Turbulent Flows within Self-gravitating Magnetized Molecular Clouds

THE ASTROPHYSICAL JOURNAL, 557 : 451È463, 2001 August 10 ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

TURBULENT FLOWS WITHIN SELF-GRAVITATING MAGNETIZED MOLECULAR CLOUDS D. S. BALSARA AND R. M. CRUTCHER National Center for Supercomputing Applications, University of Illinois, Urbana-Champaign, IL ; u10956=ncsa.uiuc.edu, crutcher=ncsa.uiuc.edu

AND A. POUQUET National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000 ; Observatoire de la Coüte dÏAzur, Centre National de la Recherche ScientiÐque, Unite Mixte de Recherche 6529, BP 4229, 06304 Nice, CEDEX 4, France ; pouquet=ucar.edu Received 1997 December 15 ; accepted 2001 March 21

ABSTRACT Self-gravitating magnetized Ñows are explored numerically in slab geometry. In this approximation, the derivatives are computed only in one dimension but all three components of vector Ðelds are retained. This is done for a range of Ðducial values for the interstellar medium at the scale of molecular clouds. The overall characteristic scale of the turbulence, its Mach number, and the initial ratio of longitudinal to transverse turbulent velocities, as well as the extent of the initial density bulges within the Ñuid, are the main parameters of the study. Simulations have been performed with and without ambipolar drift. No external forcing is included. Velocity, density, and magnetic perturbations develop selfconsistently to comparable levels in all cases. This includes those cases where the medium is initially static. However, a fully random Ñow produces substantially more density contrast with nested substructures. Collapse eventually occurs after typically three free-fall times. The magnetic Ðeld slows down the collapse as expected. For higher Mach numbers, the collapse is faster, and yet the peak densities reached in the Ðnal collapsed objects are lower. We have also modeled the e†ects of ambipolar drift in the presence of cosmic ray ionization and far-ultraviolet ionization. Because the turbulent timescales are shorter than the ambipolar drift timescales, we Ðnd that ambipolar drift does not play a signiÐcant role in gravitational collapse in a turbulent medium of the type modeled in our simulations. Subject headings : instabilities È ISM : clouds È ISM : magnetic Ðelds È MHD È turbulence 1.

INTRODUCTION

point, new satellite-borne instruments have been launched (HST , ISO, SOHO) and ground-based high-resolution telescopes are being developed, such as the BIMA array. On the other hand, as computers become more powerful and codes more sophisticated, models can be devised to address a variety of problems related to such observations. Observations clearly indicate a self-similar behavior on a huge range of scales, as exempliÐed by detailed analysis of clouds with di†erent beam apertures (Falgarone, Puget, & Perault 1992). Other signatures of turbulence abound, such as LarsonÏs laws (1981) relating velocity dispersion in a cloud with the scale of the cloud (although other explanations could be given for such laws using virial equilibrium), such as the exponential wings of velocity spectra and a wealth of disordered motions. However, a full threedimensional numerical treatment of high Reynolds number Ñows is out of reach today. For example, recent numerical simulations in the vastly simpler case (at least, from the numerical standpoint) of incompressible neutral Ñuids on a grid of 5123 uniformly distributed points using the NavierStokes equations and a pseudo-spectral code achieve a Taylor Reynolds number R (as deÐned at the end of ° 3) of j Measurements of turbulent D160 (Jimenez et al. 1993). Ñows in the laboratory and the atmosphere show that Taylor Reynolds numbers in the range of D103 are routinely found ; astrophysical Ñows have much higher Reynolds numbers still. Inviscid codes, with proper numerical dissipation in the vicinity of strong gradients, fare somewhat better but the range of length scales on which nonlinear interactions occur free of both boundary e†ects and dissipative e†ects is still limited to less than two decades. This range is still insufficient for realistic turbulent Ñows. A way to improve the range of nonlinearly interacting scales is to

Most of the stars currently forming in our Galaxy do so in molecular clouds (Myers et al. 1986 ; Mooney & Solomon 1988). With the advent of more sophisticated observational techniques, it has become possible to map out such clouds over a range of scales from the largest scales of tens of parsecs down to the size of the smallest observable cores of D0.001 pc (for a review, see Blitz 1993). Observations of line widths in molecular clouds (Crutcher et al. 1994 ; Fuller & Myers 1992 ; Williams, de Geus, & Blitz 1994 ; Williams & Blitz 1993 ; Falgarone & Puget 1986 ; Falgarone, Puget, & Perault 1992) show that the gas is anything but static ; it is a complex, turbulent, magnetized medium with prevailing supersonic and sub-Alfvenic velocities and a myriad of nested, often Ðlamentary, structures and localized condensations. For a review, see Heiles et al. (1993). Further detailed understanding of the precise physical mechanisms at play is still in demand. One major setback stems from our lack of comprehensive knowledge about turbulent Ñows and their interaction with waves. For example, how does energy distribute itself among its various modes (kinetic, magnetic, internal, gravitational) and at various scales ? Another obvious difficulty arises from the fact that the physics of the cloud itself is complex and changes with scale given the many interacting agents (supernovae remnants, stellar winds emanating from young stellar objects, cosmic ray radiation, interstellar grains, UV shielding of dense cores, ionization fraction leading to ambipolar drift at small scale, and quite possibly a Hall current in the vicinity of jets caused by substantial local ionization, to name but a few). A detailed understanding of the interstellar medium, though, may be at hand because of a combination of factors. On the one hand, from the observational stand451

452

BALSARA, CRUTCHER, & POUQUET

lower the space-dimensionality of the computations, allowing for substantially better spatial resolution while still retaining reasonable CPU and memory computer costs. In fact, a parametric study at high resolution is more easily performed when derivatives are computed in only one dimension, as done in Alecian & Leorat (1988) for the nonmagnetic case and in Gammie & Ostriker (1996) for MHD. At the small scale of dense cores (D0.01 pc and below), the ambipolar drift comes into play ; it arises from the rapid equilibrium in the momentum equation between drift of neutral particles with respect to charged particles, and the Lorentz force. The induction equation then contains nonlinear terms in B, leading to the development of sharp fronts and to a quasi forceÈfree Ñow (Brandenburg & Zweibel 1994). This ambipolar drift is important in breaking the Ñux freezing condition in the last stages of collapse (Mouschovias 1991 and references therein). Since in this paper we wish to understand the role of turbulence in modulating gravitational collapse, we choose to perform two sets of studies, one without the ambipolar term in the induction equation and one which includes it. There are several sources of energy input in molecular clouds. A list that is in no way intended to be comprehensive includes shear at the galactic scale, interclump collisions in giant molecular clouds, gravitational collapse, supernovae, molecular outÑows, and protostellar jets, and, lastly, winds and radiation from newly formed OB stars. Statistical analysis by Ward-Thompson et al. (1994) has shown that the timescale for the formation of protostellar cores is much longer than the Jeans free-fall time. Thus the cores themselves are more stable to gravitational collapse than predicted by linear analysis, with typical formation times an order of magnitude larger than that computed by Jeans criterion for a static medium. Ward-Thompson et al. (1994) have suggested that protostellar cores form on timescales that are consistent with the time required by ambipolar di†usion to remove the magnetic Ðeld from the cores. Turbulent pressure support has also been advocated as a likely candidate for slowing down the collapse of clouds (Chandrasekhar 1951 ; Bonazzola et al. 1987 ; Pudritz 1990 ; Bertoldi & McKee 1992 ; Balsara 1996). It must be pointed out that this somewhat na• ve hypothesis about the turbulent pressure support is drawn from the limiting case of incompressible hydrodynamics. It is well known, however, that line-width measurements in molecular clouds indicate that the Ñows in these clouds are supersonic. In particular, detailed mapping of molecular cloud cores by Crutcher et al. (1994) has also shown that the line widths, which track the velocity dispersion, are large and supersonic in those regions where the magnetic Ðeld is high. Furthermore, as one progresses inward to the central part of the core, the line widths become strongly subsonic. This transition takes place at scales of a few times 0.01 pc, where the gas densities exceed a few times 106 amu cm~3. Based on linear stability analysis, Balsara (1996) has shown that this is a natural consequence of ambipolar di†usion since it damps out waves (that are in the linear regime) on the same length scales on which it helps dredge out the magnetic Ðeld (see also Pudritz 1990). The signiÐcance of this dual role played by ambipolar di†usion is seen when it is pointed out that the turbulent pressure support is comparable to the magnetic pressure support for molecular cloud gas. It must also be pointed out that the linear stability analysis of waves in a

Vol. 557

medium with ambipolar di†usion does not give us a clear understanding of the behavior of these waves once their amplitudes reach the nonlinear regime. Only fully timedependent simulations can resolve this situation. The mechanisms for turbulent energy input discussed in the previous paragraphs occur on vastly di†ering length scales and drive the Ñow in di†erent forms, such as an input of heat, or of momentum, through either a purely solenoidal (shear) velocity Ðeld or one that has also a compressive component because of shocks. Numerical simulations (see Porter, Pouquet, & Woodward 1995 ; Porter, Woodward, & Pouquet 1998, and references therein) have shown that any compressible Ñow with randomly phased velocity Ðelds will soon develop compressive motions. If the initial velocity Ðeld has an rms velocity that is transonic or supersonic, then shocks will inevitably form. Since observed line widths in molecular clouds indicate that the motions are supersonic, strongly compressive motions must occur in the turbulent Ñows in molecular clouds. At higher Mach numbers and for neutral Ñuids, when strong shocks form faster than the collapse can take place, a marginal equilibrium is reached at all scales and indeed no collapse occurs (Passot 1987 ; Leorat, Passot, & Pouquet 1990 ; see also Pouquet, Passot, & Leorat 1991a, 1991b). Collapse is nevertheless induced in the vicinity of shocks, in particular in the radiative case : density gradients are strong, and the free-fall time becomes locally shorter than the hydrodynamic time. Thus shocks in a turbulent molecular cloud can play a dual role. On the one hand, they can form local condensations with extremely high densities that can go through an initial phase of gravitational collapse faster than that predicted by linear theory. On the other hand, they can suppress the collapse by raising the amount of turbulent motions in the Ñow. Which mechanism wins is a result of the competition between phenomena of di†ering timescales. The present study aims at continuing the investigation of gravitational collapse in a turbulent magnetized medium. The parameters adopted for this study are chosen to be typical of molecular clouds (Crutcher et al. 1993). Because turbulence can be input in a molecular cloud over a range of length scales and Mach numbers, we have made a parametric study in which we vary the following : (1) the Mach number of the rms velocity Ñuctuations ; (2) the Alfven speed of the rms Ñuctuations in the magnetic Ðeld ; (3) the scale at which the turbulent Ñow initially resides ; (4) the initial compressibility of the Ñow as measured by its Mach number and by the initial ratio of compressive (or longitudinal) to shear (or solenoidal) velocity ; and, Ðnally, (5) the presence or absence of ambipolar drift and the ionization mechanisms responsible for sustaining the cloudÏs ionization and thus allowing the ambipolar drift to operate. The choice of onedimensional geometry leads to a substantial improvement in numerical resolution and, therefore, in the extent of the inertial range of the turbulent Ñow that is computed uninÑuenced by boundary e†ects. Note that the work presented here is complementary to that of Gammie & Ostriker (1996), both because of its emphasis on the role played by compressibility and because of the inclusion of ambipolar drift arising from a low degree of ionization. The next section gives the physical equations we choose to model, a justiÐcation of our choice of physical parameters, and a brief description of the numerical set-up. The main results are given in ° 3, and the conclusions are given in ° 4.

No. 1, 2001

SELF-GRAVITATING MAGNETIZED MOLECULAR CLOUDS 2.

