuations in interstellar medium. The resonant interaction of
cosmic ray particles with MHD turbulence has been repeatedly suggested as the
main mechanism for scattering and isotropizing cosmic rays. In these analysis it
is usually assumed that the turbulence is isotropic with a Kolmogorov spectrum
(see ). How should these calculations be modified?
The essence of the mechanism is rather simple. Particles moving with velocity
v interact with a resonant Alfv´ en wave of frequency ω = k?vµ+nΩ (n = ±1,2...),
where Ω = Ω0/γ is the gyrofrequency of relativistic particles, µ is the cosine of
the pitch angle. From the resonant condition above, we know that the most
important interaction occurs at k? ∼ Ω/vµ ∼ (µrL)−1, where rL is Larmor
radius of the high-energy particles.
The calculations in  that made use of tensor (14) provided the scattering
efficiency of anisotropic Alfv´ enic turbulence. The results are compared in Fig. 14a
with the predictions of the scattering on isotropic Kolmogorov-type magnetic
fluctuations and also with earlier calculations by Chandran . The latter used
rather ad hoc form of the tensor to describe magnetic fluctuations within the
Goldreich-Sridhar theory. We see from Fig 14a that the scattering is substantially
suppressed, compared to the Kolmogorov turbulence that is usually used for
scattering calculations. This happens, first of all, because most turbulent energy
in GS95 turbulence goes to k⊥ so that there is much less energy left in the
resonance point k?= (µrL)−1. Furthermore, k⊥≫ k?means k⊥≫ r−1
cosmic ray particles cover many eddies during one gyration. This random walk
decreases the scattering efficiency by a factor of (Ω/k⊥v⊥)
l⊥is the turbulence scale perpendicular to magnetic field.
Thus the gyroresonance with Alfv´ enic turbulence is not an effective scattering
mechanism for cosmic rays if turbulence is injected on the large scales, since the
degree of anisotropy increases on smaller scales. However, if energy is injected
isotropically at small scales, the resulting turbulence would be more isotropic
and scattering will be more efficient. Scattering by undamped fast modes is more
efficient than the Kolmogorov theory would predict. Yan & Lazarian  per-
formed calculations taking into account the collisionless damping of fast modes
and showed that the gyroresonance scattering by fast modes is the dominant
There is another important property of turbulence that was neglected in
earlier work. When cosmic rays stream at a velocity much larger than Alfv´ en
velocity, they can excite resonant MHD waves, which in turn scatter cosmic rays.
This is the ‘streaming instability’. It is usually assumed that this instability can
confine cosmic rays with energies less than 100GeV . However, this is true
only in an idealized situation when there is no background MHD turbulence. As
noted earlier, the rates of turbulent decay are very fast and excited perturbations
2 = (rL/l⊥)
Grain size a(cm)
Grain velocity v(cm/s)
Grain velocities induced by turbulence
Fig.14. Applications. (By Lazarian & Yan). Left: Cosmic ray scattering for realistic
MHD turbulence is reduced substantially compared to scattering by isotropic turbu-
lence, but still larger than estimates in Chandran . Right: The dust acceleration by
turbulence is reduced compared to the accepted estimates in Draine .
should vanish quickly. In  we find that the streaming instability is only
applicable to particles with energies < 0.15GeV, which is less than the energy
of most cosmic ray particles. This result casts doubt on the self-confinement
mechanism discussed by previous authors.
All these findings tend to support the alternative picture of cosmic ray diffu-
sion advocated by Jokipii (see ). In this picture cosmic rays follow magnetic
field lines, but the magnetic field wanders. The rate of this wandering can be
calculated from the established turbulence scaling laws.
Knowledge of the scattering rates is essential for understanding both the first
order and the second order Fermi acceleration. The first order Fermi acceleration,
may be important for a wide range of phenomena from clusters of galaxies and
gamma-ray bursts (GRBs) to solar flares. Results obtained in  where the
discovered properties of MHD turbulence are used proved to be very different
from earlier estimates.
Turbulence induces relative dust grain motions and leads to grain-grain colli-
sions. These collisions determine grain size distribution, which affects most dust
properties, including absorption and H2formation. Unfortunately, as in the case
of cosmic rays, earlier work appealed to hydrodynamic turbulence to predict
grain relative velocities (see [74,166,33,124,167,38]).
