THE ASTROPHYSICAL JOURNAL, 562:279È296, 2001 November 20
( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.
COMPRESSIBLE MAGNETOHYDRODYNAMIC TURBULENCE IN INTERSTELLAR PLASMAS
YORAM LITHWICK AND PETER GOLDREICH
California Institute of Technology, MS 130-33, Pasadena, CA 91125; yoram=tapir.caltech.edu, pmg=gps.caltech.edu
Received 2001 June 21; accepted 2001 July 24
ABSTRACT
Radio wave scintillation observations reveal a nearly Kolmogorov spectrum of density Ñuctuations in
the ionized interstellar medium. Although this density spectrum is suggestive of turbulence, no theory
relevant to its interpretation exists. We calculate the density spectrum in turbulent magnetized plasmas
by extending the theory of incompressible magnetohydrodynamic (MHD) turbulence given by Goldreich
& Sridhar to include the e†ects of compressibility and particle transport. Our most important results are
as follows:
1. Density Ñuctuations are due to the slow mode and the entropy mode. Both modes are passively
mixed by the cascade of shear waves. Since the shear waves have a Kolmogorov spectrum,Alfve
n Alfve
n
so do the density Ñuctuations.
2. Observed density Ñuctuation amplitudes constrain the nature of MHD turbulence in the interstellar
medium. Slow mode density Ñuctuations are suppressed when the magnetic pressure is less than the gas
pressure. Entropy mode density Ñuctuations are suppressed by cooling when the cascade timescale is
longer than the cooling timescale. These constraints imply either that the magnetic and gas pressures are
comparable or that the outer scale of the turbulence is very small.
3. A high degree of ionization is required for the cascade to survive damping by neutrals and thereby
to extend to small length scales. Regions that are insufficiently ionized produce density Ñuctuations only
on length scales larger than the neutral damping scale. These regions may account for the excess of
power that is found on large scales.
4. Provided that the thermal pressure exceeds the magnetic pressure, both the entropy mode and the
slow mode are damped on length scales below that at which protons can di†use across an eddy during
the eddyÏs turnover time. Consequently, eddies whose extents along the magnetic Ðeld are smaller than
the proton collisional mean free path do not contribute to the density spectrum. However, in MHD
turbulence eddies are highly elongated along the magnetic Ðeld. From an observational perspective, the
relevant length scale is that transverse to the magnetic Ðeld. Thus, the cuto† length scale for density
Ñuctuations is signiÐcantly smaller than the proton mean free path.
5. The mode is critically damped at the transverse length scale of the proton gyroradius andAlfve
n
thus cascades to smaller length scales than either the slow mode or the entropy mode.
Subject headings: ISM: kinematics and dynamics È MHD È turbulence
1. INTRODUCTION
Di†ractive scintillations of small angular diameter radio
sources indicate that the interstellar electron density spec-
trum on length scales of 108È1010 cm is nearly Kolmogorov,
i.e., rms density Ñuctuations across a length scale j are
nearly proportional to j1@3. They also establish that there
are large variations in the amplitude of the density spectrum
along di†erent lines of sight.
Rickett (1977, 1990) and Armstrong, Rickett, & Spangler
(1995) review the observations of di†ractive scintillation
and their interpretation. They also discuss refractive scintil-
lations and dispersion measure Ñuctuations, which probe
density Ñuctuations on scales larger than the di†ractive
scales. Nondi†ractive measurements tend to indicate that
the Kolmogorov spectrum extends to much larger scales.
However, we focus primarily on di†ractive measurements
because they are much more sensitive.
Density Ñuctuations that obey the Kolmogorov scaling
occur in homogeneous subsonic hydrodynamic turbulence.
They are due to the entropy mode, a zero-frequency iso-
baric mode whose density Ñuctuations are o†set by tem-
perature Ñuctuations. Since subsonic turbulence is nearly
incompressible, the velocity Ñuctuations follow Kolmogo-
rovÏs scaling. To a good approximation, the entropy mode
is passively mixed by the velocity Ðeld, so it also conforms
to the Kolmogorov spectrum.1 Density Ñuctuations in
EarthÏs atmosphere, which cause stars to twinkle, obey the
Kolmogorov scaling. They arise from the passive mixing of
the entropy mode.