THE MODEL

2.1. Equations We write the standard nondimensional MHD equations as follows :

o

C

Lo ] + Æ (ou) \ 0 , Lt

D

A

B

o B o2 Lu ] (u Æ $)u \ [+ P ] 2 Lt

] (B Æ $)B [ +/ , LB \ + ] (u Â B) , Lt

AB

+2/ \ 4nGo ,

AB

P L P ] (u Æ $) \0 , oc Lt oc $ Æ B\0 ,

(1)

with, in the energy equation, c \ 4/3. This choice of c is made in the spirit of the work in Va`zquez-Semadeni, Passot, & Pouquet (1996), which shows that, in the presence of heating and cooling, the interstellar medium at large behaves on average as a barotropic Ñuid with a low value of c, as already obtained analytically by Elmegreen (1991) for the linear case (see also below). Note that, contrary to many authors (Mac Low 1995, 1998 ; Gammie & Ostriker 1996 ; Padoan, Jimenez, & Jone 1997), we choose not to be isothermal but rather make the choice of carrying the energy equation, consistent with the modeling of the interstellar medium at large performed in Passot, Va`zquez-Semadeni, & Pouquet (1995). Although more costly, this allows for a more detailed dynamics of such complex Ñows. In the MHD equations given above, o is the density normalized to o , P is the pressure, B is the magnetic Ðeld (in 0 fact, the induction) normalized to B with B equal to a x x constant for all runs, and u is the velocity normalized to u . 0 These equations are used to describe the dynamic evolution of a molecular cloud of a typical size of a few parsec in the presence of magnetic Ðelds, turbulence, and self-gravity. In this paper, we study separately the classical MHD case as written above and, in ° 3.5, the case with the ambipolar term included in a generalized OhmÏs law. Such an inclusion of the ambipolar drift term in the induction equation is relevant when one focuses on scales of the dense cores of the order of 0.01 pc. We impose that only L/Lx D 0 ; hence, the divergence-free condition for the magnetic Ðeld implies that B be constant. x Neither heating nor cooling functions are included in the above equation. It should be noted that such functions are poorly known at the scale of the molecular cloud, in contrast to the global galactic scale ; but it can be advocatedÈ both in the linear regime (Elmegreen 1991) as well as in the nonlinear one (Passot et al. 1995)Èthat given cooling (") and heating (!) functions with a power-law dependence both in the density and the temperature, the gas tends to behave as a barotropic Ñuid P D oceff , with an e†ective c-law exponent that depends on the functional form of both " and !. Such a behavior was observed in two-dimensional numerical simulations at the Galactic scale, with c D 0.3 eff et al. for a gas at a mean temperature of D8,000 K (Passot 1995 ; see also Va`zquez-Semadeni, Passot, & Pouquet 1996).

453

The RIEMANN code for numerical MHD used here is described at length in Roe & Balsara (1996) and Balsara (1998a, 1998b). It utilizes a characteristically designed total variation diminishing algorithm along with a linearized Riemann solver. The choice of slab geometry allows for high resolution, and all runs are performed on a grid of 2,048 uniformly spaced grid points. The simulations were done on the Cray C90 at the Pittsburgh Supercomputing Center. 2.2. A T ypical Cloud For our Ðducial cloud, we take a characteristic length scale of L \ 1 pc, a mean density of o \ 10~20 g cm~3, a 0 0 temperature of T \ 10 K, a characteristic velocity of u \ 0 0 1 km s ~1, where u is the rms velocity taking into account 0 both the transverse and longitudinal components of the turbulent velocity Ðeld, and a magnetic Ðeld B \ 16.1 kG x where B is its uniform x-component. These characteristic x scales of length, density, and velocity deÐne convenient, normalized code units that were used in the simulations. The corresponding Alfven velocity v \ B /(4no )1@2 for A xvelocity 0 :c \ this cloud is D2.5 c , where c is the sound s (kT /k)1@2 \ 1.8 ] 104s cm s~1, with k \ 2.5m , and m is sthe p p D148 mass of the proton. The L \ 1 pc cloud contains 0 solar masses. The Jeans frequency is u \ (4nGo )1@2 D 9 J D 2.2 ] 0 106 yr, ] 10~14 s~1), the free-fall time q \ 2n/u ff J and the Jeans length is L \ (2nc /u ) D 0.4 pc. The Mach number is M \ u /c and JM \ us /v J is the Alfvenic Mach 0 A equal to L /u correnumber. The unit0of stime forAthe code 0 0 sponds to 9.8 ] 105 yr or D2.8 q . ff 2.3. Initial Set-up of V ariables Locally random (or LR) initial conditions are those for which the density Ðeld is set uniform throughout the cloud with a small added perturbation, namely, o(x ; t \ 0) \ 1 ] v exp [[x2/p2] with v \ 0.1 for all runs reported here, and p \ (.5È3)L . This overdensity is imposed over one Jeans length for Jall runs. We checked that imposing it over one-half that length did not change any of the conclusions to be reported in the next section. The Ñuctuating velocity and magnetic Ðelds are either locally random (or LR)Èwhereby their Ñuctuations are limited to the bulge in density imposed at t \ 0, or the initial conditions for u and B consist in taking fully random phased (FRP) Ðelds. The velocity Fourier spectrum is initially EV(k) D k2 exp [[2(k/k )2] peaking at k \ k /2. Writing u \ u 0 spectra are decomposed 0 ] u , the Fourier as well intoS C a solenoidal (transverse, ES(k)) and a compressible (longitudinal, EC(k)) component, where ES(k) and EC(k) are the spectra of u and u , respectively, with $ Æ u \ 0 and S k andC S + ] u \ 0. Both C 0 s \ u2/u2 (2) C S are parameters of the initial conditions. We choose the ratio k /k to be sufficiently large (where k \ 2n/L is the 0 min min 0 minimum wavenumber of the computations) in order to allow the Ñow to develop somewhat independently of boundary e†ects, at least in the turbulence phase. The eddy turnover time q , i.e., the time it takes for eddies to interact NL develop small scales is proportional to substantially and 1/(k u ). In classical turbulence theory, the eddy turnover time0 is0 the time in which a randomly phased velocity Ðeld

454


with velocities initially concentrated around a length scale of 1/k would develop a Kolmogorov spectrum that reaches to the0 smallest, dissipative length scales. In one turnover time, there are D1.5 free-fall times for the runs at Mach 1 and k /k \ 10, whereas q \ 0.15q for M \ 2 and 0 min NL ff k /k \ 50. In other words, in most of the runs reported 0 min here, the e†ects of turbulence operate on the Ñuid faster than the gravitational e†ects. Table 1 gives several of the parameters of the runs described here with L \ 1 pc and no ambipolar drift, 0 whereas Table 2 gives the same information for the runs with L \ 2 pc and di†erent initial conditions, insofar as the 0 extent of the randomness of the perturbations is concerned (see ° 3.4). Finally, Table 3 gives the same data for runs at L \ 2 pc and with the ambipolar term included. The pa0 rameters of the runs are the characteristic wavenumber of the initial Ñow k , the ratio of compressive to shear motion 0 in the initial Ñow s as deÐned above, the Mach number M \ u /c , the Alfven Mach number M \ u /v , and 0 A b\ Ðnally 0thes square ratio of gas to magneticA pressure c2/(b2 /8no ). The nomenclature for such runs as given in s tables rms is0 that used in the remainder of the paper. For the those runs coined ““ static,ÏÏ initially, (1) there is no velocity, and (2) a uniform component in the transverse magnetic TABLE 1 NOMENCLATURE AND PARAMETERS FOR THE RUNS FOR A CLOUD OF L \ 1 pc WITH LOCALLY RANDOM INITIAL CONDITIONS 0 Run

M

c53 . . . c40 . . . c52 . . . c41 . . . c48 . . . c49 . . . c50 . . . c51 . . . c42 . . . c44 . . . c43 . . . c45 . . . c46 . . . c47 . . .

2 2 0 0 0.1 0.1 0.35 0.35 1 1 2 2 2 2

M A 0.8 0.8 0 0 0.1 0.1 0.35 0.35 1 1 0.8 0.8 0.8 0.8

k /k 0 min 10 50 10 10 10 50 10 50 10 50 10 50 10 50

s

b

0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.068 0.25 0.25

0.32 0.32 O O 200 200 16 16 2 2 0.32 0.32 0.32 0.32

E G 0 0 1 1 1 1 1 1 1 1 1 1 1 1

Fiducial Runs F

F F

NOTE.ÈNomenclature and main characteristic parameters for the runs computed on a grid of 2,048 regularly spaced grid points for a cloud of L \ 1 pc and random initial conditions centered on the small initial bulge of0density (locally random, or LR). No ambipolar term is included here. All runs have periodic boundary conditions ; for all of them, the constant magnetic Ðeld in the x-direction B is equal to unity (except for run c52, x which is self-gravitating and nonmagnetic), and the initial fractional overdensity is 0.1, except for runs c40 and c53, which have no overdensity initially and no self-gravity. E indicates whether the run is self-gravitating (E \ 1) or not (E \ 0). TheGMach number is M \ u /c , where u is the G velocity and cG the sound speed, and the Alfvenic 0 Mach s 0 rms number is M \ u /v with v s \ B /(4no )1@2 the Alfvenic velocity based on the conA 0 A A x 0 stant x-Ðeld and the mean density o . The characteristic wavenumber of the initial turbulent Ñow is k in units0 of the minimum wavenumber of the computation k \ 2n/L , 0and s is the initial ratio of longitudinal to min velocities. 0 For all runs (but run c41, for which B \ transverse square 0‰y B \ B \ 1), there is no constant component of the magnetic Ðeld in the x 0‰z directions : B \ B \ 0. The ratio of gas to magnetic prestransverse 0‰z8no c2/b2 , where b \ b sure at t \ 0 is denoted0‰y by b \ is the trans0 s F M indicatesM a Ðducial rms verse Ñuctuating magnetic Ðeld. Finally, run for a standard molecular cloud. A run identical to run c45 but with now 1,024 grid points has also been performed. A second series of runs with fully random initial Ñuctuations is described in Table 2, and a third series including the ambipolar drift term is given in Table 3.

Vol. 557 TABLE 2

NOMENCLATURE AND PARAMETERS FOR THE RUNS FOR A CLOUD OF L \ 2 pc WITH FULLY RANDOM INITIAL CONDITIONS 0 Run

M

c81 . . . c82 . . . c83 . . . c84 . . . c85 . . . c86 . . . c87 . . .

2 0 0.1 0.35 1 2 2

M A 0.8 0 0.1 0.35 1 0.8 0.8

k /k 0 min 10 10 10 10 10 10 50

s

b

0.068 0.068 0.068 0.068 0.068 0.068 0.25

0.32 O 200 16 2 0.32 0.32

E G 0 1 1 1 1 1 1

Fiducial Runs *1 *2

F

NOTE.ÈNomenclature and main characteristic parameters for the runs described in this paper for a cloud of L \ 2 pc and random initial condi0 tions (or fully random phase, FRP). No ambipolar term is included here. For deÐnitions of the column labels, see note to Table 1. For all runs but run c82 (see *2), there is no constant component of the magnetic Ðeld in the transverse directions. Note that (*1) run c81 is decaying, and (*2) run c82 is initially static, with B \ B \ B \ 1. As before, F stands for a Ðducial 0‰y x 0‰z run.

Ðeld is included as well (see Table 1) in order to have initially an Alfvenic Mach number of 0.8 as for the Ðducial runs indicated by ““ F ÏÏ in the tables (see below). For the highest Mach number, the choice of parameters is motivated by observations that clearly indicate the presence of supersonic sub-Alfvenic velocities in this range (see, e.g., Crutcher et al. 1993 ; Myers & Goodman 1988a, 1988b).

RESULTS

3.

3.1. T urbulent Clumps in a Nongravitating Cloud We Ðrst choose to report on two basic runs at the edge of the parameter space we are interested in, namely, magnetic runs without gravity (runs c40 and c53 of Table 1). For the Ðrst purely turbulent magnetic run, there are no initial density Ñuctuations and the turbulent velocity spectra are peaked at k /k \ 10. Initially, the Mach number is 2.0 0 nic minMach number u /v \ 0.8. No substantial and the Alfve 0 A are observed for the di†erence in the results reported below similar run c53 with k /k \ 50, except, obviously, for the 0 min range where nonlinear interlesser extent of the inertial actions and steepening prevail. As expected, smaller scales are produced through nonlinear interactions in a cascade process well documented for many Ñows (incompressible or compressible, conducting or TABLE 3 NOMENCLATURE AND PARAMETERS FOR RUNS INCLUDING THE AMBIPOLAR DRIFT TERM

Run

a

M

c71 . . . c72 . . . c73 . . . c74 . . . c75 . . . c76 . . .