The differences between the hydrodynamic and MHD calculations stem from
(a) grain charges, which couple grains to the magnetic field, (b) the anisotropy
of MHD cascade, and (c) the direct interaction of charged grains with magnetic
perturbations. Effects (a) and (b) are considered in Lazarian & Yan , while
(c) is considered in Yan & Lazarian (2002b; in preparation). As consequence the
picture of grain dynamics is substantially altered.
32Cho, Lazarian, & Vishniac
Consider grain charge first. If a grain’s Larmor period τL= 2πmgrc/qB is
shorter than the gas drag time tdrag, grain perpendicular motions are constrained
by magnetic field. Their velocity dispersion is determined by the turbulence eddy
whose turnover period is ∼ τLinstead of the drag time .
Accounting for the anisotropy of MHD turbulence it is convenient to consider
separately grain motions parallel and perpendicular to magnetic field. The per-
pendicular motion is influenced by the Alfv´ en modes, which have a Kolmogorov
spectrum. The parallel motion is subjected to compressible modes which scale
as v?∝ k−1/2
turnover time is of the order of tdamp ∼ ν−1
damped. Thus grains sample only a part of the eddy before gaining the velocity
of the ambient gas if τLor tdrag< tdamp. The results are shown in Fig. 14b.
The direct interaction of the charged grains with turbulent magnetic field
results in a stochastic acceleration that can potentially provide grains with su-
. In addition we should account for viscous forces. When the eddy
⊥, the turbulence is viscously
7.3 Turbulence in HII regions
Lithwick & Goldreich  addressed the issue of the origin of density fluctu-
ations within HII regions. There the gas pressure is larger than the magnetic
pressure (the ‘high beta’ regime) and they conjectured that fast waves, which
are essentially sound waves, would be decoupled from the rest of the cascade.
They found that density fluctuations are due to the slow mode and the entropy
mode, which are passively mixed by shear Alfv´ en waves and follow a Kolmogorov
spectrum. They also found that slow mode density fluctuations are proportional
to 1/√β. On the other hand, the entropy mode density fluctuations are sup-
pressed when cooling is faster than the cascade time. Lithwick & Goldreich 
also gave detailed discussions about density fluctuations on various scales in the
ISM, e.g. proton gyro-radius. These results are important as radio-wave scin-
tillation observations can constrain the nature of MHD turbulence in the ISM,
especially in the HII regions. Lithwick & Goldreich  argued that the turbu-
lent cascade survives ion-neutral damping only when a high degree of ionization
is present. However, the study by Cho, Lazarian & Vishniac  suggests that
the magnetic fluctuations protrude below the damping scale and the results of
 should be revised.
7.4Tiny-Scale Atomic Structures
The intermittent small scale structures in §5.2 should have important implica-
tions for transport processes (heat, cosmic rays, etc.) in partially ionized plasmas.
We also speculate that they might have some relation to the tiny-scale atomic
structures (TSAS). Heiles  introduced the term TSAS for the mysterious
H I absorbing structures on scales from thousands to tens of AU, discovered by
Dieter, Welch & Romney . Analogs are observed in NaI and CaII [108,36,1]
and in molecular gas . Recently Deshpande, Dwarakanath & Goss  an-
alyzed channel maps of opacity fluctuations toward Cas A and Cygnus A. They
found that the amplitudes of density fluctuations at scales less than 0.1 pc are
far larger than expected from extrapolation from larger scales, possibly explain-
ing TSAS. This study, however, cannot answer what confines those presumably
overpressured (but very quiescent!) blobs of gas. Deshpande  related those
structures to the shallow spectrum of interstellar turbulence.
Figure 5b indicates that while velocity decreases rapidly, but not exponen-
tially, below the viscous damping scale, the magnetic field fluctuations persist,
thereby providing nonthermal pressure support. Magnetic structures perpendic-
ular to the mean magnetic field are compensated by pressure gradients. Our
calculations so far are produced using incompressible code . In the case of
compressible media, we expect the pressure fluctuations to entail density fluctu-
ations reminiscent of the Deshpande et al.  observations.
The calculations in Cho, Lazarian & Vishniac  are applicable on scales
from the viscous damping scale (determined by equating the energy transfer
rate with the viscous damping rate; ∼ 0.1 pc in the Warm Neutral Medium
with n = 0.4 cm−3, T= 6000 K) to the ion-neutral decoupling scale (the scale
at which viscous drag on ions becomes comparable to the neutral drag; ≪ 0.1
pc). Below the viscous scale the fluctuations of magnetic field obey the damped
regime shown in Figure 5b and produce density fluctuations. For typical Cold
Neutral Medium gas, the scale of neutral-ion decoupling decreases to ∼ 70AU,
and is less for denser gas. TSAS may be created by strongly nonlinear MHD
A simple technique of estimating magnetic field was suggested by Chan-
drasekhar & Fermi  (see also review by Ostriker, this volume). According
to it, the fluctuations of magnetic field that can be measured from polarization
maps are related to velocity fluctuations measured through Doppler broadening
δb/√4πρ ∼ δv. The existence of the damped regime of MHD turbulence suggests
that this technique is not applicable to very small scales in partially ionized gas.