The electron density spectrum in the interstellar medium
cannot be explained by hydrodynamic turbulence.2 Because
the medium is ionized, magnetic e†ects must be accounted
for. This is evident since the length scales probed by di†rac-
tive scintillations are smaller than the collisional mean free
paths of both electrons and protons. If the magnetic Ðeld
1 Density Ñuctuations due to the Reynolds stress scale as j2@3. In addi-
tion, the dissipation of turbulent kinetic energy yields entropy Ñuctuations.
The ratio of the corresponding density Ñuctuations to the mean density is
comparable to the square of the Mach number at the length scale of
interest; hence, these density Ñuctuations are also proportional to j2@3.
2 Charge neutrality is maintained on di†ractive scales, so electron
density Ñuctuations include compensating Ñuctuations in the density of
positive ions.
279
280 LITHWICK & GOLDREICH Vol. 562
were negligible, freely streaming plasma would wipe out
density Ñuctuations at di†ractive scales. In the presence of a
magnetic Ðeld, electrons and protons are tied to Ðeld lines at
the scale of their gyroradii. For typical interstellar Ðeld
strengths, these gyroradii are smaller than the di†ractive
scales. A magnetic Ðeld thus impedes the plasma from
streaming across Ðeld lines and allows the turbulent cascade
and the associated density Ñuctuations to reach very small
scales across the Ðeld lines before dissipating. Therefore, a
theory for compressible turbulence in magnetized plasmas
is required to explain the observed density spectra. Our
objective is to develop this theory.
Until now, the only description of density Ñuctuations in
interstellar plasmas was by Higdon (1984, 1986). These
papers, while prescient, preceded a theory for incompress-
ible MHD turbulence and therefore did not account for the
full dynamics of the cascade. We compare HigdonÏs theory
with ours in ° 10.
Our compressible theory extends the theory of incom-
pressible MHD turbulence given by Goldreich & Sridhar
(1995) by including a slightly compressible slow mode and a
passive entropy mode. We also consider kinetic e†ects: on
sufficiently short length scales, the mean free paths of the
particles are signiÐcant, and the equations of compressible
MHD must be modiÐed. This is especially important for
damping.
In a future paper we will apply the theory developed here
to estimate amplitudes for density Ñuctuations produced in
supernova shocks, H II regions, stellar winds, and thermally
unstable regions. These are then compared to scattering
measures observed along di†erent lines of sight.
Before considering compressible turbulence, we discuss
incompressible MHD turbulence, focusing on issues that
are important for the compressible case.
2. INCOMPRESSIBLE MHD TURBULENCE
Goldreich & Sridhar (1995, 1997) propose a picture of the
dynamics of incompressible strong MHD turbulence and
describe the power spectra of waves, slow waves, andAlfve
n
passive scalars. We extend their picture to cover additional
features such as the parallel cascades of both slow waves
and passive scalars. Throughout this paper, ““ parallel ÏÏ and
““ transverse ÏÏ refer to the orientation relative to the ““ local
mean magnetic Ðeld,ÏÏ which is the magnetic Ðeld averaged
over the scale of interest. Our discussion of incompressible
MHD turbulence, while somewhat lengthy, is important for
understanding the extension to compressible turbulence
that follows.
Consider a uniform unperturbed plasma with an embed-
ded magnetic Ðeld. Turbulence is excited at the MHD outer
scale, by random and statistically isotropic forcing,L
MHD
,
with rms velocity Ñuctuations and rms magnetic Ðeld Ñuc-
tuations (in velocity units) that are comparable to the
speed,Alfve
n v
A
.3
As the turbulence cascades from the MHD outer scale to
smaller scales, power concentrates in modes with increas-
ingly transverse wavevectors. The inertial range velocity
spectrum applies to length scales below but above theL
MHD
3 The forcing Ñuctuations may also be less in which case wouldv
A
, L
MHD
be deÐned as the length scale at which the Ñuctuations extrapolate to v
A
.