3/2 3/2 3/2 1 1 1

0.30 1 2 0.30 1 2

M A 0.30 1 0.8 0.30 1 0.8

s

b

0.068 0.068 0.068 0.068 0.068 0.068

22 2 0.32 22 2 0.32

E

G 1 1 1 1 1 1

Fiducial Runs

F

F

NOTE.ÈNomenclature and main characteristic parameters for the runs with the ambipolar drift term included in the induction equation with a pre-factor o~a (see eq. [5]), for a cloud of L \ 2 pc, with k /k \ 10 and minionization, with fully random phase initial conditions0; a \ 1 for far 0UV and a \ 3/2 for cosmic ray ionization. For a description of the columns common to Table 1, see the note there ; ““ F ÏÏ stands for Ðducial values of parameters for a typical molecular cloud.

No. 1, 2001

SELF-GRAVITATING MAGNETIZED MOLECULAR CLOUDS

neutral, and in all space dimensions) ; the characteristic time of the cascade is of the order of the eddy turnover time q , NL which is much shorter than the free-fall time, in particular when taking into account the fact that q is evaluated on NL k and not k . Density Ñuctuations develop in the order of 0 min one eddy turnover time as well, at a level that is compatible with a pressure equilibrium ou2 D P ; these Ñuctuations occur at all scales (not shown) ; for that run, at q D 1, the NLthe mean density varies between 5 ] 10~2 and 4, in terms of density o . Since the Ñow is not driven, such density Ñuctua0 tions damp out with time ; at the Ðnal time of the computation (t D 18q ), they are conÐned to the range 0.8o to NL 0 1.2o . All spectra, after an initial transient, decline self0 similarly in time and follow an approximate k~2 law. As is well known, this spectral law is compatible with the scaling Ðrst noticed by Larson (1981) between the scale of a cloud and its characteristic velocity dispersion. The cusps present in such spectra have been observed already in similar onedimensional computations (Passot 1987 ; Alecian & Leorat 1988) and can be attributed to power-law prefactors in the exponential decay behavior of the solutions in Fourier space. Finally, the longitudinal modes grow to an overall value of u D 0.3u , with, however, the longitudinal and transverse CvelocitySspectra being comparable at all scales but the largest one (k D k ). min in density and velocity Ðrst Although such Ñuctuations build up at the characteristic scale of the velocity Ðeld, they migrate to larger scales as well. In particular, there is a feature noticeable on run c40, for which k /k \ 50 (not 0 minthat keeps shown) in the transverse velocity spectrum peaking around k , but the longitudinal component migrates markedly to0 larger scales together with density. 3.2. T he Static Case in the Presence of Self-Gravity In the presence of self-gravity, and in the simple case when there are no initial velocity, no density nor magnetic

Ñuctuations (runs c52 and c41), all excitations develop through the forcing caused by the gravitational potential. The initial overdensity extending over a Jeans length and placed at the center of the cloud grows with time, the Ðnal condensation having a peak density of D11 and rareÐed wings at o D .03 in units of o , with a quasi-stationary 0 structure after of the order of two free-fall times. The density spectrum peaks mildly at k D 2.5k , with a smooth and J min wide range of scales excited up to k/k D 16, at which min wavenumber a sharp cut-o† marks the limit beyond which the turbulence Ñuctuations prevail ; these density spectra are displayed in Figure 1a for run c41 at equally spaced times in units of 0.5q . ff There is a progressive accumulation of matter in the central condensation, because, in part, there are no strong shocks to drive the Ñow at Ðrst. This is to be contrasted with the evolution of condensations in a turbulent medium (see next section). Because of the infall of matter on the central condensation, velocities develop which rapidly become of the order of (but smaller than) the sound velocity. The large-scale density spectrum is related to the equilibrium structure that develops, encompassing a range of 16 : 1 in scales ; at small scales, the turbulent spectrum already observed in the nongravitating case takes over, with its multiple weaker cusps. Whereas the density spectra after t/q \ 1.5 remain very similar before the Ðrst cusp, velocitiesff still evolve substantially at large scales, beginning to decrease after t/q \ 2, as can be seen from Figure 1b, which displays the totalff velocity spectra at the same times as for Figure 1a. The longitudinal and transverse velocities develop in similar ways and with similar amplitudes (not shown), comparable as well both in amplitude and in characteristic scale to the turbulent magnetic Ðeld. An accretion shock has developed by time t/q \ 1.5, which then subsides progressively in time. This can ffbe seen on Figures 2b, 3b, and 4b. In this series of Ðgures, we display

20

18

18

6

6 66

6 6 66

16 5

5

55 55

55

66 66 66666 6666 6666 666 666

55 5555555555 5555 555

66 66

66 6 6

66

66

55 55 55

55 55

4

44

44

44 44

16 6

6

6

6

44 4444444444 44444 444

55

55

55

5

44 44

6

6

66

14 6 6 6 66

6

66

5

55555

5 4 44

6

6

66 6

4 44

55 44

5

5

4

6

55 5

5

5

6 5 55

5

6

66

6

5 55 5

5

5 6 6 6 6 66

6

5

5

66

6

8

6

4

2

0

-2

-4

-6

-8

-10

-12 .6

.8

1.0

33 33

33333333 33333 333

1.2

1.4

4

5

4

4

4

333 33 33

1.6

1.8

2.0

6 66

66 66

666666 666 6 6 6

66 6

5 444

44

2.2

444

2.4 X

4

66 6

66 6

6 6

5 5 5

5

5

2.6

6

66

3.0

55 55555 5 5 5

6 6

10

5

66

66

6 66 6 6

6

6

66 666 66 6 6

3.2

3.4

3.6

3.8

4 44 44

4

44

44

6 66 6

6 66

55555555 5

55

4 4444 4 4

44

44

6 6 6 6 66

6 5

5 5 5

55

5 5

5

55

4

444 4

4

6 6 66

66

6 66

5 55 5 55

5

55

5 5 55 5 5

5 55 55 5

5 55 5 5

66

6 6

555 5

6 6

6

5 55 5 5

6

55

6

55

6 6 6 6 6 66 6 6 66 66 66 6 6 6 6

6 6 6

5 55 5

66 6 6 6 6 6 66 6 6 6 6 6

55

6 6 55 6 5 5 5 4 4 5 5 4 4 5 5 55 5 5 5 5 5 6 4 44 4 4 4 5 6 4 66 4444 4 5 44 44 444 5 5 5 5 5 5 56 4 5 4 5 44 5 5 4 4 5 33 5 5 5 4 44 5 4 3 3 33 3 44 3 5 3 4 4 44 4 44 33 5 5 3 33 3 333 333 3 3 3 4 4 4 4 44 5 3 33 4 3 4 4 3 44 4 33 4 3 4 44 4 4 3 333 3 3 3 4 3 44 3 3 33 3 3 3 3 6 3 3 3 4 44 33 3 33 3 3 44 3 44 3 4 3 3 3 33 4 44 4 4 33 3 3 3 3 3 4 3 44 3 3 3 3 3 4 3 33 3 3 3 3 3 33 3 3 3 3 33 3 3 3 33 3 3 5 22 222 2 2 2 2 22 3 3 3 2 22 2 3 3 2 33 3 3 3 22 3 3 3 3 2 2 22 3 22 3 3 3 2 4 22 2 2 5

5

2.8

55

5 55 55

6

8

6

4 Y

3 33 33

66

12

6

6 6 6 6 6 6 5 55 6 5 5 5 5 6 6 6 66 5 6 66 5 4 33 3 5 4 4 6 6 6 4 33 4 33 44 4 33 55 5 6 5 44 5 3 55 4 3 6 6 4 4 4 3 4 55 5 4 4 3 5 5 4 3 4 45 44 5 5 5 5 5 5 5 5 4 4 3 44 4 4 4 5 4 3 3333 4 44 3 5 33 5 3 5 3 55 5 5 33 5 4 3 3 4 4 5 4 5 5 3 3 4 3 5 4 3 3 4 4 3 3 6 44 5 4 4 3 3 43 3 3 3 3 3 4 3 5 3 4 3 4 3 4 3 3 3 4 44 3 22 22222 2 2 2 3 3 22 2 2 3 44 4 3 22 33 33 3 3 22 2 3 4 3 3 22 33 22 3 3 2 3 4 4 3 3 3 3 33 3 3 3 2 334 22 4 4 5 3 3 3 3 2 3 3 2 4 4 3 22 34 3 3 3 3 2 3 22 3 2 2 2 2 4 2 2 2 2 2 2 22 2 3 2 2 11111 1 11 2 22 11 11 11 1 2 1 2 11 2 22 2 1 11 1 11 11 2 11 2 1 11 1 11 1 1 2 2 22 2 2 2 2 0000 00 1 1 1 00 2 2 00 1 00 22 0 00 2 2 2 2 1 1 0 0 00 0 00 2 2 1 11 1 11 1 00 2 22 2 0 00 0 1 1 000 11 2222222222222222222222222222222222222222222222 1 00 1 1 1 0 00 11 1 000 1 11 1 00 0 1 1 00 0 1 1 00 1 11 1 1 00 0 1 1 1 00 0 1 1 1 1 0 00 1 0 00 1 1 1 11 111111111111111111111111111111111111111111 0 00 00 0 00 000 000 0 00 0 00 0 00 00 0 0 00 0 00 0 0 00000000000000000000000000000000000 5

10

66

66 6 6

6

5 444 44 44

6 666 6 6 6

6

12

66

6

14

Y

455

2

0

4

44

44

44

44

4 4

444

4

4

4

5

44

5

4

33 33333 3

22

2 3

-2

2

222 222

2 2 2

-4 1

11

11

1 11

11 11

11 11 11

2 111

11

1

2 2 2

2 22 2 2

1

111 11 1

111 1

-6

1

2

1 11

1

1 11

-8 1

1

2

2 2

2 2 22 2 2 2 2 2 2 2

22

2 2 22 2 2222222222222222222222222222222222222222222222222222222

1 111 1 11 1 1 11 1 11 1 1 1 1 1 1 111 1 1 1

11 1 111111111111111111111111111111111111111111111111111111

-10

-12 4.0

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

X

FIG. 1.ÈDensity (left) and total velocity (right) Fourier spectra for run c41, which is a static magnetized run with k /k \ 10 and locally random initial conditions. The seven curves are labeled by their rank in a temporal series in which the time interval is 1.1 ] 106 yr or 00.5min q , with the initial time at the very ff clutter ; it represents a scale of bottom. The distance between each symbol on a given curve, as in all subsequent Ðgures of the same nature, is 8*k to avoid 0.004 pc for a cloud of L \ 1 pc (and twice that for L \ 2 pc) ; moreover, to avoid overlapping, the positive o†set of the curves on the y-axis for each time is 0 of 3 log units. Note the0development of substantial Ñuctuations in all Ðelds and at all scales after of the order of 1.5 free-fall time. 10

456


13

Vol. 557

14

6666 6666666666666 66666 6666 6 66666 6 666 6666666666666666666 6666666666666666 666666666666 6666 6666 6 666666 66 666666666666 66666666666666666666666666666 66 6 6666 6666666666 6666 6666 66 666 6 666 6666666666666666 666666666666666 66666666666666666 6666 666666 6 666 6 66666666666 66666 66

