eve c. ostriker, james m. stone and charles f. gammie- density, velocity and magnetic field...

8/3/2019 Eve C. Ostriker, James M. Stone and Charles F. Gammie- Density, Velocity and Magnetic Field Structure in Turbulent

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THE ASTROPHYSICAL JOURNAL, 546: 9801005, 2001 January 10( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DENSITY, VELOCITY, AND MAGNETIC FIELD STRUCTURE IN TURBULENT MOLECULAR

CLOUD MODELS

EVE C. OSTRIKER,1,2 JAMES M. STONE,1 AND CHARLES F. GAMMIE3Received 2000 April 26; accepted 2000 September 1

ABSTRACTWe use three-dimensional (3D) numerical magnetohydrodynamic simulations to follow the evolution

of cold, turbulent, gaseous systems with parameters chosen to represent conditions in giant molecularclouds (GMCs). We present results of three model cloud simulations in which the mean magnetic eldstrength is varied kG for GMC parameters), but an identical initial turbulent velocity eld(B

0\ 1.414

is introduced. We describe the energy evolution, showing that (1) turbulence decays rapidly, with theturbulent energy reduced by a factor 2 after 0.40.8 ow crossing times (D24 Myr for GMC parame-ters), and (2) the magnetically supercritical cloud models gravitationally collapse after time B6 Myr,while the magnetically subcritical cloud does not collapse. We compare density, velocity, and magneticeld structure in three sets of model snapshots with matched values of the Mach number MB 9,7,5.We show that the distributions of volume density and column density are both approximately log-normal, with mean mass-weighted volume density a factor 36 times the unperturbed value, but meanmass-weighted column density only a factor 1.11.4 times the unperturbed value. We introduce a spatialbinning algorithm to investigate the dependence of kinetic quantities on spatial scale for regions of

column density contrast (ROCs) on the plane of the sky. We show that the average velocity dispersionfor the distribution of ROCs is only weakly correlated with scale, similar to mean sizeline width dis-tributions for clumps within GMCs. We nd that ROCs are often superpositions of spatially uncon-nected regions that cannot easily be separated using velocity information; we argue that the samedifficulty may aect observed GMC clumps. We suggest that it may be possible to deduce the mean 3Dsizeline width relation using the lower envelope of the 2D sizeline width distribution. We analyze mag-netic eld structure and show that in the high-density regime cm~3, total magnetic eldn

H2Z 103

strengths increase with density with logarithmic slope D1/32/3. We nd that mean line-of-sight mag-netic eld strengths may vary widely across a projected cloud and are not positively correlated withcolumn density. We compute simulated interstellar polarization maps at varying observer orientationsand determine that the Chandrasekhar-Fermi formula multiplied by a factor D0.5 yields a good estimateof the plane-of sky magnetic eld strength, provided the dispersion in polarization angles is [25.

Subject headings: ISM: clouds ISM: molecules MHD methods: numerical stars: formation

1. INTRODUCTION

Since the identication of cold interstellar clouds in radiomolecular lines, observational campaigns in many wave-lengths have provided an increasingly detailed and sophisti-cated characterization of their structural properties. Theseclouds are self-gravitating entities permeated by magneticelds and strongly supersonic turbulence; the observationalproperties of giant molecular clouds (GMCs) are sum-marized, for example, by Blitz (1993), Williams, Blitz, &McKee (2000), and Evans (1999). Although it has long beenappreciated by theorists that turbulence and magnetic eldsmust play a decisive role in cloud dynamics (e.g., Mestel &Spitzer 1956; Shu, Adams, & Lizano 1987; McKee et al.

1993; Shu et al. 1999; McKee 1999), much of the theoreticalemphasis has been on evolutionary models in which theeects of turbulent magnetohydrodynamics (MHD) ismodeled rather than treated in an explicit fashion.

Recent advances in computer hardware and developmentof robust computational MHD algorithms have now madeit possible to evolve simplied representations of molecularclouds using direct numerical simulations. Fully nonlinear,

1 Department of Astronomy, University of Maryland College Park,MD 20742-2421; ostriker=astro.umd.edu, jstone=astro.umd.edu.

2 Institute for Theoretical Physics, UCSB, Santa Barbara, CA 93106.3 Center for Theoretical Astrophysics, University of Illinois 1002 W.

Green Street Urbana, IL 61801; gammie=uiuc.edu.

time-dependent, MHD integrations can test theoreticalideas about the roles of turbulence and magnetic elds incloud evolution, and also make it possible to investigatehow turbulence aects the structural properties of clouds.Progress in the rapidly developing eld of simulations ofGMC turbulence is reviewed by, e.g., etVa zquez-Semadenial. (2000).

This is the fourth in a series of papers (Gammie &Ostriker 1996; Stone, Ostriker, & Gammie 1998; Ostriker,Gammie, & Stone 1999) [Papers IIII, respectively]) inves-tigating the dynamics of turbulent, magnetized, cold cloudsusing direct numerical simulations.The previous papers pre-sented several results on energetics and overall cloud evolu-tion. They showed that (1) MHD turbulence can delaygravitational collapse along the mean magnetic eld in one-dimensional models since dissipation is slow (Paper I);however, (2) in higher dimensional models dissipationoccurs on the ow crossing timescale (Paper II); as at

fconsequence, (3) the fate of a cloud depends on whether itsmasstomagnetic ux ratio is subcritical or supercritical,independent of the initial turbulent excitation, providedthat turbulence is not steadily driven (Paper III).

Some important astrophysical implications of theseresults are that (1) star formation in turbulent clouds maybe initiated rapidly, essentially on a ow crossing timescale;and (2) models that rely on slowly dissipating turbulence tosupport GMCs against collapse do not appear to be viable.

980


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STRUCTURE OF MODEL CLOUDS 981

One is still faced with the problem of avoiding the excessiveGalactic star formation rate that would result from the col-lapse and fragmentation of the whole cold component of theISM within its gravitational free-fall time (comparable to itsow crossing time; Zuckerman & Palmer 1974). Thisrequires either (1) limitation of the star formation rate inindividual clouds (if self-gravitating clouds are long-livedafter formation), or (2) limitation of the lifetimes of self-gravitating clouds. Both of these eects may be important.

Processes that contribute to limiting star formation rates inindividual clouds include turbulent feedback from star for-mation, transmission of turbulence from the larger-scaleISM, or a large (subcritical) mean magnetization of clouds.The rst two of these processes, together with destabilizingenvironmental factors such as enhanced galactic shearoutside spiral arms, contribute to limiting lifetimes of indi-vidual clouds.

Other workers have independently used simulations todeduce the same results about the rapidity of turbulent dis-sipation under likely GMC conditions (Mac Low et al.1998; Mac Low 1999; Padoan & Nordlund 1999). Similarconclusions have also been reached concerning ongoingturbulent driving and the potential for star formation to be

initiated on a rapid timescale (see also Ballesteros-Paredes,Hartmann, & 1999 ; Elmegreen 2000 ;Va zquez-SemadeniKlessen, Heitsch, & Mac Low 2000).

In addition to studying cloud evolution, our previouswork also investigated structural properties of our modelclouds. We found that density contrasts produced by turbu-lent stresses are compatible with the typical clump/interclump ratio estimated in GMCs (Papers IIII). We alsofound that velocity and magnetic eld power spectra evolveto be comparable to power-law forms of Burgers and Kol-mogorov turbulence, regardless of the driving scale (PaperI ; see also Stone, Gammie, & Ostriker 2000). Other workershave also studied the basic structural properties of the turb-ulent gas in compressible hydrodynamic and MHD simula-

tions, concentrating on distribution functions of density andvelocity 1994 ; Passot &(Va zquez-Semadeni Va zquez-

1998; Scalo et al. 1998; Nordlund & PadoanSemadeni1999; Klessen 2000), the ability of stresses to produce tran-sient structure (Ballesteros-Paredes, &Va zquez-Semadeni,Scalo 1999), and power spectra and related functions(Passot, & Pouquet 1995 ;Va zquez-Semadeni, Va zquez-

Ballesteros-Paredes, & Rodriguez 1997; Elmeg-Semadeni,reen 1997, 1999 ; Klessen et al. 2000 ; Mac Low & Ossenkopf2000).

In this paper, we analyze decaying turbulence in self-gravitating cloud models with varying mean magnetization(i.e., masstomagnetic ux ratio).4 We begin by brieydescribing the energy evolutions of our models, which serveto conrm our earlier results on turbulent dissipation timesand the gravitational collapse criterion for magnetizedclouds. We then turn to detailed structural investigations.We analyze the density, velocity, and magnetic eld dis-tributions in our models at those stages of evolution whenthe turbulent Mach number is comparable to that in large(D510 pc scale) clouds. Our goals are (1) to provide a basicdescription of structural characteristics and how theydepend on input parameters ; (2) to make connections

4 All models reported here, and most models studied by other workers,impose the somewhat articial constraint that the initial mass-to-ux ratiois spatially uniform.

between cloud models seen in projection and their truethree-dimensional (3D) structure, so as to help interpretobservational maps; and (3) to assess whether certain sta-tistical properties of clouds can be used to estimate themean magnetization.

