The Power Spectrum of Supersonic Turbulence in Perseus

Abstract

We test a method of estimating the power spectrum of turbulence in molecular clouds based on the comparison of power spectra of integrated intensity maps and single-velocity-channel maps, suggested by Lazarian and Pogosyan. We use synthetic 13CO data from non-LTE radiative transfer calculations based on density and velocity fields of a simulation of supersonic hydrodynamic turbulence. We find that the method yields the correct power spectrum with good accuracy. We then apply the method to the Five College Radio Astronomy Observatory 13CO map of the Perseus region, from the COMPLETE website. We find a power law power spectrum with slope beta=1.81+-0.10. The values of beta as a function of velocity resolution are also confirmed using the lower resolution map of the same region obtained with the AT&T Bell Laboratories antenna. Because of its small uncertainty, this result provides a useful constraint for numerical codes used to simulate molecular cloud turbulence. Comment: 4 pages, 3 figures. ApJ Letters, in press

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arXiv:astro-ph/0611248v1 7 Nov 2006
Draft version February 5, 2008
Preprint typeset using LATEX style emulateapj v. 6/22/04
THE POWER SPECTRUM OF SUPERSONIC TURBULENCE IN PERSEUS
Paolo Padoan1, Mika Juvela2, Alexei Kritsuk1and Michael L. Norman1
Draft version February 5, 2008
ABSTRACT
We test a method of estimating the power spectrum of turbulence in molecular clouds based on the
comparison of power spectra of integrated intensity maps and single-velocity-channel maps, suggested
by Lazarian and Pogosyan. We use synthetic13CO data from non-LTE radiative transfer calculations
based on density and velocity fields of a simulation of supersonic hydrodynamic turbulence. We
find that the method yields the correct power spectrum with good accuracy. We then apply the
method to the Five College Radio Astronomy Observatory13CO map of the Perseus region, from the
COMPLETE website. We find a power law power spectrum with slope β = 1.81 ± 0.10. The values
of β as a function of velocity resolution are also confirmed using the lower resolution map of the same
region obtained with the AT&T Bell Laboratories antenna. Because of its small uncertainty, this
result provides a useful constraint for numerical codes used to simulate molecular cloud turbulence.
Subject headings: ISM: clouds – ISM: kinematics and dynamics – ISM: structure
1. INTRODUCTION
The large scale dynamics of interstellar clouds is char-
acterized by random supersonic motions with a very large
Reynolds number, meaning that inertia is much larger
than viscous forces.This interstellar turbulence is a
dominant transport mechanism in many astrophysical
processes and plays an important role in the fragmen-
tation of star-forming clouds. Random supersonic flows
in molecular clouds result in a complex network of highly
radiative shocks causing very large density contrasts and
shaping the clouds into the observed self-similar filamen-
tary structure (Nordlund & Padoan 2003).
Our understanding of turbulence is limited by the lack
of general analytical solutions of the Navier-Stokes equa-
tion.Statistical properties of turbulence, essential to
modeling many astrophysical processes, are therefore de-
rived almost entirely from numerical simulations. In the
case of incompressible hydrodynamic turbulence, numer-
ical simulations can be compared with laboratory exper-
iments. This is not possible for the isothermal, super-
sonic, magneto-hydrodynamic (MHD) turbulence regime
of molecular clouds. The best approach to validate nu-
merical models is in this case a statistical comparison
of observational data with synthetic data from numerical
simulations.
Padoan et al. (2004) found that the exponents of the
velocity structure functions of compressible and super-
Alfv´ enic turbulence follow a generalized She-L´ evˆ eque
scaling (Boldyrev 2002), depending only on the rms Mach
number of the flow. Based on that scaling formula, if
one of the exponents is known (for example the second
order that corresponds to the velocity power spectrum)
all the others can be derived, up to some order. While
the scaling formula is well constrained numerically, its
normalization, given by the actual value of one of the
exponents, is harder to measure and may depend on the
1Department of Physics, University of California, San Diego,
CASS/UCSD 0424, 9500 Gilman Drive, La Jolla, CA 92093-0424;
ppadoan@ucsd.edu
2
DepartmentofAstronomy,
T¨ ahtitorninm¨ aki, P.O.Box 14,FI-00014 University of Helsinki,
Finland
UniversityofHelsinki,
numerical method used to simulate the turbulence.