12

6666666666666 666 6 6 6 6 6 6 66 66 66 66 66 6 6666 66 666 6666 66 6666 666 66666 6666 66 6666 6666 6666 66666 66666 66 6666 666666 666666 666666 66 6666666 66666 66666 55 55 55 55 55 666666 55 555 666 666666666 555 66666666666 666666666 5 666666 666 66 6 66 5 66 66666666666666666 5 66 666666666666666 66666666666666666 5 5 66 666666666666666666 5 5 5 55 55 55 5 5 55 5555 555 5 55 55 5 5555 5555 55555 555555 555555 555555 55555 55555 555555 555555 55555555 5555555 55555 5 55555555555 5555 55555555555 4444444 44 4 4 44 44 555555555555555555555555555555555555555555555555555555 55555555555555555555555555555555555555555555555555555 4 4 4 4 4 4 4 44 44 44 44 44 444 444 4 4 4 4444 4444 4444 4444 44444 4444 444444 444444 4444444 4444444 4444444444 4444444444 4444444444444 444444444444444 44444444444444444 44444444444444444 4 3333333333 33 33 33 3 3 3 3 3 3 3 3 33 4444444444444444444444444444444444444444 33 3 44444444444444444444444444444444444444444 3 33 3

13

12

11

55 55 55 555 55555 5 555555555 5 55 55555555 55 555555555555555555555 555 55 55555555 5 5555555555 555 5555 5555555 5555 55 55 5555555555555555555 5555555555555555555555555 55 55555 55555 55555555555 555555555555555 55 55555 55 55 5 5 5555555 5 555 555 55 5 55 5555 55555555 555555555555555 5555 5555 5 55

10

11

10 9 9

444 4444444444 4444 4 444 444 4 44444444444444444444444 4444444444444444444444444444444444 44

4 444 4444444444444 44444444444 44 44 444444 444 44444444444444 44444444444 444444444444444444444 44 44 4444 4444444444 4 44444 4 44 44 4444 4 4 4 44444444 444444444444444 444444444444444 4 4444444 44 4

8

8

Y

Y

7

3 333333333333 33333333333333333333 33333333 333333 33 333 3 3333 3333333333333333 3333333333333333333333333333333333 3333 33333333333 333333333 33333333333 33333333333333333333 33 333 33333 3333333333333 333333333333333333 3 33333 3333 3333333 3333 3333333333 3 333 333333333333333 33 3

6

7

6 3333333 3333333333333333333333333333 33333333333333333333333333333333333333333333333333333333333333333333333333333

5

3333333333333333333333333 3333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333

5 2 222 222 222222 2222 222222222222 22222 22 2 22 22 22222 2 2 2 222 22 2 2 22 22 22 2222222 2222 2222222222 22 22 2 2222 2222222222222 2 2222222 2 222 22 2 22 22222 22222222222 222222 2222222 22 2 2 22222222 2 2222222222222222222222222222222222222222222222 222222 222222 22 222 22 22222222 22 222222 22 2222222 22

4

222222222222222222222222222 2222222222222222222 2222222222222 222222222222222 222222222222222 2222222222222222222 222222222222222 22222222222222222222222222222222 222222222222222222222 22222222222222222222222222 222222222222222222222222222222222222222222222222222222

4

3 3

111 1 11 1 1 111 1 1 1 1 1 1 1111 111 11111 11 1 11 11 11 11 1 11 11

11 1 1111 111 11 1 11 1 1 1 11 11111111111111111111111 11

11 1 11 1 11 11 1 11 11 1 111 11 11111 11 1111111 11 1 1 1 1 1111 1 11 11111 11111111111 11 1111111 11111 11 1111 11111 1 1 1 1111 11111 111 1 1 1 11 11111 11111111111 11 111 11 11111111 1 111 11 1111 11 11 1 1 11111111 11

2

1

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

2

1

0 -.5

-.4

-.3

-.2

-.1

0

.1

.2

.3

.4

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

0

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

.5

-.5

-.4

-.3

-.2

-.1

X

0

.1

.2

.3

.4

.5

X

13 666666666666666666666666 66666 66 6666 666 6666666 66666 66 66 6666666666 666 6 66 66 66 66 6666666 666 666 6 6 6666666 66 666666 66 66 6666 6666 666666 6666666 6 66 66666 66 66 6666666 66666666 666 6666666666 66666 6666 6666 66 66666666 66666666666 66666666666 66 66 66666 666666666 666 66666666666666666 66 55 5 55 55555 55 5555555 555 55555 555 55 555 555 55555 555 555 5555 555 55555555 55 55 55 5 5 55555 55 5 555 555 5555 55 55 555555 555 5555 55 555 555555555 55 55555555 55 55 5 555 555555555555555 55 55 555 55555555 55 5 5555555 5 555555555555 55555555555555555555555555555 5555 55555555555555 55555 55 5 555555555

12

11

10

9

4 4444 44 4 4 44 44 44444444 4444 4444 44 4 4 4444444444444 4444 444 4444 4 4 4 444 4 4 44 44444 4 44 444 444 444444444 4444444444 4 4444 444 4444 44 44444 44444 44444 4444 44444 4 44 4444 4 444444 444 44 444 44444 44 4 44 4444444 444444444444444444 444444444 4 444444444444444 444444 44444444444444444444 44 44444444 33 3333 333333 33333333 333 3 3333 3333 33333 333 33333333 33333 33333 33 3 3 333 333 3333 3333333 33333333 333 333333 33 3333 3 33333 3333333 333 33 33 33333333 33 333 3 3 333333333 3333333333 333333333333 3 3 3333 33 3333333 333 33333333333 3 333 33 33 3 33 33 33 3 3 33 33 33 33 33 33 3 333333 33 333 3333333333 33333 333 222222 22 2 2222 2 2 2 2222 2222 22 222 222 2222 22222 2222 22 222 22 2 222 2 22 22 2 22 222222 2 2 2 22222222 222222 2222222 22 22 2 2 2 2 222 22 2222222 2 22222222 2222 222222 2222222222 22 2 222222 22 22 22 22 2222 2222 222 222 22222 2222222 22222 2 222 2222222222 2222 2222 222 222 22 222 2222 22 22 222 2 22 222 222222222222222 1 1 11 111 1 111 1 1 11 11 11111 11 1 1111 1 1 1 1 1 1 111 1 1 11111111 1 111111111 111 1 1 1111111 111 111111111111111111 1 11 1 1 11 1111111 111111 11111 1 111111111111111111111111111111 11 111 11111 11 111 1 11 11111111 111111 11 1 1 1 11111111 1 11 11111 111111111 11 1111111111111 1111 11 11111111111111111 111111111

8

Y

7

6

5

4

3

2

1

0 -.5

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

-.4

-.3

-.2

-.1

0

.1

.2

.3

.4

.5

X

FIG. 2.ÈProÐles of the log of the density every 0.5 q for various initial turbulent Mach numbers and locally random initial conditions. Curves are 10 The o†set for each time ffis of 2 units on the y-axis. (upper left) Run c53 without self-gravity ; (upper right) run c41 initially labeled as in the previous Ðgures. static ; (lower middle) run c43 at Mach number M \ 2. Note the displacement of the bulge in density when the Mach number is high, and the lack of substantial core with steep boundaries for supersonic Ñows, for which, instead, exponential wings develop. For the run at Mach 2, the maximum density is D48o . 0

(see Ðgure legends) the density proÐle (Fig. 2) using log scaling, the longitudinal component of the velocity 10 3), and the y-component of the magnetic Ðeld (Fig. 4) (Fig. at various times in the course of the simulations. We show data for (1) the nongravitating run c53 already discussed in the previous section, (2) for the static run c41, and (3) for a Mach number M \ 2 (run c43), where Table 1 can be consulted for further details on the runs. In each Ðgure, the seven curves correspond to seven di†erent times, each labeled by its order in the series, and each at regular intervals of *T \ 0.5q ; time t \ 0 is at the bottom, and the Ðnal time t \ 3q ffis at the top. Furthermore, for clarity, ff

each proÐle at a given time is o†set in the vertical axis by a Ðxed positive amount, namely, do \ 2 in the density (Fig. 2), dv \ 1 for the longitudinal velocity (Fig. 3), and, Ðnally, db \ 3 for the transverse magnetic Ðeld Ñuctuations (Fig. 4), all in code units (see ° 2.2 for the relation to the physical units of the typical molecular cloud). Whereas in a turbulent medium without self-gravity, Ñuctuations develop in all Ðelds and at all scales (Figs. 3a, 4a, 5a), the static run with self-gravity is strongly ordered with the density Ðeld dominated by a central condensation (Fig. 3b) (the e†ect of an increasing Mach number is discussed in the next section). Note also in Figure 4b that at t/q \ 1.5 ff

No. 1, 2001


13

457

13

66666666666666666666666666666 666 666666666666666666666 6666666666666666666666666666666666666666666666666 666666 666 666666666666 66666666666666666666666666 666666666666666666666666666666666 666666666666666666666666666666 6666666666666666666 6666666666666666666666666

12

6666 66666666666666666666666666666666666666666666666666666 66666666666 6666666666666666666666666666666666666 6666666666 6666666666666666666 66666666666666666666666666666666666666 666666666 666666666666666 666666666666666666666666666666666666666666666666 666666666666

12

11

11 5 55555 55555555555555555555555 555555555555555555 5555 5555 555555555555555555555555555555555555555555555555555555555 555555 5555555555555555555 5555555 55 55555555555 55555555555555555555555555555555555555555 55555555555555555555555 555555555555555 55555555555555555 555

10

55555555555 55555555555 555555555555 55555 55555555555 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 5555555555555555555555555555 5555555555555 555555555 55 5555 55555

10

555 55555555 555555555 555555555555 55555555555 5555555555

9 9 444 444 44444444444 44 4 44 44 4 444444444444444444444444444444444 4444444444444444444 444444444444 4444444444 44444444444444444 444 4444444 44444444444 44444 44 444444444444444444444444444444444444444444444444444444 44 444444444444444444444444444 444444444444444 444444444444 44

8

444444444 444444444444 44444444444 444444444

444 4444444444444 44444444444 44 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 44444 444444444444444 44 4444444444 444444444444 4444444444444 4444444444444 444444444444

8 7 7 Y

3 33333333333 333333333333 3333333333333 3333333333333 333333333333333 3333333333333333333 3333333333333333333333333333 333333333333333333333333333333333

Y

3333 33333333333333 3333333 33333333 33 3333333333333 3 33333333333333333333333333333333333333333333333333 3 333333 33333333333333 33333333333333 3333333 33 33333 33333333333333333333333333333333333333333333333333333333333333333 33333333333333333333 33333 3333333333333333 33

6

6

33333 3333333333333333333333333 3333333333333333333 333333333333333 3333333333333 3333333333333 33333333333 3333333333

5 5

2 222 22 22222222222222222222222222 22 22 22 222222222 222222222222222222222 2222222 2 222 2 222222222222222 222222222 2222 2222222222 22222222222222222222222222222222222222222222222222222222 22222222222222222222222 2 22 2 22 2 222222 2222222222222222222222222222222222222222 2 22 222

4

2222222222222222222222222222222222222222222 222222222222222222222222222222222222222222 22222222222222222 22222222222222222222222222 2 22222222222222222 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222

4 3 11 1111 11 111 1111 111 11111 1 111111111111111111111 111111111111111111 11111111 11111111111111111 11 11 11 11111 111111111 111111111 1 111111 11111111 1111111 111111111111111 1111111111111111 11 111 1111 11111111111 11111111 111 111111111111111111111111 1 11 11 11 11111 1 111 11 1111 1 1111 1111

2

3

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

2

1

0

0000 00 00 0 0 0 0 0 0 000000000000000000000000000 00 0000000000000000000000000000000000000 0000000000000000000000000000000000 00000000000 0 00 00000000000000000000 000000000000000 0000 0000000000000000000000000000000000000000000 0000000000000000000000000 00 00000000000 00000000

1

0

-1 -.5

-.4

-.3

-.2

-.1

0

.1

.2

.3

.4

.5

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

-.5

-.4

-.3

-.2

-.1

X

0

.1

.2

.3

.4

.5

X

13

666 66 666666666666666 66666666 6 6666666666666666666666666666666666 666666666 66666 66 666666666 666666666 66666666666666666666666666666666666666666666666666666666666666 66666666666666666666666 6666666666666666 6666 6666666666 666666666666666666666 666 6666666666666 66666 66

