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1. Mass density PDFs measured in numerical 3-D experiments of isothermal, supersonic turbulence driven by a solenoidal random force. The PDFs are shown on a lin-log scale (a) and on a log-log scale (b). The symbols show the experimental results. The full drawn curves show Log-Normals with standard deviations equal to half the rms Mach number measured in the experiments. FIGURE
FIGURE 2. Isodensity surfaces in the low (left) and high (right) density wing of the PDF, for a snap shot from one of the numerical experiments. The density levels at which the surfaces are shown are symmetric with respect to the PDF, and correspond to a level about a factor of 103 below the maximum of the PDF.
around its maximum, isodensity surfaces in the two wings of the PDF correspond to very different structures (Fig. 2). High density is created by interaction of 3-D shocks. Structures are sheet fragments and their filamentary and knotty intersections. Low density is created by interaction of 3-D expansion waves. Structures are irregular voids. It is also remarkable that the high density wing of the Log-Normal is established very early—soon after the first shock interactions. The first shock interactions occur after about 0.2 to 0.3 dynamical times, and after this time the right hand side of the PDF is already well established. For any particular snap shot the PDF contains structure, relative to a perfect Log-Normal, but the right hand side of these early PDFs do not deviate significantly more than PDFs from later times. Features in the PDF progress from high density to low density—presumably these are individual expansion waves.
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3K
LL Q Q_
0.01 r
0.001 0.01
10 q/
FIGURE 3. PDFs from 3-D experiments with driven supersonic turbulence and a polytropic equation of state. Panel a) shows the PDFs, while panel b) shows their slopes. In both panels, the full drawn curves show analytical fits (cf. discussion in the text). The drop below the analytic fits at high densities is due to limited numerical resolution.
4. PDFs for polytropic equations of state Scalo et al. (1998) pointed out that the PDF is only Log-Normal for isothermal conditions. They argue that the PDF develops a power-law wing when the effective gamma is not equal to unity. Their figures 10 and 11 show the PDF from 1-D experiments with effective gammas equal to 1.0 and 0.3, and with varying Mach numbers (note that the Mach numbers given in their paper are not rms Mach numbers, but refer to a nondimensional parameter in their equations). It is indeed clear from the proof of the LogNormality of the PDF for isothermal flows that the PDF cannot be exactly Log-Normal for non-isothermal flows, since the dynamic equations are then no longer independent of the mean density. But does the PDF become a power-law for a polytropic equation of state? Driven supersonic 3-D turbulence experiments with effective gamma different from unity produce skewed PDFs (Fig. 3). In the case with an effective gamma less than unity, the dense gas is colder than average, and hence a certain velocity distribution corresponds to higher Mach numbers. A higher Mach number corresponds to a broader PDF, and hence for yefi < 1 the PDF has a more extended high density wing than in the isothermal case. The polytropic PDFs are reminiscent of power-laws over a limited range of densities; Scalo et al. (1998) fitted the 7eff = 0.3 PDF with a power-law over a range of about one order of magnitude in density, but made no attempt to model the PDF over a larger range of densities. It is possible to fit the numerical PDFs with an analytical expression, assuming that the logarithmic slope of the PDF is a function of the formal temperature T ~ p ^ " " 1 ) . Assuming that the slope scales with T p , one can fit the slopes for p — 5/3 (cf. Fig. 3). The resulting PDFs are neither Log-Normal, nor power-laws. As is obvious from first principles, and as illustrated by Fig. 3, the family of polytropic PDFs depends in a continuous manner on 7eff, changing gradually from the symmetric Log-Normal for 7eff = 1 to more skewed forms for effective gammas that differ from unity. But even the 7eff = 0.3 PDF differs by less than a factor of two from a LogNormal, except far out in the wings. It has a "most common density" that is about a factor of two smaller than the one expected for 7efj = 1. The continuous change of shape with 7eff means that the PDFs cannot be power-laws for any finite density (and non-zero 7eff). The PDFs for 7eff ^ 1 do have power-law asymptotes, but only because the temperature formally goes to zero at one infinity. In
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reality, the temperature of a 10 °K cloud is not expected to fall by more than a factor 3-4 before internal shielding and/or heating become significant. 5. The influence of magnetic fields The Log-Normal like shapes of the PDFs are not noticeably influenced by the presence of weak magnetic fields; as long as the mean magnetic energy remains small, the density PDFs are practically unaffected. For magnetic energies approaching, but still smaller than the mean kinetic energy, the PDFs remain Log-Normal in shape, but with reduced standard deviations, corresponding roughly to the suppression of compressive motions in two out of the three spatial dimensions. For magnetic fields approaching and exceeding equipartition, the PDFs first become truncated at small densities, and then loose their Log-Normal like shape altogether. This has important diagnostic implications, for example for extinction statistics (cf. Padoan & Nordlund, these proceedings).
6. Conclusions Based on the results of Pope & Ching (1993) one may show that the density PDF for supersonic, isothermal turbulence is exactly Log-Normal. Numerical experiments show that, with a solenoidal forcing, and in three dimensions, the standard deviation of the density is equal to half the rms Mach number. Note that, for compressional forcing at low Mach numbers (leading to an ensemble of sound waves), the standard deviation is expected to be equal to the the rms Mach number itself. The Log-Normal property is robust, in the sense that the PDFs of polytropic supersonic turbulence are not far from Log-Normal. They are skewed, because (for -yeg < 1) the dense gas is colder and hence has a larger than average Mach number. They are not power- laws for densities of interest, but formally approach power-laws as the temperature goes to zero. The presence of a magnetic field changes the standard deviation of the PDF, and modifies the shape somewhat, but a Log-Normal PDF is still a good approximation, as long as the turbulence is super-Alfvenic. Gravity changes the situation qualitatively, in that there may no longer exist stationary turbulence solutions. Empirically, the PDFs tend to approach power-laws in the dense wing. The region around the maximum of the PDF is still well described by a Log-Normal. The near Log-Normal PDF is one of several generic properties of supersonic turbulence—there is thus good hope for rapid progress in our understanding of this subject, which is again important for a better understanding of Interstellar Turbulence.
REFERENCES COLES, P., JONES, B. J. 1991, MNRAS, 248, 1 HUBBLE, E. 1934, ApJ, 79, 8
A., PADOAN, P. 1998, Phys. Fluids, (in preparation) OSTRIKER, J. P. 1984,in Galaxy Distancies and Deviations from Universal Expansion, eds Madore, B. F., Tully, R. B. (Reidel), 273 PADOAN, P., NORDLUND, A., JONES, B. 1997, MNRAS, 288, 145 PEEBLES, P. J. E. 1980, The Large Scale Structure of the Universe, Princeton Univ. Press POPE, S. B., CHING, E. S. C. 1993, Phys. Fluids A, 5, 1529 SCALO, J.M., VAZQUEZ-SEMADENI, E., CHAPPELL, D., ET AL. 1998, ApJ, astro-ph/9710075
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VAZQUEZ-SEMADENI, E. 1994, ApJ, 423, 681
ZINNECKER, H. 1984, MNRAS, 210, 43
Turbulence as an Organizing Agent in the ISM By ENRIQUE VAZQUEZ-SEMADENI1 AND THIERRY PASSOT2 ^nstituto de Astronomfa, UNAM, Apdo. Postal 70-264, Mexico D. F. 04510, MEXICO 2
Observatoire de la Cote d'Azur, B.P. 4229, 06304, Nice Cedex 4, FRANCE
We discuss HD and MHD compressible turbulence as a cloud-forming and cloud-structuring mechanism in the ISM. Results from a numerical model of the turbulent ISM at large scales suggest that the phase-like appearance of the medium, the typical values of the densities and magnetic field strengths in the intercloud medium, as well as the velocity dispersion-size scaling relation in clouds may be understood as consequences of the interstellar turbulence. However, the density-size relation appears to only hold for the densest clouds, suggesting that low-column density clouds, which are hardest to observe, are turbulent transients. We then explore some properties of highly compressible polytropic turbulence, in one and several dimensions, applicable to molecular cloud scales. At low values of the polytropic index 7, turbulence may induce the gravitational collapse of otherwise linearly stable clouds, except if they are magnetically subcritical. The nature of the density fluctuations in the high Mach-number limit depends on 7. In the isothermal (7 = 1) case, the dispersion of \n(p) scales like the turbulent Mach number. The latter case is singular with a lognormal density pdf, while power-law tails develop at high (resp. low) densities for 7 < 1 (resp. 7 > 1). As a consequence, density fluctuations originating from Burgers turbulence are similar to those of the polytropic case only at high density when 7 < 1 and M » 1.
1. Introduction One of the main features of turbulence is its multi-scale nature (e.g., Scalo 1987; Lesieur 1990). In particular, in the interstellar medium (ISM), relevant scale sizes span nearly 5 orders of magnitude, from the size of the largest complexes or "superclouds" (~ 1 kpc) to that of dense cores in molecular clouds (a few xO.Ol pc), with densities respectively ranging from ~ 0.1 cm" 3 to £ 106 cm" 3 . Moreover, in the diffuse gas itself, even smaller scales, down to sizes several x 102 km are active (see the chapters by Spangler and Cordes), although at small densities. Therefore, in a unified turbulent picture of the ISM, it is natural to expect that turbulence can intervene in the process of cloud formation (Hunter 1979; Hunter & Fleck 1982; Elmegreen 1993; Vazquez-Semadeni, Passot k Pouquet 1995, 1996) through modes larger than the clouds themselves, as well as in providing cloud support and determining the cloud properties, through modes smaller than the clouds (Chandrasekhar 1951; Bonazzola et al. 1987; Leorat et al. 1990; Vazquez-Semadeni & Gazol 1995). Moreover, another essential feature of turbulence is that all these scales interact nonlinearly, so that coupling is expected to exist between the large-scale cloud-forming modes and the small-scale cloud properties. In this chapter we adopt the above viewpoint as a framework for presenting some of the most relevant results we have learned from two-dimensional (2D) numerical simulations of the turbulent ISM in a unified and coherent fashion, as it relates to the problems of cloud formation, the phase-like structure of the ISM and the topology of the magnetic and density fields, as well as internal cloud properties, such as their virialization and scaling relations (§ 2). Next we discuss recent results from multi-dimensional simulations and a simple heuristic model of one-dimensional polytropic turbulence, as a first attempt to gain more physical insight into the mechanisms responsible for the generation of the 223
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Vazquez-Semadeni & Passot: Turbulence as Organizing Agent in ISM
1. Two views of the density fieldin a simulation of 1 kpc2 of the ISM on the Galactic 7 plane, one at a) t = 6.37 x 10 yr (left), the other at b) t = 7.80 x 107 yr (right). The large-scale structures are contracting gravitationally, while the smaller structures within them change significantly due to the turbulent motions. FIGURE
statistics of the density fluctuations in compressible turbulence (§ 3). Finally, we present a summary and conclusions in § 4.