We divide our analysis of structure into three main sec-tions. The rst ( 4) is a discussion of density structure.Numerical hydrodynamic and MHD simulations of super-sonic, turbulent ows have shown that magnetic pressure

and ram pressure uctuations produce structures with largedensity contrast that appear to resemble analogous clumpy and lamentary structures in real clouds (e.g.,Passot et al. 1995; Padoan & Nordlund 1999; Paper III;Klessen 2000; Balsara et al. 1999). Studies of densitymaxima and their immediate surroundings ( clumps )show that many are transient, as indicated by comparablevalues for the kinetic energy and kinetic surface terms in thevirial theorem (Ballesteros-Paredes, &Va zquez-Semadeni,Scalo 1999; see also McKee & Zweibel 1992). Clumpproperties in our turbulent cloud models will be examinedin a companion paper (Gammie et al. 2000, in preparation).

Analysis of the correlations of overdensityvia clumpstudies or multipoint statisticswill be needed to charac-

terize fully how a spectrum of self-gravitating condensationsis established. This process is of great interest because itmay ultimately determine the stellar IMF. A rst step inunderstanding the eect of turbulence on density structure,however, is to examine one-point statistics. Here, wecompute and compare the distributions of density andcolumn density in dierent cloud models. We consider bothbecause volume densities can be inferred only indirectlyfrom observations, whereas column density distributionscan be obtained directly from surveys of stellar extinction tobackground stars (Lada et al. 1994; Alves et al. 1998; Lada,Alves, & Lada 1999).

The second structural analysis section ( 5) considers theline widthsize relation. Observations give diering results

for the slope of this relation depending on whether thestructures involved are clearly spatially separated from thesurroundings (e.g., by a large density contrasts) or are iden-tied as coherent regions in position-velocity maps. Theformer case yields relatively steep power spectra (Larson1981; Solomon et al. 1987); the latter case yields shallowerpower spectra and larger scatter (Bertoldi & McKee 1992;Williams, de Geus, & Blitz 1994; Stutzki & 1990)Gu stenand has led to the concept of moderate-density pressure-conned clumps within GMCs. We believe the dierentslopes are a consequence of dierent denitions of clump.We use a simple binning algorithm to explore the scaling ofkinetic properties of apparent clumps within projectedclouds, and in particular to understand the consequences ofprojection eects for line widthsize relations for 2D areasand 3D volumes. We argue, consistent with the suggestionsof some other workers (e.g., Adler & Roberts 1992;Pichardo et al. 2000) that it may be difficult to identifyspatially coherent condensations from observed position-velocity maps.

The third structural analysis section ( 6) considers themagnetic eld. A topic of much interest in turbulence mod-eling is understanding how the magnetic eld aects boththe intrinsic dynamics and the observable properties of acloud. As shown in 3, a major dynamical eect of themagnetic eld is to prevent gravitational collapse in sub-critical clouds. Because magnetic eld strengths are difficult


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982 OSTRIKER ET AL Vol. 546

to measure directly, however, it is highly desirable to deter-mine if more-readily observable structural properties ofclouds could act as proxies for the magnetic eld strength.With simulations, it is possible to make comparisons ofdierent models in which the mean eld strength is varied,but other key properties (such as the turbulent Machnumber and power spectrum) are controlled.

An important theme in our analysis in 4 and 5 is to testhow the quantitative measures of structure depend on the

mean magnetization, to evaluate the potential use of suchmeasures as indirect magnetic eld diagnostics. In 6 weanalyze how more direct magnetic eld diagnostics may beaected by cloud turbulence. We evaluate the distributionof total magnetic eld strength as a function of density incloud models with dierent mean magnetization. We alsocompute the distribution of mean line-of-sight integratedmagnetic eld (one-point statistic), which is relevant forinterpreting Zeeman eect measurements of magnetic eldstrength. Finally, we study the distribution of polarizationdirections in simulated maps of polarized extinction pro-duced by turbulent clouds (one-point statistic). One of theearliest estimates of magnetic eld strength in the inter-stellar medium (Chandrasekhar & Fermi 1953) was based

on the dispersion in polarization direction, using a simpleone-wave description of the magnetic eld. We update theChandrasekhar-Fermi (CF) estimate using our simulationsas presumably more realistic descriptions of the magneticeld geometry.

The plan of this paper is as follows: We start ( 2) bydescribing our numerical method and model parameters.We then ( 3) describe our results on energy evolution, con-rming the previous results from 3D nonself-gravitatingmodels on dissipation rates and 2.5D self-gravitatingmodels on the criterion for collapse. We present our struc-tural analyses in 46, and conclude in 7 with a summaryand discussion of these investigations.

2. NUMERICAL METHOD AND MODEL PARAMETERSWe create model clouds by integrating the compressible,

ideal MHD equations using the ZEUS code (Stone &Norman 1992a, 1992b). ZEUS is an operator-split, nite-dierence algorithm on a staggered mesh that uses an arti-cial viscosity to capture shocks. ZEUS uses constrainedtransport to guarantee that $ B\ 0 to machine preci-sion, and the method of characteristics to update themagnetic eld in a way that ensures accurate propagationof disturbances (Evans & Hawley 1988; Hawley &Alfve nicStone 1995). The solutions are obtained in a cubic box ofside L with grids of 2563 zones, which permits spatialresolution over a large dynamic range at manageable com-putational cost. We apply periodic boundary conditions inall models. The simulations were run on an SGI Origin2000 at NCSA.

For the energy equation, we adopt an isothermal equa-tion of state with sound speed In the absence of a fullyc

s.

time-dependent radiative transfer, this represents a goodrst approximation for the gas at densities higher than themeancomprising most of the matterfor conditionsappropriate to molecular clouds (see discussion in PaperIII, and also Scalo et al. 1998).

The gravitational potential is computed from the densityusing standard Fourier transform methods. The k \ 0 com-ponents of the density are not included in the solution dueto the periodic boundary conditions. Rather than the usual

Poisson equation, the gravitational potential therefore/G

obeys where is the mean+2/G

\ 4nG(o [ o6), o64M/L3density (mass/volume in the box).

The initial conditions are as follows: We start withuniform density, a uniform magnetic eld and aB

04B

0x,

random velocity eld dv. As in our earlier decay models(Papers IIII), is a Gaussian random perturbation elddwith a power spectrum subject to the con-o dv

ko2P k~4,

straint so that the initial velocity eld is noncom-$ d \0pressive. This power spectrum is slightly steeper than theKolmogorov spectrum and matches the( o dv

ko2P k~11@3)

amplitude scaling of the Burgers spectrum associated withan ensemble of shocks (but diers from Burgers turbulencein that the initial phases are uncorrelated).

In conguration space, the velocity dispersion of theinitial conditions averaged over a volume of linear size Rincreases as This spectrum is comparable to thep

vPR1@2.

spectrum inferred for large-scale cold interstellar clouds(e.g., Larson 1981; Solomon et al. 1987; Heyer & Schloerb1997) and the spectrum that naturally arises from the evolu-tion of compressible turbulence that is either decaying or isdriven over a limited range of scales (Stone 1999; Stone etal. 2000, in preparation). We use an identical realization of

the initial velocity eld for all of the models, so that initialstates of the simulations dier only in the strength of the(uniform) mean magnetic eld.

This paper considers three dierent simulated cloudmodels. All are initiated with kinetic energy E

k\ 100o6L3c

s2,

corresponding to initial Mach number M4 pv/c

s\

For the purposes of comparison with observations,10J2.we shall use a ducial mean matter density (i.e., correspond-ing to the total mass divided by total volume) n

H2\ 100

cm~3 and isothermal temperature T\ 10 K in normalizingthe local simulation variables of our models to dimensionalvalues. The velocity dispersion in physical units is given by

km s~1(T/10 K)1@2, so that the initial valuepv

\ 0.19]Mis km s~1(T/10 K)[email protected]

v

\ 2.7The models dier in their initial magnetic eld strength,

parameterized by with physicalb4 cs2/v

A,02 \ c

s2/(B

02/4no6),

value given by

B0

\ 1.4] b~1@2kGA T

10 K

B1@2A nH2

100 cm~3

B1@2. (1)

We run a strong eld model with b \ 0.01, a moderateeld model with b \ 0.1, and a weak eld model withb \ 1. For characteristic ducial densities and temperaturesof molecular clouds (TD 10 K, cm~3), the corre-n

H2D 100

sponding uniform magnetic eld strengths are 14, 4.4, and1.4 kG. Of course, the evolved elds are spatially nonuni-form and can dier greatly from these initial values (see 6),although the mean magnetic eld (i.e., the volume-averagedvalue or k \ 0 Fourier component) is a constant inB

0x

time. The values of bhalf the ratio of the gas pressure tothe mean eld magnetic pressureare proportional to thesquare of the masstomagnetic ux ratio in the simulationbox ; this ratio cannot change in time.

We may identify several dierent, physically signicanttimescales in the model evolution. The sound crossing time,

is xed owing to the isothermal equation of state.ts4 L/c

s,

Another important timescale is the ow crossing time overthe box scale L, Myr] (L/10t

f4 L/p

v\ 9.8 pc)(p

v/km

s~1)~1. Because the turbulence decays (i.e., M decreases),the instantaneous ow crossing time increases relative to


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No. 2, 2001 STRUCTURE OF MODEL CLOUDS 983

the sound crossing time as Where we relatetf

\ ts/M. t

fand we use the Mach number associated with the initialt

s,

turbulent velocity dispersion, such thatpv/c

s\ 14.1, t

f\

0.07ts.

