MODIFICATION OF PROJECTED VELOCITY POWER SPECTRA BY DENSITY INHOMOGENEITIES INCOMPRESSIBLE SUPERSONIC TURBULENCE

Christopher M. Brunt1,2,3

and Mordecai-Mark Mac Low4

Received 2003 June 27; accepted 2003 November 24

ABSTRACT

The scaling of velocity fluctuation �v as a function of spatial scale L in molecular clouds can be measured fromsize–line width relations, principal component analysis, or line centroid variation. Differing values of the power-law index of the scaling relation h�v2i1=2 / L�3D in three dimensions are given by these different methods: thefirst two give �3D ’ 0:5, while line centroid analysis gives �3D ’ 0. This discrepancy has previously notbeen fully appreciated, as the variation of projected velocity line centroid fluctuations (h�v2lci1=2 / L�2D ) is indeeddescribed, in two dimensions, by �2D � 0:5. However, if projection smoothing is accounted for, this implies, inthree dimensions, that �3D � 0. We suggest that a resolution of this discrepancy can be achieved by accountingfor the effect of density inhomogeneity on the observed �2D obtained from velocity line centroid analysis.Numerical simulations of compressible turbulence are used to show that the effect of density inhomogeneitystatistically, but not identically, reverses the effect of projection smoothing in the case of driven turbulence, sothat velocity line centroid analysis does indeed predict that �2D � �3D � 0:5. For decaying turbulence, the effectof density inhomogeneity on the velocity line centroids diminishes with time so that at late times, �2D � �3D þ0:5 as a result of projection smoothing alone. Deprojecting the observed line centroid statistics thus requiressome knowledge of the state of the flow. This information can be inferred from the spectral slope of the columndensity power spectrum and a measure of the standard deviation in column density relative to the mean. Usingour numerical results, we can restore consistency between line centroid analysis, principal component analysis,and size–line width relations, and we derive �3D � 0:5, corresponding to shock-dominated (Burgers) turbulence,which also describes the simulations of driven turbulence at scales on which numerical dissipation is negligible.We find that this consistency requires that molecular clouds are continually driven on large scales or are onlyrecently formed.

Subject headings: ISM: clouds — ISM: kinematics and dynamics — ISM: molecules — methods: statistical —radio lines: ISM — turbulence

On-line material: color figures

1. INTRODUCTION

The presence of turbulence in molecular clouds is diag-nosed by the prevalence of superthermal line widths. A secondimportant diagnostic is the presence of macroscopic velocityfluctuations and the exponent, �3D, that describes the variationof rms velocity fluctuation (h�v2i1=2) with spatial scale, L, inthree dimensions: �v2

� �1=2/ L�3D . The exponent, �3D, can vary

depending on the state of turbulence in the medium; e.g., theKolmogorov (1941) case has �3D � 1

3, which applies to in-

compressible turbulence and may apply to decaying com-pressible turbulence (Falgarone et al. 1994); the compressible,shock-dominated case (Burgers 1974) has �3D � 1

2. The scale

dependence of velocity dispersion in molecular clouds will, ingeneral, be dependent on the energy injection spectrum, inaddition to the transfer of energy between scales. For a pureturbulent cascade of kinetic energy, measurement of �3D

provides fundamental physical information on the nature ofinterstellar turbulence.In this paper we describe a long-standing but largely un-

appreciated discrepancy between observational estimates of�3D. Velocity line centroid analysis, operating in (projected)two dimensions, suggests that �3D � 0 when the effects ofprojection smoothing (von Hoerner 1951; O’Dell & Castaneda1987; Stutzki et al. 1998; Miville-Deschenes, Levrier, &Falgarone 2003b; Brunt et al. 2003) are accounted for. Size–line width relations (e.g., Solomon et al. 1987) and principalcomponent analysis (PCA; Heyer & Schloerb 1997; Bruntet al. 2003) suggest that �3D � 0:5.We argue in this paper that, for driven turbulence, density

inhomogeneity induces small-scale structure on the projectedvelocity line centroid field such that the effects of projectionsmoothing are statistically but not identically reversed andthus that the observed �2D obtained from velocity line centroidanalysis does in fact closely approximate the true three-dimensional index, �3D. Some indication that this is true wasfound by Ossenkopf & Mac Low (2002).We begin with an overview of the statistical properties of

velocity fields in three dimensions, and in (projected) twodimensions as seen by velocity line centroid analysis. In x 3we describe our numerical data, and in x 4 we describe ourobservational data. The results of velocity line centroid anal-ysis applied to the numerical simulations of turbulence, bothwith and without density weighting, are discussed in x 5. In

1 National Research Council, Herzberg Institute of Astrophysics, Domin-ion Radio Astrophysical Observatory, P.O. Box 248, Penticton, BC V2A 6J9,Canada; chris.brunt@nrc.ca.

2 Department of Physics and Astronomy, University of Calgary, 2500University Drive NW, Calgary, AB T2N 1N4, Canada.

3 Current address: Department of Astronomy, LGRT 632, University ofMassachusetts, 710 North Pleasant Street, Amherst, MA 01003; brunt@fcrao1.astro.umass.edu.

4 Department of Astrophysics, American Museum of Natural History, 79thStreet and Central Park West, New York, NY 10024-5192; mordecai@amnh.org.

A

196

The Astrophysical Journal, 604:196–212, 2004 March 20

# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

x 6 we estimate the observed power spectrum of line centroidvelocity fluctuations in the NGC 7129 molecular cloud andinterpret the measured power spectrum exponent and previ-ously measured line centroid structure function exponents,accounting for both projection smoothing and density inho-mogeneity.

2. VELOCITY LINE CENTROID ANALYSIS: THEORY

There have been a number of correlation studies on pro-jected line centroid velocities in molecular clouds (Kleiner &Dickman 1985; Kitamura et al. 1993; Hobson 1992; Miesch &Bally 1994; Ossenkopf & Mac Low 2002). Most of the studieshave proceeded in direct space, relying in large part on theformalism established by Kleiner & Dickman (1985) andDickman & Kleiner (1985). Here we use the related Fourierspace representation and measure line centroid velocity fluc-tuations via their power spectrum. The power spectrum [P kð Þ,a function of wavenumber k] of velocity fluctuations isP kð Þ / k��ND , where �ND ¼ 2�ND þ N in N dimensions.5 Theabove relationship between �ND and �ND applies to the case ofexact power-law power spectra. In general, the power spec-trum and structure function of fluctuations are integrally re-lated to each other (the structure function is a simpletransformation of the autocorrelation function, which in turn isthe inverse Fourier transform of the power spectrum), anddefining the statistical relationship between them with singleindices is difficult. Our observational data do have well-defined power-law behavior over the observed range of scales,so the above simple prescription is valid.

We use the power spectrum for three reasons: (1) in thenumerical fields, increasing separations ultimately decreasebecause of field periodicity, resulting in artificially smaller �ND

for a given �ND (see Brunt et al. 2003); (2) the numerical fieldshave power spectra that are not power laws over all wave-numbers because of the scales at which the turbulence wasdriven and the effects of numerical dissipation—we use aFourier space scheme that avoids problems arising from this;(3) for the observational data, corrections for instrumentalnoise and beam smearing are easily achieved in Fourierspace—any direct space beam-smearing corrections must ingeneral proceed via Fourier space anyway. We thus charac-terize all measurements in terms of power spectra but state thedirect space equivalent according to the simple �ND-�ND re-lation given above.

In the power spectrum representation, the Kolmogorovcase has �3D ¼ 11=3, and the Burgers case has �3D ¼ 4.Previously, the power spectrum approach has been usedby Kitamura et al. (1993) and, more recently, by Miville-Deschenes et al. (2003a) for H i 21 cm line emission.

The true projected velocity line centroid field is

vtruelc x; yð Þ ¼XNpix

i¼1

vz x; y; zið Þ=Npix; ð1Þ

where vz x; y; zð Þ is the line-of-sight velocity component andNpix is the number of spatial pixels along each line of sight(this is replaced by a similar integral form for the continuouscase; our numerical and observational data are pixelized, sowe use the pixelized equations here).

Consider first an unweighted projection of the velocity fieldover the line-of-sight (z-) axis. Generation of a velocity cen-troid map, by projection over the z-direction, simply selects thewavevectors for which kz ¼ 0, since all others are oscillatoryalong the z-direction and will average out (disregarding inho-mogeneous sampling). For an isotropic power-law powerspectrum in three dimensions, the projected velocity linecentroid two-dimensional power spectrum has the same spec-tral slope as it does in three dimensions: i.e., �2D ¼ �3D (see,e.g., Stutzki et al. 1998; Miville-Deschenes et al. 2003b; Bruntet al. 2003); more precisely, the observed (projected) two-dimensional power spectrum is identical to the kz ¼ 0 plane inthe three-dimensional power spectrum. Only the transversecomponent of the velocity field is projected into two dimen-sions (e.g., Kitamura et al. 1993); in other words, the longi-tudinal contribution to vz has no amplitude on the kz ¼ 0 plane.

