fabian heitsch, mordecai-mark mac low and ralf s. klessen- gravitational collapse in turbulent...

8/3/2019 Fabian Heitsch, Mordecai-Mark Mac Low and Ralf S. Klessen- Gravitational Collapse in Turbulent Molecular Clouds II.

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THE ASTROPHYSICAL JOURNAL, 547:280291, 2001 January 20( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

GRAVITATIONAL COLLAPSE IN TURBULENT MOLECULAR CLOUDS. II.MAGNETOHYDRODYNAMICAL TURBULENCE

FABIAN HEITSCH,1 MORDECAI-MARK MAC LOW,2 AND RALF S. KLESSEN3Received 2000 March 13; accepted 2000 September 14

ABSTRACTHydrodynamic supersonic turbulence can only prevent local gravitational collapse if the turbulence isdriven on scales smaller than the local Jeans lengths in the densest regions, which is a very severerequirement (see Paper I). Magnetic elds have been suggested to support molecular clouds either mag-netostatically or via magnetohydrodynamic (MHD) waves. Whereas the rst mechanism would formsheetlike clouds, the second mechanism not only could exert a pressure onto the gas counteracting thegravitational forces but could lead to a transfer of turbulent kinetic energy down to smaller spatial scalesvia MHD wave interactions. This turbulent magnetic cascade might provide sufficient energy at smallscales to halt local collapse. We test this hypothesis with MHD simulations at resolutions up to 2563zones done with ZEUS-3D. We rst derive a resolution criterion for self-gravitating, magnetized gas: toprevent collapse of magnetostatically supported regions caused by numerical diusion, the minimumJeans length must be resolved by four zones. Resolution of MHD waves increases this requirement toroughly six zones. We then nd that magnetic elds cannot prevent local collapse unless they providemagnetostatic support. Weaker magnetic elds do somewhat delay collapse and cause it to occur more

uniformly across the supported region in comparison to the hydrodynamical case. However, they stillcannot prevent local collapse for much longer than a global free-fall time.Subject headings: ISM: clouds ISM: kinematics and dynamics ISM: magnetic elds

turbulence MHD

1. INTRODUCTIONAll star formation takes place in molecular clouds.

However, the star formation rate in these clouds is sur-prisingly low. From the Jeans argument, one would expectthat they should collapse within their free-fall time

tff

\S3n

32Go6B (1.06 ] 106 yr)

A n103 cm~3

B~1@2, (1)

where is the mean mass density of the cloud, G the gravi-o6tational constant, and the number density, withn \ o6/k

A catastrophic collapse of a giant moleculark \ 2.36mH

.cloud on this timescale would yield a single starburst event.However, molecular clouds have classically been thought tosurvive without global collapse for much longer than theirfree-fall time (Blitz & Shu 1980). Moreover, stars are nott

ffusually observed in nearby star-forming regions to form insuch a catastrophic collapse. Instead, they form in localizedregions dispersed through an apparently stable cloud.

Observations of spectral line widths in molecular cloudsshow that the gas moves at speeds exceeding the thermalvelocities by up to an order of magnitude (Williams, Blitz, &McKee 2000). These supersonic motions seem not to beordered, so that turbulent support models have often beensuggested, with the turbulence giving rise to an eectivelyisotropic turbulent pressure counteracting the gravitationalforces.

However, such models have two problems. First, simula-tions of supersonic, compressible turbulence show that it

1 Max-Planck-Institut Astronomie, 17, D-69117 Heidel-fu r Ko nigstuhlberg, Germany; heitsch=mpia-hd.mpg.de.

2 Department of Astrophysics, American Museum of Natural History,Central Park West at 79th Street, New York, NY 10024-5192;mordecai=amnh.org.

3 Sterrewacht Leiden, Postbus 9513, RA-2300 Leiden, Netherlands;klessen=strw.leidenuniv.nl.

typically decays in a time less than the clouds free-fall time(Gammie & Ostriker 1996; Mac Low et al. 1998; Mact

ffLow 1999). So, in order to be really able to support thecloud, the turbulence would have to be constantly driven.Second, although hydrodynamical turbulence can preventglobal collapse, it can never completely prevent local col-lapse except with unrealistically short driving length scale

(Klessen, Heitsch, & Mac Low 2000; hereafter Paper I).jDThe efficiency of local collapse depends on the wavelength

and on the strength of the driving source. Long-wavelengthdriving or no driving at all results in efficient, coherent starformation, with most collapsed regions forming near eachother (Klessen & Burkert 2000). Strong, short-wavelengthdriving, on the other hand, results in inefficient, incoherentstar formation, with isolated collapsed regions randomlydistributed throughout the cloud.

The model of molecular clouds being supported by turb-ulence has been widely discussed and investigated, asreviewed in Paper I and et al. (2000).Va zquez-SemadeniRecently, Ballesteros-Paredes et al. (1999), Hennebelle &

(1999), and Elmegreen (2000) have suggested thatPe raultmolecular clouds might not have to be supported for theselong timescales at all but might be transient features causedby colliding ows in the interstellar medium. This wouldsolve very naturally not only the problem of cloud supportbut also, according to Ballesteros-Paredes et al. (1999), thepronounced lack of 510 million year old postT-Tauristars directly associated with star-forming molecularclouds.

Magnetic elds might alter the dynamical state of amolecular cloud sufficiently to prevent gravitationallyunstable regions from collapsing (McKee 1999). They havebeen hypothesized to support molecular clouds either mag-netostatically or dynamically through MHD waves.