2. Cloud Formation and Properties in the Turbulent ISM In a series of recent papers (Vazquez-Semadeni et al. 1995 (Paper I), 1996 (Paper III); Passot, Vazquez-Semadeni & Pouquet 1995 (Paper II)), we have presented two-dimensional (2D) numerical simulations of turbulence in the ISM on the Galactic plane, including self-gravity, magnetic fields, simple parametrizations of standard cooling functions (Dalgarno &; McCray 1972; Raymond, Cox & Smith 1976) as given by Chiang & Bregman (1988), diffuse heating mimicking that of background UV radiation and cosmic rays, rotation, and a simple prescription for star formation (SF) which represents massive-star ionization heating by turning on a local source of heat wherever the density exceeds a threshold p±. Supernovae are now being included (Gazol-Patino & Passot 1998; see also Korpi, this Conference, for analogous simulations in 3D). The simulations follow the evolution of a 1 kpc2 region of the ISM at the solar Galactocentric distance over ~ 108 yr and are started with Gaussian fluctuations with random phases in all variables. The initial fluctuations in the velocity field produce shocks which trigger star formation which, in turn, feeds back on the turbulence, and a self-sustaining cycle is maintained. These simulations have been able to reproduce a number of important properties of the ISM, suggesting that the processes included are indeed relevant in the actual ISM. Some interesting predictions have also resulted. 2.1. Effective Polytropic Behavior and Phase-Like Structure One of the earliest results of the simulations is a consequence of the rapid thermal rates (Spitzer & Savedoff 1950), faster than the dynamical timescales by factors of 10-104 in the simulations (Paper I). Thus, the gas is essentially always in thermal equilibrium, except in star-forming regions, and an effective polytropic exponent 7e (Elmegreen 1991) can be calculated, which results from the condition of equilibrium between cooling and
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1
diffuse heating, giving an effectively polytropic behavior Peq oc p ", where p is the gas density (see Papers II and III for details). Even though the heating and cooling functions used do not give a thermally unstable (e.g., Field, Goldsmith & Habing 1969; Balbus 1995) regime at the temperatures reached by the simulations, they manage to produce values of 7e smaller than unity for temperatures in the range 100 K < T < 105 K, implying that denser regions are cooler. Upon the production of turbulent density fluctuations, the flow reaches a temperature distribution similar to that resulting from isobaric thermal instabilities (Field, Goldsmith & Habing 1969), but without the need for them. Note, however, that in this case there are no sharp phase transitions. 2.2. Cloud Formation In the simulations, the largest cloud complexes (several hundred pc) form simply by gravitational instability. Although in Paper I it was reported that no gravitationally bound structures were formed, this conclusion did not take into account the effective reduction of the Jeans length due to the small 7e of the fluid. Once this effect is considered, it is found that the largest scales in the simulations are unstable. This process is illustrated in fig. 1, which shows two snapshots of the logarithm of the density in a simulation at a resolution of 512 grid points per dimension (run 28 from Paper II), one at t = 6.37 x 107 yr (a), with minimum and maximum densities of 0.04 and 36.1 cm" 3 , and the other at t = 7.80 x 107 yr, with extrema of 0.046 and 58 cm" 3 . The two very large scale structures in the upper and lower halves of the integration box, are seen to have contracted at the later time, and the voids have expanded. Nevertheless, inside such large-scale clouds, an extrememly complicated morphology is seen in the higher-density material, as a consequence of the turbulence generated by the star formation activity. The mediumand small-scale clouds are thus turbulent density fluctuations. 2.3. Cloud and magnetic field topology
The topology of the clouds formed as turbulent fluctuations in the simulations is extrememly filamentary. This property apparently persists in 3D simulations (see chapters by Ostriker, Stone, Mac Low and Padoan). Interestingly, the magnetic field also exhibits a morphology indicative of significant distortion by the turbulent motions (Paper II; Ballesteros-Paredes & Vazquez-Semadeni 1998). The field has a tendency to be aligned with density features, as shown in fig. 2. Even in the presence of a uniform mean field, motions along the latter amplify the perpendicular fluctuations due to flux freezing, while at the same time they produce density fluctuations elongated perpendicular to the direction of compression. This mechanism also causes many of the density features to contain magnetic field reversals (e.g., the feature near the lower left corner) and bendings (e.g., the feature down and to the right off the center, with coordinates x ~ 25,y ~ 12). It happens also that magnetic fields can traverse the clouds without much perturbation, as seen for example in the feature at x = 21, y — 30. These results are consistent with the observational result that the magnetic field does not seem to vary much along clouds (Goodman et al. 1990), and in general does not present a unique kind of alignment with the density features. On the other hand, recent observations have found field bendings similar to those described here (Crutcher, this volume). Also, field reversals in clouds have been recently observed (Heiles 1997). It is important to note that the "pushing" of the turbulence on the magneticfieldoccurs for realistic values of the energy injection from stars and of the magnetic field strength, which ranges from ~ 5 x 10~3/iG (occurring at the low density intercloud medium) to a maximum of ~ 25/xG, which occurs in one of the high density peaks, although with no unique p-B correlation (Paper II). Observationally, larger values of the field occur only
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FIGURE 2. Gray-scale image of the logarithm of the densityfield,with superimposed magnetic field vectors. Shown is a subfield of 200 x 200 pc (160 x 160 pixels), from a simulation at resolution of 800 grid points per dimension (VBR.97). The minimum and maximum magnetic field intensities are 0.13 and 26.6 /iG, respectively. The axes show arbitrary units. See text for feature description.
on much smaller scales than those resolved by our simulations (1.25 pc in at resolution of 8002 grid points) (Heiles et al. 1993). Thus, the simulations suggest that the effect of the magnetic field is not as strongly dominating as often assumed in the literature. This is also in agreement with the fact that the magnetic and kinetic energies in the simulations are in near global equipartition at all scales, as shown by their energy spectra (fig. 5 in Paper II). Finally, note that the fact that the magnetic spectrum exhibits a clear self-similar (power-law) range, together with the fact that the fluctuating component of the field is in general comparable or larger than the uniform field, suggests strongly that the medium is in a state of fully developed MHD turbulence, rather than being a superposition of weakly nonlinear MHD waves. 2.4. Cloud scaling properties
An important question concerning the clouds formed in the simulations is whether they reproduce some well-known observational scaling and statistical properties of interstellar clouds, most notably the so-called Larson's relations between velocity dispersion Av, mean density p and size R (Larson 1981), and the cloud mass spectra (e.g., Blitz 1991). Vazquez-Semadeni, Ballesteros-Paredes & Rodriguez (1997, hereafter VBR97) have studied the scaling properties of the clouds in the simulations, finding that the cloud ensemble exhibits a relation Av oc R0A±°w a n d a c k m d m a s s s p e c t r u m dN(M)/dM oc M" 144 * 0 - 1 , both being consistent with observational surveys, especially those specifically including gravitationally unbound objects (e.g., Falgarone, Puget & Perault 1992). However, it was found that no density-size relation like that of Larson (p oc R~*) is satisfied by the clouds in the simulations. Instead, the clouds occupy a triangular region in a log/9-log.R diagram, as shown in fig. 8 of VBR97, with only its upper envelope being close to Larson's relation. This implies the existence of clouds of very low column density, which are presumably turbulent transients, and can be easily missed by observational surveys if
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they do not integrate for long enough times. A few observational works, however, point towards the existence of transients (Loren 1989; Magnani, La Rosa & Shore 1993) and low-column density clouds, with masses much smaller than those estimated from virial equilibrium (Falgarone, Puget & Perault 1992). An implication of Larson's relations is the so-called logatropic "equation of state" (Lizano & Shu 1989). Vazquez-Semadeni, Canto & Lizano (1998) have investigated whether this behavior is verified in numerical simulations of gravitational collapse with initially turbulent conditions. A logatropic behavior would imply a scaling AD OC p~ll2. However, a scaling Av oc pa, with 1/4 < a < 1/2 was observed, suggesting a polytropic behavior instead. This was interpreted as meaning that the logatropic equation of state was obtained by an invalid assumption, namely that Larson's relations are applicable to a thermodynamic process on a cloud of fixed mass. Instead, they seem to represent only the conditions of a (possibly relaxed) ensemble of clouds of different masses, and so are inapplicable to the former case.
3. Results on polytropic compressible turbulence 3.1. Production and stability of turbulent density fluctuations In view of the effective polytropic behavior exhibited by the simulations (§ 2.1), a natural abstraction is to consider the behavior of purely polytropic fluids, whose equation of state is P = p 7 e /7 e . For such a fluid, it has been shown in (Paper III) that the density jump X = P2/P1 in a shock in a polytropic gas satisfies X 1 + T " - ( l + 7 e M 2 ) X + 7eM2 = 0, (3.!) where M is the Mach number upstream of the shock. From this equation, we recover the fact that the compression ratio for an isothermal shock (-ye = 1) is M2, but we also see that X -> e M as 7e -^ 0, a density jump which can be much larger than the isothermal one. Turbulence-induced fluctuations can either collapse or rebound, depending on their cooling and dissipating abilities (e.g., Hunter & Fleck 1982; Hunter et al. 1986; Elmegreen & Elmegreen 1978; Vishniac 1983, 1994; Elmegreen 1993). The critical value of j e for which a turbulent density fluctuation formed by an n-dimensional compression can collapse was given in Paper III as 7 cr = 2(1 — 1/n). The same criterion was given by McKee et al. (1993) for fixed 7 e ~ 1 as an indication that 1-dimensional shock compressions cannot cause collapse. Instead, we can argue that the combination of small enough 7e and compressions in more than one dimension (shock collisions) can trigger collapse. Scalo et al. (1998) have recently discussed the possible values of 7 e in the cold ISM, finding that, although with large uncertainty, 7 e ~ 1/3 is possible at densities £, 5 x 104 cm" 3 , thus making the collapse of shock-compressed cores feasible. 3.2. Statistics of density fluctuations in polytropic turbulence The turbulent formation of clouds in the ISM must ultimately be described by the statistics of density fluctuation production in compressible turbulence. Therefore, it is of interest to investigate the probability density function (pdf) of the density fluctuations that develops in numerical simulations. Interestingly, the pdfs reported for a variety of flows show important qualitative differences. Porter, Pouquet & Woodward (1991) reported an exponential pdf for 3D, weakly compressible thermodynamic turbulence. Powerlaw pdfs have been reported for low-Reynolds number, one-dimensional Burgers flows (Gotoh & Kraichnan 1993) and for the simulations described in § 2, as well as for twodimensional Burgers flows (Scalo et al. 1998), while lognormal pdfs have been reported
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00
o
-3 -3 - 2 - 1 0 log(M 2 rms )
-3 -3 - 2 - 1 0 log(M 2 rms )
1
1
FIGURE 3. Logarithm of the variance of a) s = \np (left) and b) the density p (right) vs. the logarithm of the rms Mach number in one-dimensional simulations of polytropic turbulence. Note that a\ scales as M^ms, while a^ increases faster than M^ms.
for isothermal 2D (Vazquez-Semadeni 1994) and 3D (Padoan, Nordlund & Jones 1997) simulations. Passot & Vazquez-Semadeni (1998, hereafter PVS98) have investigated the simplest case of a one-dimensional purely hydrodynamic flow by means of a heuristic model and very high resolution (up to 6144 grid points) numerical simulations. Here, we summarize these results briefly. In order to study the production of the local density fluctuations, it is convenient to describe them as a sequence of isolated, discrete jumps (Vazquez-Semadeni 1994). Consider first the isothermal (7e = 1) case, whose governing equations read Dt
(3.2)
d (3.3) Dt where p is the fluid density, u is the velocity, s = In p and M is the Mach number of the velocity unit. Note that these equations are invariant upon the change s -t s + b, where b is an arbitrary constant, reflecting the fact that the sound speed does not depend on the density in this case. Consider now the sequence of density jumps pilp\- This is a sequence of multiplicative steps, and, consequently, additive steps in s = In p. But, because of the translation invariance mentioned above, a jump of a given magnitude must have the same probability of occurrence, independently of the initial density. Thus, the sequence involves events with the same probability distribution, and by the Central Limit Theorem it must converge to a Gaussian distribution in s, or, equivalently, a lognormal distribution in p (see Nordlund, this volume, for an alternative derivation), explaining the reported pdfs in the isothermal case. In order to fully characterize the pdf, it is necessary to determine its mean and variance. Concerning the latter, PVS98 have suggested, through an anlysis of the shock and expansion waves in the system, that for a large range of Mach numbers the typical size of the logarithmic jump is expected to be as ~ M rms , where M rms is the rms Mach number. This result is verified numerically, as shown in fig. 3a. Note that fig. 3b shows the scaling
Vazquez-Semadeni & Passot: Turbulence as Organizing Agent in ISM 2
229
Mr2ms.
of the linear density variance a p vs. An exponential behavior is observed, most noticeable at large Mach numbers, in agreement with the relation <x2 = ln(l + U2,) which holds for a lognormal distribution. This contrasts with recent claims that it is ap which scales as M r m s (Padoan, Nordlund & Jones 1997; Nordlund, this volume). This discrepancy can be understood as a consequence of the similarity between the two variances at small Mach numbers, together with the fact that the simulations from which those authors reached their conclusion were three-dimensional, implying that a significant fraction of the kinetic energy was in rotational modes (reportedly ~ 80 %), and thus not available for producing density fluctuations. Instead, in the ID simulations of PVS98, all of the kinetic energy is compressible. Concerning the mean SQ of the distribution, it can be directly evaluated from the mass conservation condition (p) = f_™ esP(s)ds = 1, where P(s) is the pdf of s, yielding so = —Cg/2. With all this in mind, we can then write the model pdf for s as
with <7g = /3M2ms, and (3 a proportionality constant. The numerical simulations confirm the dependence of the width and mean of the distribution with Mrms (PVS98). We next consider the case 7 e n e l. The governing equations can now be written as Dt ~ (1 - 7e)M 2 dx
[6 0)
-
l£ = -(l-7.)£«.