For the case of a power-law power spectrum, while thespectral slope is unchanged on going from three dimensions to(projected) two dimensions, the direct space statistical prop-erties of the field are modified. The variation of rms velocityfluctuation ( �v2

� �1=2) with spatial scale (l ) is

�v2� �1=2/ L �ND�Nð Þ=2 / L�ND ; ð2Þ

where N is the dimension on which the field is defined (hereN ¼ 2 or 3) and v, for the two-dimensional case, is understoodto be the projected line centroid field, vtruelc . The criterion �3D ¼�2D leads to �2D ¼ �3D þ 0:5; this is projection smoothing.For the Kolmogorov (1941) case (�3D ¼ �2D ¼ 11=3;�3D ¼ 1

3) the projected index is �2D ¼ 5

6(von Hoerner 1951;

see O’Dell & Castaneda 1987). For Burgers (1974) com-pressible turbulence, we have �3D ¼ �2D ¼ 4, �3D ¼ 0:5, and�2D ¼ 1 as a result of projection smoothing.

An observed �2D ¼ 0:5 thus implies �3D ¼ 0. Only in thecase that the line-of-sight depth over which the observationaltracer exists is much less than the total transverse scale ofemission observed on the sky (Miville-Deschenes et al.2003b; see also O’Dell & Castaneda 1987) is it true that�2D � �3D; i.e., this requires thin, sheetlike clouds, orientedface-on toward us.

The density-weighted projected velocity line centroidfield is

vwgtlc x; yð Þ ¼XNpix

i¼1

n x; y; zið Þvz x; y; zið Þ.XNpix

i¼1

n x; y; zið Þ;

ð3Þ

where n x; y; zð Þ is the density field. For the case of uniformlyexcited, optically thin gas, equation (3) is the appropriate formfor predicting the velocity line centroid field. As recognizedby Kleiner & Dickman (1985), the form of equation (3) sug-gests that the projected ‘‘velocity line centroid’’ field is moreakin to a measure of the line-of-sight momentum of the me-dium; however, the normalization by the projected columndensity complicates this assignment (dimensionally, the linecentroid field has units of velocity). For uniform density (e.g.,the case of fully incompressible turbulence), the density fieldamplitude is normalized out of the line centroid field. Moregenerally, for inhomogeneous density, the mean projecteddensity field is normalized out, but variations in density alongthe line of sight may induce additional structure in themeasured velocity line centroid field.

Previously established results of relevance here are thefollowing: (1) Miville-Deschenes et al. (2003b) showed that

5 Note that for �ND < N, �ND saturates to �0, while for �ND > N þ 2,�ND saturates to unity—see Stutzki et al. (1998), and see Brunt et al. (2003)for a demonstration of the latter case.

SUPERSONIC TURBULENCE 197

density inhomogeneity does not affect velocity line centroidsin the case of uncorrelated density and velocity fields (thuspreserving �2D ¼ �3D þ 0:5). However, Miville-Descheneset al. (2003b) used stochastic density and velocity fields (in-cluding some degree of nonphysical correlation betweendensity and velocity). (2) Brunt et al. (2003) found that den-sity inhomogeneity does induce additional small-scale struc-ture on the velocity line centroid field in the case of physicallycorrelated density and velocity (resulting in �2D < �3D þ 0:5).They also found that the correlations between density andvelocity are important (for a given amplitude of densityfluctuations). Brunt et al. (2003) used ‘‘spectrally modified,’’numerically simulated density and velocity fields (seeLazarian et al. 2001), in which the density-velocity coupling isalso modified to some extent. (3) Ossenkopf & Mac Low(2002) examined numerical simulations of turbulence andmeasured �3D in three dimensions and �2D from the projectedline centroid velocity field of the simulations (including den-sity weighting). They found that �3D � �2D � 0:5; they alsonoted that projection smoothing does not predict this behavior.This is an indication that the effect of density weightingcounters the effects of projection smoothing. (4) Lazarian &Esquivel (2003) have inferred that density inhomogeneitydoes not affect the power spectrum of velocity line centroidfluctuations.

Observationally, the density-weighted line centroid velocityfield may be approximated by use of an optically thin tracer:

vobslc x; yð Þ ¼XNchan

i¼1

T x; y; vzið Þvzi x; yð Þ.XNchan

i¼1

T x; y; vzið Þ;

ð4Þ

where T x; y; vzið Þ is the observed radiation temperature, vzix; yð Þ is the velocity of the ith channel at position x; yð Þ, andNchan is the number of channels along each line of sight.

3. NUMERICAL DATA

We use simulations of randomly driven hydrodynamic (HD)and magnetohydrodynamic (MHD) turbulence (Mac Low1999) and decaying HD turbulence (Mac Low et al. 1998).These simulations were performed with the astrophysicalMHD code ZEUS-3D (Clarke & Norman 1994),6 a three-dimensional version of the code described by Stone & Norman(1992a, 1992b), using second-order advection (van Leer1977), that evolves magnetic fields using constrained transport(Evans & Hawley 1988), modified by upwinding along shearAlfven characteristics (Hawley & Stone 1995). The code usesa von Neumann artificial viscosity to spread shocks out tothicknesses of three or four zones in order to prevent numer-ical instability, but contains no other explicit dissipation orresistivity. Structures with sizes close to the grid resolution aresubject to the usual numerical dissipation, however.

The simulations were performed on a three-dimensional,uniform, Cartesian grid with periodic boundary conditions inevery direction. All length scales are in fractions of the cubesize; i.e., all wavevectors run from k ¼ 1 to k ¼ Npix=2 for asimulation of Npix pixels in each dimension. An isothermalequation of state was used in the computations, with soundspeed chosen to be cs ¼ 0:1 in arbitrary units. The initialdensity and, where applicable, magnetic field were both ini-

tialized uniformly on the grid, with the initial density �0 ¼ 1and the initial field parallel to the z-axis.The turbulent flow is initialized with velocity perturbations

drawn from a Gaussian random field determined by its powerdistribution in Fourier space, following the usual procedure.As discussed in detail in Mac Low et al. (1998), the decaying-turbulence runs were initialized with a flat spectrum withpower from kd ¼ 1 to 8, because that will decay quickly to aturbulent state. Note that the dimensionless wavenumber kdcounts the number of driving wavelengths kd in the box. Afixed pattern of Gaussian fluctuations drawn from a field withpower only in a narrow band of wavenumbers around somevalue kd offers a very simple approximation to driving bymechanisms that act on that scale. For driven turbulence, sucha fixed pattern was normalized to produce a set of perturba-tions �L x; y; zð Þ. At every time step a velocity field�v x; y; zð Þ ¼ A�L was added to the velocity v, with the am-plitude A now chosen to maintain constant kinetic energyinput rate, as described by Mac Low (1999). A summary ofthe models is given in Table 1.

4. OBSERVATIONAL DATA

Our observational data are 12CO (J ¼ 1 0) and 13CO(J ¼ 1 0) spectral line imaging observations acquired withthe Five College Radio Astronomy Observatory (FCRAO)14 m telescope during the spring season of 2003. These dataform part of a large (�200 deg2) molecular line survey(in progress) within the ongoing multiwavelength CanadianGalactic Plane Survey (CGPS; Taylor et al. 2003).The FCRAO 14 m telescope was equipped with the

SEQUOIA focal plane array (Erickson et al. 1999), which wasrecently expanded to 32 pixels. The array is rectangular (4�4 pixels), with two sets of 16 pixels in different polarizationshaving the same pointing centers. Because of the expandedphysical size of the array in the dewar, it is no longer possiblefor the dewar to rotate and so maintain a constant orientationwith the celestial coordinate system (our mapping grid here isin Galactic l; b coordinates). To allow mapping in celestialcoordinates, FCRAO has developed an on-the-fly (OTF)mapping system in which the telescope is driven across themap area and spectra are dumped on the desired grid. Onepixel near the center of the array (hereafter the central pixel) isselected to define the origin of the map coordinates. The otherpixels traverse the map along paths determined by the pixelseparation in the array and the current orientation of the ce-lestial coordinate system to the telescopic (azimuth-elevation,or az-el) system. If the central pixel is scanned so that itNyquist-samples the map area, then the combination of datafrom all the other pixels results in a highly (but irregularly)oversampled map. In order to map large areas, the samplingrate of the central pixel can be relaxed slightly. Our data areobtained by dumping the correlators every 22B5 (1