Mouschovias & Spitzer (1976) derived an expression forthe critical mass-to-ux ratio in the center of a cloud for

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TURBULENT MOLECULAR CLOUDS. II. 281

magnetostatic support. Assuming ideal MHD, a self-gravitating cloud of mass M permeated by a uniform ux 'is stable if the mass-to-ux ratio

M

'\AM

'

Bcr4

c'JG

, (2)

with depending on the geometry and the eld andc'density distribution of the cloud. A cloud is termed sub-critical if it is magnetostatically stable and supercritical if itis not. Mouschovias & Spitzer (1976) determined that c' \0.13 for their spherical cloud. Assuming a constant mass-to-ux ratio in a region results in (Nakanoc' \ 1/(2n) ^ 0.16& Nakamura 1978). Without any other mechanism ofsupport, such as turbulence acting along the eld lines, amagnetostatically supported cloud will collapse to a sheetthat then will be supported against further collapse. Fiege &Pudritz (1999) discussed a sophisticated version of this mag-netostatic support mechanism, in which poloidal and toroi-dal elds aligned in the right conguration could prevent acloud lament from fragmenting and collapsing.

Investigation of the second alternative, support by MHDwaves, concentrates mostly on the eect of waves, asAlfve n

they (1) are not as subject to damping as magnetosonicwaves and (2) can exert a force along the mean eld, asshown by Dewar (1970) and Shu, Adams, & Lizano (1987).This is because waves are transverse waves, so theyAlfve ncause perturbations dBperpendicular to the mean magneticeld B. McKee & Zweibel (1995) argue that wavesAlfvencan even lead to an isotropic pressure, assuming that thewaves are neither damped nor driven. However, to supporta region against self-gravity, the waves would have to pro-pagate outwardly rather than inwardly, which would onlyfurther compress the cloud. Thus, as Shu et al. (1987)comment, this mechanism requires a negative radial gra-dient in wave sources in the cloud.

Most molecular clouds show evidence of magnetic elds.

However, the discussion of their relative strength is lively.Crutcher (1999) summarizes all 27 available Zeeman mea-surements of magnetic eld strengths in molecular clouds.He concludes that (1) static magnetic elds are not strongenough to support the observed clouds alone, with typicalratios of the mass M to the critical mass for the observedmagnetic eld (2) the ratio of thermal to mag-M/M

crB 2 ;

netic pressure (3) internal motions areb \ Pth

/Pmag

B 0.04;supersonic, with a velocity dispersion but approx-p

v? c

simately equal to the speed, and (4) that theAlfve n pv

B vA

;kinetic and magnetic energies are roughly equal, which heinterprets as suggesting that static magnetic elds andMHD waves are equally important in cloud energetics.However, McKee (1999) remarks that Crutchers data donot address the strength of the eld on large scales(threading an entire GMC) and that the data deal withdense regions in the clouds, so that ambipolar diusionalready might have altered the mass-to-ux ratio observed.Moreover, Crutchers fourth conclusion about the role ofstatic elds and MHD waves is based on the assumptionthat the kinetic energy stems mainly from MHD waves.

Nakano (1998) made two further arguments against mag-netostatic support of cloud cores. First, magnetically sub-critical condensations cannot have column densities muchhigher than their surroundings. However, observed cloudcores have column densities signicantly higher than themean column density of the cloud, indicating that they are

not magnetostatically supported. Second, if the cloud coreswere magnetically supported and subcritical, it would bedifficult to maintain the observed nonthermal velocity dis-persions for a signicant fraction of their lifetime. Mac Low(1999) conrmed this by numerically determining the dissi-pation rate of supersonic, magnetohydrodynamic turbu-lence. He concludes that the typical decay time constant isfar less than the free-fall time of the cloud.

Polarization measurements might give us a clue whether

the elds are well ordered or in a turbulent state. However,up to now, most measurements refer to the highest densityregions, thus giving information about the elds in small-scale structures but not about scales of the whole cloud.Hildebrand et al. (1999) present polarization measurementsof cloud cores and envelopes. They nd polarizationdegrees of at most 10%. More recent observations of themolecular cloud lament OMC-3 by Matthews & Wilson(2000) suggest that the magnetic eld is well ordered per-pendicularly to the lament but with a mean polarizationdegree of only 4.2%.

Passot, & Pouquet (1996) performedVa zquez-Semadeni,three-dimensional simulations including self-gravity andMHD with a resolution of 643. They found that hydrody-

namical and supercritically magnetized turbulence can leadto gravitationally bound structures. Gammie & Ostriker(1996) did simulations in dimensions, while more re-12

3cently 2.5 dimensional models were presented by Ostriker,Gammie, & Stone (1999). Mac Low et al. (1998), Stone,Ostriker, & Gammie (1998), Padoan & Nordlund (1999),and Mac Low (1999) studied decaying magnetized turbu-lence and found short decay rates with as well as withoutmagnetic elds.

We present the rst high-resolution (2563 zones) simula-tions of magnetized, self-gravitating, driven, supersonicturbulence to test the hypothesis that magnetic elds cancontribute to the support of molecular clouds. The follow-ing section describes the technique and parameters used for

the simulations. Section 3 discusses requirements regardingthe resolution needed for simulations of self-gravitatingmagnetized turbulence. In 4, we present the results, and wesummarize our conclusions in 5.

2. TECHNIQUE AND MODELS

2.1. Technique

For our computations, we use ZEUS-3D, a well-tested,Eulerian, nite-dierence code (Stone & Norman 1992a,1992b; Clarke 1994). It uses second-order advection andresolves shocks employing a von Neumann articial vis-cosity. Self-gravity is implemented via an FFT-solver forCartesian coordinates (Burkert & Bodenheimer 1993). Themagnetic forces are calculated via the constrained transportmethod (Evans & Hawley 1988) to ensure +B\ 0 tomachine accuracy. In order to stably propagate shear

waves, ZEUS uses the method of characteristicsAlfve n(Stone & Norman 1992b; Hawley & Stone 1995). Thismethod evolves the propagation of waves as anAlfve nintermediate step to compute time-advanced quantities forthe evolution of the eld components themselves to ensurethat signals do not propagate upwind unphysically.

We use a three-dimensional, periodic, uniform, Cartesiangrid for the models described here. This gives us equalresolution in all regions and allows us to resolve shocks andmagnetic eld structures well everywhere. On the other


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282 HEITSCH, MAC LOW, & KLESSEN Vol. 547

hand, collapsing regions cannot be followed to scales lessthan a few grid zones.