Numerical simulations of supersonic hydrodynamic
(HD) and magneto-hydrodynamic (MHD) turbulence
have yielded a range of values of the slope of the velocity
power spectrum, from the Kolmogorov value of β = 5/3,
to the Burgers value of β = 2 (Frisch & Bec 2001), and
beyond. The main problems with the numerical esti-
mate of the power spectrum are i) The limited extent
in wavenumbers of the inertial range of turbulence, or
even its complete absence in the case of low resolution
(or highly dissipative) simulations; ii) The emergence of
the bottleneck effect (e.g. Falkovich 1994; Dobler et al.
2003; Haugen & Brandenburg 2004) as soon as the nu-
merical resolution is large enough to generate an inertial
range; iii) The dependence of the power spectrum on
the numerical schemes used to stabilize the shocks; iv)
The dependence of the numerical resolution necessary for
convergence on the numerical method. Given these dif-
ficulties, reliable measurements from interstellar clouds
provide useful constraints to validate numerical models
and to improve our knowledge of supersonic turbulence.
Lazarian & Pogosyan (2000) have demonstrated that
the exponent of the velocity power spectrum can in prin-
ciple be derived from spectral maps of emission lines by
comparing the power spectrum of integrated intensity
with the power spectrum of single-velocity-channel inten-
sity. Their method was tested with numerical simulations
of turbulent flows, where the velocity and density fields
had to be modified to generate power law power spec-
tra, due to the limited numerical resolution (Esquivel
et al. 2003). In this Letter we present a new test of
their method. Thanks to the high numerical resolution of
our simulation (1,0243computational zones), our density
and velocity power spectra exhibit extended power laws,
and no modifications to the original turbulent fields are
required. Furthermore, the method is tested by comput-
ing synthetic CO emission lines with a non-LTE radia-
tive transfer code. We find the method allows to retrieve
the power law exponent of the velocity power spectrum
with good accuracy, using the J=1-013CO line. We then
apply the method to the Five College Radioastronomy
Observatory (FCRAO) survey of the Perseus molecular

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Fig. 1.— Compensated power spectrum of integrated intensity
(bottom curve) and average of the compensated power spectra of
single-velocity-channel intensity maps (top curve) from the numer-
ical simulation. The dotted plot (middle curve) is the 3D power
spectrum of velocity and the dashed line shows the β = 1.83 slope
predicted by the method.
cloud complex (Ridge et al.
from the COMPLETE website, and estimate the power
spectrum exponent β = 1.81 ± 0.10. The result is con-
firmed by the analysis of the lower resolution map of the
same region obtained with the AT&T Bell Laboratories
antenna (see Padoan et al. 1999).
2006), publicly available
2. SYNTHETIC SPECTRAL MAPS
Synthetic maps of the J=1-0 line of13CO are com-
puted with a new non-LTE radiative transfer code that
has been extensively tested against the older Monte Carlo
code (Juvela 1997). As a model of the density and ve-
locity fields in interstellar clouds we use the results of a
supersonic simulation of HD turbulence with rms Mach
number M = 6. Our purpose is not to simulate the con-
ditions found in a specific molecular cloud, but rather to
test a general method of estimating the velocity power
spectrum. We will study the effect of different parameter
values elsewhere. Because the method is based on an an-
alytical derivation by Lazarian & Pogosyan (2000) that
neglects correlations between density and velocity, we ex-
pect it to work equally well for a vast range of turbulence
parameters.