2beamwidth

at the 12CO frequency) along the scan direction during a totalscan length (in l ) of 0 :�5. The b-coordinate of the central pixelis then incremented by 33B75, and the next scan in l is taken.The pixel separation in the array is 88B56. In the case that thel; b system is aligned with the az-el system, then after twoscans, the first row of the array is located at 66B45 in b aboveits original position, or 22B11 below the original b-coordinateof the second row. The next scan places the first row of thearray 11B64 above the first scan of the second row of the arrayand 12B11 below the second scan of the second row. Forsubsequent scans, a similar progression of sampling ensues.Apart from at the extreme map edges, a sampling finer than

6 Available from the Laboratory for Computational Astrophysics athttp://cosmos.ucsd.edu/lca.

BRUNT & Mac LOW198 Vol. 604

the Nyquist rate is achieved, but the sampling is slightly ir-regular. More generally, the l; b system is rotated with respectto the az-el system, and an even finer sampling is achieved. AnOFF (reference) measurement is taken between every twoscans of the array (i.e., OFF, scan out and back, OFF, scan outand back, etc.), and a calibration measurement is taken afterevery four out-and-back sequences (every 9–10 minutes). Thedata are calibrated to the T�

A scale using the standard chopperwheel method (Kutner & Ulich 1981). Following recordingof the data, the spectra are converted onto a regular grid of22B5 pixel spacing using the FCRAO otftool software.

The 12CO and 13CO spectral lines were acquired simulta-neously using the recently added dual-channel correlators,which provide a total bandwidth of 50 MHz, with1024 channels at each frequency. The central velocity ofthe spectra is vlsr ¼ 45 km s�1, resulting in a total velocitycoverage of vlsr ��110 to +20 km s�1, with a channel spacingof �49 kHz (�0.13 km s�1). The spatial resolution of the data(beam FWHM) is 4500 for 12CO and 4600 for 13CO.

The otftool software was configured to remove first-orderbaselines derived from the spectra in the velocity intervals�110 to �95 and 7–20 km s�1. Spectra from individual pixelsin the array contributing to a single position in the output gridwere weighted by the reciprocal of their noise variance,derived from the above velocity intervals. Following initialgridding, we removed additional (approximately sinusoidal)baseline structure arising from a standing wave between thesubreflector and dewar, using Fourier methods. First, wedefined the ‘‘spectral window’’ between �20 and +5 km s�1,

which contains the detectable emission in the region ofinterest here. Then we interpolated over this window, using afirst-order baseline derived from the adjacent 100 channels oneither side of the spectral window. The emission (plus noise)in the spectral window at each spatial position was replacedby the interpolated line at that position, and the resultingspectrum was converted to Fourier space using a fast Fouriertransform (FFT). The first two wavevectors, of wavelength50 and 25 MHz (�130 and 65 km s�1, respectively) wereextracted from the FFTed spectrum, converted back to directspace, and then subtracted from the original spectrum. Theresulting sensitivities (1 �, T�

A) of the processed data are �0.5and �0.25 K per channel for 12CO and 13CO, respectively.

5. VELOCITY LINE CENTROID ANALYSIS:NUMERICAL RESULTS

5.1. Overview

The numerical fields may be used to evaluate the effect ofdensity weighting on the power spectrum of the line centroidvelocity field. We calculated vtruelc via equation (1) and vwgtlc viaequation (3) for all simulations. For the MHD models weexamined line centroid fields projected over two axes: paralleland perpendicular to the mean magnetic field direction. As thesimulated velocity fields have periodic boundary conditions,the kz ¼ 0 plane in the three-dimensional power spectrum of vzis identical to the two-dimensional power spectrum for vtruelc .

First, we examined the driven simulations. Examplevelocity line centroid fields are shown in Figure 1. It is

TABLE 1

Numerical Models: Parameters and Measurements

Model Npixa HD/MHD/Decay kd

b M c vA=csd te �� �cd � ln �ð Þ � ln Nð Þ

HA8................ 128 HD 7–8 1.9 0 1 �0.84 � 0.05 4.10 � 0.12 0.71 0.17

HC2................ 128 HD 1–2 7.4 0 1 �0.72 � 0.05 3.30 � 0.13 1.58 0.71

HC4................ 128 HD 3–4 5.3 0 1 �0.98 � 0.03 3.22 � 0.09 1.55 0.48

HC8................ 128 HD 7–8 4.1 0 1 �1.17 � 0.06 3.36 � 0.08 1.34 0.32

HE2 ................ 128 HD 1–2 15.0 0 0.475 �1.04 � 0.05 2.87 � 0.10 2.45 0.74

HE4 ................ 128 HD 3–4 12.0 0 0.875 �1.11 � 0.06 3.02 � 0.09 2.23 0.63

HE8 ................ 128 HD 7–8 8.7 0 1 �1.28 � 0.11 3.14 � 0.09 2.07 0.40

MC4X v? ....... 128 MHD 3–4 5.3 10 0.1 �0.91 � 0.12 3.04 � 0.09 1.68 0.56

MC4X vjj ........ 128 MHD 3–4 5.3 10 0.1 �1.21 � 0.05 2.57 � 0.09 1.68 0.46

MC45 v? ........ 128 MHD 3–4 4.8 5 0.2 �0.66 � 0.09 3.03 � 0.13 1.58 0.45

MC45 vjj ........ 128 MHD 3–4 4.8 5 0.2 �1.06 � 0.05 2.93 � 0.11 1.58 0.48

MC41 v? ........ 128 MHD 3–4 4.7 1 0.5 �0.89 � 0.05 3.18 � 0.08 1.28 0.36

MC41 vjj ........ 128 MHD 3–4 4.7 1 0.5 �0.71 � 0.05 3.20 � 0.09 1.28 0.32

MC85 v? ........ 128 MHD 7–8 3.4 5 0.075 �0.59 � 0.11 3.31 � 0.10 1.50 0.32

MC85 vjj ........ 128 MHD 7–8 3.4 5 0.075 �1.31 � 0.08 2.93 � 0.09 1.50 0.30

MC81 v? ........ 128 MHD 7–8 3.5 1 0.225 �1.00 � 0.07 3.41 � 0.09 1.18 0.27

MC81 vjj ........ 128 MHD 7–8 3.5 1 0.225 �1.09 � 0.06 3.38 � 0.08 1.18 0.28

ME21 v? ........ 128 MHD 1–2 14.0 1 0.5 �0.85 � 0.06 3.22 � 0.09 1.90 0.77

ME21 vjj ......... 128 MHD 1–2 14.0 1 0.5 �0.81 � 0.07 3.15 � 0.08 1.90 1.27

D..................... 256 HD/Decay 1–8 5.0 0 0.1 �0.76 � 0.03 4.39 � 0.08 0.68 0.19

0.15 �0.59 � 0.02 4.70 � 0.08 0.54 0.17

0.3 �0.32 � 0.02 5.13 � 0.09 0.37 0.13

0.5 �0.14 � 0.01 5.44 � 0.09 0.26 0.09

U..................... 256 HD/Decay 1–8 50.0 0 0.1 �0.97 � 0.03 4.09 � 0.07 0.97 0.35

0.15 �0.86 � 0.02 4.48 � 0.09 0.81 0.30

0.3 �0.48 � 0.02 4.93 � 0.07 0.51 0.21

0.5 �0.29 � 0.01 5.17 � 0.08 0.40 0.15

a Number of pixels in each dimension.b Driving wavenumber.c The rms Mach number (equilibrium M for driven turbulence; initial M for decaying turbulence).d Ratio of Alfven speed to sound speed.

SUPERSONIC TURBULENCE 199No. 1, 2004

immediately evident that the inclusion of density weightingresults in the insertion of additional small-scale structure intothe line centroid fields.

To quantify the effect seen in Figure 1, we calculated thepower spectra [P kð Þ, a function of wavenumber k] of all the linecentroid fields, using an FFT. Figure 2a shows representativeresults for HC8 model (kd ¼ 7 8). There is a turnover in thepower spectra at or near kd. For k < kd, the true and weightedline centroid power spectra are quite similar, while for k > kd,the power spectrum of vwgtlc is shallower. We can quantify theeffect of density weighting quite precisely by examining thedifference (in log space) between the power spectra of vwgtlc andvtruelc (in real space this corresponds to measuring the spectralindex of the ratio Pvwgt

lc=Pvtrue

lc). We characterize the ratio of power

spectra by a scaling exponent ��, where Pvwgtlc=Pvtrue

lc/ k�� (i.e.,

log Pvwgtlc

� log Pvtruelc

¼ constþ �� log k). This procedureavoids potential problems arising from non–power-law pro-jected velocity spectra.