We do not include ambipolar diusion in this work,although numerical diusion acts on scales in our simula-tions that, when translated to astronomical scales withtypical parameters, correspond to the scales on which ambi-polar diusion begins to dissipate power (see Paper I).Moreover, we do not use a physical resistivity, relying onnumerical dissipation at grid scale caused by averaging

quantities over zones. The limited resolution in high-densityregions can lead to excessive numerical diusion of massthrough the magnetic elds that must be accounted forwhen analyzing these or similar computations. In 3 wederive the appropriate resolution criterion.

2.2. Models and Parameters

We employ two sets of parameters. The rst one is thesame as in Paper I and allows us to compare its results tothe ones of this work. The second parameter set enables usto determine the numerical reliability of these results. Allparameters are given in normalized units, where physicalconstants are scaled to unity and where we consider gascubes with mass M4 1 and side length [[1.0, 1.0]. The

system can be scaled to physical units using the Jeans massand length scale In Paper I we adopt a normalizedMJ

jJ.

sound speed of which yields andcs\ 0.1, M

J\ 0.0156

such that the computed box contains M \ 64jJ

\ 0.5 MJand is L\ 4 in size. Our second parameter set is basedj

Jon a sound speed of which, in turn, gives a 10cs

\ 0.213,times larger Jeans mass and yieldsM

J\ 0.156 j

J\ 1.068,

so that follows M \ 6.4 and L\ 1.873MJ

jJ.

We use the same driving mechanism described in MacLow (1999). At each time step, a xed velocity pattern is

added to the actual velocity eld thus, so that the inputenergy rate is constant. The driving eld pattern is derivedfrom a Gaussian random eld with a given spectrum. Thisallows us to drive the cloud on selected spatial scales. Therms Mach number for the rst parameter set is M

rms\ 10,

for the second one it isMrms

\ 5.The MHD simulations start with a uniform magnetic

eld in the z-direction. As soon as the cube has evolved intoa fully turbulent state, gravity is switched on. The critical

eld value according to equation (2) is in codeBcr \ 1.56units. The initial eld strength varies between B \ 0.191.77, corresponding to covering the sub-M/M

cr\ 0.48.3,

and supercritical range. The ratio b \ Pth

/Pmag

\ 0.014.04.For an overview of the parameters used, see Table 1. Thereare four series of models: D (pure hydrodynamics, sameruns as in Paper I); E (MHD models with the same param-eter set as the hydro models); F (magnetostatic models fornumerical tests); and G (MHD models with a reducednumber of Jeans masses). The second letter in the modelnames stands for the resolution (low, 643 ; intermediate,1283 ; or high, 2563), followed by a digit denoting the drivingscale. For the MHD models, a third letter gives the relativeeld strength (low, intermediate, moderate, strong).

The dynamical behavior of isothermal, self-gravitatinggas depends only on the ratio between potential and kineticenergy. Therefore, we can use the same scaling prescriptionsas in Paper I, dening the physical timescale by the free-falltime (eq. [1]), the length scale by the initial Jeans length

jJ

\ cs[n/(Go6)]1@2

B (0.71 pc)jJ

A cs

0.2 km s~1

BA n103 cm~3

B~1@2, (3)

TABLE 1

PARAMETERS OF MODELS USED

Name Resolution kdrv MJturb b bturb M/Mcr t5

Dh1 . . . . . . . 2563 12 15 O O O 0.4Dh3 . . . . . . . 2563 78 15 O O O 0.7Eh1w . . . . . . 2563 12 15 0.9 87.2 8.3 1.4Ei1w . . . . . . . 1283 12 15 0.9 87.2 8.3 1.3El1w . . . . . . . 643 12 15 0.9 87.2 8.3 0.6Eh1i . . . . . . . 2563 12 15 0.04 3.9 1.8 0.8Ei1i . . . . . . . . 1283 12 15 0.04 3.9 1.8 0.4El1i . . . . . . . . 643 12 15 0.04 3.9 1.8 . . .Ei1s . . . . . . . 1283 12 15 0.01 1.0 0.8 1.5El1s . . . . . . . 643 12 15 0.01 1.0 0.8 1.0Ei0s . . . . . . . 1283 0 . . . 0.01 . . . 0.8 . . .El0s . . . . . . . 643 0 . . . 0.01 . . . 0.8 . . .Fl0w . . . . . . 643 0 . . . 3 ] 10~3[0.14 . . . 1.10 . . .Fl0i . . . . . . . 643 0 . . . 2 ] 10~3[0.09 . . . 0.88 . . .Fl0m . . . . . . 643 0 . . . 1 ] 10~3[0.04 . . . 0.59 . . .Fl0s . . . . . . 643 0 . . . 3 ] 10~4[0.02 . . . 0.44 . . .Gi1 . .. . . . . . 1283 12 19 O O O 1.2Gi1w . . . . . . 1283 12 19 4.04 100.0 8.3 2.1Gi1i . . . . . . . . 1283 12 19 0.23 5.9 2.0 1.0Gi1m . . . . . . 1283 12 19 0.07 1.8 1.1 (3.5)Gi1s . . . . . . . 1283 12 19 0.05 1.1 0.8 . . .

NOTE. gives the turbulent Jeans mass,MJturb \ o1@2(n/G)3@2(c

s2 ]Sv2T/3)3@2 b \ P

th/P

mag\

and denotes the time at which 5% of the total mass has been8ncs2 o/B2 b

turb\ P

turb/P

mag. t

5accreted onto cores. Times have been normalized to the global free-fall time. gives the ratioM/Mcrof cloud mass to critical mass according to eq. (2). Models El1i and El1s have been computed three

times with varying random seeds to check the inuence of the velocity eld on the results. They arenot listed explicitly. Models El0s, Ei0s, and F are nondriven runs, used for the analysis ofnumerical diusion. Model Gi1s collapses so slowly that it only reached afterM

*\ 3% t \ 3.5t

ff.