This paper concentrates on structures that form as a con-sequence of turbulence, before self-gravity becomes impor-tant. However, we also use the present models to test ourprevious results from lower-dimensional simulations (PaperIII) on the dierences in the gravitational collapse times

with strong and weak mean magnetic elds It is there-B0.fore useful to dene a gravitational contraction timescale

tg4A n

Go6

B1@2\ 9.9 Myr

A nH2

100 cm~3

B~1@2. (2)

In the absence of self-gravity, the unit of length Ldeningthe linear scale of the simulation cube would be arbitrary.In a self-gravitating simulation, an additional parametermust be chosen to represent the relative importance ofgravity and thermal pressure forces to the evolution. Auseful dimensionless measure of this is in all thet

g/ts

;models considered here this ratio is A more transparent1

3.

way of stating this is that there are three thermal Jeans

lengths across a box scale L.5LJ4 cs(n/Go6)1@2The three simulations described herein dier in the rela-

tive importance of magnetic and gravitational forces totheir ultimate evolution. As described in Paper III, a cloudwith constant masstoux ratio is super- or subcritical ift

gis smaller or larger than respectively. A supercriticalnL/v

A,

(subcritical) cloud has a ratio of masstomagnetic uxgreater (smaller) than the critical value, 1/(2nG1@2). Sub-critical clouds can collapse along the eld but not perpen-dicular to the eld ( pancake) ; in the nonlinear outcomethe peak density would be limited by the thermal pressure.Supercritical clouds can collapse both parallel and perpen-dicular to the eld, with unlimited asymptotic density. Thethree models discussed here have andt

g

v

A

/(nL) \ 0.11,0.34,1.1. Thus, the strong-eld model is subcritical and the othertwo models are supercritical. The results on long-termgravitational evolution reported in 3 conrm the expecteddierences between super- and subcritical clouds under thecondition that turbulence secularly decays.

Since self-gravity is weak for the rst portion of the evolu-tion in our simulations, the freedom of normalization of Lthat applies to nonself-gravitating models also eectivelyapplies during this temporal epoch. In particular, the struc-tural analyses of 46 are performed at stages of the simu-lations evolutions for which the kinetic energy is at least 5times as large as the components of the gravitational energy

associated with the uctuating density distribution.EG

Because of our periodic boundary conditions, the gravita-tional energy associated with the mean density (i.e., thek \ 0 Fourier component) is not included in In orderE

G.

of magnitude, the value of this lowest-order gravity iswhich equals for ourDGM2/L\ Mc

s2 n(L/L

J)2, 28Mc

s2

present models. This energy is in the middle of theL/LJ

\ 3range of kinetic energies for the M\ 9,7,5 snapshots weanalyze. Thus, as for observed clouds (e.g., Larson 1981;Myers & Goodman 1988), the lowest-order gravitational

5 Of course, the presence of strong turbulence makes the classical, linearJeans stability analysis inapplicable; the velocity eld is in the nonlinearregime from the rst instant.

energy in these snapshots would be comparable to thekinetic energy.

A useful reference length scale may be obtained by com-bining the well-known observational relations betweenvelocity dispersion, mass, and size for GMCs (see Paper III).The characteristic outer linear size scale for observedL

obsclouds scales with Mach numberM according to

L

obs B1.1

]M pcA

T

10 KB1@2

An

H2100 cm~3B

~1@2. (3)

Because the observed scale is proportional to the Machnumber, the ow crossing time for observed clouds is inde-pendent ofM, and given by

tf,obs

B 5.8 Myr]A n

H2100 cm~3

B~1@2. (4)

For observed clouds, the ow crossing time and gravita-tional contraction time are proportional, with t

f,obsB

0.59tg

.Since the turbulence (and therefore M) decays in our

models, they are comparable in their kinetic properties toincreasingly small clouds as time progresses. For example,using the relation (3), the observational scale associatedwith the initial models with M\ 14.1 would be L

obs,init\

16 pc] (T/10 cm~3)~1@2. At this size scale,K)1@2(nH2

/100the corresponding sound crossing time would be t

s\ 82

cm~3)~1@2. In the structural analyses ofMyr] (nH2

/100 46, we report on properties of model snapshots in whichMD 9, 7, and 5 ; observed clouds of linear size scaleD10, 8,and 6 pc, respectively, have kinetic energies correspondingto those of the model snapshots. To the extent that gravitymay be unimportant for much of the internal substructurein multiparsec scale observed clouds (suggested by GMCslack of central concentration, and by the weak self-gravityof substructures aside from the dense cores and largestclumps [e.g., Bertoldi & McKee 1992; Williams et al.1994]), the correspondence between the intermediate-scale( clump ) structure in real clouds and in our model snap-shots may be quite direct.

Since some ambiguity remains in associating an overallphysical length scale with our simulated models (due to theperiodic boundary conditions), we report integrated quan-tities solely in dimensionless units, giving e.g., column den-sities in units of the mean column density, ForN14 o6L.local variables (such as magnetic eld strengths), which bearno such ambiguity, we report values in dimensionless unitsand also transform to physical units based on our adoptedducial density and temperature.

3. ENERGY EVOLUTION IN MODEL CLOUDSThe early evolution in all the models follows a similar

course. Kinetic energy initially decreases as the uid worksto deform the magnetic eld. The initially noncompressivevelocity eld is transformed into a compressive eld, byinteractions with the magnetic eld and nonlinear couplingof the spatial Fourier components. This leads to the devel-opment of density-enhanced and density-decient regions,and results in the dissipation of energy in shocks. Fluctua-tions in the density cause uctuations in the gravitationalpotential that begin to dominate the dynamics at late timesand lead to runaway gravitational collapse for supercriticalmodels.


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(a) (b)

(c) (d)


To quantify the energetic evolution, we dene the kineticenergy

EK

\ 12P

d3r(vx2 ]v

y2 ]v

z2)o , (5)

the perturbed magnetic energy

dEB

\1

8n

Pd3r(B

x2 ]B

y2 ]B

z2) [ E

B,0, (6)

where is the energy in the mean magneticEB,0

\ L3B02/8n

eld, and the gravitational potential energy

EG

\ 12P

d3r(o/G

) , (7)

where is the gravitational potential computed from the/G

Poisson equation modied for periodic boundary condi-tions.

Figure 1 shows the evolution of Etot4E

K] dE

B] E

G,

and (see Figs. 1a1d, respectively). From FigureEG

, EK

, EB

1b, it is clear that, consistent with expectations and previousresults on self-gravitating cloud models with decaying turb-ulence (Paper III), all but the magnetically subcritical

b \ 0.01 model suers a gravitational runaway. Both of thesupercritical models become gravitationally bound at time

corresponding toD6 cm~3)~1@2. 6 TheD0.6tg, Myr(n

H2/102

gravitational runaway time is comparable to that found inlower-dimensional simulations.

The kinetic energies in all models decay rapidly. After oneow crossing time the kinetic energy has been reduced byt

f,

73%85% compared to the initial value (see Table 1). Thekinetic energy is reduced by a factor 2 after 0.20.4 ow

crossing times (Table 1), with this kinetic loss time decreas-ing toward lower b (stronger because of the more rapidB

0)

transfer of kinetic to perturbed magnetic energy when thefrequency is higher. For GMC parameters (see eq.Alfve n

[4]), the corresponding dimensional kinetic energy decaytime would be 12 Myr. The growth of magnetic energystored in these magnetic eld uctuations (due to advection

6 The simulations are terminated shortly after the onset of gravitationalrunaway because the coincident development of low-density regions where

is large causes the Courant-conditionlimited timestep to become veryvA

short. For the b \ 1 model, the gravitational binding time (the time toreach is an extrapolation based on the evolutions of the b \ 0.1E

tot\ 0)

and b \ 1 models shown in Figs. 1a, 1b.

FIG. 1.Energy evolution of model clouds. Models with b \ 0.01, 0.1, and 1 are shown with dotted, solid, and dashed curves, respectively.


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TABLE 1

COMPARATIVE ENERGY EVOLUTION

Model b EK

(tf

)/EK,init

a dEB/E

K(tf

)a tdec

/tfa,b t

decK /t

fa,b t

bind/t

gc

B . . . . . . 0.01 0.27 0.51 0.76 0.21 [0.9C . . . . . . 0.1 0.15 1.13 0.55 0.31 0.6

D . . . . . . 1.0 0.17 0.35 0.41 0.37 0.6

a The ow crossing time tf4 L/(2E

K,init)1@2 ] 0.07t

s.

is the time to reduce the initial energy (kinetic energy) by 50%.b

tdec (tdecK

)is the time at whichc tbind

EK

] dEB

] EG

\ 0 ; tg4 (n/Go6)1@2] 0.33t

s.

by the turbulent velocity eld) is apparent in Figure 1 d; theinitial increase is followed by decreasing or at perturbedmagnetic energy as the turbulent velocity eld decays. Thetime to reach the maximum perturbed magnetic energy liesin the range 0.10.2 times the crossing time, similarAlfve nto what was found in lower-dimensional simulations (PaperIII). At the point when is maximal, it accounts fordE

B20%50% of The fraction increases with the meanE

turb.

eld strength B0

.The total turbulent energy secu-(E

turb\ E

K] dE

B)

larly decreases in time; after somewhat more than half ofthe initial turbulent energy is lost, the decay approaches apower-law temporal behavior with (Fig. 2). ThisE

turbP t~1

late-time scaling in nonself-gravitating models of 3DMHD turbulence has been noted previously (Mac Low etal. 1998; Paper II). Most of the turbulent losses, however,occur before the onset of this behavior. The turbulent decaycan be characterized by the time to reduce the turbulentt

decenergy by a factor 2 from its initial value. We nd that thistime is in the range 0.40.8 ow crossing times (see Table 1),comparable to our results from Paper II, and consistentwith other ndings (Paper III; Mac Low et al. 1998; MacLow 1999; Padoan & Nordlund 1999) that dissipation

FIG. 2.Evolution of turbulent energy in modelEturb

\ EK

] dEB

clouds. Models with b \ 0.01, 0.1, and 1 are shown with dotted, solid, anddashed curves, respectively. The dot-dash lines indicate a slope P t~1, forreference.

times vary by only a factor D2 over the range of Machnumbers and magnetic eld strengths present in GMCs.The corresponding dimensional time for turbulent energydecay with GMC parameters is 24 Myr.