The ratio Pvwgtlc=Pvtrue

lcfor the HC8 model is shown in

Figure 2b. There is little modification to the true line centroid

Fig. 1.—Example velocity line centroid fields obtained from the numerical simulations, for the unweighted case (vtruelc ) and the density-weighted case (vwgtlc ). [Seethe electronic edition of the Journal for a color version of this figure.]

Fig. 2.—(a) Example velocity line centroid power spectra derived fromthe numerical simulations (HC8; kd ¼ 7 8), for the unweighted case(Pvtrue

lc) and the density-weighted case (Pvwgt

lc). (b) Effect of density weighting

determined by the spectral slope, ��, of Pvwgtlc=Pvtrue

lc. The dashed lines

mark the range of k used for the fit (straight line) to derive ��: from k ¼ 7(the largest driving scale) to k ¼ 32.

BRUNT & Mac LOW200 Vol. 604

power spectrum for spatial scales above the driving scale(k ¼ 7 8); Pvwgt

lc=Pvtrue

lcdoes not vary much with k at small k.

This is because there are no major density enhancements onthese large scales (Ballesteros-Paredes & Mac Low 2002). Fork > kd , Pvwgt

lc=Pvtrue

lcis a well-developed power law, with some

deviation at very high k. We fitted the exponent �� ¼ 1:17 �0:06 over the range 7 � k � 32 (i.e., from the largest drivingscale to a wavelength of 4 pixels). The largest scale at whichdissipation effects are first seen in these simulations is �10zones, or k � 12:8 (Mac Low & Ossenkopf 2002), but we donot see any spectral breaks in the power spectrum ratio beforek ¼ 32; we examine the effects of this in greater detail in x 5.2.

The other driven models gave similar results. Figure 3shows Pvwgt

lc=Pvtrue

lcfor all driven models. These spectra were

fitted between the largest driving scale and k ¼ 32 for allmodels to derive ��, which in all cases was negative andapproximately �1, with variations of up to about � 0.4. Themeasured �� are listed in Table 1. There is no obvious sys-tematic difference between the HD and MHD cases, nor anyobvious variation with rms Mach number, M, for the drivenruns. There is some evidence that �� is more negative whenthe line of sight is parallel to the mean magnetic field direc-tion, but this does not universally apply. Taking the observedrange of �� as a natural variation, we derive a mean

Fig. 3.—Variation of Pvwgtlc=Pvtrue

lcwith k for the HD and MHD fields. Fits (straight lines) to derive �� were made in the range of k delimited by the dashed lines

(k ¼ 7 32).

SUPERSONIC TURBULENCE 201No. 1, 2004

(unweighted) �� ¼ 0:96 � 0:22 from all the driven models.In other words, inclusion of density weighting results in ashallower spectral slope, with index �1 less than the trueline centroid power spectrum index. In direct space, such amodification would thus correspond to �2D � �3D, rather than�2D ¼ �3D þ 0:5. (In general, �2D ¼ �3D þ 0:5þ ��=2.)

If the driven-turbulence conditions are representative of theconditions within molecular clouds, the scaling exponentsmeasured from line centroid structure functions are approxi-mately equal to the true, three-dimensional exponents. Westress that the ‘‘cancellation’’ of projection smoothing bysmall-scale line centroid structure induced by density inho-mogeneity is statistical in nature, and the effect of densityweighting does not, of course, identically reverse the effects ofprojection smoothing.The same procedure to derive �� was then applied to the

simulations of decaying turbulence. Figure 4 shows the plots ofPvwgt

lc=Pvtrue

lcevolving as time increases. We have fitted �� over

the range 10 � k � 64 for all fields (motivated again by thelack of clear breaks in the power spectrum ratio), and theseare listed in Table 1. This shows that as the turbulence decays,the effect of density inhomogeneity on the line centroid fieldis reduced, as a result of the lessened density contrast. Ifmolecular clouds are in such a state, then projection smoothingonly applies to the observed line centroid field, and�2D � �3D þ 0:5. Over the course of 1 sound crossing time,the density contrast is reduced sufficiently, even for initiallyhighly supersonic turbulence, that �� becomes very small,because the shocks in decaying turbulence are weaker thanthose in driven turbulence (Smith, Mac Low, & Zuev 2000b;Smith, Mac Low, & Heitsch 2000a); in other words, if theturbulence is continually driven, or has only recently entereda decaying state, then we expect �� ��1, but this evolvesquickly to �� ! 0 over a sound crossing time. Figure 5summarizes the evolution of �� after the driving is turnedoff.

5.2. Regime of Validity

As noted above, we are fitting �� over a range of kthat includes the dissipation region (k greater than �12.8),

Fig. 4.—Variation of Pvwgtlc=Pvtrue

lcwith k for the decaying-turbulence simulations as time progresses. Fits (straight lines) to derive �� were made in the range of k

delimited by the dashed lines (k ¼ 10 64). Each plot is labeled by the time step in units of the sound crossing time. (Left: Model D, initial M ¼ 5. Right: Model U,initial M ¼ 50.)

Fig. 5.—Plot of �� vs. t, the time after driving is turned off, in units of thesound crossing time. For the driven runs, we have taken t ¼ 0. Filled circlesare for driven HD; open triangles are for driven MHD; open squares are fordecaying HD (model D, initial M ¼ 5); and open circles are for decaying HD(model U, initial M ¼ 50).

BRUNT & Mac LOW202 Vol. 604

motivated by the lack of distinct breaks in the power spectrumratio. To test whether dissipative effects could inducesystematic biases in our results, we repeated the fits whilerestricting the range to a maximum k ¼ 12. This was doneonly for the driven models for which kd ¼ 1 2 and 3–4, inorder that sufficient data would still be available for the fits.The resulting measurements of �� are plotted against themeasurements of �� fitted to k ¼ 32 in Figure 6. For thissample, we find that the mean �� values are �0:91 � 0:17(fitted to k ¼ 32) and �0:87 � 0:19 (fitted to k ¼ 12). Theseare not significantly different, so our visual conclusion thatthere are no breaks in the power spectrum ratios is good.

The advantage of using a relative measure, ��, in the aboveanalysis is that it circumvents potential problems arising fromsmall curvatures in the intrinsic velocity power spectra. Wenow examine actual measurements of the spectral indices ofthe projected velocity power spectra. These absolute indicesare needed to examine the range of validity of �� as derivedabove. Because of the curvature in the power spectra, likelyarising from dissipative effects at higher k, the absolutespectral slopes depend on the range in k over which they arederived. In what follows, we label the spectral slope of theunweighted projected velocity field as �2D;u and the spectralslope of the density-weighted projected velocity field as �2D;w.These are related by �2D;w � �2D;u ¼ ��.

Example line centroid power spectra for the HC2 and HC4models, driven at kd ¼ 1 2 and 3–4, respectively, are shownin Figure 7. These are fitted over two ranges in k: up to k ¼ 32(top) and up to k ¼ 12 (bottom); in both cases the smallest kused to make the fit is taken as the smaller of the drivingwavenumbers. When the fit is made to k ¼ 32, the spectra are‘‘steep’’ (�2D;u � 4:5), but when the fit is made to k ¼ 12, theindices are in good agreement with the value expected forBurgers turbulence (�2D;u � 4). Numerical dissipation willcause just the sort of steepening of the spectrum (less structureat small scale) that is seen in Figure 7.

We have derived �2D;u, �2D;w, and �� for the driven runsover the two ranges in k given above. Figure 8a shows therelation between �2D;u and �2D;w for all driven runs fitted tok ¼ 32; the corresponding variation of �� with �2D;u is shownin Figure 8b. Figures 8c and 8d show the same relationsobtained when the fits are made to k ¼ 12 for the subset ofruns with kd ¼ 1 2 and 3–4. In all these plots, the solid linedenotes �� ¼�0.96, derived in x 5.1, and the dashed linesmark uncertainties of �0.22 around this value.

These tests show that our estimated mean �� ¼�0:96�0:22 is not strongly affected by the fitting range in k, but

there is some uncertainty in the absolute range of �2D;u and�2D;w to which this value applies. Conservatively, the fits tok ¼ 12 should be used, as these are free of dissipative effects.Using only the fields for which such fits can be made, we findthat �� ¼�0:87 � 0:19, and this applies to �2D;u � 3:5 4:2and �2D;w � 2:7 3:4.

However, we also note that the lack of dependence of �� onthe fitting range in k is indirect evidence that a single value of�� is applicable at all �2D;u, but this interpretation should beexamined in more detail in future. In what follows we assumethat �� ¼�0:96 � 0:22 holds for all �2D;u and �2D;w.