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No. 1, 2001 TURBULENT MOLECULAR CLOUDS. II. 283

and the mass scale by the Jeans mass

MJ

\ o6jJ3 B (20.8 M

_)

]MJ

A cs

0.2 km s~1

BA n103 cm~3

B~1@2, (4)

where and are expressed in code units. We includejJ

MJthe magnetic eld B via the pressure, using P \ oc

s2 \

This yieldsB2/(8n).

B \ (4.44 ] 10~6G)BA cs0.2 km s~1

BA n103 cm~3

B1@2, (5)

with again in code units.B

3. NUMERICAL RESOLUTION CRITERION

We must consider the resolution required to accuratelyfollow gravitational collapse. To follow fragmentation in agrid-based, hydrodynamical simulation of self-gravitatinggas, the criterion given by Truelove et al. (1997) holds. Theystudied fragmentation in self-gravitating, collapsing regionsand found that the mass contained in one grid zone mustremain signicantly smaller than the local Jeans massthroughout the computation to accurately follow the frag-mentation. Bate & Burkert (1997) found a similar criterionfor particle methods. Applying it strictly would limit oursimulations to the very rst stages of collapse. We thereforeonly apply this criterion to the resolution of initial collapse.Thereafter, we study only the gross properties of collapsedcores, such as mass and location, but not their under-resolved internal details.

The Truelove analysis does not include magnetic elds,which must also be sufficiently resolved to determinewhether initial collapse occurs. Numerical diusion canreduce the support provided by a static or dynamic mag-netic eld against gravitational collapse. Increasing thenumerical resolution decreases the scale at which numericaldiusion acts. In this section we attempt to determine theresolution necessary to adequately resolve magneticsupport against collapse in the presence of supersonic turb-ulence. Two regimes of eld strength concern us. Forstrong, subcritical elds, the resolution should ensure thatnumerical diusion remains unimportant even for thedense, shocked regions ( 3.1). For weaker, supercriticalelds, the resolution should enable us to evolve MHDwaves within the shocked regions ( 3.2).

3.1. Numerical Diusion in Magnetostatic Congurations

We begin by considering magnetostatic support. Figure 1demonstrates that numerical diusion can dominate thebehavior in this case. The left-hand panel displays the peakdensity and magnetic eld amplitude for the low-o

max

B

maxand intermediate-resolution undriven models El0s and Ei0s,while the right-hand panel contains the same quantities forthe driven, but otherwise identical, models El1s and Ei1s(see Table 2). All these models have initial magnetic eldstrength sufficient to support the region, with M/M

cr\ 0.8.

Starting with a sinusoidal density perturbation in theundriven case, both the driven and undriven models rstcollapse into sheet structures. The dotted lines in the densityplots show the density corresponding to having all theo

sheetmass in the box in a layer one zone thick. In a volume lledwith otherwise unperturbed gas of initially uniform density,reaching peak densities higher than this thresholdo

maxmeans that numerical diusion across eld lines must have

FIG. 1.Peak densities and maximum magnetic eld strengths forstrong eld driven (El1s and Ei1s), and undriven (El0s and(M/M

cr\ 0.8),

Ei0s) runs. The dotted lines (upper N \ 1283, lower N \ 643) denote thesheet densities, i.e., the densities corresponding to all mass concentrated ina layer of one grid zones height. Gravity is turned on at t \ 0.0 with t inunits of the global free-fall timescale The time interval at t \ 0.0 ist

ff.

necessary for the driven models to reach a state of fully developed turbu-lence. In the lower panels, the thin lines denote the magnetic eld strengthrequired to support a region of density according to eq. (2).o

max

begun, as can be seen in the top left-hand panel of Figure 1.However, in driven, isothermal, supersonic turbulence, thepeak densities in shocked regions scale with the MachnumberM as Thus, can reach values orderso

pk

PM2. opkof magnitude higher than the mean density, easily exceeding

even in the absence of diusion.osheetThe magnetic eld provides an alternate diagnostic. In

the absence of numerical diusion, mass should be tied tothe eld lines running vertically through the cube, so themass to ux ratio along any given eld line should notchange. In the lower panels, we plot the magnetic eldstrength required to support a region of densityB

sup(o) o

max(thin lines) according to equation (2). The magnetic eldstarts out signicantly stronger than as the mass isB

sup,

insufficient for collapse to occur. If grows more slowlyBmaxthan then density must be diusing across eld lines. IfB

sup,

the two values cross, then numerical diusion has allowedcollapse to occur unphysically. This happens in theundriven models at while in the driven modelst \ 4.7tff,

TABLE 2

PEAK JEANS L ENGTHS FOR ALL TURBULENT MODELS

Model cs

nJ

M jJpk 643 1283 2563

D/E . . . . . . 0.10 64 10 0.05 1.6 3.2 6.4G . . . . . . . . . 0.213 6.4 5 0.11 3.4 6.8 13.6

NOTE.Peak Jeans lengths for all turbulent models, where isjJpk

determined via the peak density The second column listsopk

\M2o0

.the sound speed, followed by the Mach number,M, and the numberc

s,

of thermal Jeans masses in the box, The last three columns containnJ.

the Jeans lengths in zones for the resolution denoted in the top row.


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with their greater density contrasts, collapse already occursat We conclude that in these low- andt \ 0.5t

ff.

intermediate-resolution models, the resolution is not suffi-ciently high for the magnetic eld to follow the turbulence,especially in shocked regions.

We can use this example to derive a criterion for theresolution of a magnetostatically supported sheet. We canalso conrm that we are using the correct numerical con-stant in the mass-to-ux criterion given by equation (2). We

did a suite of models F (see Table 1) varying the soundspeed and the mass in the cube while holding the magneticeld strength constant, thus varying and the numberM/M

crof zones in a Jeans length For we certainlyjJ

. jJ

\ 1.0,cannot expect to get reliable results, as this Jeans lengthdoes not even fulll the hydrodynamic criterion of Trueloveet al. (1997). In Figure 2 we present the results. We ndunphysical collapse occurring for physically supportedregions until zones. We also nd that at thisj

J 4.0

resolution, a model with collapses (thin line inM/Mcr

\ 1.1rst panel) while a model with does not col-M/M

cr\ 0.88

lapse, conrming equation (2). We conclude that for a self-gravitating magnetostatic sheet to be well resolved, its Jeanslength must exceed four zones.