Magnetic reconnection is the fundamental process that allows magnetic fields
to change their topology, despite being ‘frozen’ into highly conducting plasmas.
It is the key process for solar flares, the magnetic dynamo, the acceleration of
energetic particles, etc. According to the Lazarian & Vishniac model  (see
also review  and Vishniac, Lazarian, & Cho, this volume) of stochastic recon-
nection, this process is controlled by the turbulent wandering of magnetic field.
The exact properties of the turbulent cascade are especially important for the
viscously damped regime present in partially ionized gas. However, it is shown in
Lazarian, Vishniac & Cho  that the reconnection rates are sufficiently high
in this case. The implications of the finding for the removal of magnetic flux
during star formation is to be evaluted yet.
34Cho, Lazarian, & Vishniac
7.6 Support and Compression of Molecular Clouds
To understand the dynamics of molecular clouds and star formation it is nec-
essary to understand turbulence. In a recent review  McKee pointed out
that the fast damping of MHD turbulence observed in numerical simulations is
difficult to reconcile with the fact that “a GMC such as G216, which has no
visible star formation, can have a level of turbulence that exceeds that in the
Rosette molecular cloud, which has an embedded OB association”. He pointed
out that the conclusions obtained on the basis of numerics should be treated
with caution as they do not resolve the microscales.
In typical astrophysical conditions the sources of turbulence are localized
both in space (stars) and time (stellar outflows; supernovae), and the outgoing
waves have much larger amplitudes than the background waves (we call this
situation “imbalanced cascade”). Fig. 5a shows that the turbulent damping could
be substantially reduced in this situation. Moreover, even in a balanced regime,
we expect fast modes to be subjected to slow non-linear damping.
At the same time, it worth mentioning, that turbulence can not only sup-
port, but also compress molecular clouds. Clouds can be compressed by external
turbulence feeding into them and depositing energy and momentum. Myers &
Lazarian  explained observed infalling motions of molecular gas surround-
ing dense cores [164,94] in this way, based on ion-neutral damping. The infall
rate is proportional to the rate of turbulence damping. Therefore, fast non-linear
damping associated with the Alfv´ enic turbulence should enhance the infall.
7.7 Heating of Diffuse Ionized Gas
The “Diffuse Ionized Gas” (DIG), or equivalently the “Reynolds layer” within
the Milky Way, is detected by rather faint but ubiquitous Galactic Hα emission
[144,145]. Such emission is found in several other spirals as well [142,143,127,128].
In the Galaxy, the Reynolds layer contains a substantial portion of the H+in
the ISM. Current models generally involve photoionization from the OB stars,
although how the Lyman continuum radiation from OB stars can penetrate the
neutral H layer remains controversial.
The observations show strong [SII] λ6717 and [N II] λ6583 that increase rel-
ative to Hα with distance z above the planes of various galaxies, including the
Milky Way . The only reasonable conclusion is that there is an additional
source of heating in the ISM that dominates over photoionization heating at low
densities. It has been proposed that carbonaceous molecules provide the excess
heating through the photoelectric effect , but this explanation is not unique.
Heating by turbulence, surely present, may dominate. Minter & Spangler 
suggested a heating rate that is adequate to explain the [S II]/Hα and [N II]/Hα
ratios, but did not take nonlinear interactions into account, thereby underesti-
mating the heating. A new study that would capitalize on the new understanding
of MHD turbulence (damping of Alfv´ en and fast modes, imbalance etc.) is on
8 Observational Tests
Comparing numerics with observations is a challenging problem. Ostriker (this
volume) discusses PDF, clump identification, and linewidth-scale relations as
possible diagnostics and outlines problems with any of those approaches. A use
of spectral line data cubes and application to it of different techniques (e.g. spec-
tral correlation function, principal component analysis, wavelets, etc.) can be
found in the review . Here we shall concentrate on comparing spectra from
observations with our theoretical expectations (see also review ).