(3-6)
Interestingly, these equations return to the form of eqs. (3.2) and (3.3) upon the densitydependent rescaling M —t M(s;je) = M e ' 1 " 7 ' ' ^ 2 . Thus, we formulate the ansatz that the form of the pdf also remains the same, provided the above replacement is made. After relocating the term in s 0 from inside the exponential function to the normalization constant, we can write the model pdf as r_ s 2 e ( 7 e -i)s
P(s; 7e)ds = C(7e) exp [
^
,
afre)*] ds.
(3.7)
Note that this is a particular form of the pdf valid only in a range of s-values (PVS98), but for illustrative purposes it suffices here. This equation shows that when (7,, - l)s < 0, the pdf asymptotically approaches a power law, while in the opposite case it decays faster than a lognormal. Thus, for 0 < j e < 1, the pdf approaches a power law at large densities (s > 0), and at low densities for j e > 1. The former case is in agreement with the pdfs reported for our 2D simulations, which in general have 7 e < 1. The ID simulations also verify this result for j e > 1 (fig. 4). See Nordlund (this volume) for a parallel treatment of this problem. It is important to note that even at very small values of 7 e (~ 0.01), the fast drop of the pdf at low densities is still observed, due to the factor e^"r"~l^s in the exponential in eq. 3.7, which in turn implies that there is always a range of s-values in which the pressure is not negligible in the hydrodynamic case, for any j e . This leads to the speculation that the pdf for Burgers flows, which are strictly pressureless, should exhibit power laws at both large and small densities. This speculation is also verified numerically (PVS98). Thus, the Burgers case appears to be singular, not being the limit of hydrodynamic flows as 7 e -> 0, at least as far as the pdf is concerned.
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1
_
FIGURE
i
4. Density pdfs for two ID simulations of polytropic turbulence, one with -ye = 0.3 (left), and the other with j e = 1.7 (right), both with M = 3.
4. Conclusions In this Chapter we have discussed a scenario in which turbulence plays a fundamental role in the production and determination of interstellar cloud properties. Large-scale turbulent modes intervene in the former, while small-scale modes seem to participate in determining cloud scaling relations. ISM features that appear naturally in our simulations are the phase-like appearance (a consequence of turbulent density fluctuation production together with an effective polytropic exponent 7 e < 1), density and magnetic field topologies and field strength ranges, and the velocity dispersion-size relation and cloud mass spectrum. However, the suggestions are made that the Larson (1981) densitysize scaling relation may be an artifact of surveys which do not integrate for long enough times, and that the logatropic equation of state (Lizano & Shu 1989) is not verified in highly dynamic situations. Given the possible nature of clouds as turbulent density fluctuations, their production in polytropic flows was also discussed. The density jump across shocks and a criterion for the collapse of these fluctuations were advanced. Finally, a model for the probability density function of the fluctuations was described, which satisfactorily explains the pdf shapes observed in isothermal, polytropic and Burgers cases. Future work in this area will address the fully thermodynamic and magnetic cases, aiming at explaining other forms of the pdf, which have been observed numerically, but not produced by the model. We gratefully acknowledge fruitful conversations with Annick Pouquet and Susana Lizano. The numerical simulations have been performed on the Cray Y-MP 4/64 of DGSCA, UNAM, and the Cray C98 of IDRIS, France. This work has received partial funding from grants UNAM/DGAPAIN105295 and UNAM/CRAY SC-008397 to E. V.-S and from the PCMI National Program of C.N.R.S. to T.P.
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Turbulence and magnetic reconnection in the interstellar medium By ELLEN G. ZWEIBEL JILA and Department of Astrophysics and Planetary Science, University of Colorado, Boulder, CO 80309, USA Magnetic reconnection is often assumed to occur at an enhanced rate in the interstellar medium because of the effects of small scale turbulence. This effect is not modelled directly in numerical simulations, but is accounted for by explicitly assuming the resistivity is large, or assuming that numerical resistivity mimics the effect of small scale turbulence. The effective resistivity really is large only if the field can rapidly reconnect. In this paper I discuss two physical mechanisms for fast magnetic reconnection in the interstellar medium: enhanced diffusion at stagnation points, and formation of current sheets.
1. Introduction Numerical experiments are making important contributions to the study of turbulence in the interstellar medium (ISM). Since any numerical simulation is restricted in the range of spatial and temporal scales which it can describe, it is important to develop a prescription for treating the effects of turbulence at the smallest scales, which are generally omitted from this range. Although very little energy resides at the smallest scales, the small scale motions dramatically increase momentum and magnetic flux transport in the ISM, and can also produce rapid thermal and chemical mixing. The most common way to account for these subgridscale effects is to simply assume that the viscosity, electrical resistivity, and other transport coefficients are much larger than their molecular values. The difficult problem of justifying this approach and calculating the so-called eddy diffusivities has received more attention in the atmospheric and stellar turbulence communities than it has, so far, among interstellar turbulence theorists. The magnetic reconnection problem acutely brings out these subgridscale modelling issues. The magnetic Reynolds number RM: the ratio of the Ohmic diffusion time to the dynamical time, is very large under interstellar conditions: RM ~ 3 X lQl0LpcVkmT3/2, where Lpc is the magnetic lengthscale in parsecs, Vkm is the characteristic velocity in km/sec, and T is the gas temperature in degrees Kelvin. This means that if Ohmic processes occur anywhere in the ISM, it must be at locations where the magnetic lengthscale is very small. On the other hand, RM is small enough in numerical simulations that the magnetic field is seen to diffuse and change topology quite readily. Therefore, simulations can only reproduce the evolution and structure of the magnetic field in the interstellar medium if the fieldlines can reconnect at a rate that is nearly independent of RM- Such a process is called fast reconnection. The well known Sweet-Parker steady state recon— 1/2
nection theory predicts that the reconnection rate scales as RM , i.e. reconnection is slow. The maximum reconnection rate in the Petschek model scales as (log.RM)~\ i-e. reconnection is fast (see Parker 1979 for a review). In §2 of this paper I describe some ways in which the topology of the interstellar magnetic field influences interstellar processes and contains clues as to the origin of the field itself. In §3 I describe a model for fast reconnection. In §4 I describe a mechanism for producing current sheets, which might rapidly reconnect. Neither of these treatments necessarily solves the problem. Section 5 is a summary and discussion. 232
Zweibel: Reconnection in the ISM
233
2. Magnetic topology and reconnection It is known from observations that the Galactic magnetic field has a mean direction when averaged over at least a few kiloparsecs, and that it also has a random component with magnitude equal to or slightly greater than the mean component (see Zweibel &; Heiles 1997 for a review). In discussing magnetic topology, it is necessary to distinguish between direction and orientation. For example, a magnetic loop which has been stretched in the azimuthal direction by Galactic differential rotation would contribute little to the mean field as measured by Faraday rotation, which is sensitive to direction, but would contribute substantially to the mean field as measured by synchrotron polarization, which is sensitive only to orientation. It is possible that small scale reversals acted upon by rotational shear account for the fact that the ratio of mean to random field is smaller when measured by direction - sensitive methods than when measured by orientation sensitive methods, but the discrepancy is not large ( Zweibel &c Heiles 1997). Since there is evidence for a mean field direction, theories of the origin and maintenance of the field which rely on flux injection at small scales (e.g. by stars) require that the field reconnect (Blackmail 1998); this is true of dynamo theories in general (see Beck et al. 1996 for a review). Although direct observational probes of Galactic magnetic topology are thus rather coarse, there are some indirect arguments, which are at least suggestive. The first is the nature of cosmic ray propagation in the Galaxy. Both the amount of material traversed by cosmic rays detected at Earth (3-5 gm/cm2) and the time elapsed since their acceleration (~ 2x 107 yr) are quite successfully explained by a theory in which cosmic rays propagate away from their sources along coherently directed magnetic fieldlines with little cross field diffusion, but substantial scattering up and down their home magnetic flux tubes, (see Cesarsky 1980 for a review). If, instead, the Galactic magnetic field consisted of many disconnected loops, most of them unconnected to cosmic ray sources, there would have to be profuse cross -fieldpropagation. In that case it would be difficult to see why cosmic rays are confined to the Galactic disk for several million years instead of being freely scattered into intergalactic space, unless the loops are highly anisotropic. But it might be possible to produce a convincing cosmic ray propagation model without invoking a coherent Galactic magnetic field. The second examples are based on the physics of molecular clouds and star formation. Recent observations and modelling of the diffuse Galactic 7-ray emission shows no evidence that the cosmic ray flux in molecular clouds is reduced below the value in the ambient ISM (Digel et al. 1996, Hunter et al. 1997), although a reduction of the flux in small dense cores cannot be ruled out by such analysis. This suggests that magnetic fieldlines in dense clouds are connected to the general Galactic magnetic field. Second, the theory of magnetic braking of clouds assumes that the magnetic fieldlines which thread the cloud are connected to, at a minimum, the cloud's own mass of external material. Braking of a cloud with, for example, a dipole magnetic field, would be very slow (see McKee et al. 1993 for a review). These examples argue against substantial magnetic reconnection in molecular clouds. But there are also arguments in favor of reconnection. At some point in the star formation process, a protostar must become magnetically detached from its parent cloud. And, more generally, unless the field is strong enough to control turbulent motions in the cloud, it seems inevitable that a cloud which survives for several crossing times must have a highly tangled magnetic field. At this point, it is tempting to say that when the field is "sufficiently" tangled, it reconnects. But how tangled is that? This brings us to the physics of magnetic reconnection.
234
Zweibel: Reconnection in the ISM
3. Fast reconnection of weak magnetic fields The magnetic field evolves according to the induction equation =Vx(VixB)
+ XohmVB,
(3.1)
where Vi is the velocity of the plasma component and Xohm is the Ohmic diffusivity. One can derive from (3.1) the dynamical timescale tdyn and Ohmic timescale
''dyn —
i
'-ohm — >
>
\"""/
where space derivatives have been replaced by the inverse magnetic lengthscale Lg1. The magnetic Reynolds number RM is then tohm/tdyn = LBVi/XOhmThe kinematic fast reconnection problem consists of finding solutions of (3.1) in which magnetic flux is destroyed at a rate nearly independent of RM when the velocityfieldVi is prescribed. It is a reasonable approach when the magnetic field is weak. The question is whether the field remains weak. In order for Ohmic diffusion to operate, LB must become very small. This could lead to large Lorentz forces which modify the flow so that the reconnection is quenched. I have recently studied this problem by classifying the action of all possible incompressible stagnation point flows on arbitrary magnetic fields (Zweibel 1998). Here I extend some of these results to compressible flows and time dependent flows by way of the following example. Consider the stagnation point flow Vi = i{-x,-y,az),
(3.3)
where 7 is constant and positive and a is a dimensionless parameter. Fluid converges in the (x, y) directions and, assuming a > 0, is ejected in the z direction. Let the initial magneticfieldbe B = Bo(smkoy,0,0).
(3.4)
The solution of (3.1) for the flow (3.3) with initial condition (3.4) is B = B(t)(smky,0,0),
(3.5)
where fc = fcoe7t; B(t) = B0exp((l-a)'yt-\ohm(k2-k2Q)/(21)).
(3.6)
The exponential increase of k oc L^1 in (3.6) is caused by convergence in x, while the factor exp((l - a)jt) reflects the balance between stretching and shrinking along the axes of this 3D flow. It can easily be shown from (3.6) that the magnetic flux through a Lagrangian surface in the (y, z) plane decays as exp(-Ao/im(fc2 — fcg)/(27). This yields an e-folding time for the flux
( ? I )
(3.7)
Because trec depends only logarithmically on the resistivity, this is a model for fast reconnection. Whether it is a self consistent model depends on the magnitude of the
Zweibel: Reconnection in the ISM
235
Lorentz force, which in this case is a magnetic pressure gradient force acting only in the x direction. Using (3.5) and the definitions just below it, it can be shown that
FL = - ^
exp ((3 - 2a)jt - Xohm(k2 - fco2)/7) sin 2%.
(3.8)
Using (3.7), (3.8) we see that FL(trec)/FL(0)
~ (27/Ao/imfc02)15-Q ~ R1*-".