3.2. MHD W aves in High-Density Regions

As we want to investigate whether MHD turbulence canprevent gravitationally unstable regions from collapsing, wehave to be sure to resolve the MHD waves, which are themain agent in this mechanism. The same argument holds asfor the criterion to prevent local collapse in the pure hydro

case (Paper I). If the wavelength of the turbulent pertur-bation is smaller than the local Jeans length stabilizationj

J,

should be possible, at least in principle. To sample a sinus-oidal wave on a grid, we need at least four zones. Polygonalinterpolation of a sine wave with evenly spaced supportswould then yield an error ofB21%. This decreases to B9%with six zones and B5% using eight zones. We choose toset the minimum permitted Jeans length to six zones, admit-tedly somewhat arbitrarily.

In Table 2, we list the local Jeans lengths for all modeltypes and their resolution. Models of type E start to beresolved at a resolution of N \ 2563, whereas the ones oftypeG can already be regarded as resolved at N \ 1283.

4. RESULTS

In the previous section, we considered under what cir-cumstances numerical eects could allow unphysical gravi-tational collapse. In this section, we consider adequatelyresolved models in order to determine whether magnetizedturbulence can prevent the collapse of regions that are notmagnetostatically supported. We begin by demonstratingthat supersonic turbulence does not cause a magneto-statically supported region to collapse and then demon-strate that in the absence of magnetostatic support, MHDwaves cannot completely prevent collapse, although theycan retard it.

4.1. Magnetostatic SupportIn a subcritical region with the cloud isM \ M

cr,

expected to collapse to a sheet, which in turn should be

FIG. 2.Peak densities for all test models of type F. is the Jeans length in units of grid zones, when all the mass is collected in a sheet of a single zonesjJheight. The horizontal dotted line denotes the density reached if all mass is collected in a sheet of a single zones height. Note that for zones, we getj

J\ 4.0

collapse in the supercritical case but not in the subcritical one. Thus, we verify the mass-to-ux criterion (eq. [2]).(M/Mcr

\ 1.1)


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FIG. 3.Two-dimensional slice of increased Jeans mass model Gi1swith velocity eld vectors (upper panel ) and magnetic eld vectors (lowerpanel) displaying the whole computational domain. The initial magneticeld is oriented along the z-direction, i.e., vertically in all plots presented.Driving happens at k \ 12. The eld is strong enough in this case notonly to prevent the cloud from collapsing perpendicular to the eld linesbut even to suppress the turbulent motions in the cloud. The turbulenceonly scarcely aects the mean eld. The picture is taken at t \ 5.5t

ff.

stable. Figure 3 shows the corresponding model Gi1s. Theseruns have been computed with a lower Mach numberM\ 5.0 in order to demonstrate the behavior of a magne-tostatically supported cloud. The initially uniform magnetic

eld runs parallel to the z-axis. The eld is strong enough toforce signicant anisotropy in the ow, although the densesheets that form do not always lie perpendicular to the eldlines as the driving can shift the sheets along the eld lineswithout changing the mass-to-ux ratio. The sheets do notcollapse further because the shock waves cannot sweep gasacross eld lines and the cloud is initially supported magne-tostatically.

Figure 4 demonstrates that this result is reasonably well

resolved numerically. As in Figure 1, we show peak den-sities and magnetic eld strength for two models that dieronly in their sound speeds and thus by the number of zonesin a Jeans wavelength Whereas model Ei1s, withj

J. j

J\

3.2 zones does collapse, although it should be supported,model Gi1s, with zones, behaves as expected physi-j

J\ 6.8

cally, just as our resolution criterion predicts. Note that theactual eld strength in model Gi1s always exceeds theB

sup,

eld strength necessary to support the region.

4.2. MHD W ave Support

A supercritical cloud with is not magneto-M [ Mcrstatically supported and could be stabilized only by MHD

wave pressure, assuming ideal MHD. In this section we

show that this appears to be insufficient to completelyprevent gravitational collapse, although it can slow theprocess down.

4.2.1. Morphology

In the upper panels of Figure 5 we compare the morphol-ogy of hydrodynamical (Dh1), weakly magnetized (Eh1w),and strongly magnetized (Eh1i) supercritical models at aresolution of 2563 zones. The gure presents two-

FIG. 4.Peak densities and maximum magnetic eld strengths forstrong eld model Ei1s (solid line) and model Gi1s (dashed line). The dottedline denotes the sheet density, i.e., the density corresponding to all massconcentrated in a layer of one grid zones height. Gravity is turned on att \ 0.0 with t in units of the global free-fall time. In the lower panel, thethin lines denote the magnetic eld strength required to support a region ofdensity according to eq. [2]. Whereas model Ei1s shows unphysical,o

maxnumerical collapse, model Gi1s is well resolved.


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FIG. 5.Two-dimensional slices of the high-resolution models Dh1, Eh1w, and Eh1i and the corresponding models at intermediate resolutionDi1, Ei1w,and Ei1i, displaying the whole computational domain. Slices are taken at the location of the zone with the highest density at the time when 10% of the totalmass has been accreted onto cores. The plot is centered on this zone. Arrows denote velocities in the plane. The length of the largest arrows corresponds to avelocity of vD 2.0. Gray scale stands for density, where highest density regions are darkest. All slices are scaled equally, using the same scale as Fig. 3.Driving, as in Fig. 3, happens at k \ 12.

dimensional slices through the three-dimensional simula-tion volume centered on the locations of the most massive

clumps. To compare the models at similar stages of theirevolution, we took the snapshots at a time when roughly10% of the total mass had been accreted onto collapsingclumps. All three runs show well-developed turbulence,rareed regions, shocked regions, and at least one clump.However, model Eh1w, with seems to containM/M

cr\ 8.3,

more power on small scales than the pure hydro run, modelDh1 We discuss this more quantitatively(M/M

cr\ O).

below. In model Eh1i, with the verticallyM/Mcr

\ 1.8,oriented mean eld (in the plane of the gure) starts produc-ing some anisotropy. This model represents a morphologi-cal transition from the pure hydrodynamical model Dh1with completely randomly oriented motions to the magne-tostatically supported model Gi1s with its ordered struc-tures.