dissipation scale. It is anisotropic and is characterized by
v
j
M
\ v
A
A
j
M
L
MHD
B
1@3
, (1)
"
A
\ j
M
2@3 L
MHD
1@3 . (2)
The inertial range magnetic Ðeld spectrum is identical. Here
is the length scale transverse to the local mean magneticj
M
Ðeld, is the rms velocity Ñuctuation across and isv
j
M
j
M
, "
A
the length scale parallel to the local mean magnetic Ðeld
across which the velocity Ñuctuation is We interpretv
j
M
. "
A
as the elongation along the magnetic Ðeld of an ““ eddy ÏÏ that
has a size transverse to the magnetic Ðeld; it is not anj
M
independent variable but is a function of Deep withinj
M
.
the inertial range, where eddies are highly elon-j
M
> L
MHD
,
gated along the magnetic Ðeld: In the following"
A
? j
M
.
subsections we explain the physics underlying the spectrum
and consider some of the implications.
Wave Spectrum2.1. Alfve
n
Arbitrary disturbances can be decomposed into Alfve
n
waves and slow waves. Appendix A summarizes the proper-
ties of these waves in the more general case of compressible
MHD. In incompressible MHD, waves and slowAlfve
n
waves are usually referred to as shear waves andAlfve
n
waves, but the former designation is morepseudo-Alfve
n
convenient for making the connection with compressible
MHD.
Our understanding of the MHD turbulence is based on
two facts: (1) MHD wave packets propagate at the Alfve
n
speed either parallel or antiparallel to the local mean mag-
netic Ðeld, and (2) nonlinear interactions are restricted to
collisions between oppositely directed wave packets. These
facts imply that in encounters between oppositely directed
wave packets, each wave packet is distorted as it follows
Ðeld lines perturbed by its collision partner. A wave packet
cascades when the Ðeld lines along which it is propagating
have spread by a distance comparable to its transverse size.
waves have quasi two-dimensional velocity andAlfve
n
magnetic Ðeld Ñuctuations that are conÐned to planes per-
pendicular to the local mean magnetic Ðeld. As their more
complete name ““ shear implies, they dominate theAlfve
nÏÏ
shear of the mapping of planes transverse to the local mean
magnetic Ðeld produced by Ðeld line wander. Thus, Alfve
n
waves control the dynamics of MHD cascades; slow waves
may be ignored when considering the dynamics of Alfve
n
waves.
In strong MHD turbulence the cascade time of an Alfve
n
wave packet is comparable to its travel time across the
parallel length of a single oppositely directed waveAlfve
n
packet of similar size. Goldreich & Sridhar (1995) refer to
this balance of timescales as ““ critical balance.ÏÏ It relates the
parallel size of a wave packet, to its transverse size,"
A
, j
M
.
Wave packets of transverse size cascade when the Ðeldj
M
lines they follow wander relative to each other by a trans-
verse distance Critical balance implies that this occursj
M
.
over a parallel distance "
A
.
The wave spectrum is given by equations (1) andAlfve
n
(2), with referring to the velocity Ñuctuations of thev
j
M
waves. It is deduced from two scaling arguments: (1)Alfve
n
KolmogorovÏs argument that the cascade time t
jM
^j
M
/v
jM
leads to an energy cascade rate, which isv
j
M
2 /t
j
M
^ v
j
M
3 /j
M
,
independent of length scale; and (2) the ““ critical balance ÏÏ
No. 1, 2001 MHD TURBULENCE IN INTERSTELLAR PLASMAS 281
assertion that the linear wave period that characterizes the
waves in a wave packet is comparable to the nonlin-Alfve
n
ear cascade time of that wave packet, i.e., v
A
/"
A
^ 1/t
j
M
.
Before considering slow waves in MHD turbulence, we
discuss two topics that are governed by the dynamics of
waves only: eddies and passive scalars.Alfve
n
2.2. Eddies
Because of their transverse polarization, waves areAlfve
n
responsible for the wandering of magnetic Ðeld lines. A
snapshot of wandering Ðeld lines is shown in Figure 1. Each
of these Ðeld lines passes through a localized region of size
in one plane transverse to the mean magnetic Ðeld. Awayj
M
from this plane the bundle of Ðeld lines diverges as a result
of the di†erential wandering of the individual lines. At a
second plane, the bundleÏs cross-sectional area has approx-
imately doubled. Critical balance implies that the distance
to this second plane is comparable to the parallel wave-
length that characterizes the bundle, As the bundle"
A
.
spreads, other Ðeld lines, not depicted, enter from its sides.