8.1 Is the Big Power Law real?
We have mentioned above that observations suggest that the Kolmogorov power
law should span from AU to kpc scales. Kolmogorov scaling is exactly what one
would expect from the GS95 picture when the observations sample magnetic field
in the system of reference aligned with the mean magnetic field (see Fig. 15 for
examples of observable quantities). Indeed, it is obvious from Fig. 1 in the this
system of reference (i.e. global system of reference) the locally defined scalings
of k?with k⊥ are not valid. Indeed, one can easily see that the fluctuations
perpendicular to the local direction of magnetic field dominate both the statistics
measured perpendicular to the mean magnetic field and parallel to it. As the
result in the reference system aligned with the mean magnetic field k′
Fig. 15a,b,c) and according to equation (11) E(k) ∼ k−5/3will be measured.
Ambiguities in measurements reviewed in , however, make it uncertain
whether or not the Big Power Law should be taken at face value. Still, we note
that agreement between an observed power spectrum and theoretical expecta-
tions is far more significant than just an approximate fit between observations
and numerics. The latter is definitely not unique and is a priori suspect in view
of the huge difference in terms of Re and Rm between any numerical simulation
and the ISM.
To test the Big Power Law properly, it is important to extend the theory
of Velocity Channel Analysis (VCA) [84,86] by including self-absorption in the
analysis of turbulent emission lines. It can then be applied to regions of HI in
the inner Galaxy  or CO [163,162]18. Lazarian & Pogosyan  provided
results consistent with observations. For instance, the study predicts that in
the presence of absorption the emitted power in the line is proportional to k−3,
exactly what is seen in Dickey et al.  for HI in the inner region of the Galactic
An application of the VCA to different emission lines (e.g. Hα, [N II], [S II]
etc.) would help to answer the question of whether or not ISM turbulence is
18Brunt & Heyer  applied Principal Component Analysis (see ), to simulated
CO data and found empirical relations between the statistics of velocities, eigen-
vectors and eigenimages. However, they note that somehow their relation does not
depend on the absorption coefficient for the limited range of absorptions they tested.
Unfortunately, their analysis does not seem to be applicable to Galactic HI and the
applicability of their technique to the correlated velocity and density fields is unclear.
36Cho, Lazarian, & Vishniac
a large scale cascade with various phases of the ISM interconnected through a
dynamically important magnetic field. A contrasting possibility is that various
phases form their own cascades.
HI is rather smoothly distributed across the sky. Therefore, the effects of
image edges does not pose a problem for the statistical analysis. Results by
Stanimirovic (private communication) show that the Fourier analysis of the SMC
image and a more laborious wavelet analysis19provide identical results. However,
when dealing with molecular clouds we might expect that the cloud edges become
important. Therefore the incorporation of wavelets (see ) within the VCA
is a natural step to make. The only difference would be to apply the wavelets
instead of Fourier transforms to the channel maps.
Synchrotron fluctuations and fluctuations of polarized radiation arising from
aligned dust should be used to study magnetic field statistics. As we mentioned
earlier Cho & Lazarian  has shown that those fluctuations are consistent with
Kolmogorov scaling. More studies in this direction are necessary. The fact that
those fluctuctuations interfere with the CMB studies garantees that in the near
future we shall have a lot of relevant data.
One should remember, however, that the measured power index of fluctua-
tions may not correspond to the spectral index of the underlying turbulence. For
instance, it is shown in  that while the actual turbulence in SMC is close
to being Kolmogorov, depending on the thickness of the slice, the spectral in-
dex of intensity fluctuations within channel maps span from ∼ −2.8 to ∼ −3.4.
Similarly, it is shown in  that for Kolmogorov turbulence the spectral in-
dex of observable fluctuations may vary from ∼ −1 to −3.7 depending on how
observations sample turbulence.
8.2Does turbulence reveal magnetic field direction?
Anisotropy of Alfv´ enic turbulence is a definite prediction of the GS95 theory. We
mentioned earlier that the scale-dependent anisotropy can only be revealed in the
local frame of reference, which in practical terms require direct measurements,
e.g. with spacecraft. Measurements of the Solar wind magnetic turbulence have
failed so far to reveal the differential scaling of the turbulence in terms of k?and
k⊥, but these measurements are inconclusive . If measurement are performed
in a global system of reference, as is the case with observations, they should reveal
anisotropy in the direction of the mean magnetic field.