4. DENSITY AND COLUMN DENSITY DISTRIBUTIONS

A basic statistical property of a real or model cloud is thedistribution of density in its constituent parts. This distribu-tion may be described either by its fractional volume perunit density (dV/do) or by its fractional mass per unitdensity (dM/do). Previous analyses of the density distribu-tions in compressible hydrodynamic turbulence simulations(before gravity becomes important) show that when theequation of state is approximately isothermal, the densitydistribution is close to a log-normal (Va zquez-Semadeni1994; Padoan, Jones, & Nordlund 1997; Passot &

1998; Paper III). Scalo et al. (1998),Va zquez-SemadeniPassot & (1998), and Nordlund &Va zquez-SemadeniPadoan (1999) also show that in a medium where the tem-perature decreases (increases) with increasing density, anextended tail in the density distribution function develops atdensity higher (lower) than the mean density. In the presentmodels we assume an isothermal equation of state. This is areasonable approximation here since most of the gas is con-tained in condensations at density larger than the meanvalue where the temperature likely varies by less than afactorD2 (see Paper III ; also Scalo et al. 1998).

As described in 1, a key question is whether it is possibleto discriminate the magnetic eld strength in a cloud fromits structural properties. Using our present models, we cantest how the strength of the mean magnetic eld aects theobservable density and column density statistics. For thesetests, we choose sets of model snapshots from the threedecay models in which the Mach number M4 Sv2/c

s2T1@2

(or kinetic energy) matches in the three models; because theenergy evolves at somewhat dierent rates in the runs withdierent b, these times of the snapshots vary. The sets ofmodel snapshots haveMB 9,7,5.

Figure 3 shows an example of the distributions of volumeand mass as a function of volume density, o, where o ismeasured in units of the mean density in theo6 \ M/L3simulation cube. The volume-density distributions are wellapproximated by log-normal functions, i.e., volume andmass distributions in of the formy4 log (o/o6)

fV,M

(y) \1

J2np2exp [[(y ^ o k o )2/2p2] , (8)

where the upper/lower sign on the subscript applies to thevolume/mass distribution; is the fraction of thef

V,Mdy

volume or mass with y in the interval (y, y ] dy). It is


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0.1 10

0.5

1

0.1 10

0.5

1

0.1 10

0.5

1


FIG. 3.Comparative statistics of volume density in three model snap-

shots (B2, C2, D2 from Table 2) with matched Mach numbers MD 7.Solid curves show fraction of volume as a function of density relative to themean dashed curves sho w fraction of mass as a function of(o/o6) ; o/o6 .Dotted curves show lognormal distributions with the same mean anddispersion as in each model snapshot.

straightforward to show that the mean k and dispersionp are related by ok[y] o\ ln (10)p[y]2/2 for a log-normaldistribution, so Table 2 (fthp[y] \ 0.93Jk[y].7and sixth columns) gives the values of k

V[y] (4SyT

V)

and for three sets of models at dierentkM

[y] (4SyTM

)Mach numbers (where the subscript on the angle brackets

denotes weighting by volume or mass). In all cases,consistent with a lognormal distribution.kV

[y]D[kM

[y],For a log-normal distribution, the weighted mean and

dispersion of the density itself are related to the mean of the

7 Elsewhere, distributions are sometimes given as a function of x4in that case, ok[x] o\ p[x]2 /2 since k[y] \ k[x]/ln (10) andln (o/o6) ;

p[y] \ p[x]/ln (10).

logarithmic density contrast k[y] using

logTo

o6

UM

\ 2T

logo

o6

UM

4 2 o k[y] o , (9)

TAoo6

B2UV

\To

o6

UM

\ 102@k*y+@ , (10)

and

TAoo6B2U

M

\Too6UM

3 \ 106@k*y+@ , (11)

where V and M subscripts denote weighting byvolume and mass, and From Table 2, k[y] is inSoT

V4 o6.

the range D0.20.4 for M\ 59, implying from equation(9) that the typical mass element has been compressed by afactor compared to its unperturbed initialSo/o6T

MD 2.56

value. Because of the log-normal form of the distribution,two-thirds of the matter is within a factor 100.93S@k*y+@(D2.73.8) above or below the value 10@k*y+@o6D (1.62.5)o6,and 95% is within a factor (D7.214) above101.86S@k*y+@or below this value. The volume-weighted rms standarddeviation in is (102k*y+ [1)1@2 (D1.32.2), and theo/o6mass-weighted rms standard deviation in iso/o6102k*y+(102k*y+ [1)1@2 (D313).

The above results on density contrast may be comparedwith previous work. In Paper III, we found that for 2.5-dimensional models of decaying turbulence withb \ 0.01,0.1,1.0, the mean logarithmic density contrasto k[y] o\ 0.20.5 for Mach numbers in the range 510, witha weak trend toward an increase in the contrast withincreasing Mach number, and the largest contrast in thestrong-eld (b \ 0.01) group. For the quasi-steady forced-turbulence models withMB 5 reported on in Paper II, themean logarithmic density contrasts o k[y] o are in the range0.200.28, increasing from b \ 1.0 to 0.01. Thus, overall, wend comparable values of the density contrast in all ouranalyses of turbulence in those stages where self-gravity isnot important.

Nordlund & Padoan (1999) and Padoan et al. (1997)report ndings implying that, for 3D unmagnetized forcedturbulence, k[y] is related to the Mach number M byo k[y] o\ (1/2) log (1 ] 0.25M2). For the range of Machnumbers (MD 59) in our Table 2, the correspondingvalues of o k[y] o would beD0.40.7, somewhat larger than

TABLE 2

COMPARATIVE D ENSITY AND COLUMN DENSITY

Snapshot b t/ts

M kV

[y]a kM

[y]a kMx

[Y]b kMy

[Y]b kMz

[Y]b

B1 . . . . . . . 0.01 0.03 8.8 [0.25 0.30 0.024 0.033 0.029C1 . . . . . . . 0.1 0.03 8.8 [0.27 0.28 0.030 0.037 0.047D1 . . . . . . . 1.0 0.03 9.4 [0.42 0.37 0.044 0.047 0.061B2 . . . . . . . 0.01 0.07 7.4 [0.38 0.38 0.038 0.037 0.046C2 . . . . . . . 0.1 0.04 7.6 [0.27 0.29 0.034 0.046 0.048D2 . . . . . . . 1.0 0.05 7.2 [0.37 0.34 0.060 0.054 0.065B3 . . . . . . . 0.01 0.19 4.9 [0.23 0.21 0.015 0.028 0.021C3 . . . . . . . 0.1 0.09 4.9 [0.35 0.37 0.047 0.050 0.056D3 . . . . . . . 1.0 0.09 4.9 [0.31 0.33 0.048 0.057 0.063

a Volume-weighted or mass-weighted average of the logarithmic density contrast, y4 log (o/o6) ;expected sampling error isD10~4.

b Mass-weighted average of logarithmic column density contrast, for projectionY4 log (N/o6L),along or expected sampling error isD10~3.x , y, z ;


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those we found; however, Nordlund & Padoan (1999)remark that they nd lower density contrasts when themagnetic eld is nonzero, which would yield better agree-ment with our results for magnetized turbulence. Analysisof simulations of compressible, isothermal, unmagnetizedturbulence in one dimension by Passot & Va zquez-

(1998) suggest a linear rather than logarithmicSemadeniscaling for k[y] with Mach number M and much largervalues of the contrast than those found in 3D simulations.

This may be due to the purely compressive velocity eld in1D.