12

11

55 55 555 55555555555555555555555555555555555555 55 55 5 5555555 55555 5555 55555555 555 555555555555555555555555 55 5555555 55555555555 55555 55555555555555555555555 5555555555555555555555 55555555555555555555555555555555 5 555 555555555555555555555555 5555555555555555555555 555

10

9

8

44 44444444444444 4 44 44 44444444444 4444444444444444444444444 44444444 44444444444 44444444 4 444444444 444 4 44 4 44444444444 444 444444444444444444444444444444 444 44 4444444444444444 4444444444 4 44 44444444444444444444444444444444444444444444444444444444444 44444444444444 44 44

7

Y

6

333333 333 3 3 3333333 3333 333333 333333 333333333333333333 3333333333 33333333333333333333333333333 33333333 3333 33 33333 3333 3333 33 33333 333 3 3333333333333 333333333333333333333 33333333 333 3 3333333333 3 333 33333333333 33 33 33 33 333 3 3333 33 3333 3333333 33 33333333333 3333 33333333 33

5

4

3

2

22 22 22222 2222222222 2222 2222222222 2222 222222 222222 22 222 2222 222 2222222222222222 22 2222 2222222222 222 2222 22 22 2222222222 2 2222222 222 222 2 22222 222222 222222222222222222222 22 2222 2 22222222 222222222222 22222222

2222222222222 2 22222222222 222222222 22 2222 2 2 222222 222222222 222

1 1111 1 1111 111111111111111 1 11111 111 1 1111111 1 11111111 111 1111111111 11 11 1111111 11 1111111 11 111 1 11111111111 1111 11111 1111 11 11 11 11 11 11 11 1111111111111 1 1 11 11 111111 111111111111 111111111 11 1111111 1111111 11111 1 11111111111111111 11 11 1 111 11 1 11 1 11 1111 111111 111111 1 111111 11 11

1

0

0000 00 00 0 0 0 0 0 00 000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0 0000000000000000000 00000000000000000000 00000000000000000 0000 00000000000000000000000 00000000 00

-1 -.5

-.4

-.3

-.2

-.1

0

.1

.2

.3

.4

.5

X

FIG. 3.ÈLongitudinal velocity proÐles at the same times and for the same runs as in Fig. 2. The o†set for each time is of 2 units on the y-axis. Note the development of a strong accretion shock at intermediate times, which later on subsides.

(curves labeled by ““ 3 ÏÏ), the magnetic Ðeld has developed two sharp gradients, and reconnection may take place at these locations when transverse perturbations are allowed. We have run as well a nonmagnetic static run. In this case (run c52), the temporal evolution is slightly di†erent, but the end condensation is similar to that in the magnetic case. Noticeable, though, are the following di†erences : (1) whereas a transverse velocity Ðeld develops in the B D 0 case because the Lorentz force acts as a solenoidal source, here v remains zero (recall that, in the slab approximation, M no transverse pressure gradient) ; (2) the longitudinal there is velocity is smooth for the nonmagnetic case, whereas it is turbulent in the magnetic run because of its interaction with

the transverse component of the velocity (in fact, u D b ) ; S q ,turb (3) the nonmagnetic run condenses faster : at t \ the peak density is D2 for B D 0 and D12.6 for B 4 0 ;ffsimilarly, at that same time, the compressive component of the velocity is four times larger when B 4 0. 3.3. V ariation with Mach Number Because the interstellar medium is observed to be in a turbulent state, it is likely that ab initio condensations will occur in a medium with sizable velocities. In fact, strong condensations can develop in the presence of heating and cooling because the medium behaves as a highly compressible Ñuid (Va`zquez-Semadeni et al. 1996). To investigate the

458


26

28 66 666 66 6 666666666666 66666 6666 66666666666666666666666 666 666666666666666666 666 666666 6666 66 6666666 66 6666666 666 6666 6666 666666666 66 6666 66666 6 6666 666666666666666 66 6666 666666666666666666 6666666 6 66 66 66666666666 666 666666666666666666666 666666 66 66666666 666666666 66666

24

666666666666666666666 6666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666 666666666666 66666666666666666666666666666666666666666666666666666666666666666666666666 666666666666666666 666666666666666666666666666666666666 6

26

22

24 55 555 55 55 55 55 5 55 5 55 5 555 55555555555555555 55555 555555 5 5555555 5 5 555 55 55555 5 5 55555 555555555555 5555 55 55 555555555555555555555555555555555555555 55 55555555555555555 55555 5555555555555555 5555 55555555555 5555555555555555555555555555 5555555555555 5 5 55 5 55 555 5 55 5 55555 5 555

20

55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 5555 555555555555555555555555 5555 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 5555 55 55 555

22

18

20 444 44 4444 44 444 4444 444444 4 4 44 444 444444 44444444444 4 444 4 4 4 44 4444 4 44444444444444444444444444 4 4 44 44 44 444 4 444 4 44444 44 444 44444 444444444444 444444 4 44444444444444444444444 444 44444444444444444444444444444444444444 444 4 44444 44 4 44444444 444444444 444444444444444 4 4 444 4444 4

16

4444444444444 444444444444444444 44 4444 444 4444444444 4444 4 44 44444444444 444444 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444

18

14

16 33 3 33 3 33333333 3 33 33333333 33 3333333333333 3333 3333 333333 3 33 3 3333333 3 33 33 33333 3 333 333333333 3 33 33 3 333 3 33 33 33 333 333333333333333 333333 333 3 33333333333333 333333333333333333333333333333333333 333333 33 33 33 3 33333 333 33 33 3 33 3 3 33 3 33 3 3333 3333 333 3 3333 3 3333 3333 3333 33333 3333

12

10

2 2

Y

Y

Vol. 557

22 2 222222222222 2 22222222 2 22 22 22222222 2 22 222 22222 2 22 22 222222222 22 2 2 222 222222 222 2 22 22 22 22 222222222222222 22 22 22 2 2 22 2222222222222222 2 222 2222222 22222222 22222 222 2 00 2222222222222 2222 22 2 0 22 22 222 2 2 222 2222 2 0 2 22 2 0 2 2 22 22222222 11 11 0 1 1111 111111 1111 0

2 2 2 2222222222222 2 2 222222222222 2 2 22222 2 2 222 222

8

6 11 11 111

4

2

0

-2 -.5

-.3

-.2

-.1

0

.1

.2

.3

.4

333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

12

33333 3333 33 3333 33 33 33 33 333 3333333

10

2222222222222222222222222222222222222222222 2222222222222222222222222 222222222222222222222222222 22222222222222222222222222222222222222222 22222222222222222222222222 2222222222222222222222222222 222222222222222222222222222222222222222222222222222222222222222222

8

1 11 0 111111 1 1111 111111 11 11 1 111 11 1111 1 1111 1 111111111111 11 1 1111111 111 0 11 1111 1 11 1111 11 1 1 11111111 11111 111 1111 11 11111111111111 111111111111 1 111 1 1 1 1 11111111111111111 1 1 11 11 11 111 11 11 111 11 1 11 1 11 1 11111 11111 1111 1 111111 11 1111 1 111 11 1 111 1 0 1 1 1 1 0 1 11111 0 0 0000 00 00 0000 00 0 000 000 00 0 000 00 00 000000000 00 000 0 0000 0 0 0 00 000 00 0 00 000 00 0 0000000 00 00 00 0 0 0 000 0 00 0 0 00 000 00 0 0 0 0 0 0000000000000 000 00 000 000000 0 0 00000 00 0 0 00000 0 0 00000000000000000000000 0000 0 0000 0 00 00 00 00 00 0 00000 0 0 0000 0 0000 00000 0 0 0 0 00000 0 0 0 0 000 0 0 0 00 00000 00

-.4

3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

14

6

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

4

2

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

0

.5

-.5

-.4

-.3

-.2

-.1

X

0

.1

.2

.3

.4

.5

X

26 66 666 66666666666666 6666 6666 6 666666 666 66 666666666666666666666 6666666 666 666666666 66666666 6 6666666666666666666666666666666666 666666666666666666666666666666666666666666666666666666666666 66666 66666 666666666 666 666 6 6666 6 6666666666666 6 6666666666666 6 6 6666 66 66 6 66 6 66

24

22 55 5555 5555555555555555 55555 55 5 55 5555 555555555555555 55 5 5 5 555555555555555 55555555 555 5555 5555 5555555 5555 5 555 555555 55555555555555555 555555 5 55 55555 5555555 55 55555555555555555 55 555555 55 555 555 5 55 55 5 5 5 555 55 5 5 5555 55555555555555555555555555555 5 5 555 5 5 555555 555 555555 55

20

18

4 4 4 4 4 4 4 4444 4444 4 44444 44 444 4 44 44444444444444444444444444 444 444 44 44 4 444 4 444 44 44 44444 44 44 444 444444444444 44 44 44 44 44 4 44 44 4 44 44 44 44 44 444444 4 4444444444 444 44444444444444 44444444444444444444444444444444 44 4444 444444444 44444444444444444444444444 444 4444 44444444444

4 444

16

Y

14 33 3 33 3333 333333 3 3 33 333333333333333 3 333333333 3333 333 33 3333333333333 33 33 33 3 3 3333 33333333 33 333333 33 3 3333 33 3333333333333333333 33333333333333333333333333 3 333333333333333 3 3 3 3 33333 3333333333 33333333 33 33 3 3333 3333333333 33333333 3 33 3333 33333333333 3333 333 33 3333 3 33333 3

12

10 2 22 2 2 22 22222 222 2 222 2 22222 2222 2222 22 2222 2 2 222 2 2 2222222222222 2 2222222 2 2 2 22 2222222 22222222222 22222 22 2 2 2 2222222 2222222222 2 22 222 222222 22 222 22222 22 2 2 22 22222 222 2222222222222222222222 2222222 22222222222 222222222222222222222 2 22 2 222222222222222200 2 22 0 222 11 2 22 111 222 2 2 0 2 1 1 1 1 22 222 0 2 2 1 11 1 0 1 1 1 111 0 1

8

6

0 1 1111111111111111 1111111111111111111111111111 1111 111 11111 111111111111111111111111111 11111111111111 111111111110111111 111 1111 111111111111111 1111111111111111 111111 11 1 1111 1111 1111111111 11 11111 11 1111 111 11 111 1 111 1 1 11 11 1 111111111 0 0 1 1

4

2

1

0 0

0

1 1 11 0 000 000 00000 00 000 0 1 0000 0000 0000000 0 00 0 000 000 00 0 0 0 0000000000000 000 000000 00 0 0 0 0 00 000 0 0 00 00 00 00 00 00 1 0000 0 0 0 00000 0 0 00 0 0 00 00 0000

0000 00 00 0000 00 00 0 00 00 00000 0 0 0 0 00 00 0000000 00 000 0 0 0 00 000 0 0 00 000 0 00 00000 0 0 0000 00000000000000000000000 0 00 00 00 0 0000 0 00000 0 0 0 000

-2 -.5

-.4

-.3

-.2

-.1

11 1 11 1

0

.1

.2

.3

.4

.5

X

FIG. 4.ÈProÐles of the transverse (b ) Ñuctuating component of the magnetic Ðeld at the same times and for the same runs as in Figs. 2 and 3. The o†set y that in the presence of self-gravity, the magnetic Ðeld is quasi-uniform except within the collapsing condensation. for each time is 4 units on the y-axis. Note

e†ect on the collapse of such an initial Ñow, we perform now a parametric study in terms of the Mach number of the initial turbulence. For the computations described in this section, the randomness of the initial Ñow velocities and magnetic Ðeld is constrained to reside in the vicinity of the density perturbation (locally random, or LR) ; for the case of a fully randomly phased (FRP) Ñow with initial perturbations throughout the cloud in space, see °° 3.4 and 3.5. Comparing results from the static run c41 and runs at low Mach number (c48 at Mach 0.1, and c50 at Mach 0.35), several di†erences arise in the evolution of these Ñows : 1. Transverse velocities grow faster and are more turbulent for the Mach D 0 runs. This is caused by the presence

in the static run of a uniform y-component of the magnetic Ðeld with B /(4no )1@2 \ 2.5c , leading to the development 0,y Ñuctuating 0 s of comparable transverse velocity and magnetic Ðeld on the longest length scales in the simulations. On the other hand, in the M D 0 cases, the initially impressed Ñuctuations are on smaller scales leading to more rapid growth. 2. The peak of the density pulse is stronger by a factor three in the M D 0 runs, and slightly o†-centered (by roughly a quarter wavelength in units of k ) from the initial 0 carefully the peak in the density Ñuctuation. Examining velocity Ðeld in the vicinity of the peak of density, we observe that it forms where the longitudinal velocity gradient is negative and strongÈthus pushing matter togetherÈa location that is not governed so much by the