In general, the neighboring Ðeld lines of any individual Ðeld
line within a region of transverse size change substan-j
M
tially over a parallel distance of order It is natural to"
A
.
think of as the parallel size of an ““ eddy ÏÏ that has trans-"
A
verse size Two eddies with the same transverse lengthj
M
.
scale that are separated by a parallel distance greater than
FIG. 1.ÈWandering of magnetic Ðeld lines. A Ðeld line bundle of trans-
verse size diverges after a parallel distance where is the parallelj
M
"
A
, "
A
size of an eddy (eq. [2]) as determined by critical balance.
their incorporate di†erent Ðeld lines and hence are sta-"
A
tistically independent. Eddies are distinct from wave
packets. The former are rooted in the Ñuid, whereas the
latter propagate up and down magnetic Ðeld lines at the
speed.Alfve
n
Aside from their anisotropy, eddies in MHD turbulence
are similar to those in hydrodynamic turbulence. They are
spatially localized structures with characteristic velocity
Ñuctuations and lifetimes. The rms velocity di†erence
between two points is determined by the smallest eddy that
contains both. Di†erent eddies of a given size are sta-
tistically independent. The three-dimensional spectrum for
rms velocity Ñuctuations across transverse length scales j
M
and parallel length scales isj
A
v
j
M,jA
\ v
A
]
4
5
6
0
0
A
j
M
L
MHD
B
1@3
for j
A
> "
A
,
A
j
A
L
MHD
B
1@2
\
A
j
M
L
MHD
B
1@3
A
j
A
"
A
B
1@2
for j
A
? "
A
.
(3)
There is negligible additional power within an eddy on
parallel length scales smaller than so for"
A
, j
A
> "
A
(j
M
),
For the smallest eddy that con-v
j
M,jA
\ v
j
M
. j
A
? "
A
(j
M
),
tains both and has a transverse length scale thatj
M
j
A
j
M
@
satisÐes The velocity Ñuctuation of this eddy is"
A
(j
M
@ ) \ j
A
.
obtained by solving this equation for (eq. [2]) and insert-j
M
@
ing this in equation (1). Contours of the three-j
M
@
dimensional spectrum are plotted in Figure 2. Each contour
represents eddies of a characteristic size.
Maron & Goldreich (2001) give the three-dimensional
spectrum in Fourier space. Since eddies that are separated
by more than are statistically independent, the power"
A
spectrum at a Ðxed transverse wavenumber is indepen-k
M
FIG. 2.ÈThree-dimensional spectrum. Contours of constant (eq.v
jM ,jA
[3]), labeled by the value of Along thev
j
M,jA
/v
A
. j
M
-axis, v
j
M,0
/v
A
\
along the(j
M
/L
MHD
)1@3 ; j
A
-axis, v
0,jA
/v
A
\ (j
A
/L
MHD
)1@2.
282 LITHWICK & GOLDREICH Vol. 562
dent of the parallel wavenumber in the correspondingk
A
region of Fourier space, i.e., where k
A
~1 Z "
A
(k
M
~1).
The turbulent cascade is generally viewed as
““ proceeding ÏÏ from larger eddies to smaller eddies as this is
the direction of energy transfer. However, smaller eddies
cascade many times in the time that a large eddy cascades.
This is particularly important in turbulent mixing. Consider
the evolution of two Ñuid elements whose initial separation
is larger than the dissipation scale. On cascade timescale t
j
M
,
their transverse separation will random walk a distance j
M
as the result of the cascade of smaller eddies. Therefore, on a
timescale comparable to an eddyÏs cascade time, the trans-
verse locations of its component Ñuid elements, whose sizes
may be considered to be comparable to the dissipation
scale, are completely randomized. Moreover, since mixing
at the dissipation scale causes neighboring Ñuid elements to
be rapidly homogenized, transverse smoothing of the eddy
occurs on the timescale that it cascades. Rapid transverse
mixing in MHD turbulence is similar to the more familiar
isotropic mixing in hydrodynamic turbulence.