The present analysis suggests that for nonsteady magne-tized turbulence, there is no one-to-one relationshipbetween the density contrast and the sonic Mach number orother simple characteristic of the ow. There does, however,appear to be a secular increase in the minimum value of thecontrast with the eective Mach number for magnetizedow, the fast magnetosonic Mach number dened byM

FIn Figure 4, we plot the logarithmicM

F24 Sv2T/Sv

A2 ] c

s2T.

contrast factors against the value of the lowerlog (1 ]MF2) ;

envelope of the contrast is found to be tted by ok[y] o\for the models studied. There is no0.2[log (1 ]M

F2) ] 1]

similar secular relationship between the density contrast

and the ordinary sonic Mach numberM.Applying similar reasoning to the argument of Passot &

(1998), the weak relation between theVa zquez-Semadenieective Mach number and the density contrast may beM

Funderstood heuristically as follows. From equations (9) and(11), we may write the mean logarithmic contrast in terms ofthe mass-weighted dispersion in density amplitude as

Tlog

Aoo6

BUM

\1

6log

CTAoo6

B2UM

D. (12)

In strong, unmagnetized, isothermal shocks, which wouldoccur for ow parallel to the eld, the preshock and post-

FIG. 4.Mass- and volume-weighted mean values of the logarithmicdensity contrast as a function of mean-square fast magnetosonic Machnumber from model snapshots in Table 2. Pentagons, squares, and tri-angles correspond to b \ 0.01,0.1, and 1, respectively. Dashed lines showSlog (o/o6)T

M,V\ ^0.2[log (1 ]M

F2) ] 1].

shock densities and have Foro1

o2

(o2/o

1) [ 1 BM2.

strong isothermal shocks magnetized parallel to the shockfront and is linear rather than quadraticb[ 1, (o

2/o

1) [ 1

in approaching If the typical shockMF

, J2v/vADJ2M

F.

jump compression factor determines the rms dispersion inthe density, then the term in square brackets in equation(12) would scale between quadratically and quartically in

for a range of b and shock geometries (noting thatMF

for b large). The real situation is of course moreMF]M

complicated. It is interesting, however, that the slope D0.2of the lower envelope of the versusSlog (o/o6)T log (1 ]M

F2)

relation does fall in the range between 0.17 and 0.33 sug-gested by this heuristic argument. The fact that this lowerenvelope lies closer to the (shallower) slope correspondingto parallel-magnetized shocks indicates that the modelturbulent clouds do not invariably evolve to be dominatedby (more compressive) ows aligned with the mean mag-netic eld.

Because of the potential for direct comparison withobservation, it useful to examine the distributions ofcolumn density N. In particular, we would like to ascertainif the distribution depends on the mean magnetization. Thedistribution of column densities can be described by the

fractional area, or fractional mass, perdA/dN(s), dM/dN(s)unit column density, where is the orientation of the line ofssight through the cloud. In Figure 5 we compare the dis-tributions of projected area and mass as a function ofcolumn density for model snapshots (B2, C2, D2 from Table2) with matched Mach numbers and dierent values of themean magnetic eld strength. Although the statistics arepoorer than for the distributions of volume density, thecolumn density distributions are also approximately log-normal in shape. Thus, the column density distributions can

FIG. 5.Comparative statistics of column density in three model snap-shots (B2, C2, D2 from Table 2) with matched Mach numbers. Projectionis along the axis (perpendicular to the magnetic eld). In each frame,zleft-displaced curves show fraction of projected area as a function of columndensity relative to the mean right-displaced curves show(N/o6L4N/N1 ) ; fraction of mass as a function of Dotted curves show lognormalN/N1 .distributions with the same mean and dispersion as in each model snap-shot.


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be described using the same form as equation (8), but repla-cing where The mean and dis-y ] Y4 log (N/N1 ), N14 o6L.persion may depend on the projection direction sos,

and To the extent that the dis-k[y] ] ks[Y], p[y] ] p

s[Y].

tributions follow log-normal forms, the formal relations (9)(11) would apply, with and area weightingo/o6 ] N/N1replacing volume weighting.

In Table 2 we list the values of the mass-weighted meanof the logarithmic column density contrast ( ok

Ms[Y] o4

for the dierent model snapshots in each ofSlog (N/N1)TMs)three projection orientations From the data ins \ x, y, z.the table, the projections in the various directions for anymodel snapshot yield somewhat dierent statistics (mostly10%20% dierences in p[Y] and twice that in k[Y]); theprojection along the magnetic eld tends to give slightlylower contrast than the two perpendicular projections. Dif-ferences between the two perpendicular directions and(z y)are simply a result of specic realizations of random initialconditions. Because the models have the same initial veloc-ity perturbation realization, they will have similar evolvedstructure to the extent that the magnetic elds only weaklyaect the dynamicsthis explains, for example, why modelsC and D both have larger contrast for projections thanz y

projections.Notice that models with the strongest magnetic eld tend

to have lower column density contrasts than models withthe same Mach number and weaker mean (20%50%B

0dierences in p[Y] for most sets). This eect is most pro-nounced for the Mach-5 set MB3, C3, D3N ; this set has afactor 2 (3) dierence in the p[Y] (k[Y]). Padoan & Nor-dlund (1999) previously pointed out that column densitycontrasts may be larger in weaker models. Ourmean-B

0results conrm this tendency, although we nd that theeect is relatively weak in magnitude, and does not hold inall cases (see e.g., the results for snapshots B2 and C2 in theTable).

Overall, the range of mean logarithmic column density

contrasts in Table 2 is k[Y]D 0.0150.065, correspondingto typical mass-averaged column density in the range

i.e., only a modest enhancement overSN/N1TM

\ 1.071.35,the average in a uniform cloud. The range of logarithmiccolumn density contrasts is an order of magnitude lowerthan the range of mean logarithmic density contrasts. Thisis understandable, since each column contributing to thedistribution samples a large number of over- and under-densities along the line of sight. The column density dis-tributions still require a density correlation length over asignicant fraction of the box size L along the line of sight,however; otherwise the dispersion in column densitieswould be wiped out by line-of-sight averaging.

This can be seen more quantitatively as follows. Each ofcolumns that contributes to the distribution is created byn

Ataking the sum of densities in cells along the line of sight.n

sFrom the central limit theorem we know that if the densityin each cell along the line-of-sight were an independentrandom variable, then for large, the distribution ofn

scolumn densities would approach a Gaussianrather thanlog-normalshape, with (area-weighted) mean of N/N1equal to 1 (where and (area-weighted) standardN14o6L),deviation in equal to times the (volume-N/N1 n

s~1@2

weighted) standard deviation in For a log-normalo/o6.volume-density distribution obeying equation (10), anassumption of independent sampling along the line-of-sightwould therefore predict an (area-weighted) rms deviation of

from unity given byN/N1

pN@N1Gauss \

1

Jns

(102@k*y+@ [ 1)1@2 , (13)

with typical sampling error in determining theDpN@N1

/JnA

mean and dispersion of For ok[y] oD 0.2 [ 0.4, theN/N1 .expected standard deviation in would be D0.080.14,N/N1with sampling error D0.00030.0006, if the line-of-sight

cells were all independent. In fact, using the area-weightedequivalent of relation (10) for the log-normal (not Gaussian)column density distribution that is evidently produced, thearea-weighted standard deviation in isN/N1

pN@N1logvnorm \ (102@k*Y+@ [ 1)1@2 , (14)

or approximately M2ln(10)k[Y]N~1@2 for k[Y]> 1. For ourtabulated values, this is in the range 0.27-0.59, signicantlylarger (by hundreds of times the sampling error) than wouldbe predicted by assuming uncorrelated values of the densityalong any given line of sight. Thus, both the non-Gaussianshape and the breadth of the dispersion of the columndensity distributions argues that the volume densities arenot independent but are correlated along any line of sight

as indeed should be expected since there are large coherentregions of density created by the dynamical ow.

We speculate that it may be possible to understand thedynamical process behind the development of the log-normal column density distribution following similarreasoning to the argument of Passot & Va zquez-Semadeni(1998) for the development of a log-normal volume densitydistribution. They argue that if consecutive local densityenhancements and decrements occur with independentmultiplicative factors due to independent consecutive veloc-ity compressions and rarefactions, then the log of thedensity in some position is the sum of logs of independentenhancement/decrement factors; this would yield a lognor-mal density distribution if there are many independent

compressions/rarefactions, each sampling independentlyfrom the same distribution of enhancement/decrementfactors.

Suppose, similarly, that the gas along any line of sight issubject to multiple independent compression/rarefactionevents; since the compression/rarefaction axes are not ingeneral along the line of sight, column density on a givenline of sight is not conserved. Each compression/rarefactionevent which produces a local change in the volume densityby a factor X aects only a fraction f of the column of gas,resulting in an eective enhancement/decrement factor forthe column closer to unity than X. A simple model wouldbe to suppose that each event i independently produces achange in the column density by a factor X

i

@ \ (1 [fi

)(taking the fraction] fi Xi \ Xi [ (1 [fi)(Xi [ 1) 1 [fi

of the gas in the column at unchanged volume density andthe fraction at volume density enhanced/decreased by af

ifactor If (respectively, thenX

i). X

i[ 1 X

i\ 1), X

i@ \ X

ii

(respectively The logarithm of the column densityXi@ [ X

i).

contrast would then be a sum of terms taking theselog Xi@ ;

as random variables, the resulting distribution would belog-normal (assuming a large number of [spatiallyoverlapping] successive events). Since each is closer toX

i@

unity than the mean and dispersion of the logarithmicXi,

column density distribution are expected to be smaller thanthose of the volume-density distribution. Although it wouldbe interesting to test in detail whether this sort of heuristic


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model could be rened and used to relate projected densitydistributions to volume-density distributions, the potentialfor nding a unique inversion (even in a statistical sense) islimited by the many degrees of freedom associated with theextended spatial power spectrum producing the compres-sions.

For the power-law input turbulent spectrum that weadopt, the spatial correlations which produce the columndensity distribution occur at sufficiently large scale that the

distributions are not, except at columns N much larger thanthe mean, very sensitive to the observers resolution. Forexample, Figure 6 shows the statistics of column density forone model at the full resolution of the simulation, and forresolving power reduced by factor of 4 by averaging thecolumn density values within squares of edge size four timesthat of simulation cells (so that each pixel has sixteentimes the area of a projected simulation cell). The overallshapes and mean values of the distributions are quite com-parable. At column densities much larger than the mean, ofcourse, the distributions become sensitive to resolutionbecause of the scarcity of regions with the highest columndensity; averaging these with their lower column-densityneighbors results in a cuto of the distribution at lower N.