5.3. Origin of the Spectral Modification

5.3.1. Effect of Density Spectral Index

Lazarian & Esquivel (2003) have suggested that (in ourterminology) �� may have a larger amplitude for ‘‘shallow’’density field power spectra. The density field power spectrumslope may be estimated from the power spectrum of the pro-jected column density field, Pcd kð Þ, using the same argumentsas presented in x 2 (see also Lazarian & Pogosyan 2000; Stutzkiet al. 1998). If �� is related to the spectral slope, �cd, of thecolumn density power spectrum [where Pcd kð Þ / k��cd ], thenthis would indeed provide a means of inferring �� observa-tionally. In other words, the observed column density powerspectrum slope is used as a probe of the (relative) densityfluctuation spectrum as a function of scale within the medium.

To investigate this, we produced column density maps forall the simulations (including projection along two axes—parallel and perpendicular to the magnetic field—for the MHDfields). We then fitted �cd for each field, using the same rangein wavenumber that was used to derive ��. The fitted values of�cd are given in Table 1. (The column density power spectraare also slightly curved, so �cd would be lower if fitted tok ¼ 12; thus, the �cd-axis is uncertain by a roughly linearshift, as in Fig. 8 for �2D;u.)

Fig. 6.—(a) Plot of �� derived from fits to the power spectrum ratio fittedup to k ¼ 12 vs. �� derived from fits to the power spectrum ratio fitted up tok ¼ 32, for a subset of the driven runs (see text). The solid line is a line ofequality. (b) Difference between the �� values shown in (a) (fit to k ¼ 32minus fit to k ¼ 12). The solid line marks a difference of zero.

Fig. 7.—Velocity line centroid power spectra for models HC2 and HC4.The dotted lines delimit the range of k used in the power-law fits (straightlines) to �2D;u.

SUPERSONIC TURBULENCE 203No. 1, 2004

In Figure 9, we plot the variation of �� with �cd. There is aclear variation of �� with �cd, demonstrated by the decaying-turbulence simulations. As time proceeds, �cd increases (i.e., apreferential reduction in small-scale density contrast), and theamplitude of �� diminishes (i.e., a lessened effect of densityinhomogeneity on the velocity line centroid field). Forthe driven simulations, the �cd are typically �3. These are inreasonable accord with observational estimates of �cd � 2:8for molecular clouds (e.g., Bensch, Stutzki, & Ossenkopf2001). For our �cd � 3, �� varies around the mean value of�1. Thus, observational estimates of �cd suggest thataccounting for density inhomogeneity is very important forvelocity line centroid analysis of molecular clouds, implying(with �� � �1) that �2D � �3D, counter to the expectationsbased on projection smoothing alone. This provides an ex-planation for the �2D � �3D result observed in these simu-lations by Ossenkopf & Mac Low (2002). Lazarian &Pogosyan do not provide a physical motivation for whetherthe medium is density-dominated (i.e., a ‘‘shallow densityspectrum’’ having �cd � 3) or not: it is simply a question ofwhat �cd is, regardless of why. In our scenario, this regime isthe driven-turbulence regime.

5.3.2. Amplitude of Density Fluctuations

A note of caution regarding the use of �cd to infer �� shouldbe made: �cd is a nonunique measure of the density field. Toinvestigate this more deeply, we analyzed modified versionsof the HE2 field. First, we randomized the Fourier spacephases of the density and velocity fields to produce newfields with the same power spectra as the original fields. Thismodification has two effects: (1) it removes the density-

velocity correlations, and (2) it drastically reduces the ampli-tude of density fluctuations. Following phase randomization ofthe density field, we found that we had to recenter the meandensity because of the creation of negative densities by thephase randomization process. After renormalization back to amean density of unity, the phase-randomized density field hada standard deviation of 0.21 and a range of 1� 10�6 to 2.02.For comparison, the original density field, normalized to amean of unity, had a standard deviation of 2.53 and a range of1:2� 10�6 to 92.1. The logarithm of the phase-randomizeddensity field had a standard deviation of 0.22, compared tothe standard deviation of the logarithm of the original densityfield of 2.49. Using the phase-randomized density fieldsand velocity fields, we derived �� ¼�0:18 � 0:03. This issubstantially lower than the original �� ¼�1:04 � 0:05,obtained with the original HE2 density and velocity fields. Thisshows that the ‘‘shallowness’’ of the density power spectrum,as measured by �cd , does not uniquely allow inference of ��.We conducted a second test, this time randomizing the

phases of the logarithm of density and then exponentiating thephase-randomized log(density) field. This procedure againremoves the density-velocity correlations but does not lead tothe creation of negative densities, and it allows a greateramplitude of density contrast. We found that the resulting�cd � 2:41 for the density field produced by this method wasslightly shallower than the original �cd � 2:87 for the HE2model. The standard deviation of log(density) was the same forthe phase-randomized density field and the original densityfield (2.49). Using the exponentiated, phase-randomizedlog(density) field in conjunction with the phase-randomizedvelocity field, we found �� ¼�1:16 � 0:05.These two tests together suggest that it is the amplitude of

density fluctuations, measured as the standard deviation oflog(density), that is controlling the value of ��. To make surethat this is the case and that density-velocity correlation doesnot play a major role, we took the original velocity field andthe original density field, shifted by 64 pixels from its originalposition. For a shift along the line of sight, we derived

Fig. 9.—Plot of �� vs. �cd for all simulations. Symbols as in Fig. 5. Thearrow shows the direction of evolution of �� as time (since driving is turnedoff ) increases; see Fig. 5.

Fig. 8.—(a) Density-weighted velocity line centroid spectral index, �2D;w,vs. the unweighted (true) velocity line centroid spectral index, �2D;u, for alldriven runs, with the indexes fitted to k ¼ 32. (b) Plot of �� (=�2D;w � �2D;u)vs. �2D;u for all driven runs, with the indexes fitted to k ¼ 32. (c) Same as (a),but for a subset of the driven runs (see text), with the indexes fitted to k ¼ 12.(d) Same as (b), but for a subset of the driven runs (see text), with the indexesfitted to k ¼ 12. In all plots the solid line denotes the relationship �� ¼ �0:96,and the dashed lines mark uncertainties of �0.22 around this relationship.

BRUNT & Mac LOW204 Vol. 604

�� ¼�0:90 � 0:04, and for a shift transverse to the line ofsight we derived �� ¼�0:97 � 0:06. These are slightly lowerin magnitude than the original �� ¼�1:04 � 0:05, but thedifference is marginally significant.

We thus conclude that the amplitude of density contrast ismost important in determining ��. To examine this directly,we measured the standard deviation of log(density), �ln �, forall the numerical fields (see Table 1), and we plotted thisversus �� (Fig. 10a) and �cd (Fig. 10b). This indicates thatthere is a fairly good correlation between �� and �ln � and that,with the assumption that the numerical fields are a betterrepresentation of real molecular clouds than the phase-randomized fields, �cd may be used as a surrogate for �ln �.We find that the driven simulations are characterized by�cd � 3 and �ln � exceeding �unity, but that there is noobvious dependence of �� on �cd or �ln � within this regime.

Ideally, one would like to measure �ln � to infer ��, but it isof course impossible to measure �ln � directly. However, itmay be possible to infer �ln � indirectly using the columndensity field. Figure 11a compares the variation in thestandard deviation of log(column density), �ln N , to �ln �.The dashed line is fitted—�ln N ¼�0:02 � 0:08þ 0:33�ð0:06Þ�ln �—as a guide. The values of �ln N are listed in Table 1.Unfortunately, it is also impossible to use �ln N observation-ally, as the small N are dominated by noise. Instead, weconsider simply the standard deviations of density, ��=�0 , andcolumn density, �N=N0

, where both fields are normalized totheir mean values, �0 and N0, respectively. The dependenceof �N=N0

on ��=�0 is shown in Figure 11b, along with thefitted line, �N=N0

¼�0:06�0:07þ 0:34 � 0:04ð Þ��=�0 . Theseresults show that the variation in column density may be usedto infer the variation in (three-dimensional) density, but thedynamic range in N is compressed as a result of line-of-sight

averaging (�N=N0� ��=�0=3). However, in general, it should

be recognized that, at any finite resolution, the mean densityand column density are well estimated, but their standarddeviations are continuously variable with resolution in therespective dimensions. It may be possible to attempt toconstruct a resolution-independent measure to compare two-and three-dimensional standard deviations, but this is beyondthe scope of the current work.

