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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

Department of Astronomy,

T¨ ahtitorninm¨ aki, P.O.Box 14,FI-00014 University of Helsinki,

Finland

University ofHelsinki,

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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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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