4.2.2. Resolution Study

We must address the question of whether our magneto-dynamic simulations are indeed well resolved. The parame-ters for models E yield a global Jeans length of j

JB 0.5,

corresponding to a local, post-shock Jeans length of jJ

B0.05 for isothermal shocks with Mach number MB 10 (seeTable 2). At N \ 1283 zones, this results in a local Jeanslength of only 3.2 zones, but at 2563 the local Jeans length is6.4 zones, satisfying our resolution criterion. Instead ofincreasing the resolution, we increased the Jeans length inthe models G discussed below in 4.2.4 by increasing the

sound speed. In these models, we used a global Jeans lengthof corresponding to a local Jeans length of 6.8j

JB 1.1,

zones at N \ 1283.Figure 5 compares high-resolution 2563 zone models

with the corresponding lower resolution 1283 zone modelsin the same dynamical state when 10% of the mass has beenaccreted onto cores. The lower resolution makes itself felt inbroader shocks in all cases, so that the peak densities arelower than in the high-resolution runs. In the MHD models,decreasing the resolution also leads to thicker collapsedsheets. Thus, unstable regions form at later times in both theMHD and hydrodynamical cases, as can be seen in Figure6. There, we show the core mass accretion history for threeweakly magnetized models varying only in their resolutionEl1w (643), Ei1w (1283), and Eh1w (2563) (lower panel) andthe corresponding hydrodynamical models (Dl1, Di1,Dh1upper panel). Cores were determined using the modi-ed CLUMPFIND algorithm of Williams, De Geus, &Blitz (1994) as described in Paper I.

Collapse occurs in both cases at all resolutions. However,increasing the resolution makes itself felt in dierent ways inhydrodynamical and MHD models. In the hydrodynamicalcase, higher resolution results in thinner shocks and thushigher peak densities. These higher density peaks formcores with deeper potential wells that accrete more massand are more stable against disruption. If we increase theresolution in the MHD models, on the other hand, we canbetter follow short-wavelength MHD waves, which appearto be able to delay collapse although not to prevent it.


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FIG. 6.Comparison of the mass accretion behavior for runs driven atk \ 12 with varying resolution. Pure hydro runs are shown in the upperpanel (models Dl1 [dotted], Di1 [dashed], and Dh1 [solid]), and MHDruns are shown in the lower panel (models El1 [dotted], Ei1 [dashed], andEh1 [solid]). denotes the sum of masses found in all cores determinedM

*by the modied CLUMPFIND (Williams et al. 1994; see also Paper I).Times are given in units of free-fall time as dened in eq. (1). Although thecollapse rate varies, we get collapse in all cases.

The turbulent formation of gravitationally condensedregions via shock interactions is a highly stochastic process.As in Paper I, we demonstrate this by choosing dierentrandom realizations of the driving velocity eld with the

same characteristic wavelengths. Figure 7 shows the coremass accretion history for a pure hydrodynamical model setand a MHD set. In the upper panel, we plotted the modelsDl1,Di1, andDh1, where the low-resolution model Dl1 hasbeen repeated multiple times (solid thin lines). The dottedthick line denotes the average of these runs. We nd thatresolution eects are exceeded by statistical variationscaused by random variations of the driving elds.

In the MHD case (lower panel), the thickness of the linesstands for the strength of the eld, expressed in terms of theratio Dotted lines denote low-resolution runs com-M/M

cr.

puted with varying driving velocity elds, as in the upperpanel. The high-resolution run with wasM/M

cr\ 1.8

stopped at because the time step becamet \ 1.0tff

Alfve n

prohibitively small. Increasing the resolution makes itselffelt for the runs with stronger elds in the same way as forthe ones with weak elds. The higher the resolution, thebetter the small-scale MHD waves are resolved and, thus,the slower the collapse. Collapse does always occur,however.

4.2.3. Core Distribution

Although MHD waves cannot prevent local collapseentirely, the resulting collapse appears qualitatively dier-ent from collapse in the hydrodynamic case with corre-sponding global and driving strength. In thej

Jhydrodynamical case driven at shocks are widelyjD

[ jJ,

separated and sweep up substantial mass, producing iso-

FIG. 7.(Upper panel ) Core mass accretion rates for 10 hydro runswith equal parameter set (model Dl1) but dierent realizations of the turb-ulent velocity eld. The thick line shows a mean accretion rate, calcu-lated from averaging over the sample. For comparison, the higherresolution runs Di1 and Dh1 are shown. The latter one (N \ 2563) can beregarded as an envelope for the low-resolution models. (L ower panel ) Massaccretion rates for runs with identical magnetic elds but dierent drivingelds and runs with identical driving elds but dierent magnetic elds(models El1w, Ei1w, Eh1w and El1i, Ei1i, Eh1i). The eects of magneticelds are covered by variations caused by the turbulent velocity eld.Identical line styles stand for models with identical parameters but dier-ent driving velocity elds. Models El1w andEl1i have been computed threetimes with varying driving velocity elds.

lated clusters of cores. In the presence of a weak super-

critical eld, the shock structure appears to have moresmall-scale structure, resulting in cores distributed moreuniformly across the simulation volume, as shown in themiddle panel of Figure 8. In fact, the weakly magnetizedmodel driven on large scales with rather morej

D[ j

Jresembles the hydro model driven on small scales withshown in Figure 11 of Paper I.j

D\ j

JFigures 9 and 10 try to quantify this dierence. Figure 9shows the histogram of core distances for each panel ofFigure 8, multiplied with the mean core mass. A clusteredensemble of high-mass cores should result in a peaked dis-tribution at small distances, whereas a spread-out ensembleof low-mass cores should have a broader distribution.Figure 9 hints at such a behavior, although we are well

aware of the fact that the statistics barely suffice. Neverthe-less, the distribution for the magnetized runs is shifted tolarger radii. Note that the total mass in the cores found iswithin 10% the same for all three models. Figure 10 showsthe weighted means of core distances, with their standarddeviations as error bars. Again, we see a slight shift towardlarger separations, suggesting a more uniform distributionof cores in the MHD cases, although larger simulationswith greater core numbers will be needed to conrm thisresult.