In Figure 12, we plot combinations of ��=�0 , �N=N0, ��,

and �cd to gauge the utility of different measures. Insummary, some guidelines concerning �� may be establishedobservationally by measuring �cd and �N=N0

. Note that ameasurement of �cd and �N=N0

may also be used to distin-guish the phase-randomized fields from the numericallysimulated fields: the phase-randomized density field has amuch lower �N=N0

, 0.065, than the numerical field (�N=N0¼

0:786) for the same �cd ¼ 2:87. Some caution in comparing�N=N0

with Figure 12 is needed if the observations are atsignificantly different spatial dynamic range than the simu-lations used here.

5.4. Comparison with Previous Results

Our results appear inconsistent with those of two previousstudies. Brunt et al. (2003) found that �� � 0:4, from thesimulation used in their paper (M � 1). The density field usedin Brunt et al. (2003) had � ln � ¼ 0:45. Comparison withFigure 10a shows that these numbers are not discrepant withthe current results. The conclusion of Brunt et al. (2003) that,for a given amplitude of density fluctuations, the correlationsbetween density and velocity are important was based on the asingle simulation and is outweighed by our new results here.Our results are inconsistent with the results (�� � 0) found byLazarian & Esquivel (2003) for their simulation (M � 2:5).This may be due to the solenoidal driving in their simulations(Cho & Lazarian 2002); certainly, this discrepancy deservesfurther attention.

Fig. 10.—Plots of (a) �� and (b) �cd vs. �ln � for all simulations. Symbols asin Fig. 5. The phase-randomized density model has �ln � ¼ 0:22, �� ¼ 0:18,and �cd ¼ 2:87.

Fig. 11.—(a) Plot of �ln N vs. �ln � for all simulations. (b) Plot of �N=N0vs.

��=�0 for all simulations. Symbols as in Fig. 5.

Fig. 12.—For all simulations, (a) �� vs. ��=�0 , (b) �cd vs. ��=�0 , (c) �� vs.�N=N0

, and (d) �cd vs. �N=N0. Symbols as in Fig. 5. The phase-randomized

model has ��=�0 ¼ 0:21, �N=N0¼ 0:065, �� ¼ 0:18, and �cd ¼ 2:87.

SUPERSONIC TURBULENCE 205No. 1, 2004

5.5. Resolution Tests

Finally, we examined the effect of limited spatial dynamicrange on our results. The driven numerical fields implicitlycontain realizations at different effective resolutions (i.e., dueto the varying driving scales). These show no obvious trendof �� with driving wavelength; however, there is a largeamount of scatter in the derived �� that may hide such atrend. In addition, we have explored the effect of limiting thefits to �� to different ranges of k. We conducted further, moredirect tests, using realizations of model D (decaying turbu-lence) at sizes of 1283 and 643 and a realization of modelHC2 (driven turbulence) at 2563 resolution. These realizationsutilize the same parameterization but are not identical in detail(i.e., they were not initialized with the same random numberseeds and so are subject to the natural scatter in �� evidencedin Table 1).

Snapshots of model D at 1283 and 643 sizes are available attime steps of 0.1, 0.15, and 0.3 sound crossing times. Weconducted the same analysis to derive �� from these loweredresolution simulations. At 1283 size, we find �� ¼�0:78�0:08, �0:57 � 0:04, and �0:22 � 0:03 for times 0.1, 0.15,and 0.3, respectively. At 643 size, we find �� ¼�0:71 � 0:21,�0:33 � 0:22, and �0:11 � 0:25 for times 0.1, 0.15, and0.3, respectively. As the resolution is increased, the �� areconverging to the values of �� ¼�0:76 � 0:03, �0:59 � 0:02,and �0:32 � 0:02 found at 2563 size at the same time steps.The driven model HC2, with a higher Mach number, showed asomewhat more pronounced shift, from �� ¼�0:72 � 0:05 at1283 to �� ¼�1:01 � 0:03 at 2563. In sum, this resolution

study suggests that the magnitude of �� is decreased slightly bylimited spatial dynamic range. Therefore, our derived ��magnitudes should probably be considered to be lower limits(i.e., the true, high-resolution values may be a little morenegative than found here).

6. VELOCITY LINE CENTROID ANALYSIS:OBSERVATIONAL RESULTS

6.1. Overview

An overview of the CGPS CO data in the vicinity ofNGC 7129 is given in Figure 13. We have selected a region(64� 64 pixels, 32� 32 beams, 4096 spectra) for velocityline centroid analysis, indicated by the superposed square inFigure 13. This region is located away from the main star-forming area (it contains no IRAS point sources) and hassimple, singly peaked spectral lines. To derive velocity linecentroid maps, we utilized Gaussian fitting to the spectrallines, rather than a direct projection via equation (4), asGaussian fitting was found to be less noisy (see below).Example fits to both the 12CO and 13CO lines are shown inFigure 14. These spectra are taken from a 16� 16 pixel blockat the center of the analysis region and are block-averagedover 4� 4 pixels. The 12CO lines have roughly twice the peaktemperature of the 13CO lines, resulting in comparable signal-to-noise ratios for each tracer.Fits to the 13CO lines could not be made for the entirety of

the analysis region, because of weak or absent emission. Useof the 13CO line centroid field for analysis would then requirevalues for the missing data to be assumed before calculation of

Fig. 13.—Overview of the region near NGC 7129 selected for velocity line centroid analysis. Left: 12CO peak-temperature image. Right: 13CO peak-temperatureimage. The wedges show antenna temperature in K. [See the electronic edition of the Journal for a color version of this figure.]

BRUNT & Mac LOW206 Vol. 604

the power spectrum. A possible choice would be to set themissing data to the mean value of the measurable data.However, a better procedure is to create a ‘‘hybrid’’ linecentroid field, using the 12CO centroid velocities as the bestestimate of the missing 13CO centroid velocities. We thusmasked off all pixels in the analysis region for which thepeak temperature T �

A of the 13CO line was less than 1.4 K (i.e.,�5 �) and replaced the absent or noisy 13CO measurementswith the 12CO centroid velocities. For the hybrid field, 40% ofthe line centroid velocities are from 13CO, and 60% are from12CO.

For the lines of sight where we have good fits to both 13COand 12CO, we compared the derived line centroid velocities, asshown in Figure 15a. The two measurements are very similarover the entire variation of centroid velocity (roughly �11to �9.5 km s�1), but there is a slight average offset betweenthe two of �0.075 km s�1 (� 1

2channel width), as seen in

Figure 15b. This can arise from greater saturation on one sideof the 12CO line profiles or from depth-dependent abundance

variations in the presence of a line-of-sight velocity gradient.The effects of opacity on our measured 12CO velocity linecentroid map are thus very small. With the offset understood,the width of the distribution in Figure 15b (about� 0.1 km s�1)can be otherwise attributed to noise on the measurements. Tomeasure the noise on the line centroid fields, we took the fittedGaussian line profiles as the best estimate of the true lineprofiles and added actual instrumental noise from the samespatial positions, taken from a signal-free area of the baselineregion. We then carried out the same Gaussian fitting proce-dure to derive the noise distribution. For comparison, wederived the direct line centroid field from the ‘‘renoised’’ fieldsusing equation (4). The estimated errors on the 12CO linecentroid field are �0.093 km s�1 (Gaussian fitting) and�0:122 km s�1 (using eq. [4]). The Gaussian-fitting procedureis less noisy, and as the lines in the original data are welldescribed by Gaussians (Fig. 14), we used the line centroidsderived by Gaussian fitting for the power spectrum analysisbelow. The resulting velocity line centroid fields derived fromGaussian fitting are shown in Figure 16. The observed fieldsare qualitatively more similar to the simulations driven at largescale (the kd ¼ 1 2 case in Fig. 1) than to those driven atsmall scale.

6.2. Large-Scale Gradients and Edge Effects

Prior to the full analysis, we investigated the effect of fil-tering out the ‘‘large-scale’’ line centroid velocity fluctuations.Whether this should be done or not is a matter of somecontroversy (Miesch & Bally 1994; Ossenkopf & Mac Low2002). We estimated the large-scale gradients by fitting aquadratic surface to the 12CO line centroid map. Then wesubtracted this from the original line centroid map andcalculated the power spectrum before and after subtraction,using FFTs. We found that only the low-frequency (k ¼ 1)power is strongly affected by this procedure, as may have been

Fig. 14.—Example fitted Gaussians (heavy lines) to the observed spectra (lighter lines) in the analysis region

Fig. 15.—(a) Comparison of the velocity line centroids derived from 12COand 13CO where both measurements can be made. (b) Histogram of thedifferences in measured v lsr (12CO velocity minus 13CO velocity).