The simulations are carried out with the Enzo code,
developed at the Laboratory for Computational Astro-
physics by Bryan, Norman and collaborators (Norman
& Bryan 1999).
Enzo is a public domain Eulerian
grid-based code (see http://cosmos.ucsd.edu/enzo/) that
adopts the Piecewise Parabolic Method (PPM) of Colella
& Woodward (1984). We use an isothermal equation
of state, periodic boundary conditions, initially uniform
density and initial random large scale velocity. The tur-
bulence is forced in Fourier space only in the wavenumber
range 1 ≤ k ≤ 2, where k = 1 corresponds to the size of
the computational domain that contains 1,0243compu-
tational zones (for details see Kritsuk et al. 2006).
The radiative transfer calculations assume a box size
Fig. 2.— Convergence plot for the exponent of the velocity power
spectrum as a function of the channel width.
of 5 pc, a mean density of 103cm−3, a mean kinetic
temperature of 10 K, an rms Mach number M = 6 (con-
sistent with the HD simulation) and a uniform abun-
dance of13CO molecules. These values were chosen as
a generic reference model, not tailored to the Perseus
molecular cloud complex. The original density and ve-
locity data-cubes are re-sampled to a resolution of 2563
zones. This has three advantages: i) It leaves density and
velocity fields with power spectra that are power laws
almost up to the new Nyquist frequency; ii) It speeds
up enormously the radiative transfer calculations; iii)
It generates a map of synthetic spectra with a range of
scales comparable to that of the rebinned FCRAO map
of Perseus (see below).
The result of the radiative transfer calculations is a
map of spectral profiles of the line intensity, T(v,x),
over 280 velocity channels, v, and across all positions,
x, within the map of 2562positions. The sum of the
line intensity of all velocity channels, multiplied by the
channel width, ∆v = 0.025 km/s, gives the integrated in-
tensity at each position, I(x) = ΣvT(v,x)∆v. The map
I(x) may be used to derive a rough estimate of the col-
umn density. We do not address the issue of converting
integrated intensities into accurate estimates of column
density, because we derive the velocity power spectrum
directly from the line intensity data.
3. THE METHOD
We call PI(k) the two-dimensional spatial power spec-
trum of the integrated intensity map, I(x), and look for
a power law fit such that, PI(k) ∝ k−βI. The integrated
map is the sum of all the single-velocity-channel maps.
We then call PT(v,k) the two-dimensional spatial power
spectrum of the individual channel maps, T(v,x), where
we have left the velocity dependence to stress that there
are as many of these maps and power spectra as there
are velocity channels. We look for a power law fit to
the average of these power spectra as well, such that
PT(k) = ?PT(v,k)?v ∝ k−βT. Finally, we assume the
three dimensional spatial power spectrum of the velocity

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Fig. 3.— Same as in Figure 1, but for the FCRAO survey of
Perseus (Ridge et al. 2006).
field is also a power law (a reasonable assumption for
the cloud turbulence, given the huge Reynolds number),
Pv(k) ∝ k−β.
The result of Lazarian & Pogosyan (2000) can be ex-
pressed as β = 1 + 2(βI− βT), if βI < 2, and β =
1 + 2(2 − βT), if βI > 2. In this Letter we test their
result only for the case of βI < 2. The single-velocity-
channel power spectra are shallower than the integrated
intensity power spectrum, βT< βI, because they contain
small-scale structure originating from both the density
and the velocity fields. It is this difference that makes it
possible to derive the velocity power spectrum exponent,
based on the above formula.
In this Letter we refer to power spectra as the total
power within shells of wavenumber between k and k+dk.
In Lazarian & Pogosyan (2000), the power spectra are
defined as the average power within shells of wavenumber
between k and k + dk. Furthermore, we do not absorb
the negative sign in the definition of βI as they do. Our
exponents are related to their n and m exponents in the
following way: βI= 1−n, β = 1+m. In our convention,
the velocity power spectrum corresponds to the usual
turbulent energy spectrum, for example k−5/3for the
Kolmogorov case.