A related point for observed 13CO data was discussed byBlitz & Williams (1997). They showed that the distributionof the number of cells in position-velocity space as a func-tion of in the cell attens as the linear resolutionT

A/T

A,maxscale increases, due to the smearing-out of the highest-column regions. We have veried that the distribution ofnumber of projected cells with similarly becomesN/N

maxatter if the map of projected density is averaged over gridswith increasing cell size.

Because the periodic boundary conditions introduce aneective correlation in the density along the line of sight atscale DL, a potential concern might be that the typicalcolumn density contrast might be enhanced by introducing

FIG. 6.Comparative statistics of column density at full simulationresolution (a) and at resolution a factor four larger in linear scale (b) ;simulation data is from snapshot B2.

FIG

. 7.Same as in 5, except line-of-sight integration is only overz [ L/2.

articial coherence along lines of sight.8 To investigatethis eect, we have evaluated sets of half-column densitydistributions by summing only over distances L/2 along theline of sight. In general, the resultant half-column distribu-tions are still lognormal in form (although noisier), withlarger means and dispersions than those found for the full-column integrations. Figure 7 shows one such set of dis-tributions, obtained from the z [ L/2 front half of thevolume snapshots B2, C2, D2. This result suggests that the

coherent volume-density regions responsible for the lognor-mal column density distribution in fact have intermediatescalethey are much larger than the cell size, but signi-cantly smaller than the overall size of the box. In this situ-ation, one would expect that a factor 2 decrease in thenumber of (multicell) correlated regions along the line ofsight would produce a factor increase in corre-J2 p

N@N1,

sponding to a factor D2 increase in ok[Y] o (cf. eq. 14).Indeed, we nd the half-column values of o k[Y] o are typi-cally larger than the full-column values by a factor D1.52,supporting this interpretation.

The robustness of the column density distribution toresolution changes makes it a viable statistic for comparingsimulations to the observable properties of turbulent

clouds. Such comparisons are a test of the idea that much ofthe moderate-density clumpy structure in molecularclouds may be produced by turbulent stresses. Preliminaryresults are promising; for example, we have compared thedistribution of the extinction data values from the darkcloud IC5146 (Lada, Alves, & Lada 1999) with columndensity distributions from our simulation snapshots. Figure8 shows that the cumulative distributions are indeedremarkably similar in form (although this particular realcloud has a slightly larger dispersion than our models have).Unfortunately, however, the column density distribution is

8 We thank E. for noting this point.Va zquez-Semadeni


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FIG. 8.Comparison of (a) cumulative column density distribution forsimulated cloud model (data from Fig. 7, with hexagons, squares, andtriangles marking the b \ 1,0.1,0.01 model distributions) and (b) cumula-tive extinction distribution for the cloud IC5146. Dotted curves in bothpanels show cumulative log-normal ts.

determined by more than just a few simple global param-eters. In some circumstances, there may be as much varia-tion in the column density distributions between the samecloud viewed at dierent orientations as there is in twoclouds with the same turbulent Mach number but a factor10 dierence in the mean magnetic eld. This large cosmicvariance, and the relatively weak variation with parame-ters of k and p compared to their scatter, make it unlikelythat it will be possible to estimate individual clouds meanmagnetic eld strengths, for example, from column densitydistributions alone.

5. LINE WIDTHSIZE RELATIONS ANDPROJECTION EFFECTS

An important way of characterizing the kinetic structurein turbulent clouds is to measure the distribution of thevelocity dispersion versus the physical size or mass of theregions over which it is averaged. Means over these dis-tributions then represent line widthsize relations. Theregions over which velocity dispersions are averaged inobserved clouds are often apparent clumps . At the mostbasic level, an apparent clump in a cloud or projectedcloud is a spatially connected, compact region that standsout against the surrounding background. In any hierarchi-cal structure, clumps will contain other smaller clumps, andin general the identication of clumps is a resolution-dependent procedure. Starting from the fundamentalconcept of a clump as a region of contrast (ROC) on agiven spatial scale, we have developed a simple algorithm toidentify and characterize the ensemble of projected ROCs atmultiple scales, so as to explore the scaling of kineticproperties with physical size.9

The procedure is as follows: First, we choose a size scale s(here, a factor 2n times the simulation grid scale, wheren \ 28). We then divide the projected cloud into zones ofarea s2. Within each zone, we compute the mean projectedsurface density as the total zone mass divided by s2 ; we alsocompute the mass-weighted mean surface density &[s] forthe set of zones on scale s. We label a zone as a ROC onscale s if its surface density is at least a factor times &[s].f

c

9 In Gammie et al. (2000), we use an alternative approach to deneclumps and characterize their properties.

Typically we use but the results are not qualitativelyfc

\ 1,sensitive to this choice; we note that (1) regions above themean column density at a given scale occupy less than halfthe area owing to mass conservation, and (2) since &[s]increases with decreasing s, the ROCs on a given scalewould appear by eye to stand out against the back-ground even with For each projected ROC, we alsof

c\ 1.

compute the (mass-weighted) dispersion of the line-of-sightvelocity this represents the line width for a region ofp

v;

projected area s2.We are now in a position to examine the correlations

among line width mass M, and spatial size s for ourpv

,ROC collections. In a data set based on molecular lineemission, the contributions from any local region woulddepend on the local excitation rather than simply beingproportional to the amount of matter present. For theanalysis described below (except as noted), we only includecontributions from material if its local density (mass/volume) is at least equal to as a simple way of o

min\ 3o6,

selecting material in the range of densities that contribute tocommon molecular lines.10

Figure 9 shows an example of how the ROCs at multiplescales are distributed on the map of model snapshot B2

projected in the direction (Fig. 22 shows a color-scalezimage of the column density for the same snapshotprojection). For the ROC ensemble shown in the gure, wecompute masses, velocity dispersions, and values of the so-called virial parameter (Bertoldi & McKeea4 5p

v2 s/GM

1992). In Figure 10, we plot the values as a function of(linear) size scale and/or mass. We also evaluate least-

squares linear ts to d log (M)/d log (s),d log (pv)/d log (s),

and d log (a)/d log (s); the respectived log (pv)/d log (M),

values in this example are 0.09, 1.87, 0.06, and [0.4.From Figures 10a and 10c, it is clear that although the

there is a mean increase in velocity dispersion with mass andlinear size, there is a great deal of scatter as well. The upperenvelopes of the velocity dispersion distributions in fact

even decrease as a function of increasing M and s ; the lowerenvelopes increase more steeply. The distribution of aversus M also shows large dispersion, with a nearly-atlower envelope and an upper envelope showing a decreasein a with M. The M versus s distribution has a relatively lowdispersion.

Many of the features evident in Figure 10 can be under-stood by reference to the scaling properties of the under-lying three-dimensional distribution, together with theeects of projection onto a plane. First consider the

distribution. The procedure we have used tolog (pv)log (s)

identify ROCs in the projected plane also can be used in the3D data cube itself; we can then compute the mass andvelocity dispersion for each 3D cell of edge size s that meetsthe contrast criterion. In Figure 10a, we show how the meanvelocity dispersion for these 3D cells depends on size scale.Interestingly, this curve traces fairly closely the lowerenvelope of the distribution of versus projected size forp

vROCs on the projected plane. Thus, for nearly all projectedregions of area s] s, the majority the velocity dispersion

10 Realistically, of course, the contribution to observed lines depends onmore than the local density; because of radiative transfer eects, it mighteven be possible for lower density material to contribute more efficientlythan higher density material if its emission occurs in line wings and suersless absorption.


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(h)

(a)

(b)

(c)

(d)

(e)

(f)

(g)


FIG. 9.Identication of regions of contrast (ROCs) as a function ofspatial scale for data from model snapshot B2. Figs. 1a1g outline regionsthat meet the contrast criterion at increasingly ne spatial resolution. Agray-scale representation of the projected density is shown in Fig. 1 h forcomparison.

can be attributed to the superposition along the line of sightof many regions of volume s] s] s with dierent meanvelocities. The relatively weak dependence of mean linewidth on projectedsize (or mass) simply reects the ubiquityof contamination by foreground and backgroundmaterial. Previously, Issa, MacLaren, & Wolfendale (1990)and Adler & Roberts (1992) have made a related point thatinferred broad line widths of apparently quite massiveGMCs may arise from overlapping in velocity space ofnarrower velocity distributions from individual smallerclouds superimposed along the line of sight. Relatively steep

increases of line width with size, as reported by Larson(1981) and subsequent authors, may be obtained in obser-vations provided that a structure is distinguishable from itssurrounding by a sufficient density or chemical contrast;these steeper laws correspond to what we measure with our3D ROC procedure (solid lines in Figs. 10a and 11a).

In Figure 10b we show the distribution of mass withprojected size for the ROCs; the mean logarithmic slope isnearly equal to 2rather than 3, as would be the case for

compact objects with three comparable dimensions. Themean slope is close to 2 simply because each ROC samplesalong the entire line of sight so that mass is nearly pro-portional to projected area; note, however, that at smallscales, the masses can lie considerably above the mean t.