(3.9)
Evidently if a < 1.5, Lorentz forces grow enormously, and the reconnection scenario must be strongly modified. This can be related to the compressibility of the flow: according to (3.3), V • Vi = 7(a —2). Therefore it is possible to have a self consistent, fast reconnection model in a slightly compressive stagnation point flow; 1.5 < a < 2, or in an expansive flow (a > 2). These results hold because compressive flows tend to increase the field, and thereby the Lorentz force. The self consistent fast reconnection model always fails for this class offlowsif the field happens to have a component in the z direction. For example, if the x and z magnetic field components in (3.4) are interchanged, the field amplitude grows as exp(27i) and LB shrinks as exp(-7<), so Fi ~ exp(57^), irrespective of the value of a. These results are easily extended to the case that 7 in (3.3) depends on time; 7 = j(t). Equation (3.6) is modified to
ds-y(s)j; B(t) = Boexp Ul - a) j ds-y(s) - \ohm j dsk2(s)J (3.10) Equation (3.10) predicts that the rate of magnetic diffusion is not significantly enhanced above its initial value unless the coherence time for 7 is almost as long as the reconnection time given in (3.7). The examples presented here are consistent with the more general analysis of incompressible flows presented previously (Zweibel 1998). Although it is not difficult to make models of fast magnetic reconnection in which Lorentz forces are not amplified (provided the flow is not too compressive), the models require special conditions, such as a field entirely in the (x, y) plane in the example presented here. In general, it is possible to show that a stagnation point flow amplifies the Lorentz forces by a factor ranging from K>hm t 0 ^ohm> depending on the initial orientation of the field. This means that even an initially weak field cannot be passively annihilated by stagnation point flow, if the field is randomly oriented. If one approximates turbulent flow as a collection of stagnation points with random relative orientation, then magnetic fields tend to be amplified by the turbulence. This appears to be an argument against the operation of turbulent resistivity, at least in its most naive incarnation. It has also been argued elsewhere, on the basis of both numerical simulations and analytical techniques, that turbulent resistivity is quenched by Lorentz forces (Cattaneo & Vainshtein 1991; Cattaneo 1994; Gruzinov & Diamond 1996). The results of the kinematic reconnection study support these arguments. It is worth noting that Zel'dovich et al. (1984) used arguments very similar to those in Zweibel (1998) to show that a random ensemble of stagnation points would be a kinematic dynamo.
236
Zweibel: Reconnection in the ISM
4. Formation of current sheets by ambipolar drift The rate of reconnection may be enhanced if current sheets or filaments form. In these current sheets the rate of Ohmic dissipation is large, which suggests that the field might rapidly reconnect. It is not a foregone conclusion, however, that fast reconnection occurs under these circumstances. In fact, there are at least two counterexamples. Hahm & Kulsrud (1985) considered a sheared magnetic field which is driven out of equilibrium by a boundary perturbation. Of the two states accessible to the perturbed system, one has a current sheet but the same magnetic topology and the other has no current sheet but a different topology. The system evolves toward the current sheet equilibrium, but then, through reconnection, reaches the second equilibrium. However, this occurs on a slow timescale proportional to A~h7^ (which is also the growth time of tearing mode instabilities in a slab). Chiueh & Zweibel (1987) considered resistive instabilities of a sheared magnetic field with a current sheet. They found that the reconnection time is proportional to K>hL i w m c h is still slow. In both of these examples, there are only two timescales: the Alfven time and the resistive time. It is quite possible that reconnection in current sheets can only be fast if the structure of the reconnection layer is controlled by other effects which introduce their own timescales, such as compressibility, radiative cooling and ion-neutral drift (Dorman & Kulsrud 1995). Fortunately, all of these effects are present in the interstellar medium. Therefore, let us assume that current sheet formation is an interesting process and look for evidence that it might occur. Numerical studies of incompressible, 2D turbulence which begin from a state with a magnetic neutral line (Matthaeus & Lamkin 1986) or with neutral points (Politano, Pouquet, & Sulem 1989) show that the electric current becomes highly intermittent, and concentrated into thin sheets. These results can be applied to the ISM if it is assumed that there is a strong component of magnetic field perpendicular to the plane of the turbulence, in which case the turbulence becomes 2D and nearly incompressible, and the neutral line and neutral points are nulls only in projection. Although the magnetic field is seen to reconnect in these simulations, the effect of turbulence on the overall reconnection rate is not yet clear, especially at large RMIn the past few years I have explored another mechanism for generating current sheets in weakly ionized gases, namely ambipolar drift (Brandenburg & Zweibel 1994, 1995; Zweibel & Brandenburg 1997). The ambipolar drift rate vD = Vi - vn is found by equating the Lorentz on the ions to the frictional force due to ion-neutral collisions. For an ion-neutral collision rate Vin,
For a derivation of (4.11) see Shu (1983). Using (4.11), the magnetic induction equation (3.1) becomes (4.12) ^ = V x („ x B) + Vx ( ( V * f l ) x * x B\ + XohmV2B, V 4irpiUin J dt where v « vn is the bulk velocity of the medium. The second term on the rhs of (4.12) can also be written, using Ampere's Law, as (4.13) -V x ( * k ) , \CPiVinJ where Jx is the component of the current density J which is perpendicular to B. Equa-
Zweibel: Reconnection in the ISM
237
tion (4.13) suggests that ambipolar drift acts like nonlinear diffusion, with diffusivity ^ -
(4-14)
From eqn. (4.14) one can define an ambipolar Reynolds number RAO ^ ^
-
(4.15)
Although ambipolar drift, unlike Ohmic diffusion, does not actually break the magnetic fieldlines, one expects the magnetic field to be decoupled from the bulk medium (while still frozen to the ions) if RAD < 1- This is the condition, for example, for critically damped Alfven waves (Kulsrud & Pearce 1969). When the magnetic topology is simple and variations in the fieldstrength are not large, ambipolar drift acts much like classical diffusion - the best known example being the problem of the removal of a laminar field from a self gravitating cloud, which is the context in which ambipolar drift was first studied (Mestel & Spitzer 1956). However, when the magnetic field reverses or even undulates strongly, (4.12) has solutions in which the current density is singular. A very simple example is the steady state magnetic field B oc x y 1 / 3 , v = 0, Xohm = 0 (Brandenburg & Zweibel 1994). This magnetic profile can be formed in a magnetic field with a null point, because the ion neutral drift (4.11) is directed toward the null, which is, however, a stagnation point of the ion flow. In this case, the y~2^3 singularity in the current is resolved by the inclusion of resistivity and ion pressure (Brandenburg & Zweibel 1995). However, in the weakly ionized regions of the interstellar medium, the current density is still expected to become extremely large, with the B oc y 1 / 3 solution holding everywhere except in a thin layer. Current sheets have also been seen in cases where there is no magnetic null point. Examples include a field twisted up by a rotating eddy (Brandenburg & Zweibel 1994; Zweibel & Brandenburg 1997), accretion disks displaying Balbus-Hawley instability (Mac Low et al. 1995), large amplitude Alfven waves (Suzuki &; Sakai 1996), and the nonlinear stages of the Wardle instability of C shocks (Mac Low & Smith 1997). J. Stone (personal communication) has pointed out that a magnetic neutral sheet tends to collapse even without ambipolar drift, because of the inward directed magnetic pressure gradient. This is quite true: if the gas pressure is rigorously zero, a neutral sheet evolves to a step-like magnetic profile, which is the only way a field with a reversal can sustain constant magnetic pressure. This is a much more severe singularity than the y1/3 profile formed by ambipolar drift. However, gas pressure removes the singularity. If the initial ratio of gas pressure to magnetic pressure is /?o, then 2D relaxation of a linear field reversal will shrink the magnetic lengthscale by a factor of order /3Q. Under interstellar conditions, this generally does not bring the field into the resistive regime. Further reduction of the gas pressure might occur through "squirting" in the third dimension or through radiative cooling. In any case, the the gas pressure must be reduced by a fractional power of RM, which is always a large factor. Ion pressure broadens current sheets less effectively than neutral pressure because the ion pressure is very low to begin with, but more importantly because ions are removed from the stagnation point region by recombination. The numerical simulations themselves do not distinguish directly between current sheets formed by ambipolar drift, with their y~2^3 current profile, and current sheets formed by conventional MHD collapse, because the sheets are not resolved by the grid. It may be possible to discriminate between these two mechanisms indirectly, however, because ambipolar drift leads to relaxation of the magnetic field to a force free state.
238
Zweibel: Reconnection in the ISM
This is clearly seen in the 3D computations of Brandenburg et al. (1995), in which the mesh is too coarse to support actual current sheets, and in unpublished computations by Mac Low & Smith (private communication). The force free state minimizes the rate of energy dissipation E, E = -PiVin | vD | 2 ,
(4.16)
where VD is given by (4.11). (Even though VD is formally infinite at a current singularity, the resulting dissipation rate is integrable, so the presence of current sheets does not vitiate this argument). Single fluid MHD processes should not lead to a force free state unless the fieldstrength is well above equipartition. Chiueh (1998) has studied resistive tearing in current sheets produced by ambipolar drift. He finds that the reconnection time depends on the ambipolar drift time, not on the resistive time. His calculation is based on a "blowup" solution of (4.12) and it is not clear that the conclusion would apply to the current sheets with B oc t/1/3.
5. Conclusions Magnetic diffusion occurs quite readily in numerical simulations of turbulent processes in the ISM, because the global value of the magnetic Reynolds number RM - the ratio of the Ohmic diffusion time to the dynamical time - rarely exceeds ~ 104. This is typically more than ten orders of magnitude less than RM in the ISM. Therefore, the simulations can only predict the structure of the interstellar magnetic field if the effective resistivity of the ISM is large - that is, if the fieldlines easily break and reconnect. What is required is so called "fast" reconnection, which proceeds at a rate that is independent, or nearly independent, of RM- The large ranges of timescales and lengthscales which occur in the reconnection problem make it difficult to study numerically. In this paper I discussed magnetic reconnection from two viewpoints. The first involves simply assuming that the field is weak and can be reconnected in a stagnation point flow which brings the field in, (and reduces its lengthscale), in some directions, and expels it in others. However, it turns out to be difficult to find examples of fields and flows in which the Lorentz forces remain small. That is, it is difficult to find self consistent kinematic models. This implies that a random, weak field in a turbulent flow cannot be treated self consistently by the kinematic approach, and also that fast magnetic reconnection is probably fully dynamical. I also discussed current sheets as fast reconnection sites. Current sheets appear in simulations of 2D incompressible turbulence. They can also form by ambipolar drift. However, there are at least two known examples of slow reconnection in current sheets. Therefore, current sheet formation does not, by itself, guarantee the presence of fast reconnection. I conjectured that systems with only two important timescales, the Alfven time and the Ohmic diffusion time, do not rapidly reconnect, but rather that another timescale, such as the radiation cooling time, the ambipolar drift time, the magnetosonic compression time, or another dynamical timescale must enter the problem. All these effects, of course, come into play in the ISM. I am happy to acknowledge useful discussions with Steven Cowley and Remy Indebetouw. The work reported here was supported in part by grants from the NASA Space Physics and Astrophysical Theory Programs.
Zweibel: Reconnection in the ISM
239
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The Evolution of Self-Gravitating, Magnetized, Turbulent Clouds: Numerical Experiments By EVE C. OSTRIKER Department of Astronomy, The University of Maryland, College Park, MD 20742-2421 [email protected] Interstellar giant molecular clouds and dark clouds are observed to have comparable kinetic and gravitational energies, and low enough temperatures that their internal turbulent velocity amplitudes are highly supersonic. It has been believed for some time that the presence of magnetic fields can have important consequences for the properties of turbulence in these clouds, and for cloud's gravitational stability. In this paper, I outline how the physical parameters of clouds can be translated to dimensionless ratios (the Mach number, the Jeans number, and the plasma (3), describe a series of numerical experiments underway to evaluate how the character of the turbulence depends on these parameters, and present selections from our results to date.