No. 1, 2001


initial perturbation in density than by the phase of the random velocity Ðeld, a remark that in fact holds for all runs of this series. 3. The peak of the density spectrum in run c48 is roughly a 100 times larger than in the static case at t/q \ 1.5, and ff so is the peak in the velocity spectrum, an e†ect mostly caused by initial conditions. But even with a faster development of the density clump for the M D 0 cloud, all Ñows evolve to the same Ðnal condensation, with equivalent density and velocity proÐles (and spectra), except in the wings far away from the centrally condensed peak. 4. Whereas for early times (up to 0.2q ), the density specff trum builds up at the scale of the turbulence in the M D 0 runs, by t \ q the peak has markedly shifted over to k \ ff k D 2.5k . A similar observation can be made for the J min velocity spectra. As we increase the Mach number of the initial velocity Ðeld, increasing as well the Alfvenic Mach number (see Table 1), a change in behavior is observed. Above a critical rms Mach number slightly above 0.35 (see also Passot & Pouquet [1987] for a similar transition in the nongravitational case, with transition linked to the development of shocks), the condensation of density will take place where the strongest velocity gradients occur as they develop self-consistently in the Ñow, as opposed to developing in the immediate vicinity of the initial bulge of density. Obviously, for a strong subsonic turbulence (and for supersonic Ñows as well), the Ðnal condensations occur randomly. Furthermore, at the location of clumps, gravity washes out the turbulence as already noticed in Alecian & Leorat (1988) in the neutral case. However, far from the central clump, substantial but still subsonic velocities obtain (recall that, in code units as used on the axis of the plots, c \ 0.2), but this s could be, in part, because of the periodic boundary conditions leading to the collision of two shocks traveling in opposite directions. At higher Mach numbers, the peak in density is yet at another location (Figs. 2cÈ4c) ; it is also less intense and with broader wings. The velocities are more random and decorrelated from the density condensation as opposed to a core with steep boundaries. One key result of this study is that the higher the Mach number, the faster the formation of density clumps. This is linked to the fact that the nonlinear time, in terms of the free-fall time, is shorter the higher the Mach number. All things being equal, q is inversely proportional to the NL that, for all times, at high Mach velocity. We also observe number, the transverse Ðelds (velocity and magnetic Ñuctuations) remain approximately constant in the core, whereas the longitudinal velocity that is directly inÑuenced by the infalling matter changes drastically. Indeed, it is straightforward to observe based on the basic equations that both transverse Ðelds (normalizing the magnetic Ðeld to an Alfven velocity) evolve in identical ways provided $ Æ u D 0, a circumstance fulÐlled in the collapsing core. Finally, we note that changing the initial peak wavenumber of the turbulent Ñow (speciÐcally, from k /k \ 10 to 0 min longituk /k \ 50) or changing the initial ratio between 0 min dinal and transverse velocities (speciÐcally, from s \ 0.068 to s \ 0.25 ; see Table 1) does not alter in either case the conclusions reached before and, as such, appears irrelevant as to the Ðnal outcome of the collapse except that the Ðrst cusp in the density spectra at k/k D 16 is more marked min

459

than in the nongravitating case. We checked that it is not grid-dependent : a run identical to c45 but on a smaller grid of 1,024 points places the cusp at the same wavenumber. We Ðnally deÐne R(V) \ [j(V) /j(V)]2 ; (3) ~1 2 R(V) is the square of the ratio of the integral scale j(V) for the ~1 total velocity Ðeld at any given time to the Taylor scale j(V) 2 for that Ñow. Such scales are based on the computation of moments of the velocity spectrum EV(k), which are deÐned for any n as : j(V) \ n

CP

knEV(k)dk

NP

D

EV(k)dk

~(1@n)

.

(4)

Similar deÐnitions can be written for all other spectra, such as that of the magnetic Ðeld. The integral scale (n \ [1) emphasizes the large scales of the Ñow, whereas the Taylor scale (n \ 2) is characteristic of velocity gradients, both shocks and vortices. The parameter R(V) deÐned above is thus, in some sense, a measure of the extent of the inertial range actually achieved in the computation, i.e., of the ratio between energy-containing eddies at scale D j and ~1 As larger, and dissipative eddies at scale Dj and smaller. 2 not measure, already mentioned in ° 1, this ratio does however, the rate at which the large scales decay because of dissipative (viscous and ohmic) processes since Euler codes as the one used in this paper are quasi-inviscid at such large scales ; furthermore, for most of the computations of this paper, self-gravity injects energy in the system. But another important feature of turbulent Ñows is the amount of selfsimilarity that is achieved in the cascade process ; it can be measured by the above ratio R(V). We plot in Figure 5 the temporal evolution of R(V) for runs c43 at a Mach number M \ 2 (Fig. 5a) and run c48 at M \ 0.1 (Fig. 5b) ; the time in the abscissa is in units of 0.5q . After an initial peak correff sponding to accelerated velocities at the infall of matter, these ratios settle to values in the range 50 to 90. However, as mentioned before, one can note again the faster development of small scales for the run with a higher Mach number, together, however, with a less developed range of scales both at peak time and at later times. 3.4. L ocalized versus Global Initial Randomness We discuss here the e†ect of changing the character of the randomness of the initial Ñow. For that purpose, several runs have been performed which are characterized succinctly in Table 2. Here, the scale of the cloud is L \ 2 pc, and the random perturbations in the phases of the0 velocity and magnetic Ñuctuations are maximal, with the perturbations scattered throughout the cloud as opposed to the previous sections where the velocity and magnetic Ðeld perturbations were concentrated on the initial density bulge at the center of the cloud of otherwise uniform density. The main conclusions of the previous section persist, but we note the following changes : 1. There is a substantial increase of a factor 10 in the density contrast *o \ o /o when comparing to the localized random runs, a max resultmin holding throughout the range of Mach numbers tested here. 2. In the FRP case, at all Mach numbers except for the most supersonic Ñow, the peaks are narrower, and they are centered on the initial bulge of density. In the case of locally

460


200

Vol. 557

240 0

0

220

0

180

200

160

0

180

0 0

140

0

0

160 0

0

120 140

100

Y

Y

0

0

120

0 0 0

100

0

80

0 0

0

60

0

0

0

0

0

80

0

0 0

0

0

40

0

0

0

0

0 0

0

0 0

0

40

0

0

0 0

0

20

0

20

0

0

0

0

0

0

0

0

.5

1.0

1.5

2.0

0

0

0

0

0

0

60

0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0

0

6.5

0

0

.5

1.0

1.5

0

0

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

X

X

FIG. 5.ÈAmount of small-scale production as a function of time using as an indicator the extent of the inertial range given by the function R(V) (see text, eq. [3] for deÐnition) ; the time unit is 1.1 ] 106 yr or 0.5 q ; (left) run c43 with an initial Mach number M \ 2 and (right) run c48 with an initial Mach ff number M \ 0.1. Note again the faster yet less productive development of small scales at higher Mach numbers and the strong Ñuctuations at later times.

density spectrum for the ISM that is probably random throughout the cloud. The transverse magnetic Ðeld is also stronger than at lower Mach numbers and, on occasion, shows very strong current structures. 6. At Mach 2, the density peak becomes more roundish, i.e., wider density condensations form (instead of narrow small cores). The transverse magnetic Ðeld is now stronger 14 66666

13

12

11

10

9

8

Y

random (LR) initial conditions, as soon as the Mach number is sufficiently high, it is the shocks that drive the condensations. 3. In the static case, the narrow density peak seems to rebound with a temporal variation in its width. The magnetic Ðeld develops a singular structure in the core, possibly in the form of a cusp. Note that in the extension of the Burgers equation to MHD (Thomas 1968), it has been shown that cusps can develop in the magnetic Ðeld (Passot 1987 ; Galtier & Fournier 2001). Here, the longitudinal velocity is driven by pressure gradients, whereas the transverse velocity remains close to zero since it does not feel a Lorentz force. On the other hand, examining in detail the one-dimensional MHD equations, one observes that because a uniform Ðeld is present in this computation in the transverse directions, this uniform Ðeld acts as a source of transverse magnetic Ðeld in regions where the divergence of the velocity is signiÐcant. This e†ect depends linearly on the amplitude of the uniform transverse component and also will be smeared out as soon as transverse Ñuctuations are present, Ñuctuations which develop through the Lorentz force. This e†ect is particularly prominent in the onedimensional geometry treated here since in this case there are no pressure gradients to drive the transverse velocity Ðeld components. 4. At a low Mach number, one strong density peak forms, with exponentially decreasing wings in the core envelope. Both transverse Ðelds v and b are negligible, M longitudinal M whereas a strong shock forms in the velocity and is driven by pressure gradients. 5. As the Mach number is increased, transverse velocities develop and smaller density clumps form in the wings of the main core, as can be seen in Figure 6 showing the density proÐle for run c85 using log scaling. This development of nested self-similar structures10is reminiscent of observations of molecular clouds, and is thus indicative of an initial

7

6

5

4

3

2

6 6 6 6 6 6 6666 6 6 66 666 66666666 666 6 6 6 66 6 66 6 6666 6 66 666 66666 66 6 66 66 66 66 66 666 6 66666 666 66 6 66 666 66 66 6 6666 55 555 5 666666 66 6666 666 66 666 66 666666 5 666666 6 6 6 66666 5 6666666 6 5 6 5 66 66 66666666 66 666 66666666 6 666 666 6 6 66 66 5 66666 555 5 666 5 5 666666666 66666666 55 555 666666666666666666 5555555 66 6 666 5 5 5 5 66666666 666666 6 5 5 666 6 55 5555 55 555 5 55555555555 55 55 55 5 5 55 55 555555 555 55555 555555 5 5 5 555 5 55555555 44444 55 5 55 5 4 5 55555 5 5 55 5 55 55 5 55 5555 55 44 4 55 4 5 55 5 55 4 555555 555 555 555555 555 4 4 4444 55 5555555555 444 4444 444444 44 4 4 44 4 44 55 55555555 4 5 5 55 55 55 55 5 555555 55 4444 55 444 44444 55555555555555555555555 5555555 4444 4 5555555555555555555 44 4444444 4444 44444 4444444 44444444444 44 4 44 44 4444444 33333 4 4 3 44 44 44 444 44 4 4 44 3 4 44 3 4444 4 4 4 4444 44444 3 44444444 4 4 44444444 44444444 44444 44 44 4 4 444 44 4 4 44 3 44 44 44444 4 4 4 33 44444 444444 4444444 44444 3 444444 44 44444 3 333 3333 4 4 4 3333333 33 3 44444 4444444 3 4444 3 3333 333 3 333 333 3333 333 33333 33 3 33 3 3 3333333 333 333333333 333 3 3333333 3 3 333 33 3 2222222 3 3 3 2 3 3 333 3 33 22 2 33 33 3 3 2 3 3 333 3 3 3 33 2 33 3 3 2 33 3 3 333 3 3 333 33 22 333 3 2 333 2 2 2 222 3 333 3 3 3 33 3 33 22222 3 3 222 222 22 33 2222222 33 3 33 3333 3 33 333 3 3 3 3 3 33 3 333 3 2 333 3 3 3333 3333 33 3333 33 33 3 22222 3 3 33 333 3 222 2222222 3 333 3 222222222222222 2 22 22 22 22 22 22 222 22 22 22 22 22 22 22 2222 2222 2222233 222222222222 33333 3 2 2222 2222222222222222222222222222222222 22222222222 22 22 22 3322 22 3 22 22222222222 222 333 3 22222222222222222 333 33 2222 222222222222222222222 222222222 111 1 11 1111 1 111 1 1111 111 11 1 111111 11 1 11111 111111 11111111111 11111 11 1 1 111111 111 11 11 111111 1111 1 11111111 111 11111 1111 1 1111111 1 1111 1 11 1111 111 11 1111 11 1111111111 111111 1 11 1 11111 1111 11 1111 11 111 11111 1 1111 111 1 11 1 1111 111 111 11 11 11 1111 1 1 1111 1 11 1 11 1 11 11 11 11111 11 11111 11111111

1

0 -1.0

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

-.8

-.6

-.4

-.2

0

.2

.4

.6

.8

1.0

X

FIG. 6.ÈProÐles of the log of the density every 0.5 q for run c85 (see 10 ff Table 2) with fully random initial conditions. The labeling of curves is identical to that of Fig. 1. Contrasted with the case of locally random (LR) initial conditions, note the appearance in this case of nested substructures in the low-density wings of the central condensation.