It is interesting to compare the virial parameter a versusM distribution shown in Figure 10 with the analogous plotpresented by Bertoldi & McKee (1992) analyzing theproperties of apparent clumps in four dierent observedclouds. For the data sets considered in that work, linear tsto the log a versus log M relation gave slopes between [0.5and [0.68. For the model data shown in Figure 10 (and forour other snapshots as well), the mean t has a somewhatshallower slope. But the upper envelope of this distribution

(and those for other snapshots) has slope D[0.5 to [0.6.We can understand this upper envelope as follows: First,the largest velocity dispersions at a given projected scale (cf.Fig. 10a) are nearly independent of scale (typical logarith-mic slope is D0 to [0.1). With this, together with the massscaling nearly as s2, the result is an upper envelope ofaPM~0.5 to M~0.6. The relatively at lower envelope ofthe a versus M distribution can be explained by the project-ed ROCs that sample the lower-envelope of thep

vP s0.5

line widthsize distribution (following the true 3D linewidthsize relation), together with the approximate MP s2scaling.

All of the other model snapshots show qualitativelysimilar distributions of the kinetic parameters for ROCs to

those shown in Figure 10. For example, we show the samedistributions obtained for a weak magnetic eld modelsnapshot (D2) in Figure 11; qualitatively, all of the kineticscalings are quite comparable to those obtained for thestrong magnetic eld model. In general, for the model snap-shots in Table 2, the projections parallel to the magneticeld axes yield slightly stronger increase of line width withsize than do the other projections. For projections perpen-dicular to the mean eld, the ranges in the ts for thedierent snapshots are d log (p

v)/d log (s) \ 0.070.12,

and d log (a)/d log (s) \d log (pv)/d log (M) \ 0.030.08,

[0.45 to [0.34 (using the same minimum surface densitycontrast factor and For projections paral-f

c\ 1 o

min\ 3o6).

lel to the mean eld, the respective ranges for these ts are0.110.19, 0.060.12, and [0.40 to [0.29. The ts tod log (M)/d log (s) have a very small range, 1.831.89, for allprojections (using andf

c\ 1 o

min\ 3o6).

The results depend weakly on the denition of a ROC,and in particular on Reducing tends to yieldo

min. o

minatter slopes for andd log (p

v)/d log (s) d log (p

v)/d log (M)

(because velocity is anticorrelated with density, so thatadditional low-density material along the line of sightincreases the dispersion closer to the maximum), andsteeper slopes for d log (M)/d log (s) (approaching 2, thelimiting form for uniform column density), and for d log (a)/d log (s) (approaching [0.5, the limiting form for velocitydispersion independent of size and uniform column


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(a) (b)

(c) (d)


FIG. 10.Scale dependence of kinetic quantities in a projected map. In Figs.ad, each point represents one of the square regions of contrast (ROCs)identied in Fig. 9, with edge size s. In (a), we plot vs. size s (normalized to the box length L) the dispersion of the line-of-sight velocity in each projectedp

vsquare ROC, and also show (solid line) the mean dispersion in line-of-sight velocity for 3D cubes of side s. In (b), we plot vs. s/L the mass M of each ROC(normalized to the total mass in the simulation box In (c) we show vs. M. In (d) we show the virial parameter a vs. M. In each frame, dashed lineso6L3). p

vrepresent linear least-squares ts to the data; dotted lines represent 1p deviations from the t. In (c) and (d), we plot points from dierent-sized regions withdierent expansion factors.

density). Increasing has the opposite eect. Theomin

changes in slopes come about mainly from variations in theloci of the lower envelopes of the distributions when o

minvaries; the upper envelopes change very little, since theyreect the kinetic properties of ROCs that sample throughthe largest possible portion of the model cloud.

Because the projected ROC identication algorithm doesnot take into account any line-of-sight information for thematerial in any projected region, it should not be surprisingthat the velocity dispersions for projected regions can bemuch larger than the velocity dispersions for 3D cubes withthe same projected size. One might argue that foregroundand background material extraneous to a principal conden-sation could easily be removed based on velocity informa-tion, so that structures identied as contrasting regions inobserved molecular line l-b-v data cubes would truly rep-resent spatially coherent structures. Examination of theline-of-sight velocity and line-of-sight position distributionsfor individual projected ROCs, however, suggests that it

may in fact be difficult to eliminate foreground/backgroundcontamination.

To illustrate the problem, Figure 12 shows histograms ofline-of-sight velocity (equivalent to a line prole for an opti-cally thin tracer uniformly excited above foro

min\ 3o6)

regions of projected linear scale s \ L/8. Although some ofthe line shapes are irregular, none of those meeting theROC criterion (in this example) are clearly multicomponentdistributions. For comparison, in Figure 13, we show thedistribution of mass with position along the line of sight.Evidently almost every regionboth ROCs and non-ROCshas multiple spatial components along the line ofsight. Figures 14 and 15 show the same distributions forspatial regions at higher resolution; again, almost all veloc-ity proles are single-component, while spatial distributionsare multicomponent. By dividing our data cubes in half andcomputing velocity histograms separately for the front and back halves, we have checked that the ubiquity ofsingle-component velocity distributions is not an artifact of


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(a) (b)

(c) (d)


FIG. 11.Same as in Fig. 10, but for model snapshot D2

periodic boundary conditions. We have also checked thatthe phenomenon of multispatial component/ single-velocitycomponent ROCs is still prevalent even when the densitythreshold is set higher; for example, Figures 16a ando

min16b shows the velocity and position distributions for thesame model as before, but now with A comple-o

min\ 10o6.

mentary phenomenon that we have also identied in severalmodel projections is that multiple velocity components in agiven ROC may correspond to a single extended spatialcomponentas a consequence, for example of two colliding clumps being viewed during a merger along theline of sight.

The general lack of correspondence between structures inposition and velocity space in ISM models has previouslybeen noted, based on various sorts of analyses. Forexample, Adler & Roberts (1992) analyze the model galacticdisks generated from two-dimensional N-body cloud-uidsimulations, and show that apparent single clouds inlongitude-velocity space are often highly extended along theline of sight, and that what appears to be a single GMC in aspatial plot may be assigned to multiple clouds inlongitude-velocity space. Pichardo et al. (2000) show thatthe morphology of structures in position-position-velocity

space (equivalent to channel maps) in their 3D MHD simu-lations is more strongly correlated with velocity structuresin physical space than with density structures in physicalspace.

The current analyses and previous work on this questiondo not treat molecular excitation and radiative transfer indetail. Studies that do include these complex eects will berequired to reach denitive conclusions on the relationbetween maps of molecular lines and 3D physical density-temperature-velocity cubes. If spatially compact regionshave substantially higher molecular excitation than morediuse surroundings due to line trapping, then it is stillpossible that velocity information could be used to separatespatially connected clumps from foreground and back-ground material. Large amplitude rotation of clouds, ifpresent, would also help to dierentiate superposed line-of-sight clumps in the velocity domain. Potentially, methodsthat use specic information about spectral line shapes (e.g.,Roslowsky et al. 1999) may also be adapted to discriminatespatially separated regions. The present simplied analysissuggests, however, that foreground and backgroundmaterial may at least signicantly increase the dispersion inthe line widthsize distributions for clumps identied from


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FIG. 12.Distribution of mass with line-of-sight velocity for model snapshot B2 projected in the direction, for regions of linear size L/8. Regions meetingzsurface density contrast criterion (cf. Fig. 9) are indicated with heavy outlines.

molecular emission data cubes (e.g., Williams, de Geus, &Blitz 1994; Stutzki & 1990).Gu sten

6. MAGNETIC FIELD DISTRIBUTIONS ANDSIMULATED POLARIZATION

We now turn to magnetic eld structure and begin byconsidering how the distribution of the magnetic eld variesfor models with dierent mean magnetization. As seen inFigure 1d, the rms magnetic eld strength initially increases,

due to the generation of perturbed eld by velocity shearand compression. The distribution of the individual com-ponents of B for matched Mach number model snapshotsB2, C2, D2 is shown in Figure 17. The dimensionless eldstrength that we report can be convertedB4B/(4no6c

s2)1@2

to a physical value using

B \ 1.4]B] (T/10 K)1@2(nH2

/102 cm~3)1@2 kG . (15)

As illustrated by the gure, the component distributions aremore nearly Gaussian for the case of stronger magneticelds; this is true for all of the model snapshots, althoughthe distributions in the high-b (low- cases do becomeB

0)

more Gaussian in time. For the weak-eld models, the dis-

persion in each component of the magnetic eld is largerthan the mean eld component.

Because magnetic elds are measured via the Zeemaneect with dierent atomic and molecular tracers in dier-ent density regimes, it is interesting to analyze how themean eld strength in simulations may depend on density.Since the magnetic eld is weaker and less able to resistbeing pushed around by the matter in the b \ 0.1,1 (C andD) simulations, one expects that the eld strength will havestronger density dependence for these models than for the

b \ 0.01 simulation. This is indeed the case, as can be seenin Figure 18. Particularly at densities below the mean, themagnetic eld strengths in the high-b models are stronglydensity dependent; the low-density slope of d log B/d log ofor these models is near the value associated with a con-2

3stant ratio of mass to magnetic ux and isotropic volumechanges.