4.2.4. Energy Distribution

Further evidence for a qualitative dierence betweenhydrodynamic and MHD collapse comes from Figure 11.


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FIG. 8.Projected coordinates of clumps found when 10% of the total mass has been accreted onto cores. All simulations (models Dh1, Eh1w, and Eh1i)are driven at wavenumbers k \ 12. For the pure hydro case, we get strongly clustered collapse, whereas for supercritical elds, the cores are more evenlydistributed. For slightly supercritical elds model Eh1i), the cloud tends to collapse along the eld lines, so that the extent of the core(M/M

cr\ 1.8,

distribution is reduced in direction parallel to the initial eld (vertically oriented in the plots).

FIG. 9.Histogram of core distances weighted with the mean core massfor the projected cores of Fig. 8. The box length is L\ 2. The weightedmeans and their standard deviations are shown in Fig. 10. Although thestatistics are not sufficient, the magnetized models tend to show a moreuniform distribution.

FIG. 10.Weighted means and their standard deviations for the coredistances of Fig. 9. The eect of low number statistics is clearly to be seen(between six and ten cores were found).

Here, we show the time evolution of the ratio of kinetic topotential energy decomposed into contributions from fourspatial scales. (The time resolution is somewhat coarse asthese models with 2563 zones were dumped only every t \

Whereas the hydrodynamic model (Dh1) driven at0.3tff

.)k \ 12 collapses within less than the weakly magne-0.5tff

,tized model Eh1w is supported until(M/M

crit\ 8.3) tZ 1t

ffbefore also collapsing. This is at least qualitatively similarto the behavior of model Dh3, which is driven at wavenum-bers k \ 78 and thus has a denser network of shocks. Onthe other hand, Figure 11 suggests that the stronger eldmodel Eh1i collapses even more thoroughly(M/M

cr\ 1.8)

than its hydrodynamical counterpart because of the order-ing inuence of the strong mean eld.

We use the model series G to follow the collapse to latertimes and with more frequent time sampling. These 1283models still resolve the Jeans length in the densest clumps,as discussed in 4.2.2. We reduced the Mach number toM\

5 in order to maintain an energy input comparable to

FIG. 11.Time evolution of the ratio of kinetic to potential energy,T/V split up according to wavenumber k \ 1,2,4,8 (solid, dashed, dash-dotted, and dotted lines) for models Dh1, Dh3 (driven at k \ 78), Eh1w,and Eh1i. The horizontal line at T/V\ 1.0 indicates the instability bound-ary. Time is normalized to units of global free-fall time Note that at

ff.

strong subcritical eld leads to faster collapse than a weak subcritical one.


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FIG. 12.Ratio of kinetic to potential energy against time for models Gwith reduced number of Jeans masses split into contributions(n

J\ 6.4),

from four spatial scales (k \ 1,2,4,8). All models show collapse except forthe magnetostatically supported one (Gi1s). Note that models Gi1w andGi1i behave as their counterparts Eh1w andEh1i shown in Fig. 11.

models E. This decreases the rms post-shock density, so weactually expect the G series to form cores with somewhatmore difficulty than the E series. Figure 12 shows the ratioof kinetic to potential energy of the magnetized models inthe G series. (The hydrodynamical model Gi1 collapseswithin half a free-fall time after gravity is turned on att \ 0.0.) With increasing eld strength (models Gi1w andGi1i), the collapse is delayed but never prevented. This evenapplies for Gi1m, where the eld strength is only marginallysupercritical In this model, bound cores(M/M

crit

\ 1.1).

form but are then destroyed by passing shocks, probably forthe unphysical reason that they cannot continue collapsingto sizes smaller than a few zones (see discussion in Paper I).Increasing the eld strength further leads to model Gi1s

where the eld supports the cloud magne-(M/Mcrit

\ 0.8),tostatically and gravitationally bound cores do not form.

4.2.5. Energy Spectra

We next examine Fourier spectra of the energy. Figure 13presents the spectra of kinetic, potential, and magnetic ener-gies at times t \ 0.0 and of the high-resolutiont \ 1.5t

ffmodels Dh1 (hydrodynamic), Eh1w and(M/Mcr

\ 8.3),Eh1i all driven at wavenumber k \ 12. The(M/M

cr\ 1.8),

spectra at time t \ 0.0 represent fully developed turbulencejust before gravity is switched on. Here we nd anotherreason for the fast collapse of the more strongly magnetizedmodel Eh1i. The density enhancements caused by shockinteractions are larger for models Dh1 and Eh1i than forEh1w, as can be inferred from comparing the potential ener-gies at t \ 0.0. Although still supercritical, the eld in modelEh1i is already strong enough to suppress motions perpen-dicular to the mean eld, so that the eld strength perpen-dicular to its initial mean direction is small, while strongshocks/waves can be formed parallel to the mean eld, asobserved as well by Smith et al. (2000). Thus, somewhatsurprisingly, the density enhancements are larger for thestronger eld model Eh1i than for the weak-eld model

FIG. 13.Kinetic, potential, and magnetic energies for models Dh1(k \ 12, Eh1w (k \ 12, and Eh1i (k \ 12,M/M

cr\ O), M/M

cr\ 8.3),

at the time t \ 0.0 at which gravity is turned on in a state ofM/Mcr

\ 1.8)fully developed turbulence (upper row). For comparison, we includedpower spectra ofP(k) P k~5@3 and P(k) P k~2 (dotted lines). The lower rowcontains the same models but for time t \ 1.0t

ff.