No. 1, 2001


and concentrated mostly within the clumps. The run c87, with M \ 2 and initial turbulent Ñuctuations peaked at k /k \ 50, evolves similarly : gravity is strong enough to 0 min erase the initial scale of the turbulence. 7. In the nongravitational case (run c81), a striking result is the growth of the characteristic scale of the magnetic Ñuctuations. This is probably linked to the invariance of the magnetic helicity Sa Æ BT, where B \ + ] a, a result that holds in the compressible case as well. This has important consequences for the dynamo problem, but this matter goes beyond the present analysis (see Balsara & Pouquet 1999). In the presence of self-gravity, large-scale enhancement comes also from the formation of density clumps. To disentangle the two e†ects would necessitate putting the Jeans length at a substantially smaller scale, a problem outside the scope of the present analysis. 3.5. T he Role of Ambipolar Di†usion We now discuss the e†ect of including the ambipolar term in the strong coupling approximation when the collision time between neutrals and ions is short and a balance is reached between the Lorentz force and the drift term in the ion momentum equation ; the induction equation now reads :

G

H

1 LB [(J ] B) ] B] , (5) \ + ] (u ] B) ] A + ] a oa Lt where a reÑects the degree of ionization of the medium (see below), and A is a coefficient that depends on the physical a cloud ; J \ + ] B is the current density. properties of the As discussed in Balsara (1996), and following the approach in McKee (1989), two main regimes arise according to what is the main mechanism of ionization in the cloud, leading to di†erent density dependencies of the ionization fractions m \ o /o , where o and o are the ion and i n n of ionization neutral density, respectively. In ithe case induced by cosmic rays (CR), this leads to a \ 3/2, whereas for far-UV (FUV) ionization, a \ 1. The a-dependent part of the ambipolar term can in fact be written as : A 1 a\ , oa 4nc o o D n i where c \ 3.5 ] 1013 g~1 s~1 is the drag coefficient (Draine, DRoberge, & Dalgarno 1983) ; here o \ 2.3m n p n with m the mass of the proton and n D 0.6n nthe number p n H density of neutral particles in terms of the number density of hydrogen atoms, taking it to be 9 times that of helium. For cosmic ray ionization, we take A \ 1.208 ] 10~2 3@2 A \ 3 (where in code units, with a choice of G \ 10 and 0 V G is the ratio of mean FUV intensity at the cloud bound0 ary to that in the solar vicinity, and A is the visual extinction measured from the location of Vthe core out to the surface of the cloud), whereas for FUV ionization, a value of G \ 1 is chosen, leading to A \ 5.05 ] 10~3. The values for0 A given above lead to a 1sti† problem. The solution a here is to subcycle on the fast-evolving induction adopted equation with a Ðrst-order scheme, sufficient since this local time step is a ratio of typically 1 : 100 (and at times, of up to 1 : 1000) compared to the time step used for the remaining Ñuid equations. This is also the method of choice suggested by Mac Low et al. (1995). Finally, we note that the choice of G is made such that for both the CR runs and the FUV 0

461

runs, the dissipation length scale is identical, corresponding roughly to the small scales of the computation. Denoting the ambipolar damping length scale for small amplitude waves as l8 \ v /c o , we Ðnd l8 D 0.07 pc for the choice of A D i,0 coefficients taken above for the CR case, and l8 D 0.03 pc for the FUV case. For an Alfven velocity of 1 km s~1, this gives an ambipolar time at that scale of roughly 3 ] 105 yr for the CR ionization and twice that for FUV. The characteristic features of the runs now described are summarized in Table 3, with deÐnitions similar to that of Table 1. The cloud is L \ 2 pc wide and initial conditions are fully random, as in 0 the series of runs described in ° 3.4. For the three Mach numbers that have been run, namely, 0.3, 1, and 2, no signiÐcant di†erences in the results between CR and FUV ionization were observed. Figure 7 gives the time evolution of the proÐles for (1) density and (2) the transverse velocity for run c71 corresponding to a Mach number of 0.3. There are slightly stronger Ñuctuations in b within the central density clump in comparison with they identical run c43 without ambipolar drift, which is displayed in Figure 4c. Note also that, at the last time of the computation, the transverse Ñuctuations are smaller (by a factor of D2 at a Mach number of 0.3) for the CR case than they are for run c84 of Table 2, which is the corresponding run without ambipolar di†usion. For earlier times, the differences are not signiÐcant. This is so because ambipolar di†usion operates over very long times and so the di†erences show up only at later times in the simulation. As the Mach number is increased, the di†erences in the runs with ambipolar di†usion and runs without ambipolar di†usion disappears. This is seen by comparing runs c72, c75, and c85, all of which have a Mach number of 1 or by comparing runs c73, c76, and c86, all of which have a Mach number of 2. This is explained by the fact that the stronger nonlinearities that develop on all scales within one eddy turnover time in the higher Mach number simulations dominate the ambipolar di†usion e†ects. The eddy turnover time itself decreases as the Mach number is increased while the ambipolar di†usion time stays Ðxed. This further helps the nonlinear terms in the ideal MHD equations to become more important than the ambipolar di†usion terms. We further note that our assessment of the ambipolar di†usion time is based on linear stability analysis calculations that do not apply to Ñows where the turbulence has become strongly nonlinear. Furthermore, Brandenburg, & Zweibel (1994) have shown that the ambipolar di†usion terms can themselves generate narrow structures when treated as fully nonlinear terms, i.e., ambipolar di†usion terms in the MHD equations may not act so much like di†usion terms as the linear stability analysis would lead us to expect. Insofar as the inÑuence on turbulent gravitational collapse in a molecular cloud is concerned, the following points stand out : 1. The density contrast as measured by *o is somewhat larger at a given Mach number in the presence of the ambipolar drift term and, as noted in the previous section, it is substantially larger the lower the Mach number (see Table 4). 2. The transverse component of the magnetic Ðeld is strong within the density clumps, both with and without AD. Indeed, at high densities, the AD term becomes weak, although to what extent this e†ect will persist for higher space dimensionality remains to be seen.

462


14

13 66 6 6

12

11

10

9

8

7

6

5

4

6 6 6 66 66 66 66 6 66 66 66 66666 6 66 66 6666666666 66 66 6666 6 6 6666 66666 666 6 66 6 666666 6666 6 5 5 66 5 666666 6666 6666 66666 66 5 66 666666 666 66666666 5 66666666 66 66 6 5 55 666666666 666 6 6 5 6 66 55 6 55 66 66 6 55 66666666 5 6 66 6 55 66 555 55 555555555 6 5 55555555555 5 55 66666666 55 55 6 6 66666666 6 55 6 66 55 5555 66666 55 55 6666 5 555 5 666 55 555 66666 66666 5 6 44 55 4 555 66 66666666 55 555555 666 6666 555555 5555555 666666 55 66 66 55 555 555 66666666 6 55555 4 4 5555 55 55 55 55 5 55 55 6 5 66 555555 5 5 55 4 4 5 5 55 4 4 5 55 5 44 5555 5 5555555555555 44 44 5 555555 4 44 4 55555 55555 55 44 55 4444 44 5 4 44 44 55 5 4444444444 44 44 555 4444 5 5555555 555555555555 4 4 5555555555 4 44 5555 55 555555555555 555 44444 4 444 5 555 5555 444 4444 444444 444444 333 444444 4444444 44 44 44 44 444 444444 3 3 444444 4 444444 44 44 44 4444444 4 3 3 44444444 44 4 3 4 3 444 4 33 4444444 4 33 33 3 444 44444 44 33 333 4444 44 444 33 333333333 44 33333333333 33 4 4 333 3 4444 4 3 33 4 444444444444 3 3 3 44 4 3 3 33 3 3333 33 444 33 333 33 3 33 3 333333 3333 222 3333333 333333 33 2 33 33 4444444444444444 333 44444444444444444444 333333 444444 2 444 2 2 2 22 22 222 22 222 33333333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333 22 2 2 22 2222 22 222222 22222 2 2222222222222222222222222222222222222222222222222222222222222222222222222 222222222222222222222 22222222222

11

55555 5555 5555555 555 555 5 55555555555 555

9

8

7

6

5

4

3

2

222222222222222222222222 2222222222222222222222222 22222222222222222222222222222222 2222222222222222222222222

2

11111111 111111111 111111111111111111111111 111111111111111111111111111111111 111111111111111111111 11 1111111111111111111111111 11111 11111 1111111111111111111111 111111111111111111111111111111 1111111111111111111111111 11111111111111111111111111111111111111111111 111

0

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

-1.0

-.8

-.6

-.4

-.2

0

.2

.4

.6

.8

1.0

44 4444 4444 4444 4444 4444 444444444444444444444444444444444444444444444444444444444444444444 444444444444444444444444444444444444444444444 444 44 444444444444 44444444444 4444 444444444444444444444 44 4444 4 44 44 44 44 44444 444444444444444 4444 444444 4444 333 44444 333333 33 33 3 333333 3333 3333 33333 333 333 3333 33333 3 33 3333 3333 333 33333 333 333333333333333333333333333333333333333333333333333 33333333 333333333333333333333333 3 33 33 3333 33333333333333333 3333 33333333 33333 3333 222 3333 33 222222 3333 222222 22222 33333 222222 3333 3 22 3 22 3333 2222222 222222 3333 222222 33333 3333 2222222 3333 222222 33333 222222 2 3333 2222222 33333 222222 2222222 22 22 222222 2222222 22222 22222222 22222 222222222222222222222222222222 222222 2222222 222222 2 222222 2222222 222222 22 22 22222 222222 222222 22222 222222 222222 22222 222222 222222 2 22 1111111111111111111111111111111111111111111111111111111111111111 22222 1111111111111 11111111111111111 1111111111111111111111111111111111 1111 11111 11 1111111111111111111111111111 11 11111111111111111111111 11111111111111111111111111111111111111111111111111111111111111 11 4 444444 4444 44444

1

1

0

5 55555 55 5 555 5555 55555 55555 55555 55555 55555 555 555 555555 55555555 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 5555 55555 555555

10

3

2

66 6666 6666 6 6 6666 66666 6666 66666 6 6666 6666 66 6666 666666 6666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666 6666666 6 66666666 6666 6666 66 666 66666 6666 66 666666 6666 66 66 666

12

Y

13

Y

Vol. 557

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

-1 -1.0

-.8

-.6

X

-.4

-.2

0

.2

.4

.6

.8

1.0

X

FIG. 7.ÈProÐles every 0.5 q of (left) the density and (right) the transverse velocity for run c71 (see Table 3) with the ambipolar term included in the ff the y-axis is, respectively, do \ 2 for the density and dv \ 1 for the velocity. induction equation. The o†set on y

3. The Ñuctuations in the transverse velocity have a lesser numerical amplitude than the Ñuctuations in the transverse magnetic Ðeld. However, the Ñuctuations in the transverse velocity are comparable to the Ñuctuations in the transverse Alfven velocity. Thus we see that the two types of velocity are in rough equipartition with each other, a fact that is consistent with intuition about turbulent modes as well as observations of magnetized molecular clouds. 4. The turbulent spectra, in density as well as in transverse velocity and magnetic Ñuctuations, all extend toward larger scales as the Mach number increases, the larger scales feeling progressively more the e†ect of turbulence competing with the self-gravitational structure which spans a range of scale of the order of 80 in the case computed here at M \ 0.3, of 40 at M \ 1, and half that at M \ 2. An identi-

TABLE 4 DENSITY CONTRAST OF CLOUD FOR RUNS AND THEIR GIVEN MACH NUMBERS

Run

M

c43 . . . c86 . . . c73 . . . c50 . . . c84 . . . c71 . . .