SUPERSONIC TURBULENCE 207No. 1, 2004

expected. The measured spectral slopes, between k ¼ 2 and 13(see below) are �2D ¼ 3:38 � 0:10 (before subtraction) and�2D ¼ 3:42 � 0:09 (after subtraction). Thus, the spectrumsteepens slightly (but insignificantly) after subtraction of thelarge-scale gradients, most likely because of the suppressionof high-frequency structure resulting from mismatch in theline centroids at the field edges (i.e., the FFT routine assumesperiodic fields). As there is little difference between the linecentroid power spectrum before and after subtraction of thelarge-scale gradients, and as it is not clear whether they shouldbe subtracted, we do not employ the subtraction in the analysisbelow.

We also examined the effect of apodization (Miville-Deschenes, Lagache, & Puget 2002). We centered the 12COline centroid field to a zero mean, then applied a cosine taperto a strip 2 pixels wide around the edge of the field. Theedges of the line centroid field now match at zero velocity.For the apodized field, we applied the same taper function tothe noise estimates, to ensure that the noise floor is still wellestimated. At large k both the apodized line centroid powerspectrum and the apodized noise floor are reduced, resultingin comparable signal-to-noise ratios before and after apod-ization. After subtraction of the respective noise floors (seebelow), the fitted spectral slopes are �2D ¼ 3:70 � 0:13(before apodization) and �2D ¼ 3:76 � 0:12 (after apodiza-tion). There is little to distinguish these measurements overthe measurement uncertainty, so we do not utilize apodizationin the analysis below. As the apodization routine simplyimposes arbitrary smooth structure in the line centroid field, itis not surprising that the measured power spectrum slopesteepens slightly (i.e., a reduction of small-scale power).Alternatively, the untapered field contains unphysical sharpvelocity structure at the field edges, so the actual noise floor–subtracted spectral slope most likely lies between the twoslope estimates above.

6.3. Correction for Noise and Beam Smearing

The raw measured indices stated above (�2D � 3:4) are notthe actual true, measured indices, as we have not yet accountedfor the effects of measurement noise and beam smearing on themeasured power spectra. The total power spectrum in thepresence of noise is, simply, Ptrue kð Þ þ Pnoise kð Þ (where Ptrue isthe power spectrum that would be observed in the absence of

noise), assuming independence of the signal and noise. Thenoise power spectrum must thus be subtracted from the rawmeasured power spectrum prior to analysis. The finite size ofthe telescope beam induces correlations between closelyspaced pixels; for a beam pattern B x; yð Þ in direct space (i.e.,roughly a Gaussian form with 4500 FWHM), the observed linecentroid map (neglecting noise for now) is vobslc ¼ vobs;truelc B,where denotes convolution and vobs;truelc is the line centroidfield that would be observed without beam smearing. Thebeam-smeared power spectrum is then Pvobs

lckð Þ ¼ Pvobs;true

lc

kð ÞPB

kð Þ, where PB kð Þ is the power spectrum of the beam pattern.The final output power spectrum, corrected for noise and beamsmearing, is

Poutput kð Þ ¼ Pvobslc

kð Þ � Pnoise kð Þh i�

PB kð Þ: ð5Þ

Note that the division out of the beam pattern is appliedafter the noise floor subtraction, as the noises in our data arespatially independent (in contrast to the measured signal).The noise power spectra can be derived from the renoised

fields discussed above; we did this for both the 12CO linecentroid field and the hybrid line centroid field. We assumedthat a Gaussian beam pattern, B x; yð Þ, of 4500 FWHM, appliedto both the 12CO line centroid field and the hybrid line cen-troid field. A summary of the relevant terms in equation (5) isprovided in Figure 17. Output power spectra were thenderived via equation (5), and these are shown in Figure 18.The resulting scaling range covers k ¼ 2 13, limited at lowk by a possible turnover and the uncertainty of large-scalegradient subtraction and limited at high k by the amplificationof measurement noise by beam pattern division. The resultingfits to the range k ¼ 2 13 are �2D ¼ 3:11 � 0:10 and2:93 � 0:13 for the 12CO line centroid field and the hybrid linecentroid field, respectively. These are jointly consistent with�2D � 3, giving �2D � 0:5.The range of the fit appears deceptively large in Figure 18;

the spatial dynamic range achieved is only 6.5, betweenwavelengths of 32 and 4.9 spatial pixels (�2.5 beamwidths).Similar spatial dynamic range was achieved by Miesch &Bally (1994), who measured a typical value of �2D � 0:43.Around 3 orders of magnitude in spatial dynamic rangewas achieved for the Polaris Flare molecular cloud byOssenkopf & Mac Low (2002) by combining three data sets

Fig. 16.—Observed velocity line centroid maps. [See the electronic edition of the Journal for a color version of this figure.]

BRUNT & Mac LOW208 Vol. 604

at different angular resolutions and spatial scales; they alsomeasured �2D � 0:5.

6.4. Interpretation

If we interpret the measured index of �2D � 3 (�2D � 0:5),allowing for projection smoothing but without taking intoaccount the effect of density weighting, then the inferredthree-dimensional spectral index is �3D � 3. We would thusobtain �3D � 0 (via eq. [2]), or no dependence of ‘‘turbulent’’velocity dispersion on scale. If correction for projection

smoothing is applied to the indices measured by Miesch &Bally (1994) and Ossenkopf & Mac Low (2002), this againgives �3D � 0. A similar result (i.e., �3D � 0) was found byO’Dell & Castaneda (1987) for H ii region turbulence, aftercorrecting for projection smoothing. The correction for pro-jection smoothing, however, has been the exception ratherthan the rule for velocity line centroid analysis, and mostauthors have taken their projected �2D as being equivalent tothe intrinsic �3D. This is not correct, as the selection of thekz ¼ 0 plane of the three-dimensional line-of-sight velocitypower spectrum by projection onto the two-dimensionalplane of the sky is purely a geometrical effect: projectionsmoothing thus always applies (unless the cloud is actuallytwo-dimensional; see below), being modified only by theeffects of density inhomogeneity (and radiative transfer).Hobson (1992) interprets, e.g., �2D ¼ 3 (i.e., �2D ¼ 0:5) asbeing applicable to shock-dominated turbulence (�3D ¼ 0:5).Miesch & Bally (1994) directly compare their �2D to theo-retical predictions of �3D with no modification. Lazarian &Esquivel (2003) derive from their numerical data (in ourterminology) �� ¼ 0, but then interpret the �2D � 0:43 ofMiesch & Bally (1994) as meaning also that �3D � 0:43.

Our numerical results here show that the (statistical)equivalence of �2D and �3D is actually valid in the case ofdriven turbulence but only because of the accidental cancel-lation of two competing factors: projection smoothing, whichsuppresses small-scale velocity line centroid structure, anddensity inhomogeneity, which creates small-scale velocity linecentroid structure.

The interpretation of existing results from line centroidanalysis applied to molecular clouds in general will depend onthe inferred, or assumed, state of turbulence. If molecularclouds are in an evolved state of decaying turbulence, thenonly projection smoothing applies, and we would infer �3D �0 for all clouds studied to date. If, however, the clouds are stillbeing strongly driven or have only recently formed, then the�� ��1 induced by density inhomogeneity should apply, and

Fig. 17.—Overview of the relevant terms necessary to construct the output line centroid power spectra. Left: 12CO field; raw observed power spectrum, noisefloor, and beam pattern (4500). Right: Same for the hybrid field; the dashed line is the 4600 beam pattern applicable to the 13CO data; no uncertainty arises fromapplying the 4500 beam pattern to the hybrid field.

Fig. 18.—Output velocity line centroid power spectra after noise floorsubtraction and beam pattern division (eq. [5]). The heavy lines are for the fitsof �2D to these spectra in the range k ¼ 2 13.

SUPERSONIC TURBULENCE 209No. 1, 2004

we thus can find a better consistency between line centroidanalysis and other measures of �3D. To show this, we appliedPCA to the analysis region, according to the prescriptiongiven in Brunt & Heyer (2002), to derive a sequence ofcoupled characteristic scales, �v and L, in velocity and space,respectively. The results of this, yielding slightly better thanan order of magnitude in spatial dynamic range, are shown inFigure 19. The �v and L pairs were fitted as �v / L� . Wederive � ¼ 0:58 � 0:01 (13CO) and � ¼ 0:59 � 0:02 (12CO),resulting in �3D ¼ 0:44 � 0:01 (13CO) and �3D ¼ 0:46 � 0:04(12CO), according to the Brunt (1999) calibration. Somecaution is necessary here, however, on two counts. First, � ismore directly dependent on the first-order velocity fluctuationspectrum, not the second-order spectrum that is traced by thepower spectrum of the line centroid field; second, estimates of� are uncertain (around the mean calibration line) by �0.1(Brunt et al. 2003). Neither of these factors can reconcile thePCA results with �3D � 0, however. If, as suggested by PCAin the limit of weak intermittency (as also supported by thelack of strong exponential tails in the line profiles),�3D � 0:45, then we infer that �� ��0.9 would be needed toreconcile the line centroid results with PCA. This value of ��is consistent with �� ¼�0:96 � 0:22, derived from thenumerical simulations of driven turbulence.