1. Introduction The turbulence in Galactic molecular clouds has a rather different character from the other forms of interstellar turbulence discussed at this meeting. Strong molecular cooling brings the ambient temperatures to the range T = 10 - 30 K, which renders turbulence with velocities of a few km s - 1 not just nonlinear, but hypersonic. Most observational evidence on magnetic field strengths suggest that Alfven speeds are of same order than the turbulent speeds or a few times larger, and in any case unlikely to be smaller than the sound speed. Thus, the turbulence in molecular clouds is highly compressible, and strongly magnetic. In addition, although high-latitude unbound molecular clouds exist, most of the molecular material in the Galaxy resides in much more massive, self-gravitating, giant molecular clouds and cloud complexes (GMCs). The tendency for gravitational contraction of clouds must influence, and be influenced by, the evolution of the turbulent flows inside. In addition, phenomena associated with gravity (either local instabilities, or winds arising from the star formation process - which is effectively a self-gravitational runaway) may be responsible for exciting turbulence in the first place. Understanding the evolution of turbulent flow under these conditions is a fascinating hydrodynamics problem in itself, and is also crucial to developing the theory of star formation. The rate of star formation within a cloud, the efficiency of conversion of gas to stars in a cloud before it is destroyed, and the correlations among individual properties of the stars formed (i.e. clustered vs. dispersed star formation, distribution of stellar masses in the IMF) must all depend on the properties of the turbulence in GMCs. To understand cloud properties and to predict their evolution, we must be able to characterize how the "background state" of a cloud - e.g. its mean temperature, density, magnetic field strength, and ionization state, as well as its overall size - affects the character of its internal turbulence (including its dissipation rate, its modes of dissipation via shocks, ambipolar diffusion, or turbulent cascades, and its distributions, power spectra, and spatial correlations of density, magnetic fields, and velocity fields, etc.). This is obviously an extremely complex problem. However, even fairly simple computational models (using, e.g., an isothermal equation of state, periodic boundary conditions, and arbitrary initial or forced velocity fields) can be used in the initial attack on these dy240
Ostriker: Numerical MHD Studies of Turbulent, Self-Gravitating Clouds
241
namical questions. The quantitative results and insight gained from such experiments will help inform ideas about the yet-more-complicated process of creating a collection of stars within a turbulent cloud. Our research group has, over the last few years, begun a campaign of using numerical experiments with the ZEUS code for MHD (Stone & Norman 1992a, Stone & Norman 1992b) to simulate turbulent molecular clouds and to characterize how their properties depend on input parameters. In this paper, I provide background on how the relevant physical parameters can be modeled, and on expectations for self-gravitating cloud evolution based on linear theory. I also present highlights from some of our numerical surveys, addressing in particular questions of whether turbulent clouds collapse under self-gravity, and how magnetic fields affect the clumping of matter.
2. Cloud Model Parameters and Expectations from Linear Theory 2.1. Dimensional Scales for Cloud Dynamics In modeling molecular cloud dynamics, we start by identifying and parameterizing physical properties that are expected to have significant effects on cloud evolution. As in other astrophysical situations, the relative importance of different effects can by characterized by corresponding time scales. Four important dynamical timescales are those associated, respectively, with the times for gravitational contraction,
thermal pressure (sound) wave crossing,
magnetic pressure/tension (magnetosonic/Alfven) wave crossing, = L__ _ L ~ vA ~L
and fluid flow crossing!, t(
=A av
=
o.98 x 107 yr (—) (-^-A ~'; 1O C lkm J \ P / V /s/
(2.4)
all crossing times are defined over a global scale within a cloud. Here, p is the total mass density (n#2 is the number density), L is the characteristic length of the cloud, T is the cloud temperature (cs = y/kT/p = 0.19km/s(T/10 K) 1 / 2 is the sound speed), B is the mean magnetic field strength, and av is the (3D) velocity dispersion. Alternative definitions, with factors of order unity difference, are given by other authors; the basic scalings with density, temperature, cloud size, magnetic field, and fluid velocity, are standard. The simulations described herein take as their fundamental units the mean cloud density p, cloud edge size L = njLj (we use periodic boundary conditions with equal lengths t t{ is often referred to as an "eddy turnover time" in incompressible turbulence; for compressible turbulence, the present terminology is preferred.
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in each dimension), and the thermal sound speed c s . Here nj is the number of Jeans wavelengths
across the box, so that the total mass in the simulated region is M = nZjMj in terms of the Jeans mass
Time is described either in units of ts or ts; the corresponding physical times depend on the mean cloud properties via equations (2.1) and (2.5). In MHD models the magnetic field strength is conveniently described by the ratio (3 = c2/t>A.t The corresponding physical field strength is given in terms of the cloud temperature and density as
2.2. Cloud stability criteria A self-gravitating equilibrium supported by thermal pressure satisfies cs ~ (GM/L)1/2, or equivalently, cloud stability requires L < Lj (i.e. ts < ts). Molecular clouds, with their extremely low temperatures, fail to meet this (Jeans) criterion by an order of magnitude. In the absence of any other mitigating factors, a uniform, cold, quiescent cloud with L 3> Lj (i.e. ts 3> tg) collapses in a time ss 0.3t3 (the exact factor depend on the symmetry - e.g. spherical, cylindrical, slab - of the collapse). Since the Galactic star formation rate would be more than two orders of magnitude larger than observed if all Galactic GMCs collapsed and converted their material to stars in this time (assuming 100% efficiency), one of the main goals of understanding GMC internal dynamics is to identify the "mitigating factors" that prevent overall cloud collapse, and that limit the efficiency of star formation once portions of a cloud have lost their support. A number of mitigating effects may arise from the fact that the ISM in general, and self-gravitating GMCs in particular, are magnetized. Consider first a cold, homogeneous cloud in which a uniform magnetic field is embedded. In determining stability to compressions transverse to the mean field (i.e. k ± B), Chandrasekhar & Fermi (1953) showed that the thermal Jeans criterion is replaced by a magneto-Jeans criterion by substituting the magnetosonic speed for the sound speed in Lj; for I»A ^ cs stability requires 1 2 £A < tg, i-e. nj = L/Lj < I>A/C5 = f3~ / . The magneto-Jeans stability criterion can be translated to a maximum for the column density pL in which transverse gravitational instabilities can be suppressed by the embedded field B: Lp < B/(2\/G), equivalent to
NH7 < 4.1 x i o » W (JL)
.
(2.8)
Field strengths in the range 7—24/iG, consistent with observations from Zeeman measurements (see the paper of Crutcher in this volume), would be adequate to make observed clouds (NH2 = 3 - 10 x 1021cm~2) magneto-Jeans stable. However, since magnetic fields exert no force parallel to themselves, the stability along the field (for the uniformfield case) is subject to Jeans's original criterion; for nj > 1 all the matter in a cloud f Our definition of/3 differs from the usual plasma /3P = Pgas /Pm.agne.tic = 2CS/DA by a factor of two.
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would simply slide along the field to make a thin, self-gravitating pancake. The stability of a cloud pancake to perturbations with k in its plane can be described in terms of a critical ratio of column density along the field (i.e. surface density in the pancake S; or pL in terms of the original mean density and length) to magnetic field strength (Nakano & Nakamura 1978, Tomisaka & Ikeuchi 1983), with stable configurations having E < B/(2KVG) t For a given column density, the necessary field strength for stability of a thin pancake is a factor TT larger than that specified by the magneto-Jeans criterion (eq. 2.8), i.e.
While smaller dark clouds may be sub-critical, the strong fields required for stabilization at high column density make it likely that at least the most massive GMCs are supercritical. In addition, even when a cloud is subcritical on average, if there is variation of the mass load on field lines (i.e. the mass-to-flux ratio S/B) within it, then instability could arise on those field lines which exceed the critical condition. The criteria discussed above apply to stability about a quiescent state, although, in fact, molecular clouds are quite turbulent. The potential for time-varying magnetic fields to help stabilize self-gravitating clouds - particularly along their mean field directions - has been discussed by a number of authors (see Shu et al 1987, McKee et al 1993, and references therein). In the idealized configuration of a cloud containing a uniform mean field and Afven waves propagating along it, a quasilinear treatment of the time-dependent terms shows that they correspond to a wave pressure 8B\ rms/(8np) (McKee & Zweibel 1995); a Jeans-type analysis then predicts that contraction along the mean field can be stabilized as long as pL < 5Brms/(4\/G), i.e.
8Brms > 15MG (-£-) {^-^)
= 49/xG f T # ^ ) • 3
22
(2.10)
2
\ 10 p c / V100 cm" / \10 cm- / Since Alfven waves have Sv = SB/y/Airp, another way to phrase the "wave-stabilization" criterion is crv/cs = ts/t{ > 2L/Lj — 2nj = 2ts/ts (i.e. ts > 2tf). The fact that observed clouds are known to have av/cs — 1.7nj (i.e. t% = 1.7£f), and may well have sufficient fluctuating field strengths to meet the criterion (2.10), has led to the hypothesis that GMCs are indeed supported (in all directions) over times £ i g by a combination of mean fields and fluctuating fields. Since nonlinear MHD waves are dissipative, any stabilizing effects from the time-dependent fields will decrease over time; thus, whether the quasilinear-theory expectation is realized or not must depend on the nonlinear dissipation rate of the turbulence. Over very long times, stabilization per force requires mechanical energy replenishment (which can be supplied in astronomical contexts by the star formation process). Direct numerical MHD simulations are required to evaluate these ideas quantitatively; some of our results are highlighted in the next section. 3. Numerical Simulations - Selected Results Our group has performed a variety of simulations to systematically explore compressible MHD turbulence in self-gravitating cloud models in 1 2/3, 2 1/2, and 3 dimensions.f X For self-gravitating magnetized cloud equilibria to exist, the ratio of the central column density to central field strength Y,/B must be smaller than the same value l/(27r\/G) (Mouschovias & Spitzer 1976, Tomisaka, Ikeuchi, & Nakamura 1988). f In "1 1/2 D" and "2 2/3 D" restricted geometry, there are respectively one or two independent spatial variables, but all components of v and B evolve in time (subject to V • B = 0).
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These models either start with a turbulent velocity field with a spectrum dE = (l/2)v2(k)dk oc k~2dk which freely decays, or else they include stochastic impulsive forcing on wavelengths small compared to the box, such that the kinetic energy input rate is constant. In the latter case, a quasisteady state is achieved. These numerical surveys have allowed us to address a number of questions about the fundamental nature of these turbulent flows, including: • How do the survival times for self-gravitating clouds depend on their initial kinetic energy, magnetic field strength, and size (described by the dimensionless parameters M = av/cs - ts/tf, j3 = (tA/ts)2, and nj = ts/te)? • Is it possible to support a cloud indefinitely, given sufficient mechanical energy inputs? • How do the temporal profiles for decaying turbulence, and the dissipation rates for quasisteady turbulence, depend on M and /?? • How do the density, velocity, and magnetic field distributions and power spectra depend on M, (3, and nj? Full presentations of our surveys and results to date appear in Gammie & Ostriker (1996), Ostriker (1997), and Ostriker, Gammie, & Stone (1998). Stone's contribution to this volume discusses recent results on comparative dissipation rates and power spectra with varying (3 in high-resolution 3D simulations. Other work in preparation analyzes the distributions of axis ratios and orientations (relative to the magnetic field) of 3D clumps and their 2D projections (Gammie, Stone, & Ostriker 1998). Here, we will highlight recent findings in two areas: comparisons of the survival times of cloud against gravitational collapse for varying A4, (3, and nj, and differences in the density structure that arise in clouds when varying the field strength. 3.1. Simulation results on cloud support Our first set of surveys, in 1 2/3D restricted geometry, confirmed the quasilinear-theory predictions for cloud support along the meanfieldby nonlinear analogues of Alfven waves. We found that low-turbulence model clouds (i.e. av/cs < 2nj) collapsed along the mean field in times t < £g, higher-turbulence models have delayed collapse, while in the highestturbulence cases, collapse was forced by the strong compression from nonlinear waves. Overall, survival times increased with the strength of the mean magnetic field, because the turbulent dissipation rate was lower. We also performed a survey of forced-turbulence self-gravitating experiments in 1 2/3 D, and found that quasi-steady, non-collapsed states can be attained as long as the input power (applied mainly at a scale 8 times smaller than the box) exceeded Ewave > 24/3°-24nj pc^. The corresponding energy replenishment timescale to maintain a steady state would be Ewave/E w 0.2/3~1//4ig ~ 0.3/3~1//4if. In 2 1/2D geometry, where the fluid is free to contract in the direction transverse to the mean field (this motion is restricted in 1 2/3D), we find that turbulent motions are unable to prevent cloud collapse unless the field is strong enough to make the clouds magnetically subcritical. Unmagnetized clouds collapse at « 0.5£g (corresponding to 5 Myr at typical conditions) regardless of the initial kinetic energy level; weakly-magnetized (supercritical) clouds generally last up to 0.5 - l£g before collapsing (some small, magneto-Jeans stable clouds survive up to 1.5tgbefore collapsing), while more-strongly magnetized (subcritical) clouds can last beyond 1.5tg with no signs of collapse. The subcritical models, which are most similar to the 1 2/3D models in that they are unable to collapse perpendicular to the mean field, confirm the conclusion that time-dependent fluctuating magnetic field can prevent gravitational collapse along the mean field for times 3> 0.3£g. However, since the majority of real molecular clouds are probably supercritical, our models imply that they are unlikely to last for times greater than 5 — 10 Myr before some parts of their
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interior undergo collapse and initiate star formation. Further studies in 2 1/2D geometry are now in progress to asses the continuous energy input rate, as a function of /3, required to sustain clouds against collapse. We have also performed a number of self-gravitating simulations in 3D with varying (3 and nj, and found that, similar to the results in 2 1/2D, supercritical clouds collapse at times ~ 0.5ig even when their initial perturbation energies are large, whereas subcritical clouds can survive to later times. Further 3D simulations are planned to expand the parameter space, and further test these ideas. We can compare our results so far to the expectations of linear and quasilinear theory for cloud support. Our numerical experiments in 2 1/2D and 3D verify that magnetically supercritical (nj/?1 / 2 ^ > 1) clouds typically gravitationally collapse in times ,$, £g, whereas subcritical clouds can survive much longer without collapsing. In subcritical clouds, matter eventually slides along the field and forms thin sheets which do not themselves fragment (but may oscillate with respect to one another for some time). However, for lower field strengths the turbulent dissipation is rapid enough that long-term cloud support along the mean field is not possible. That is, even if clouds are magneto-Jeans stable and have initial turbulence levels exceeding crv/cs > 2nj, they have sufficient dissipation that they can collapse in all directions in times shorter than ts (contrary to what quasilinear theory would predict if dissipation is ignored). 3.2. Simulation results on density distributions An important way to make comparisons between simulated clouds and real clouds, so as to discriminate among the possible values of the input model parameters, is to analyze density distributions. In all of our models, we find that the cloud density structure becomes very clumpy and filamentary, due to the combination of turbulent Reynolds (i.e. ram pressure) and magnetic stresses. Figure 1 shows examples of snapshots of cloud structure in two 2 1/2 turbulent-decay models. Both models have initial turbulent energies with M2 = (av/cs)2 = 200. One has a stronger (3 — 0.01 mean field; the other has a weaker /? — 0.1 mean field (corresponding to BQ = 14/^G and B$ — 4.4/iG for fiducial GMC parameters, cf. eq. 2.7). The two models' snapshots are taken when the
FIGURE 1. Snapshots of density (contours, starting at log(p/p) = 0 with logarithmic increments 0.1), velocity field (vectors), and magneticfieldlines in ft = 0.01 (left) and P = 0.1 (right) 2.5D decay simulations when crv/cs ss 10. nj = 3 for both models.
