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.