2.3. Passive Scalar Spectrum
A passive scalar, p, satisÐes the continuity equation (L/Lt
and does not a†ect the ÑuidÏs evolution. It]¿ Æ $)p \ 0
could represent, for example, the concentration of a con-
taminant. We consider the spectrum of a passive scalar
mixed by the wave cascade. These considerations areAlfve
n
important for our subsequent investigation of compressible
turbulence. They are also helpful for understanding the slow
wave spectrum. We discuss the passive scalar spectrum both
in the inertial range and also below the scale at which the
wave spectrum is cut o†.Alfve
n
2.3.1. Passive Scalar Spectrum in the Inertial Range
As we show in this subsection, the transverse spectrum of
the passive scalar in the inertial range is
p
j
M
P j
M
1@3 , (4)
where is the length scale transverse to the local meanj
M
magnetic Ðeld and is the rms Ñuctuation in the passivep
j
M
scalar across The passive scalar parallel spectrum is thej
M
.
same as the wave parallel spectrum given in equationAlfve
n
(2), where is now to be interpreted as the length scale"
A
parallel to the local mean magnetic Ðeld across which the
passive scalar Ñuctuation is p
j
M
.
Mixing of the passive scalar is due to waves. SlowAlfve
n
wave mixing is negligible. This is because transverse veloc-
ity gradients are much larger than parallel ones in MHD
turbulence. Thus, waves, whose velocity ÑuctuationsAlfve
n
are perpendicular to the magnetic Ðeld, are much more
e†ective at mixing than slow waves, whose velocity Ñuctua-
tions are nearly parallel to the magnetic Ðeld. The trans-
verse cascade arises from the shuffling of Ðeld lines as Alfve
n
waves propagate through the Ñuid.
The transverse spectrum (eq. [4]) follows from the
Kolmogorov-like hypothesis that the cascade rate of the
““ energy ÏÏ in the scalar Ðeld is independent of length scale,
i.e., is constant, where is the passive scalarp
j
M
2 /t
jM
t
jM
cascade time, which is assumed to be proportional to the
cascade time of waves. Comparing this with the con-Alfve
n
stancy of the kinetic energy cascade rate, we con-v
j
M
2 /t
jM
,
clude that which implies equation (4). A similarp
jM
P v
jM
,
argument holds for the cascade of a passive scalar in hydro-
dynamic turbulence (e.g., Tennekes & Lumley 1972).
The parallel cascade of a passive scalar is more subtle. It
might appear that a passive scalar cannot cascade along
Ðeld lines since, neglecting dissipation, both the scalar and
the magnetic Ðeld are frozen to the Ñuid, and thus the scalar
must be frozen to Ðeld lines. In that case there certainly
could not be a parallel cascade. If the scalar were injected
on large scales, then Ñuctuations with smaller wavelengths
along the magnetic Ðeld would not be generated. However,
dissipation cannot be neglected. It is an essential part of
MHD turbulence, as it is of hydrodynamic turbulence. For
example, the description of turbulent mixing in ° 2.2
depends crucially upon small-scale dissipation.
Perhaps the best way to understand the parallel cascade
is to consider mixing on the transverse length scale j
M
within two planes that are perpendicular to the local mean
magnetic Ðeld and are separated by a parallel distance
larger than Velocity Ñuctuations within the two planes"
A
.
are statistically independent. This is evident because a
bundle of Ðeld lines cannot be localized within a transverse
distance over a parallel separation greater than Evenj
M
"
A
.
a pair of Ñuid elements, one in each plane, that are initially
on the same Ðeld line are mixed into two regions with di†er-
ent values of passive scalar concentration. It follows that the
parallel cascade of the passive scalar also obeys equation
(2).
2.3.2. Passive Scalar Spectrum below Wave Cuto†Alfve
n
A passive scalar cascade may extend below the transverse
scale at which the MHD cascade is cut o†. Mixing on these
scales is driven by Ñuid motions at which results in aj
cutoff
,
scale-independent mixing time equal to the cascade time at
This yieldsj
cutoff
.
p
j
M
\ constant , j
M
\j
cutoff
. (5)
A similar argument applies in hydrodynamic turbulence.