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kinetic energy has declined to half of the initial value, i.e. to M = 10 (corresponding to 3D velocity dispersion 2-3km sounder GMC conditions); this is before self-gravity has become very important (the specific gravitational binding energy is only -l.lCg, -1.2Cg for the (3 — 0.01, 0.1 models, respectively).
FIGURE 2. Density distributions for snapshots shown in Figure 1. Solid-line histograms show fractional volume as a function of log(p/p); dotted-line histograms show fractional mass as a function of log(p/p);
For each of these snapshots, we have tabulated the fraction of the total volume, and the fraction of the total mass, as a function of log(p/p), the logarithmic density contrast relative to the mean (p = Mtot/L3). These volume and mass distributions are shown in Figure 2. In both models, the mass distribution is centered at log(p/p) > 0, while the volume distribution is centered at log(p/p) < 0, as a consequence of matter clumping: most of the mass is in clumps which have densities higher than the mean for the whole cloud, whereas most of the volume is filled with matter at less-than-average densities. The (3 = 0.01 model has mass-averaged mean contrast (A-)M = 0.49 (with median contrast 0.52), and volume-averaged mean contrast (A-)y = ~0-50 (with median contrast -0.60). The (3 = 0.1 model has mass-averaged mean contrast (A-)M = 0.33 (with median contrast 0.31), and volume-averaged mean contrast (ir)v = -0.28 (with median contrast -0.34). The distributions are very roughly log-normal, which accounts for the values of (ir)v and (ZT)M being almost equal and opposite. We note that the density contrast in the (3 = 0.01 models is closer to the values ~ ±0.4 - 0.6 inferred for real clouds based on studies of clumping in 13CO (e.g. Bally et al 1987, Williams, Blitz, & Stark 1995). The difference shown in these examples in density contrast between models with different (3 at the same Mach number is characteristic of all of the models examined so far: we generally find greater contrast in models with /3 = 0.01 than in the corresponding models with /3 = 0.1. We have also found that the density contrast in (3 = 1 models is intermediate between the /? = 0.1 and /? = 0.01 cases. Physically, we believe that the increase in contrast toward high (3 can be attributed to stronger compressions arising directly from the compressive part of the velocity field (V • vne0) when the magnetic pressure B2/8n is smaller, whereas the increase in contrast toward low /? can be attributed to stronger compressions arising nonlinearly from the shear part of the velocity field - at kinks in
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2
the magnetic field lines - when B is larger. Further research aimed at clarifying these issues is now underway. I am grateful to my collaborators Charles Gammie and Jim Stone for their contributions to this work, and for comments which improved my presentation. Support for this work is provided by NASA under contract number NAG53840. REFERENCES BALLY, J., LANGER, W. D., STARK, A. A., & WILSON, R. W. 1987, ApJ, 312, CHANDRASEKHAR, S. & FERMI, E. 1953, ApJ,
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GAMMIE, C. F., STONE, J. M., & OSTRIKER, E. C. 1998, in preparation McKEE, C. F. & ZWEIBEL, E. G. 1995, ApJ, 440, 686 MCKEE, C. F., ZWEIBEL, E. G., GOODMAN, A. A., & HEILES, C. 1993, in Protostars and Planets III, ed. E. Levy & J. Lunine (Tucson: University of Arizona Press), 327 MOUSCHOVIAS, T., & SPITZER, L. 1976, ApJ, 210,
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Super-Alfvenic Turbulent Fragmentation in Molecular Clouds By PAOLO PADOAN1 AND AKE NORDLUND2 x
Instituto Nacional de Astroffsica, Optica y Electronica, Apartado Postal 216, Puebla 72000, Mexico 2 Astronomical Observatory and Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
The dynamics of molecular clouds are often described in terms of magneto-hydro-dynamic (MHD) waves, in order to explain the super-sonic line widths and the fact that molecular clouds do not seem to be efficiently fragmenting into stars on a free-fall time-scale. In this work we discuss an alternative scenario, where the dynamics of molecular clouds are super-Alfvenic, due to a lower magnetic field strength than usually assumed (or inferred from observations). Molecular clouds are modeled here as random MHD super-sonic flows, using numerical solutions of the three-dimensional MHD equations. A Monte Carlo non-LTE radiative transfer code is used to calculate synthetic spectra from the molecular cloud models. The comparison with observational data shows that the super-Alfvenic model we discuss provides a natural description of the dynamics of molecular clouds, while the traditional equipartition model encounters several difficulties.
1. Introduction Molecular clouds (MCs) are recognized to be the sites of present day star formation in our galaxy. The description of their dynamics is an essential ingredient for the theory of star formation. A lot of work has been devoted to understand i) how super-sonic random motions in MCs can persist for at least a few dynamical times and ii) why MCs do not collapse, or fragment gravitationally into stars, on a free-fall time-scale. The magnetic field has been advocated as the solution for both problems. Magneto-hydrodynamic (MHD) waves were believed to dissipate at a significantly lower rate then super-Alfvenic and super-sonic random motions. Moreover, the magnetic pressure could at the same time support a cloud against its gravitational collapse. Therefore, models of magnetized clouds have been proposed, where the magnetic energy is of the order of the internal kinetic or gravitational energy of the cloud. If the motions observed in MCs have velocities of the order of the Alfven velocity, the magnetic field strength should be about 25 fiG ubiquitously. However, many upper limits on the field strength in MCs are available from OH Zeeman splitting, which seem to indicate in most cases a value significantly lower than 25 fxG (eg Crutcher et al. 1993), although high density regions, favorable to the field detection, are usually selected by OH observations. The Zeeman splitting can only detect the field component in the direction along the line of sight, but the field orientation cannot be claimed to explain the majority of the low upper-limits, and statistically can account only for a factor 2 in the field strength. Field tangling is often used as a possible explanation for the lack of detection through OH Zeeman splitting, but field tangling is not significant in cloud models where the magnetic energy is in approximate equipartition with the kinetic energy of the random motions. In such models, the magnetic field is too strong to be significantly tangled by the flow. It has also been recently shown that MHD waves dissipate at almost the same rate as 248
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super-Alfvenic and super-sonic random motions (Mac Low et al. 1988a, 1998b, 1998c, Padoan & Nordlund 1998), and so the magnetic field is not anymore a particularly good candidate for explaining the molecular spectral line-width. In recent years, compelling evidence for fragmentation of MCs have accumulated. Structures are found in MCs down to extremely small scales (Falgarone et al. 1992, Langer et al. 1995). Such a fragmented structure is apparently not due to the gravitational fragmentation (see the next section), and should be explained by a model for the dynamics of MCs, together with the nature of the super-sonic random motions. While the old question was: "Why do not MCs collapse or fragment on a free-fall time-scale?", the more modern question should rather be: "Why are MCs so strongly fragmented?". In the context of a traditional equipartition model, where the magnetic field is dynamically strong, it is rather hard to understand the origin of the fragmentation, and the explanation must rely on the combined effect of gravity and ambipolar diffusion. Clearly, the existence of a density related time-scale and length-scale for the process of ambipolar diffusion, generates a great difficulty. On the other hand, in a model where the random motions are super-Alfvenic the answer to our question ("Why are MCs so strongly fragmented?") is trivial: the random super-sonic and super-Alfvenic motions generate a complex system of criss-crossing shocks, and therefore a network of strong density enhancements, due to the highly radiative nature of the gas. Since gravity enters this mechanism of fragmentation only in a second stage (collapse of dense unstable fragments) we refer to this process as turbulent fragmentation. In a series of papers we have discussed the turbulent fragmentation from different points of view, using numerical simulations of super-sonic MHD turbulence (Padoan et al. 1997a, 1998b, Padoan & Nordlund 1998). The description of the method and the setup of different numerical experiments can be found in those works, while here, for reason of space, we focus on the latest results. The idea that the observed random super-sonic motions might be at the origin of the complex structure of MCs and might play a role in the star formation process, is found in previous papers (Larson 1981, Hunter 1979, Hunter & Fleck 1982, Leorat et al. 1990, Falgarone et al. 1992, Elmegreen 1993, Vazquez-Semadeni 1994, Vazquez-Semadeni et al. 1995, 1997, Scalo et al. 1997). The present work concentrates on the discussion of the basic assumption of the turbulent fragmentation mechanism, that is the super-Alfvenic nature of the random motions.
2. Turbulent Fragmentation Once the hypothesis of a strong magnetic field is abandoned, super-sonic random motions can generate large density contrasts, due to the highly radiative nature of the gas. The super-Alfvenic model thus offers a simple solution to the problem of the observed fragmentation of MCs. We have run many numerical simulations of super-Alfvenic and super-sonic MHD turbulent flows, and studied their statistical properties. The probability density function (pdf) of the gas density is well approximated by a Log-Normal distribution (Padoan et al. 1997b, Scalo et al. 1997, Nordlund & Padoan 1998). A Log-Normal distribution implies that most of the mass concentrates in a small fraction of the volume, reminiscent of the small volume rilling fraction of the dense gas in MCs. A significant fraction of the mass of a typical MC model is in cores and filaments with densities as high as 105-106 cm" 3 , as found in real clouds. The turbulent fragmentation also offer an explanation for the filamentary and cobwebby morphology of the gas in MCs, since the natural topology of random compressible flows is made of filaments and cores distributed around voids.
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Padoan & Nordlund: Super-Alfvenic Turbulent Fragmentation in MCs 200
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FIGURE 1. Pdf of extinction (upper panels) and dispersion versus mean extinction (lower panels). From left to right: Lada et al. (1994), super-Alfvenic model, equipartition model.