At high densities (above the relatively at slope ofD10o6)the b \ 0.01 model increases, becoming comparable to theslopes of the b \ 0.1,1 models. Figure 19 shows the high-density B versus o dependence, for various model snap-shots ; ts for ducial density in the range 103104 cm~3n

H2(i.e., to 100) yield slopes 0.30.7 for d log B/o/o6 \ 10


16/26


FIG. 13.Distribution of mass with line-of-sight position for model snapshot B2 projected in the direction, for regions of linear size L/8. Regionszmeeting surface density contrast criterion (cf. Fig. 9) are indicated with heavy outlines.

d log o. Only the Mach-9 b \ 1 model yields a high-densityregime slope as steep as the isotropic contractionlimit. The other snapshots have slopes 0.30.5, which maybe compared with the slope 0.47 found from a compilationof Zeeman measurements at high densities n

H2\ 102107

cm~3 (Crutcher 1999). The values of the mean B2 in anydensity regime generally increase with increasing mean netmagnetic ux (i.e., decreasing b), but because there isB

0signicant dispersion about the mean B2, there is consider-able overlap of the 1 p deviation regions among the dierentmodel snapshots (Fig. 18).

The numerical results on the B-versus-o relation present-ed by Padoan & Nordlund (1999) (see their Fig. 7) arequalitatively similar to our results, with some dierencesapparent at the high-density end. The lower Mach numberin their low-b model compared to their high-b model likelyaccounts for its relatively weaker increase of B with o athigh density, compared to our results. We also dier withthose authors regarding the astronomical implications ofthe numerical results. In particular, we do not attempt acomparison of the low-density end of the B-versus-o dis-tributions with observations made in the diuse ISM,

because (1) the physical regime modeled by the simulationsis not appropriate for the diuse ISM (where thermal pres-sure is comparable to, rather than much smaller than,Sov2T) ; and (2) the transformation from simulation tophysical variables for local magnetic eld values involvesmultiplying by the mean magnetic eld on the largestB

0scale, and this need not be the same in the diuse and coldISM (b parameterizes this mean eld strength). We con-clude that the B versus o relations obtained from simula-tions do not at present constrain the value of b. At highdensities, all models (either weak or strong on the largeB

0scale) yield slopes which are consistent with high-densitymolecular Zeeman observations. At very low densities,where the predictions of models with varying b do dier,estimating B within clouds would be difficult, since H IZeeman observations probe the high columns of foregroundand background material, rather than the low column ofcloud material (although velocity information may help ; seeGoodman & Heiles 1994). The eld strengths in the low-density regions within molecular clouds may in fact be sys-tematically higher than those at comparable density in thediuse ISM.


17/26

996 OSTRIKER ET AL

FIG. 14.Same as Fig. 12, but for projected region size s \ L/16

For all the snapshots, there is signicant dispersion in thetotal magnetic eld strength. In addition to this overalldispersion in magnitude, there is a dispersion in the mag-netic eld vector direction, which increases with decreasingstrength of the mean eld component simply becauseB

0,

xed amplitude uctuations have larger relative amplitudescompared to a weak mean magnetic eld. The dispersion ineld directions has important consequences for any obser-vational measurement of the mean magnetic eld viaZeeman splitting. Observations of Zeeman splitting at anyposition on a map yield the line-of-sight average value forthe line-of-sight magnetic eld, weighted at each pointalong the line of sight by the local excitation. When a givenline of sight has many uctuations in the direction of themagnetic eld, the average value of will be small, evenSB

losT

if individual local components of the eld have large magni-tudes.

To demonstrate how the averaged line-of-sight eld com-ponents vary with mean eld strength and observer orienta-tion, we depict in Figure 20 an overlay of on theSB

losT

column density for three model snapshots with matchedMach number. In Figure 21 we plot the values of SB

losT

versus column density of dense gas. The gures show,

unsurprisingly, that the line-of-sightaveraged magneticeld strengths are greatest when the mean eld is largestB

0and is oriented along the observers line of sight (top left

panel). For the weaker-eld models, the average line-of-sight eld is lower, and there is larger dispersion. For all thesnapshots, there is considerable dispersion in the values of

on the map, and the largest values do not correspondSBlos

Tto the positions of highest column density; in fact, there issome tendency of line-of-sightaveraged eld to anti-correlate with column density. Thus, although the local eldstrength oB o increases with density (see Figs 18, 19) andmay be much larger than the volume-averaged mean eld

for the entire box, line-of-sight superpositions of non-B0

aligned vector components produce average line-of-sighteld strengths closer to the large-scale volume-averagedvalue.

It is well known that it is difficult to detect the Zeemaneect in molecular clouds (e.g., Heiles et al. 1993) becausethe frequency splitting is small when the eld is weak. This,coupled with the possibility (see Fig. 21) that an impracti-cally large number of measurements might be required toobtain statististically signicant results for the large-scaleeld, underscores the importance of supplementing pro-


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FIG. 15.Same as Fig. 13, but for projected region size s \ L/16

FIG. 16.Same as Figs. 12 and 13, but for omin

\ 10o6


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-10 0 100

0.05

0.1

0.15

0.2

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0.05

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0.05

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0.15

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0.25


FIG

. 17.Comparative statistics of magnetic eld components for threemodel snapshots (B2, C2, D2 from Table 2) with matched Mach numbers.Solid, dotted, and dashed curves show fraction of mass as a function ofB

x,

and respectively. Mean eld strengths are nonzero only in theBy

, Bz, x

direction, with 3.16, and 1, for the top, middle, andB0/(4no6c

s2)1@2 \ 10,

bottom panels.

FIG. 18.Dependence of the total magnetic eld strength on density inthree model snapshots (B2, C2, D2 from Table 2) with matched Machnumbers. Triangles, squares, and pentagons show the mean value ofB2 ineach density bin for Mach-7 models with b \ 0.01, 0.1, and 1, respectively.Shaded regions surrounding each curve corresponds to the 1 p departuresfrom the mean (errors in the means from counting statistics are smallerthan the symbols shown). Dashed horizontal lines show the values of thesquare of the mean magnetic eld, for the three models. Left and bottomscales give magnetic eld strength B2 and density o in dimensionless units ;right and top scales give corresponding ducial values of oB o and n

H2assuming T\ 10 K and cm~3 for the temperature and volume-n6

H2\ 100

averaged density.

grams of direct detection with other methods for estimatingthe mean eld strength. Long before direct Zeeman detec-tions were rst made, Chandrasekhar & Fermi (1953) esti-mated mean spiral-arm eld strengths from the mean gasB

0density, line-of-sight velocity dispersion, and the dispersionin orientations of the magnetic eld in the plane of the sky.The eld line orientation is taken to be traced by the polar-ization direction for background stars, which occurs pro-vided that the dust grains producing the intervening

extinction are aligned with short axes preferentially parallelto B, and so preferentially extinguish linear polarizationsperpendicular to B.

The Chandrasekhar & Fermi (1953) (hereafter CF) esti-mate is based on the fact that for linear-amplitude trans-verse MHD waves,(Alfve n) B

p\ (4no6)1@2 o d o/( o dBo/B

p).

Here is the projection of the mean magnetic eld on theBp

plane of the sky, and dB and are the components of thedmagnetic and velocity perturbations in the plane of the skytransverse to If the interstellar polarization is parallel toB

p.

the local direction of then the ratio in theBp

, o dBo/Bp

denominator may be replaced by the dispersion d/ inpolarization angles (for small-angle/low-amplitudeperturbations). With the further assumption that the true

velocity perturbations are isotropic, then the dispersion inthe transverse velocity is equal to the rms line-of-sighto d ovelocity We thus obtaindv

los.

Bp

\QJ4no6dvlos

d/~1 , (16)

where, for CFs eld model, Q\ 1. Modications to the CFformula allowing for inhomogeneity and line-of-sightaveraging are discussed by Zweibel (1990) and Myers &Goodman (1991), respectively. Both of these eects (andothers; see Zweibel 1996) tend to reduceQ.

Potentially, the CF polarization-dispersion methodcan be used to estimate plane-of-sky magnetic eldstrengths on scales within turbulent interstellar clouds. Itmay also be possible to combine these results with Zeeman

measurements to estimate the total magnetic eld strength(Myers & Goodman 1991; Goodman & Heiles 1994). Inorder to evaluate the ability of the CF method to measuremean plane-of-sky eld strengths, we provide a rst(simplied) test of it using our model turbulent clouds. Forthis test, we have created simulated polarization maps foreach cloud by integrating the Stokes parameters along theline of sight over a projected grid of positions, assuming thepolarizability in each volume element is proportional to thelocal density. The details of this procedure, together with amore extensive discussion of simulated polarization dis-tributions, will appear in a separate publication (Ostriker etal. 2000, in preparation).

For two projected model snapshots (B2 with b \ 0.01and D2 with b \ 1 projected along Figures 22 and 23z),show examples of the polarization maps overlaid on colorscale column density maps. The analogous map (not shown)for the model C2 (b \ 0.1) looks quite similar to Figure 23.From the gures, it it immediately clear that the model witha stronger mean magnetic eld has more ordered polar-B

0ization directions and larger typical values of the fractionalpolarization, compared to the model with a weaker meanmagnetic eld. These trends are as expected: a weakermean eld has lower tensile strength, so that for a givenlevel of kinetic energy the Reynolds stresses will producelarger fractional perturbations in the magnetic eldcorresponding to larger uctuations in projected line-of-


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eve c. ostriker, james m. stone and charles f. gammie- density, velocity and magnetic field...

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