2 2 2 0.35 0.35 0.30

M A 0.8 0.8 0.8 0.35 0.35 0.30

L

0 1 2 2 1 2 2

AD

t\0

*o

no no yes no no yes

LR FRP FRP LR FRP FRP

44 530 780 1,600 31,250 37,000

Fiducial Runs F F F

NOTE.ÈDensity contrast *o \ o /o , where o (resp., o ) is the max minÐeld over the max whole cloud. min The maximum (resp., minimum) of the density nomenclature of the runs and their initial Mach and Alfvenic Mach numbers are given, together with the fact that the runs are ambipolar or not (in the latter case, with a \ 3/2 ; see Table 3). LR (resp., FRP) gives for t \ 0 an indication of the extent of the randomness of the initial velocity and magnetic Ñuctuations, with LR standing for ““ locally random ÏÏ and FRP for ““ full random phase ÏÏ (see text). L is the overall size of the molec0 runs. ular cloud, in parsec ; ““ F ÏÏ stands for Ðducial

cal evolution with Mach number obtains for the runs of Table 2, i.e., without ambipolar drift. 4.

DISCUSSION AND CONCLUSION

This paper investigates the dynamical evolution of a molecular cloud embedded in a magnetic Ðeld and pays particular attention to the e†ect of the compressibility of the Ñow, to the extent of the initial random perturbations, and to the inÑuence of ambipolar drift at a scale of D0.01 pc. This leads us to a vision of a complex Ñow with a hierarchy of structures, as exempliÐed by the extended spectra of density, velocity, and magnetic Ðeld and as observed with detailed maps of molecular clouds. The main results are brieÑy summarized below. Selfgravity can lead to the development of a turbulent Ñow in the absence of any other source (see also Gammie & Ostriker 1996). This makes it possible to explain the velocity widths observed in a very young molecular cloud such as the Thaddeus-Maddalena cloud (Maddalena & Thaddeus 1985), which seems to have few or no OB stars in it to supply the mechanical energy through winds needed to produce the velocity widths. At later stages of local contraction, the longitudinal velocity is strongly damped, as already noticed in the non-MHD case in Alecian & Leorat (1988) ; on the other hand, the transverse variables, both velocity and magnetic Ðeld, remain quasi-constant throughout the condensation phase and in quasi-equipartition. For low Mach numbers, the initial phase follows the Jeans picture of local and quasi-static condensation, whereas at higher Mach numbers, this initial phase di†ers from the initially static case on two grounds : it takes place faster and at a random location that depends on the initial randomness of the Ñow, in particular on the existence of strong shocks. Within the collapsing structure, transverse perturbations persist in all cases at levels comparable to that of the beginning phase, with v D b equilibrated as observed M M

No. 1, 2001


in molecular clouds. At later times, at a low Mach number, a core develops with steep density gradients at its rim, whereas at a high Mach number a core cannot form because random motions, shocks, and random pressure gradients all concur to prevent it, leading to the formation of less intense local density clumps with typically a width of 0.12 pc and containing of the order of 9 M . In order to _ check further whether condensations at a high Mach number can actually collapse into cores, we would have to take into account the fact that we deal with a renormalized Jeans length ; this would lead us to perform another set of computations with L > L , which is outside the scope of J 0 the present analysis. We have varied other parameters that pertain to the detailed structure of the turbulence, such as the characteristic scale of the initial turbulent Ñow and the initial ratio of compressive to shear velocities. The calculations show us that they do not a†ect the structure of the Ðnal collapsed object as strongly as the Mach number of the turbulence. This is a signiÐcant point pertaining to gravitational collapse in a turbulent medium because it indicates that the detailed structure of the turbulence does not strongly a†ect the Ðnal collapsed state. The role of ambipolar di†usion, noticeable at subsonic Mach numbers, is not so prominent in the nonlinear regime, where, in particular, it loses its purely dissipative character as the Mach number increases and produces strong gradients together with other steepening agents in the Ñow ; narrow proÐles in the presence of ambipolar drift are exempliÐed in Brandenburg & Zweibel (1994 ; see also Henon 1981, 1984). An improvement on the computations presented here would be to model a larger ratio of the Jeans length to the

463

characteristic scale of the turbulence. In particular, this would allow a check as to whether turbulent pressure gradients can stop the collapse at the Jeans length. On the other hand, a better understanding of the physics of molecular clouds would require the inclusion of a plausible model for heating and cooling in the cloud ; if such a model is in terms of power laws in the density and temperature, then, as computed in Elmegreen (1991) in terms of a linearization of the basic equations, and in Va`zquez-Semadeni et al. (1996) in the nonlinear case, the Ñow may behave as a barotropic medium with P D oceff with a locally varying c that can be eff in some cases substantially smaller than unity, at least in the context of the kiloparsec scale (Va`zquez-Semadeni et al. 1996), thus leading to a clumpy medium even in the absence of self-gravity. This would, in turn, inÑuence the role of turbulent pressure gradients to be included in a modiÐed Jeans analysis. These points clearly deserve further study. Finally, higher dimensionality will allow for other physical e†ects to come into play, such as the presence of shear layers or vortex tubes as well as magnetic structures possibly shaped by helicity. In that light, a series of threedimensional computations are being performed with resolutions up to 5123 grid points (see Balsara, Crutcher, & Pouquet 1996 and Balsara et al. 1999 for preliminary results). This work has received partial Ðnancial support from NASA NAG 5-2667, from the Programme National (CNRS) ““ Physique et Chimie du Milieu Interstellaire ÏÏ and from two grants from the Observatoire de la Coüte dÏAzur. Computing support from the Pittsburgh Supercomputing Center (PSC) is gratefully acknowledged.

REFERENCES Alecian, G., & Leorat, J. 1988, A&A, 196, 1 Leorat, J., Passot, T., & Pouquet, A. 1990, MNRAS, 243, 293 Balsara, D. 1996, ApJ, 465, 775 MacLow, M.-M. 1995, Nature, 377, 287 ÈÈÈ. 1998a, ApJS, 116, 119 Mac Low, M.-M., Klessen, R. S., Burkert, A., & Smith, M. D. 1998, Phys. ÈÈÈ. 1998b, ApJS, 116, 133 Rev. Lett., 80, 2754 Balsara, D., Crutcher, R. M., & Pouquet, A. 1996, Numerical Simulations MacLow, M.-M., Norman, M. L., Konigl, A., & Wardle, M. 1995, ApJ, of MHD Turbulence and Gravitational Collapse, Extended Abstract, 442, 726 AAS Fall Mtg., Maryland Maddalena, R., & Thaddeus, P. 1985, ApJ, 294, 231 Balsara, D., & Pouquet, A. 1999, Phys. Plasmas, 6, 89 McKee, C. F. 1989, ApJ, 345, 782 Balsara, D., Pouquet, A., Ward-Thompson, D., & Crutcher, R. 1999, in Mooney, T. J., & Solomon, P. 1988, ApJ, 334, L51 Interstellar Turbulence, ed. J. Franco & A. Carraminama (New York : Mouschovias, T. C. 1991, in The Physics of Star Formation and Early Cambridge Univ. Press), 261 Stellar Evolution, ed. C. J. Lada & N. D. KylaÐs (Dordrecht : Kluwer), Bertoldi, F., & McKee, C. 1992, ApJ, 395, 140 61 Blitz, L. 1993, in Protostars and Planets III, ed. E. H. Levy & J. I. Lunine Myers, P. C., Dame, T. M., Thaddeus, P., Cohen, R. S., Silverberg, R. F., (Tucson : Univ. of Arizona Press), 125 Dwek, E., & Hauser, M. G. 1986, ApJ, 301, 398 Bonazzola, S., Falgarone, E., Heyvaerts, J., Perault, M., & Puget, J. L. Myers, P. C., & Goodman, A. A. 1988a, ApJ, 326, L27 1987, A&A, 172, 293 ÈÈÈ. 1988b, ApJ, 329, 392 Brandenburg, A., & Zweibel, E. 1994, ApJ, 427, L91 Padoan, P., Jimenez, R., & Jone, B. 1997, MNRAS, 285, 711 Chandrasekhar, S. 1951, Proc. Roy. Soc. London A, 210, 26 Passot, T. 1987, Thesis, Univ. Paris VII, France Crutcher, R. M., Mouschovias, T. Ch., Troland, T. H., & Ciolek, G. E. Passot, T., & Pouquet, A. 1987, J. Fluid Mech., 181, 44 1994, ApJ, 427, 839 Passot, T., Va`zquez-Semadeni, E., & Pouquet, A. 1995, ApJ, 455, 536 Crutcher, R. M., Troland, T. H., Goodman, A. A., Heiles, C., Kazes, I., & Porter, D., Pouquet, A., & Woodward, P. R. 1995, in Small-scale StrucMyers, P. C. 1993, ApJ, 407, 175 tures in Three-dimensional Hydrodynamic and Magnetohydrodynamic Draine, B. T., Roberge, W. G., & Dalgarno, A. 1983, ApJ, 264, 485 Turbulence, ed. M. Meneguzzi, A. Pouquet, & P. L. Sulem (New York : Elmegreen, B. G. 1991, ApJ, 378, 139 Springer), 51 Falgarone, E., & Puget, J. L. 1986, A&A, 162, 235 Porter, D., Woodward, P., & Pouquet, A. 1998, Phys. Fluids, 10, 237 Falgarone, E., Puget, J. L., & Perault, M. 1992, A&A, 257, 715 Pouquet, A., Passot, J., & Leorat, J. 1991a, in IAU Symp. 147, FragmentaFuller, G. A., & Myers, P. C. 1992, ApJ, 384, 523 tion of Molecular Clouds and Star Formation, ed. F. Boulanger, Galtier, S., & Fournier, J. D. 2001, J. Plasma Phys, in press G. Duvert, & E. Falgarone (Dordrecht : Kluwer), 101 Gammie, C., & Ostriker, E. 1996, ApJ, 466, 814 ÈÈÈ. 1991b, in Advances in Turbulence, vol. 3, ed. A. Johansson & Heiles C., Goodman, A. A., McKee, C. F., & Zweibel, E. G. 1993, in P. Alfredsson (New York : Springer), 343 Protostars and Planets III, ed. E. H. Levy & J. I. Lunine (Tucson : Univ. Pudritz, R. E. 1990, ApJ, 350, 195 of Arizona Press), 279 Roe, P., & Balsara, D. S. 1996, SIAM J. Applied Math., 56, 57 Henon, M. 1981, Nature, 293, 33 Thomas, J. 1968, Phys. Fluids, 11, 1245 ÈÈÈ. 1984, in IAU Colloq. 75, Planetary Rings, ed. A. Brahic (Paris : Va`zquez-Semadeni, E., Passot, T., & Pouquet, A. 1996, ApJ, 473, 881 IAU & Cepadeus), 363 Ward-Thompson, D., Scott, P. F., Hills, R. E., & Andre, P. 1994, MNRAS, Jimenez, J., Wray, A., Sa†man, P. G., & Rogallo, R. 1993, J. Fluid Mech., 268, 276 255, 65 Williams, J. P., & Blitz, L. 1993, ApJ, 405, L75 Larson, R. B. 1981, MNRAS, 194, 809 Williams, J. P., de Geus, E. J., & Blitz, L. 1994, ApJ, 428, 693