Turbulence and gravity can act together, in the sense that the turbulent fragmentation can produce dense regions that are large enough to be gravitationally unstable and collapse into protostars. On the other hand, turbulence can also slow down the process of star formation, by producing dense fragments that are too small to be gravitationally bound and collapse (Padoan 1995). Observations of MCs on very small scale seem to confirm the presence of a fragmentation mechanism alternative to the gravitational instability. Falgarone et al. (1992) found gravitationally unbound and probably transient structures on very small scale in different MCs, that could not be the consequence of gravitational instability. Langer et al. (1995) found that the Taurus MC is fragmented into clumps with mass of 0.01 MQ and density of about 105 cm" 3 , also too small to be gravitationally bound. Turbulent fragmentation is certainly a candidate to explain the presence of such small unbound clumps in MCs. Is the turbulent fragmentation efficient enough to be the main fragmentation mechanism also inside dense star forming cores? If the answer is affirmative, then gravitational instability would be only the ultimate cause of the collapse/accretion of single protostars, and the study of the statistical properties of turbulent flows could be a viable way to formulate a theory for the origin of the stellar initial mass function (IMF) (Padoan et al. 1997b). This is the physical motivation behind mathematical models of the stellar IMF based on the assumption of the existence of scaling properties in the mass distribution inside MCs (Elmegreen 1997, 1998).
3. Stellar Extinction Stellar extinction measurements can be used to map the column density distribution of dark clouds. It can also be used to infer some statistical properties of the 3-D structure
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of dark clouds. Lada et al. (1994) proposed to use the plot of dispersion of extinction versus mean extinction to discriminate between different models of the structure of MCs. Padoan et al. (1997a) used the same plot to show that the gas density distribution in MCs is consistent with a Log-Normal distribution, with the same properties as the density distribution in super-sonic turbulent flows of an isothermal gas. Here we reproduce the plot of dispersion of extinction versus mean extinction using our numerical simulations as models of the mass distribution inside MCs. We simply project onto a 2-D plane the 3-D density field of a snapshot of one of our runs, select randomly a number of points on the 2-D plane, as to simulate the random positions of stars, and superpose a 2-D regular grid. On each cell of the grid a few "stars" are found, and the mean surface density and the dispersion around the mean are measured, using only the value of surface density where the "stars" are found. The results are plotted in Fig. 1, where the observational plot by Lada et al. (1994) is compared with the plots for a super-Alfvenic model and for an equipartition model. While the super-Alfvenic model compares rather well with the observations, the equipartition model does not. The reason is that the super-Alfvenic model is able to develop a very large density contrast with very low density regions (voids). The equipartition model instead, behaves much more like an elastic medium, due to the large magnetic pressure, and therefore is not able to produce a sufficient density contrast, with randomly distributed deep voids. Moreover, in the equipartition model, the high density regions tend to accumulate over 2-D structures perpendicular to the magnetic field direction, since significant gas compressions are possible only along magnetic field lines. Such more regular and sheet-like structure found in the equipartition model also contributes to the low contrast in the density field projected along a random direction. Stellar extinction measurements show therefore that the turbulent fragmentation mechanism is a good candidate to explain the observed fragmentation of MCs, and that the observed random motions are more likely to be significantly super-Alfvenic, rather than MHD waves.
4. Distribution of Magnetic Field Strength and B — n relation For conditions typical of MCs, the magnetic field is well coupled to the neutral gas through ion-neutral collisions. Only in the densest regions, on small scales, and assuming a low fractional ionization, can the ambipolar diffusion time-scale be comparable with the dynamical time-scale. The hypothesis of flux-freezing might therefore be thought to allow an approximate description of the evolution of the magnetic field topology and statistical distribution. Under this hypothesis, and assuming isotropic compressions, the magnetic field strength should depend on the local density a s B a n 2 / 3 . In our numerical simulations of the super-Alfvenic model, we indeed find a correlation between B and n, but the B - n relation has a very large dispersion. In 1283 runs, we find a range of values of B covering 2 or 3 orders of magnitude, at any given value of n. On the other hand, the B — n scatter plot has a well defined power law upper envelope. The slope of the upper envelope is initially close to unity, B oc n, and later decreases until B oc n 0 3 " 0 - 4 . The B — n relation is initially close to linear because 1-D compressions perpendicular to the direction of the magnetic field are initially dominant. Later on the magnetic field tends to align with the velocity field, and therefore the gas density tends to grow due to motions parallel to the field lines, that do not affect B. This causes the flattening of the B — n relation. The magnetic field and velocity vectors align because of the stretching of field lines by the flow. This partial alignment of the field lines with the flow, expressed by the flattening of the B — n envelope, is an important effect, because it allows even
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2. The B — n relation. The over-plotted contour lines are for the super-Alfvenic model (thick lines) and for the equipartition model (thin lines).
random isotropic kinematics to concentrate gravitational energy faster than magnetic energy, in regions of high density. In other words, the turbulent fragmentation in the super-Alfvenic model proceeds in such a way that dense cores tend to accrete mass along magnetic field lines and reduce their magnetic flux to mass ratio efficiently, even in the absence of ambipolar diffusion. The distribution of magnetic field strength in the super-Alfvenic model is characterized by a long exponential tail, which means that regions with field strength much larger than the mean exist with a finite probability. Because of the intermittent distribution of B, Zeeman splitting measurements may detect a field strength much larger than the mean field strength. This is the reason why we suggest that the detection of a 10-50 \xG field strength in some dense cores does not mean that the mean field in MCs is close to the equipartition value (about 25 fiG). In fact many low upper limits on the field strength have been obtained (eg Crutcher et al. 1993) in favor of a lower mean field strength. In Fig. 2 we present a compilation of observational results that seems to confirm the existence of a B - n relation, but also the presence of a scatter of about 2 orders of magnitude in B, for density values between 10 and 104 cm" 3 . Both the slope and the scatter are consistent with the prediction of the super-Alfvenic model (thick contour lines). Instead, the equipartition model (thin contour lines in Fig. 2) has an almost flat B - n envelope, and covers a too small range both in density and in magnetic field strength. Note that in 1283 simulations the largest density that can be reached in a MC model is about 105 cm" 3 , due to the limited resolution. With a larger resolution and the same rms Mach number of the model flow, the largest density could reach 106 cm" 3 , even without the assistance of gravity. The discrepancy between the equipartition model
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4
FIGURE 3. Left panel: line width versus integrated antenna temperature in the equipartition model, for a line of sight parallel to the magnetic field direction. Right panel: 12CO mean spectra; the J=2—)•! spectra are divided by 0.62.
and the observations would then be more obvious, while the super-Alfvenic model would account for the latest CN Zeeman splitting measurements (Crutcher 1998). The super-Alfvenic model and the turbulent fragmentation mechanism are therefore good candidates to interpret the Zeeman splitting observations of dark clouds as well, while the equipartition model would predict a too small range of values of B and n, and almost no correlation between the two.
5. Synthetic Molecular Spectra We have used our 1283 MHD runs to compute grids of 90 x 90 spectra of different molecular transitions. The radiative transfer calculations are performed with a non-LTE Monte Carlo code, described in Juvela (1997). Several statistical properties of the synthetic spectra are discussed in Padoan et al. (1998b) and compared with observational data in Padoan et al. (1998a). Here we present some recent results concerning the relation between the line width and the integrated antenna temperature, and the line width and line intensity ratios. Heyer et al. (1996) proposed to use the observed growth of line width with integrated temperature as a test for the magnetic field strength in MCs. Using synthetic spectra from the super-Alfvenic model we find that the line width grows with integrated temperature, and also its dispersion around the mean, as in the observations (Heyer et al. 1996, Padoan et al. 1998a). In the equipartition model, instead, for lines of sight perpendicular to the magnetic field, the growth of the line width is very limited, and its dispersion does not grow with integrated temperature. For a line of sight parallel to the magnetic field, the line width in the equipartition model does not grow at all with integrated temperature, as shown in Fig. 3, left panel. Falgarone k Phillips (1996) find the line intensity ratio RCo (2-1/1 - 0 ) = 0.62 ±0.08, constant in space and also across line profiles, in a region situated at the edge of the Perseus-Auriga complex. In Fig. 3 (right panel) the average 12CO J=l->0 and J=2-»1 spectra are plotted for two super-Alfvenic models representative of MCs on the scales of 5 and 20 pc. The J=2-»l spectra, divided by 0.62, are almost perfectly coincident with the J=l—>0 spectra, in agreement with the observations. The left panel of Fig. 4 shows the i ? c o ( 2 - l / l - 0 ) ratio, but for velocity integrated temperature of single lines of sight. The plot is again consistent with the observations. A good agreement with the results of
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10 20 30 40 /T[12CO(1-0)]dv (K km/s)
10 20 30 40 /T[ 12 C0(1-0)]dv (K km/s)
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FIGURE 4. Left panel: Rco{2 - 1/1 - 0) = 0.62 ± 0.08 ratio, for velocity integrated temperature of single lines of sight. Right panel: 12CO J=2—>1 to 13CO J=2—»1 line width ratios.
Falgarone & Phillips (1996) is also found for the ratio of 12CO J=2->1 to 13CO J=2->1 line widths, plotted in the right panel of Fig.4 The comparison of synthetic spectra with the observations confirms again the validity of the super-Alfvenic model, and therefore supports the scenario of the turbulent fragmentation of MCs.
6. Conclusions In this work we have argued that the internal dynamics of MCs are super-Alfvenic and that MCs are primarily fragmented by the observed super-sonic random motions, rather than by the gravitational instability. We have referred to this process as turbulent fragmentation. Using numerical simulations of highly super-sonic MHD turbulent flows, and solving the radiation transfer problem through the numerical datacubes with a non-LTE Monte Carlo radiative transfer code, we have shown that the turbulent fragmentation process provides a natural explanation for i) the highly fragmented and filamentary structure of MCs; ii) the statistical properties of the mass distribution in MCs, as probed by stellar extinction measurements; iii) the Zeeman splitting measurements of the magnetic field strength; iv) the slope and the dispersion of the B — n relation; v) the molecular line intensity ratios; vi) the line width ratios; vii) the relation between the line width and the integrated antenna temperature. On the other hand, the equipartition model cannot easily account for i) the stellar extinction results; ii) the many low upper limits on B from Zeeman splitting measurements; iii) the slope and the scatter of the B — n relation; iv) the growth of the line width with integrated temperature. Moreover, it has recently been confirmed that equipartition MHD turbulence is approximately as dissipative as super-Alfvenic motions (Mac Low et al. 1988a, 1998b, 1998c, Padoan & Nordlund 1998), even if the highly dissipative ion-neutral friction is not considered. We conclude that the super-Alfvenic model we have proposed offers a reasonable description of the dynamics of MCs, since it provides natural explanations for all the observed properties of MCs that we have analyzed so far. The same is not true for the equipartition model. We thank all participants to the conference for useful discussions, and Jan Johannes Blom for reading carefully the manuscript and suggesting corrections.
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Decay Timescales of MHD Turbulence in Molecular Clouds By MORDECAI-MARK MAC LOW1, RALF S. KLESSEN1, ANDREAS BURKERT 1, AND MICHAEL D. SMITH2 1
Max-Planck-Institut fur Astronomie, Konigstuhl 17, Heidelberg, Germany
2
Astronomisches Institut der Universitat Wiirzburg, Am Hubland, Wiirzburg, Germany
We compute 3D models of supersonic, sub-Alfvenic, and super-Alfvenic decaying turbulence, with initial rms Alfv^n and Mach numbers ranging up to five, and an isothermal equation of state appropriate for star-forming interstellar clouds of molecular gas. We find that in 3D the kinetic energy decays as t~v, with 0.85 < 77 < 1.2. In ID magnetized turbulence actually decays faster than unmagnetized turbulence. We compared different algorithms, and performed resolution studies reaching 2563 zones or 703 particles. External driving must produce the observed long lifetimes and supersonic motions in molecular clouds, as undriven turbulence decays too fast.
1. Introduction Molecular cloud lifetimes are of order 3 x 107 yr (Blitz & Shu 1980), while free-fall gravitational collapse times are only iff = (1.4 x 106 yr)(n/10 3 cm" 3 )" 1 / 2 . In the absence of non-thermal support, these clouds should collapse and form stars in a small fraction of their observed lifetime. Supersonic hydrodynamical (HD) turbulence is suggested as a support mechanism by the observed broad lines, but was dismissed because it would decay in times of order t^. A popular alternative has been sub- or trans-Alfvenic magnetohydrodynamical (MHD) turbulence, which was first suggested by Arons & Max (1975) to decay an order of magnitude more slowly. (Also see Gammie & Ostriker 1996). However, analytic estimates and computational models suggest that incompressible MHD turbulence decays as t~v, with a decay rate 2/3 < r) < 1.0 (Biskamp 1994; Hossain et al. 1995; Politano, Pouquet, & Sulem 1995; Galtier, Politano, & Pouquet 1997), while incompressible HD turbulence has been experimentally measured to decay with 1.2 < T) < 2 (Comte-Bellot & Corrsin 1966; Smith et al. 1993; Warhaft & Lumley 1978). The difference in decay rates between incompressible HD and MHD turbulence is clearly not as large as had been suggested for compressible astrophysical turbulence. In this work we compute the decay rates of compressible, homogeneous, isothermal, decaying turbulence with supersonic, sub-Alfenic, and super-Alfvenic root-mean-square (rms) initial velocities vrms, and show that the decay rates in these physical regimes, 0.85 < 77 < 1.2, strongly resemble the incompressible results. These results are also presented in Mac Low et al. (1998).