Fig. 1 shows PI(k) (bottom plot) and PT(k) (top plot)
for the simulated data-cube “observed” along the x di-
rection.The error bars are one standard deviation
above and below the mean values and depend on the
statistical sample size (number of Fourier modes inside
each wavenumber shell), so they decrease with increasing
wavenumber. The linear least square fits, plotted as solid
lines, are computed in the wavenumber range 1 ≤ k ≤ 30.
Based on the estimated exponents of these two power
spectra, we obtain β = 1 + 2(βI− βT) = 1.83 ± 0.17.
Values of β from synthetic maps of the data-cube ob-
served in different directions fall within the estimated 1-
σ uncertainty. The value of the velocity power spectrum
exponent, computed directly from the original three di-
mensional velocity field, is β = 1.8 ± 0.1 (dotted plot in
Fig. 1), showing that the method retrieves the correct
exponent.
In order to verify the dependence of β on the velocity
channel width, we have applied the method to synthetic
maps at different velocity resolutions. The result, plotted
in Fig. 2, shows that the value of β is completely con-
verged only for a channel width of the order of the ther-
mal width. This was to be expected, because all velocity
fluctuations above the thermal width may in principle
affect the velocity power spectra of the single-velocity-
channel maps.
This method is based on an analytical derivation by
Lazarian & Pogosyan (2000) that neglects the correla-
tions of density and velocity in turbulent flows. We in-
terpret this test as a confirmation of the validity of their
method in the case of βI< 2, rather than as an empirical
calibration of the value of β based on βIand βT. The un-
certainty of the method, when applied to observational
data, is then determined by the error bars of the obser-
vational power spectra, independently of the uncertainty
of our numerical test. The final error bar is dominated
by the uncertainty in the power spectrum exponent of
the integrated intensity, because the uncertainty in the
single-velocity-channel spectrum is reduced by averaging
the power spectra of many velocity channels.
4. THE POWER SPECTRUM OF PERSEUS
We have applied the method to the J=1-013CO sur-
vey of the Perseus molecular cloud complex carried out
with the FCRAO 14 m antenna by Ridge et al. (2006).
The grid spacing of the survey is 23”, and the beam size
46”. The velocity-channel size is 0.06 km/s. The power
spectra we compute are corrected for the effect of beam
and noise, by simply dividing by the power spectrum of
a gaussian beam, and subtracting the power spectrum of
the noise. Spatial correlations in the noise arising from
the “On-the-Fly” mapping mode are neglected. At the
largest wavenumbers, the power spectra are sensitive to
the noise subtraction, and realistic error bars account-
ing for that would make such wavenumbers essentially
useless for estimating the power spectra. We therefore
prefer to regrid the map to a resolution of 92”, which
has the advantage of increasing the signal-to-noise by a
factor of 4.
The power spectra are shown in Fig. 3.
square fits are computed in the range 5 ≤ k ≤ 80, and
yield β = 1.81 ± 0.10. Although this range is less ex-
tended than that used with the synthetic data, it includes
larger wavenumbers (between k = 30 and k = 80) than
the synthetic fit (the synthetic power spectra are still
affected by numerical dissipation at large wavenumbers
even after rebinning from 1,0243to 2563). This reduces
the final uncertainty, because the statistical sample size
is much larger at larger wavenumbers (∝ kdk in two di-
mensions). As a result, the 1-σ uncertainty of β is 9%
for the synthetic data, and 5% for the observations.
The value of the slope of the projected density power
spectrum, βI= 1.99±0.05 is similar to values previously
found in different regions observed in HI (e.g. Green
1993; Stanimirovich et al. 1999) and CO (e.g. Bensch,
Stutzki, Ossenkopf 2001; Padoan et al.
tice that in those previous works, with the exception of
Padoan et al. (2004a), the power spectrum is not inte-
grated over wavenumber shells, so its slope is equivalent
to βI+1. Furthermore, the power spectrum of the13CO
The least
2004a).No-

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4
integrated intensity is slightly steeper than the gas den-
sity power spectrum due to radiative transfer effects. Ac-
counting for such effects, the slope of the projected gas
density power spectrum in the Perseus region was esti-
mated to be consistent with that of super-Alf´ enic turbu-
lence simulations (Padoan et al. 2004a).