In isotropic turbulence, correlations depend only on the distance between
sampling points. Contours of equal correlation are circular in this case. The
presence of a magnetic field introduces anisotropy and these contours become
elongated with a symmetry axis given by the magnetic field. To study turbulence
anisotropy, we can measure contours of equal correlation corresponding to the
data within various velocity bins. The results obtained with simulated data are
shown in Fig. 15.
19Wavelet analysis involves determining the deviation of each pixel from a weighted
average of the pixels at a particular projected distance from it.
Fig.15. Observational test of the synthetic data by Cho, Esquivel & Lazarian. Con-
tours of equal correlation obtained with Centroids of Velocity ((a)) and with Spectral
Correlation Function (SCF)((b)). The direction of anisotropy reveals the direction of
projected magnetic field. Combined with the anisotropy analysis, the SCF (introduced
by Alyssa Goodman) is likely to become even more useful tool. (c) Contours of equal
correlation obtained for synthetic synchrotron intensity map. (d) shows the 2D genus of
the Gaussian distribution (smooth analytical curve) against the genus for the isother-
mal compressible MHD simulations with Mach number ∼2.5 (dotted curve).
Since the degree of anisotropy is related to the strength of the magnetic
field, studies of anisotropy can provide the means to analyze magnetic fields. It
is important to study different data sets and channel maps for the anisotropy.
Optical and infrared polarimetry can benchmark the anisotropies in correlation
functions. We hope that anisotropies will reveal magnetic field structure within
dark clouds where grain alignment and therefore polarimetry fails (see  for a
review of grain alignment).
Not only velocity statistics can be used for such an analysis. Lazarian &
Shutenkov  (see review ) showed that the mean magnetic field must lead
to anisotropies in the synchrotron statistics. Lazarian & Chibisov  pointed
out that using HI regions as screens for radiation at the decameter wavelength
it should be possible to study the 3D distribution of the magnetic field. Fig. 15c
shows the anisotropy of synchrotron statistics available through simulations.
38Cho, Lazarian, & Vishniac
8.3How else can we compare observations and simulations?
Velocity and density power spectra do not provide a complete description of
turbulence. Intermittency (variations in the strength of the turbulent cascade)
and its topology in the presence of different phases are not described by the
power spectrum. Use of the higher moments is possible (see discussion of the 3
point statistics, or bispectrum, in Lazarian ), but is limited by the noise in
the observational data.
“Genus analysis” is a good tool for studying the topology of turbulence (see
the review ). This tool has already been successfully applied to cosmology
. Consider an area on the sky with contours of projected density. The 2D
genus, G(ν), is the difference between the number of regions with a projected
density higher than ν and those with densities lower than ν. Fig. 15d shows the
2D genus as the function of ν for a Gaussian distribution of densities (completely
symmetric curve), for MHD isothermal simulations with Mach number ∼2.5. It
is shown in  that the genus of the Small Magellanic Cloud is very different
from that in Fig. 15d, while the spectra in both cases are similar.
Recently there have been significant advances in the field of MHD turbulence:
1. The first self-consistent model (GS95) of incompressible MHD turbulence
that is supported by both numerical simulations and observations is now
available. The major predictions of the model are scale-dependent anisotropy
2. There have been substantial advances in understanding compressible MHD.
Simulations of compressible MHD turbulence show that there is a weak cou-
pling between Alfv´ en waves and compressible MHD waves and that the
Alfv´ en modes follow the Goldreich-Sridhar scaling. Fast modes, however,
decouple and exhibit isotropy.
3. Contrary to general belief, in typical interstellar conditions, magnetic fields
can have rich structures below the scale at which motions are damped by the
viscosity created by neutral drag (the ambipolar diffusion damping scale).
4. These advances will have a dramatic impact on our understanding of many
fundamental interstellar processes, like cosmic-ray propagation, grain dy-
namics, turbulent heating and molecular cloud stability.
5. New techniques, e.g. VCA, allow observational tests of the theory.
⊥) and a Kolmogorov energy spectrum (E(k) ∝ k−5/3).
Acknowledgments: We thank Peter Goldreich, John Mathis, Steven Shore,
Enrique Vazquez-Semadeni, and Huirong Yan for helpful discussions. AL and JC
acknowledge the support of the NSF through grant AST-0125544. ETV acknowl-
edges NSF grant AST-0098615. This work was partially supported by National
Computational Science Alliance under AST000010N and AST010011N and uti-
lized the NCSA SGI/CRAY Origin2000. AL thanks the LOC for their financial
support. The authors thank the editors for their patience with the manuscript.
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