Tennekes & Lumley (1972) call this regime in hydrody-
namic turbulence the ““ viscous convective subrange.ÏÏ
2.4. Slow W ave Spectrum
The slow wave spectrum is the same as that of the Alfve
n
waves. It is given by equations (1) and (2), with referringv
j
M
to the velocity Ñuctuations of the slow waves. This is a
consequence of the similar kinematics of slow waves and
shear waves and the fact that both are cascaded byAlfve
n
shear waves.Alfve
n
Slow waves obey the same linear wave equation as Alfve
n
waves, and to lowest nonlinear order they travel up and
down the local mean magnetic Ðeld lines at the speedAlfve
n
just as waves do. However, the dynamics of theAlfve
n
MHD cascade is controlled by the waves (GoldreichAlfve
n
& Sridhar 1997; Maron & Goldreich 2001) because their
velocity and magnetic Ðeld Ñuctuations are perpendicular
to the local mean magnetic Ðeld, whereas those of the slow
waves are nearly parallel to it. Since perpendicular gra-
dients are much larger than parallel ones in the MHD
cascade, waves are much more e†ective at mixingAlfve
n
than are slow waves. Hence, waves cascade bothAlfve
n
themselves and slow waves, whereas slow waves cascade
neither.4
The transverse mixing of the slow waves by wavesAlfve
n
is analogous to the mixing of a passive scalar. As discussed
4 We are assuming that and slow waves have comparableAlfve
n
strengths at a given length scale.
No. 1, 2001 MHD TURBULENCE IN INTERSTELLAR PLASMAS 283
in ° 2.3, a passive scalar assumes the same inertial range
spectrum as that of the velocity Ðeld that is responsible for
its mixing. Thus, equation (1) is also applicable to the veloc-
ity Ñuctuations of the slow waves.
Similarly, the parallel cascade of slow waves is analogous
to the parallel cascade of a passive scalar. Since Alfve
n
waves cascade in the time they move a distance slow"
A
,
waves separated by this distance are independently mixed.
Thus, equation (2) also applies to slow waves. There is,
however, a conceptual di†erence between the parallel cas-
cades of the passive scalar and of the slow mode. In the
absence of dissipation a passive scalar is frozen to Ðeld lines,
whereas slow mode wave packets travel along them at the
speed. A passive scalar has a parallel cascadeAlfve
n
because Ñuctuations are statistically independentAlfve
nic
within two transverse planes frozen in the Ñuid and separat-
ed by The parallel cascade of slow waves occurs because"
A
.
two transverse planes separated by that are travelling at"
A
the speed in the same direction experience uncor-Alfve
n
related sequences of distortions su†ered as a result of inter-
actions with oppositely directed waves. Nevertheless,Alfve
n
these two requirements are both satisÐed in the MHD
cascade when the transverse planes are separated by a dis-
tance greater than and so the passive scalar and the"
A
,
slow mode have the same parallel spectrum. Whereas
passive scalar mixing is due to eddies, slow wave mixing
may be thought of as due to ““ travelling eddies,ÏÏ i.e., eddies
that travel up and down the magnetic Ðeld at the Alfve
n
speed.
2.5. Numerical Simulations
Numerical simulations o†er some support for the above
description of incompressible MHD turbulence. Those by
Cho & Vishniac (2000b) support both equations (1) and (2),
and those by & Biskamp (2000) support equationMu
ller
(1). However, although the simulations of Maron & Gold-
reich (2001) support equation (2), they yield v
j
M
P j
M
1@4
instead of equation (1). Because the simulations of Maron &
Goldreich (2001) are stirred highly anisotropically, whereas
those of Cho & Vishniac (2000b) and & BiskampMu
ller
(2000) are stirred isotropically, it is not clear whether these
disparate results conÑict. Maron & Goldreich (2001) specu-
late that the discrepancy between their spectrum and the
scaling prediction of Goldreich & Sridhar (1995) results
from intermittency. In any case, we expect that the physical
picture of a critically balanced cascade, which underlies
Goldreich & SridharÏs (1995) description of MHD turbu-
lence, remains valid. Even if the spectrum is proportional to
we expect that the results of this investigation, whichj
M
1@4,
assumes a spectrum proportional to would not be sig-j
M
1@3,
niÐcantly altered.