The value of β estimated for the Perseus region as a
function of the velocity resolution is shown in Fig. 2 (solid
line). The channel width of 0.06 km/s is close to the
thermal line width and the value of β seems to be al-
most converged, at least within its 1-σ uncertainty. As
an independent test, we have applied the method also
to the AT&T Bell Laboratories map of the same region
(see Padoan et al. 1999). The Bell Laboratories 7 m an-
tenna has a beam twice the size of the FCRAO 14 m an-
tenna. We did not regrid this map to a lower resolution,
so the map resolution and size are comparable to those
of the FCRAO map, but its velocity resolution is only
0.273 km/s, so it is not expected to yield a converged
value of β. The values of β for the Bell Laboratories
map is shown in Fig. 2 as a dotted line, showing very
good agreement with the FCRAO result, well within the
estimated 1-σ uncertainty.
5. DISCUSSION AND CONCLUSIONS
This result has very interesting implications.
numerical simulations giving power spectra with slope
significantly larger than β = 1.8 ± 0.1 may be ruled out
as correct descriptions of molecular cloud turbulence (at
least for the Perseus region). Burgers exponent, β = 2.0,
is 2σ larger than the Perseus exponent (assuming this
is converged as a function of velocity resolution). The
slope of the power spectra of the SPH simulations in
Ballesteros-Paredes et al. (2006), β ≈ 2.7 in the case of
a turbulence rms Mach number ≈ 6, is 9σ larger than the
present estimate, and their grid based simulations have
β ≈ 2.2, 4σ too large. Second, we can now derive the
absolute values of the velocity structure function expo-
nents in Perseus. Padoan et al. (2004b) have determined
numerically the relative values of those exponents, so the
First,
knowledge of one of them, for example the second order
exponent given by β−1, allows to determine the absolute
values of all the others.
A different method of estimating the scaling of the tur-
bulence from molecular clouds surveys was developed by
Brunt & Heyer (2002a,b), based on the principle compo-
nent analysis (PCA). They analyzed 23 molecular clouds
in the outer Galaxy and estimated a value β = 2.2±0.3,
if the structure function exponents of order p are as-
sumed to scale linearly as p/3. This value is significantly
larger than the one we estimate in Perseus. However, the
PCA method is dependent on a calibration with numer-
ical simulations. Based on such a calibration, it appears
that the PCA method estimates the exponent of the ve-
locity structure functions of order p = 0.5 or lower (Brunt
et al. 2003). Taking this into consideration, the result of
Brunt & Heyer (2002b) would be roughly consistent with
ours, if a very intermittent scaling of the structure func-
tion is assumed, consistent with numerical simulations of
supersonic turbulence.
We have shown in this Letter that the method for es-
timating the velocity power spectrum slope proposed by
Lazarian and Pogosyan (2000) works well in the case of
βI < 2, and for a velocity resolution not much larger
than the thermal line width. However, regions with lower
turbulence Mach number than Perseus, for example the
Taurus region, may have steeper density power spectra,
and hence βI> 2. Such regions will be studied in a future
work, where the method will be tested also with numer-
ical simulations generating synthetic data with βI> 2.
P.P., A.K., and M.L.N. were partially supported by
the NASA ATP grant NNG056601G, the NSF grants
AST-0507768 and AST-0607675 and the NRAC alloca-
tion MCA098020S. We utilized computing resources pro-
vided by the San Diego Supercomputer Center and by the
National Center for Supercomputing Applications. M.J.
was supported by the Academy of Finland Grants no.
206049 and 107701.
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