To determine the observationally motivated value of ��,without recourse to PCA, we examined the power spectra ofthe integrated intensity fields for both isotopes (it would bevery difficult to attempt this with a hybrid integrated intensityfield). Here we assume that the integrated intensity field pro-vides a good estimate of the column density field. To do this,we estimated the noise floor by integrating signal-free regionsof the baseline over the same number of channels used togenerate integrated intensity maps. The measured ‘‘columndensity’’ power spectra were then corrected for the noise floorand beam pattern (eq. [5]), and the resulting output powerspectra are shown in Figure 20. We have fitted �cd over therange k ¼ 2 13 (the same as used for the velocity linecentroid power spectra), to estimate �cd � 2:5 consistentlybetween the two isotopes. This is in general agreement with

typical �cd measured in molecular clouds (Bensch et al. 2001),but slightly smaller than can be accounted for by our nu-merical fields. Using Figure 9 as a rough guide, we interpretthe measured �cd as motivating �� ��1. In doing this, we areconsidering the numerically simulated fields to be a morerealistic representation of real molecular clouds than thephase-randomized versions. We do not see a turnover at smallk in the integrated intensity power spectra or in the velocityline centroid power spectra, except perhaps at k ¼ 1, which isquite uncertain. Our observed power spectra are thus more inaccord with the numerical models driven at small wavenumber(see also Ossenkopf & Mac Low 2002). Finally, we used the13CO data to provide an estimate of �N=N0

, by assuming thatthe 13CO is optically thin. To do this, we created an integratedintensity image of the emission and an integrated intensityimage of instrumental noise over the same number of channels(64); the data were left in T�

A units, since the absolute scalingis unimportant. The mean integrated intensity of the datais I0 ¼ 1:034 � 0:004 K km s�1. The total (signal+noise)standard deviation of the integrated intensity image is �I ;tot ¼0:661. The contribution of the standard deviation of the noiseis �I ;noise ¼ 0:253 K km s�1. Thus, we measure �I ;signal=I0 ¼0:591 � 0:004, which we take as being ��N=N0

. This isclearly inconsistent with �N=N0

¼ 0:065, seen in the phase-randomized density field (obtained with �cd ¼ 2:87). Theobservational data (�cd, �N=N0

) are much more consistentwith the unmodified numerical data than with the phase-randomized numerical data (see Fig. 12d ). The effective spatialdynamic range in the simulations for computing �N=N0

will liebetween the nominal spatial dynamic range (128 per axis) andthe most stringent interpretation of the maximum dissipationscale (�12.8 per axis). These compare reasonably well withthe data, at a spatial dynamic range of 32 beams (64 pixels)per axis. We interpret the consistency found between �cd and�N=N0

as motivating the use of the numerically simulated fieldsto infer �� ��1.Our results regarding the effect of density inhomogeneity

indicate that the measured �2D � 3 is thus shallower than theFig. 19.—PCA results for the selected field. The solid lines are for the fits

to the 12CO (upper) and 13CO (lower) �v, L pairs (�v / L� ).

Fig. 20.—Output integrated intensity power spectra after noise floorsubtraction and beam pattern division. The heavy lines are for the fits of �cd tothese spectra in the range k ¼ 2 13.

BRUNT & Mac LOW210 Vol. 604

true �3D by approximately �� ��1, and we then infer �3D � 4(see Fig. 8c) and thus �3D � 0:5. In more detail, we derive(using �� ��0:96 � 0:22) �3D ¼ 4:07 � 0:24 (12CO) and�3D ¼ 3:89 � 0:26 (hybrid). These convert to �3D ¼ 0:54 �0:12 (12CO) and �3D ¼ 0:45 � 0:13 (hybrid). These are ingood agreement with �3D � 0:45, determined from PCA, andare consistent with compressible Burgers turbulence. The un-certainty in the inferred �3D is large, mainly because of theintrinsic variability of �� (about � 0.22). We thus cannot en-tirely rule out the Kolmogorov case (�3D � 3:67, �3D � 0:33)from velocity line centroid analysis alone, particularly if weinclude the extra uncertainties arising from subtraction oflarge-scale gradients and edge effects (estimated to be about� 0.1 in �2D).

We interpret the conditions required for consistencybetween velocity line centroid analysis, PCA, and size–linewidth analysis as providing indirect support for molecularclouds’ being only recently formed and/or continuallydriven on large scales (Elmegreen 1993; Ballesteros-Paredes,Vazquez-Semadeni, & Scalo 1999).

Projection smoothing does not apply when the line-of-sightdepth over which the observational tracer exists is much lessthan the total transverse scale of emission observed on the sky,requiring sheetlike clouds oriented face-on toward us. For aface-on sheet, the effect of density weighting would alsovanish, as there is no longer any line-of-sight variability (seeeq. [3] in the limit that Npix ! 1 as a representation of thisregime). While this solution is quite unsatisfactory in general, itrepresents a limiting case: no projection smoothing and nodensity-weighting effects, resulting in the direct measurementof intrinsic statistical properties; i.e., the field is actuallytwo-dimensional. Some modified version of this limiting casemay be operating in our numerical fields; i.e., if a singledensity enhancement (not necessarily confined to a thin sheet,but having some spatial coherence) is dominant along the lineof sight, then this will approximate the Npix ! 1 limit ofequation (3).

In general, for the full three-dimensional (isotropic) drivencase, our numerical results show that projection smoothing iscountered by the effects of density inhomogeneity, againresulting in recovery of intrinsic statistical scaling properties,but in a less direct way.

7. SUMMARY

We have examined numerical simulations of supersonicturbulence in molecular clouds in order to understand theeffects of projection smoothing and density inhomogeneityon observable velocity line centroid fields [vlc x; yð Þ]. Wefind that the observed index, �2D, describing the variationof projected line centroid velocities ( �v2lc

� �1=2 / L�2D ) is re-

lated to the intrinsic index, �3D (where �v2lc� �1=2/ L�3D ), via the

following relation:

�2D ¼ �3D þ 0:5þ ��=2;

where the increase of 0.5 is a direct consequence of projectionsmoothing and �� is an empirically determined correction forthe effects of density inhomogeneity.

For driven turbulence, both with and without magneticfields, �� ��1 (with variations of about � 0.22), thus im-plying �2D � �3D as a result of the accidental cancellationof the effects of projection smoothing and small-scale struc-ture induced by density inhomogeneity on the line centroidfield. For decaying turbulence, �� tends to �0 at late times,so that �2D � �3D þ 0:5 entirely as a result of projectionsmoothing.

The variations in �� observed in high–Mach number driventurbulence have no obvious dependence on Mach number, M.As M ! 0 (the incompressible case), one should also expectthat �� ! 0, as is indeed seen at late times in the decayingruns. We have inferred that �� is determined mostly by theamplitude of density fluctuations. The scatter in �� in thesupersonic regime emphasizes the need for continued highspatial dynamic range surveying of the molecular interstellarmedium in order to provide a statistically useful sample offields.

Deprojection of observed scaling relations in velocity linecentroid fields requires some knowledge of the state of the flow.This information can be inferred from the spectral slope of thecolumn density power spectrum and a measure of the standarddeviation in column density relative to the mean (�N=N0

). Ournumerical results show that �3D is now consistently estimatedat �0.5, using velocity line centroid analysis, PCA, and size–line width relations. This consistency requires that molecularclouds be continually driven or only recently formed.

We acknowledge useful comments from Volker Ossenkopfand particularly the anonymous referee. The Dominion RadioAstrophysical Observatory is a National Facility operated bythe National Research Council. The Canadian Galactic PlaneSurvey is a Canadian project with international partners and issupported by the Natural Sciences and Engineering ResearchCouncil (NSERC). C. M. B. is supported by a grant fromNSERC to the Canadian Galactic Plane Survey. M.-M. M. L.is supported by NSF career grant AST 99-85392 and NASAAstrophysical Theory Program grant NAG5-10103. Compu-tations analyzed here were performed at the RechenzentrumGarching of the Max-Planck-Gesellschaft. The Five CollegeRadio Astronomy Observatory is supported by NSF grantAST 01-00793.

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