The simulations of Maron & Goldreich (2001) conÐrm
that a passive scalar has the same transverse spectrum as
that of waves, although both are proportional toAlfve
n
They also indicate that the parallel cascade of a passivej
M
1@4.
scalar conforms to equation (2).
Maron & Goldreich (2001) present results from simula-
tions of the interaction between oppositely directed slow
and waves; the slow waves cascade, whereas theAlfve
n
waves do not. They also compute the spectrum ofAlfve
n
slow waves in a simulation of MHD turbulence and Ðnd
that its transverse and longitudinal behavior matches that
of the wave spectrum.Alfve
n
2.5.1. Kolmogorov Constants
Scaling arguments do not yield values for the
““ Kolmogorov constants,ÏÏ the order-unity multiplicative
constants of the spectrum. However, they can be obtained
from simulations. We deÐne them such that equation (1)
remains valid, i.e., we take to be the separation atL
MHD
which the rms velocity di†erence is equal to, or extrapolates
to, Two Kolmogorov constants, and are neededv
A
. M
A
M
t
,
in this paper:
M
A
4
v
A
v
j
M
j
M
"
A
F "
A
\ M
A
~1 j
M
2@3 L
MHD
1@3 , (6)
M
t
4
v
A
t
j
M
"
A
. (7)
In these deÐnitions, and are the inverses of the wave-j
M
"
A
numbers transverse and parallel to the local mean magnetic
Ðeld, and is the cascade time of waves with transverset
j
M
wavenumber From the numerical simulations ofj
M
~1.
Maron & Goldreich (2001),
M
A
^ 3.4 , M
t
^ 1.4 . (8)
Because these simulations yield a transverse spectrum that
is proportional to instead of the resultingj
M
1@4 j
M
1@3,
““ Kolmogorov constants ÏÏ are not truly constant.
3. COMPRESSIBLE TURBULENCE : OVERVIEW
Our primary concern is interstellar scintillation, which is
a†ected by electron density Ñuctuations on very small
scales, typically 108È1010 cm for di†ractive scintillation. In
the remainder of this paper we calculate the spectrum of
density Ñuctuations that results from compressible turbu-
lence in magnetized plasmas with parameters appropriate
to the interstellar medium. Throughout, we consider
plasmas that have more ions than neutrals and that have
where b is the ratio of the thermal pressure to1 [ b\O,
the mean magnetic pressure: Hereb 4 8np/B2\2c
T
2/v
A
2 .
is the isothermal sound speed, is thec
T
4 (p/o)1@2 v
A
Alfve
n
speed, p is the thermal pressure, B is the magnetic Ðeld
strength, and o is the mass density.5 The incompressible
limit corresponds to b \ O.
On the length scales that we consider, compressible
MHD is a good approximation for the dynamics of the
ionized interstellar medium. Kinetic e†ects, where impor-
tant, may be accounted for by simple modiÐcations to the
MHD equations. Therefore, we turn our attention to turbu-
lence in compressible MHD.
The turbulent velocity spectrum in compressible MHD is
approximately the same as the turbulent velocity spectrum
in incompressible MHD because the mode remainsAlfve
n
incompressible in a compressible medium and the slow
mode is only slightly compressible. Thus, the velocity spec-
trum for both of these modes is given by equations (1) and
(2). Appendix A summarizes the properties of the relevant
modes in compressible MHD.
There are two additional modes in compressible MHD
that are not present in incompressible MHD. One of these
is the fast mode. However, as long as the fast modeb Z few,
is essentially a sound wave. Its phase speed is larger than
the phase speed of either the mode or the slow mode,Alfve
n
5 We brieÑy discuss plasmas with b\1in° 11.
![FIG. 2.ÈThree-dimensional spectrum. Contours of constant (eq.vjM,jA[3]), labeled by the value of Along thevjM,jA/vA. jM-axis, vjM,0/vA \along the(j M /L MHD)1@3 ; jA-axis, v0,jA/vA \ (jA/L MHD)1@2.](/figures/fig-2-ethree-dimensional-spectrum-contours-of-constant-eq-2ogt9o0q.webp)