Eh1w, leading to earlier collapse, as seen in Figure 11. Inmodel Eh1w, the weaker, more turbulent magnetic eld pro-duces a more isotropic magnetic pressure that cushions andbroadens the shocks, thus decreasing the density enhance-ments and delaying collapse.

We illustrate this eect in Figure 14, where we plot the x-y- and z-components of the magnetic energy against timefor the magnetized models Eh1w and Eh1i. In the weak-eldcase, the eld is quickly tangled by the ow, so that it has nopreferred direction by t \ 0.0, when gravity is switched on.The magnetic energy, and so the magnetic pressure, is iso-tropic. There is no secular increase of any component withtime, thus supporting the picture of local collapse. In a

globally collapsing environment, the magnetic eld lineswould follow the global gas ow and lead to a noticeableincrease of magnetic energy. In the stronger eld case, on

FIG. 14.Magnetic energy components for magnetic models Eh1w(thick lines) and Eh1i (thin lines) against time. Solid lines denote the energyof the z-component (initial mean eld orientation), dashed lines the x- andy-components. Gravity is turned on at t \ 0.0. In the weakly magnetizedmodel Eh1w, the eld has ceased to show a net orientation at t \ 0.0. It hasdeveloped a fully turbulent state. The strong eld (Eh1i) continues to keepits mean orientation.


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the other hand, the ow is dominated by the mean eldoriented along the z-axis. The eld allows matter to movemore freely in the parallel than in the perpendicular direc-tion. Matter thus collapses preferentially along eld linesrst and then globally collapses.

After one free-fall time, all the models have collapsed(Fig. 13). Note that, as in Paper I, for all kE

pot(k) [ E

kin(k)

does not necessarily mean that the model becomes globallyunstable. With increasing time, becomes constant forE

pot

(k)

all k just because this is the Fourier transform of a d func-tion, signifying that local collapse has produced pointlikehigh-density cores. A similar argument applies for thekinetic energy: the at spectrum stems from local concen-trations of kinetic energy around the collapsing regions.Here, the spectral analysis no longer yields information onglobal stability.

4.2.6. Conclusions

We conclude that the delay of local collapse seen in ourmagnetized simulations is caused mainly by weakly magne-tized turbulence acting as a more or less isotropic pressurein the gas, decreasing density enhancements caused byshock interactions. We feel justied in claiming that mag-

netic elds, as long as they do not provide magnetostaticsupport, cannot prevent local collapse, even in the presenceof supersonic turbulence.

We note that once bound cores form, we take this asevidence for local collapse, although subsequent shockinteractions may destroy these cores again. In a real cloud,ambipolar diusion would set in at the length scale oftransient cores, so that any internal turbulence would bequickly dissipated, allowing further collapse, as discussedin Paper I.

A large-scale driver, such as interacting supernova rem-nants or galactic shear, together with magnetic elds, seemsto act like a driver with a smaller eective scale in the sensethat both yield a more uniform core distribution and a

somewhat slower collapse rate. In weakly magnetized turb-ulence, a more or less isotropic magnetic pressure reducesthe density enhancements behind shocks and thus slowsdown the process of isolated collapse. In strongly magne-tized turbulence, however, the mean magnetic eld domi-nates. The magnetic pressure is not isotropic any more, sothe shocks perpendicular to the mean eld direction causehigh enough density enhancements for the regions to col-lapse within a free-fall time.

Thus, for small eld strength, the eective additionalpressure may be represented by a simple pressure term.However, in the regime of eld strength interesting formolecular clouds, the eld, although supercritical, is strongenough to result in an anisotropic magnetic pressure. Mag-

netic turbulence is an all-scale nonisotropic phenomenon,

and the compression and perturbations on large scalesmake the cloud nally collapse.

5. SUMMARY

In this paper, we investigated whether magnetized turbu-lence can prevent collapse of a Jeans-unstable region. Fromour high-resolution simulations we conclude that:

1. In order to resolve self-gravitating MHD turbulence

using a grid-based method such as ZEUS-3D, the localJeans length should not fall short of at least four grid zonesfor magnetostatic support and six grid zones for magneto-dynamic support.

2. Local collapse cannot be prevented by magnetizedturbulence in the absence of mean-eld support. Stronglocal density enhancements caused by shock interactionsstart collapsing at once.

3. However, the magnetic elds do delay local collapseby decreasing local density enhancements via magneticpressure behind shocks.

4. Weakly magnetized turbulence appears qualitativelysimilar to hydrodynamic turbulence driven on a slightlysmaller scale, while stronger elds close to but under the

value for magnetostatic support tend to organize the owinto sheets and allow more clustered collapse.

5. The strength and wavelength of turbulent drivinggoverns the behavior of the cloud, overshadowing theeects of magnetic elds that do not provide magnetostaticsupport.

6. MHD turbulence cannot prevent local collapse formuch longer than a global free-fall time. Stars begin to format a low rate as soon as local density enhancements con-tract. This result favors a dynamical picture of molecularclouds being a transient feature in the ISM (Ballesteros-Paredes et al. 1999; Elmegreen 2000) rather than living formany free-fall times.

We thank A. Burkert, H. Lesch, S. Phleps, E. Va zquez-and E. Zweibel for valuable discussions. ThanksSemadeni,

go to the referee, whose criticism helped clarify importantpoints in the paper. M-M. M. L. is partially supported by anNSF CAREER fellowship, grant number AST 99-85392.Computations presented here were performed on SGIOrigin 2000 machines of the Rechenzentrum Garching ofthe Max-Planck-Gesellschaft, the National Center forSupercomputing Applications (NCSA), and the HaydenPlanetarium. ZEUS was used by courtesy of the Labor-atory for Computational Astrophysics at the NCSA. Thisresearch has made use of NASAs Astrophysics Data

System Abstract Service.

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