2. Numerical Techniques We use both a finite difference code and an SPH code for our HD models, while for our MHD models we use only the finite difference code. Thisfinite-differencecode is the well-tested MHD code ZEUS (Stone & Norman 1992a, 1992b), which uses second-order Van Leer (1977) advection, and a consistent transport algorithm for the magnetic fields (Evans & Hawley 1988). It resolves shocks using a standard von Neumann artificial viscosity, but otherwise includes no explicit viscosity, relying on numerical viscosity to provide dissipation at small scales. This should certainly be a reasonable approximation 256
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1. ID, isothermal, M = 5 models with ZEUS, and comparisons to 3D models. Resolutions, physics, and dimensionality are given in the figure.
for shock-dominated flows, as most dissipation occurs in the shock fronts, where the artificial viscosity dominates in any case. Our resolution studies show that our major results are, in fact, independent of the resolution, and thus of the strength of numerical viscosity. SPH is a particle based approach to solving the HD equations described, for example, by Benz (1990) and Monaghan (1992), in which the system is represented by an ensemble of particles, each carrying mass, momentum, and fluid properties. We used the specialpurpose processor GRAPE (Ebisuzaki et al. 1993), to accelerate computation of nearestneighbor lists for the SPH algorithm (Steinmetz 1996). We chose initial conditions for our models inspired by the popular idea that setting up velocity perturbations with an initial power spectrum P(k) oc ka in Fourier space similar to that of developed turbulence would be in some way equivalent to starting with developed turbulence, as adopted by, among others, Padoan & Nordlund (1997), and Porter, Pouquet, & Woodward (1992, 1994). Observing the development of our models, it became clear to us that, especially in the supersonic regime, the loss of phase information in the power spectrum allows extremely different gas distributions to have the same power spectrum. For example, supersonic, HD turbulence has been found in simulations by Porter, Pouquet, & Woodward (1994) to have a power spectrum a = - 2 . However, any single, discontinuous shock wave will also have such a power spectrum, as that is simply the Fourier transform of a step function, and taking the Fourier transform of many shocks will not change this power law. Nevertheless, most distributions with a — — 2 do not contain shocks. After experimentation, we decided that the quickest way to generate fully developed
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FIGURE 2. Comparison of density and velocity profiles of ID hydro and MHD models demonstrates why ID MHD models dissipate faster: they have more dissipation regions, due to their more complex physics.
turbulence was with Gaussian perturbations having a flat power spectrum a = 0 with a cutoff at kmax — 8. In all of our models we take c s = 0.1, initial density p0 = 1, and we use a periodic grid with sides L = 2 centered on the origin.
3. One-Dimensional Results To verify our numerical methods, we reproduced the ID, MHD results of Gammie & Ostriker (1996). The first panel of Fig. 1 shows the results of a resolution study comparable to their Figure 1, with initial rms Mach number M = 5, initial uniform field parallel to the z-axis, and initial rms Alfven number A = urms/i>A = 1, where v\ = B2/47T/90- Note that t — 20 in our units corresponds to t = 1 in theirs. Aside from a rather faster convergence rate in our study, attributable to the details of our choice of initial conditions, we reproduce excellently their result: a decrease in wave energy -Ewave = EK + (By + B%)/8n by a factor of five in one sound-crossing time L/cs. We then extended our study by examining the equivalent HD problem, as shown in the third panel of Fig. 1, only to find that the decay rate of HD turbulence in ID is significantly slower than that of MHD turbulence. This appears to be due to the sweeping up of slower shocks by faster ones in the HD case, resulting in the pathological case of pure Bergers turbulence, as shown on the left in Fig. 2, as predicted by, e.g. Lesieur (1997). As a result there are very few dissipative regions, and energy is only lost very slowly. In contrast, multiple wave interactions occur in the MHD case shown on the right in Fig. 2, producing many dissipative regions and so faster dissipation.
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-0.10 -
0.01 10.0 FIGURE 3. 3D resolution studies for M—5, isothermal models. Linear resolutions vary by a factor of two between lines for the ZEUS runs, while particle number varies by seven. "Weak field" corresponds to A = 5, while "strong field" corresponds to A — 1.
Finally we compared ID models with 256 zone resolution to equivalent 3D models with 2563 zones. The 3D model loses energy far faster than the ID model in both the the MHD and HD cases, as shown on the right in Fig. 1. The increased number of degrees of freedom available in 3D presumably produces more shocks and interaction regions, again resulting in increased energy dissipation.
4. Three-Dimensional Results We next performed resolution studies using ZEUS for three different cases with M — 5 and no field, weak field and strong field as shown in Fig. 3. The initial ratio of thermal to magnetic pressure is /3 = 2 for the weak field and /? — 0.08 for the strong field. We ran the same HD model with the SPH code to demonstrate that our results are truly independent of the details of the viscous dissipation. We also ran HD and MHD models with adiabatic index 7 = 1.4, as well as an isothermal model with initial M = 0.1. For each of our runs we performed a least-squares fit to the power-law portion of the kinetic energy decay curves shown in Fig. 3. A full description of the results is given by Mac Low et al. (1998). We find the results for the power laws appear converged at the 5-10% level, and, reassuringly, that the different numerical methods converge to the same result for the HD case, r] ~ 1. We find that highly compressible, isothermal turbulence decays somewhat more slowly, with rj = 0.98, than less compressible, adiabatic turbulence, with r/ = 1.2, or than incompressible turbulence, with 77 = 1.1 (also see Smith et al 1993; and Lohse 1994).
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Adding magnetic fields decreases the decay rate somewhat further in the isothermal case to 1) ~ 0.9, with very slight dependence on the field strength or adiabatic index. Even strong magnetic fields, with the field in equipartition with the kinetic energy, cannot prevent the decay of turbulent motions on dynamical timescales far shorter than the observed lifetimes of molecular clouds. The significant kinetic energy observed in molecular cloud gas must be supplied more or less continuously. If turbulence supports molecular clouds against star formation, it must be constantly driven. Some computations presented here were performed at the Rechenzentrum Garching of the MPG. ZEUS was used courtesy of the Laboratory for Computational Astrophysics at the NCSA. MDS thanks the DFG for financial support. REFERENCES ARONS, M., & MAX, C. 1975, ApJ,
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HOSSAIN,
STEINMETZ, M. 1996, MNRAS, 278, 1005
J. M. & NORMAN, M. L. 1992a, ApJ, 80, 753 J. M. & NORMAN, M. L. 1992b, ApJ, 80, 791 VAN LEER, B. 1977, J. Comput. Phys., 23, 276 WARHAFT, Z. & LUMLEY, J. 1978, J. Fluid Mech., 88, 659 STONE, STONE,
Numerical Magnetohydrodynamic Studies of Turbulence and Star Formation By D. S. BALSARA1, A. POUQUET 2 , D. WARD-THOMPSON3 AND R. M. CRUTCHER 1 'N.C.S.A., University of Illinois at Urbana-Champaign, Illinois, U.S.A. 2
Observatoire de la Cote d'Azur, France
3
Royal Observatory, Blackford Hill, Edinburgh, U.K.
In this paper we examine two problems numerically. The first problem concerns the structure and evolution of MHD turbulence. Simulations are presented which show evidence of forming a turbulent cascade leading to a self-similar phase and eventually a decay phase. Several dynamical diagnostics of interest are tracked. Spectra for the kinetic and magnetic energies are presented. The second problem consists of the formation of pre-protostellar cores in a turbulent, magnetized molecular clouds. It is shown that the magnetic field strength correlates positively with the density in keeping with observations. It is also shown that the density and magnetic fields organize themselves into filamentary structures. Through the construction of simulated channel maps it is shown that accretion onto the cores takes place along the filaments. Thus a new dynamical process is reported for accretion onto cores. We have used the first author's RIEMANN code for astrophysical fluid dynamics for all these calculations.
1. Introduction The conference for which this paper is being written has been instrumental in opening the eyes of astronomers to the need for understanding turbulent processes in astrophysics. While several astrophysical environments where turbulent processes could be important were identified by numerous contributors in this conference, the pulsar scintillation measurements and the study of lines in molecular clouds provide two environments where the need for magnetohydrodynamic (MHD) turbulence is observationally well-founded. Since the MHD equations are highly non-linear analytical approaches sometimes prove to be of limited utility. As a result, the numerical study of MHD turbulence in non-selfgravitating and self-gravitating environments becomes a very useful tool. This has been aided by the availability of very accurate and reliable numerical methods that use higher order Godunov methodology for numerical MHD, see Roe and Balsara (1996) and Balsara (1998a,b) . Such methods have been implemented in the first author's RIEMANN code for astrophysical fluid dynamics. Several non-self-gravitating and self-gravitating simulations have been carried out by these authors. In Section II we discuss MHD turbulence decay. In Section III we discuss pre-protostellar core formation. In Section IV we give some conclusions.
2. Decay of MHD Turbulence The computation is done on a uniform grid of 2563 points, with periodic boundary conditions, adequate for homogeneous flows. Initial conditions are centered in the large scale, with a random distribution of Fourier modes with an exponential fall-off in the smallest scales. The initial ratio of longitudinal to transverse velocity fluctuations is ~ 0.08%. In the computation that is reported on here, there is no uniform magnetic field, and the turbulent magnetic energy is initially in statistical equipartition with the 261
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FIGURE 1. (a) shows the evolution of the kinetic , "V" , and magnetic energies, "M" , as a function of time that is normalized by the turn-over time, (b) does the same for the corresponding enstrophies.
kinetic energy. Density is normalized to unity, the initial sound speed is 0.24 and the initial r.m.s. Mach number is equal to unity. The computation is done for roughly one and a half acoustic (and nonlinear eddy turn-over)times r ac (r oc = 4.0 in code units). In this study we : (i) examine the overall temporal dependence of relevant variables and (ii) examine the statistical properties of the fluid at the level of Fourier mode spectra. The Mach number, initially unity, decreases to a final value of 0.36 based on the r.m.s. velocity. The speed of sound has increased by 9% because of heating due to shocks. Fig la shows the evolution of the kinetic and magnetic energies as a function of time that is normalized by the turn-over time. Fig lb does the same for the corresponding enstrophies. Thus the line denoted by "M" in Figlb indicates the magnetic enstrophy which is just the current and is, therefore, a measure of the dissipation in the field. Similarly, the line denoted with "V" is the vorticity. In this evolution, a plateau can be observed which is due to acoustic exchanges between kinetic and internal energy. We observe that shocks develop rapidly as exemplified by the growth of the second moment of the compressible part of the velocity field, with a maximum at t = 1 . ( Times are either given in code units of roughly 0.25rac or as times that are normalized by rac . ) The existence of magnetic shocks imply the early development of small-scale currents as well, as observed. The development , through mode-coupling and cascading to small scales, of the vorticity follows but is slower, with a maximum of the enstrophy reached around t = 3 which corresponds to 0.75rac in Fig lb . During that second phase (1. < t < 3.), the compressible excitation at small scale decreases, whereas the electric current, subject to mode coupling through the induction equation, continues to grow and reaches its maximum at t = 3 , with a total increase by a factor of 16 (and of 10 for the vorticity). In the last phase of development (3. < t < 6.) the self-similar decay of energy begins, with the spectra preserving their power-law shape. The flow initially dominated by vortices remain in that regime, with a peak in the ratio of compressible to solenoidal
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FIGURE 2. (a) and (b) show stacked sequences of the kinetic energy and magnetic energy spectra. A unit offset in the vertical direction is put between successive spectra which are shown at equal intervals of time.
energy of 0.19 at t = 1. This peak is stronger when stressing small scale properties, measuring the ratio of r.m.s. divergence to vorticity, which equals 0.7 at t=l. In apparent contradiction with these observations - since the current is the dissipation of the magnetic energy - the magnetic energy grows during that phase. This can be seen from Fig la. This is due to the nonlinear coupling between velocity and magnetic field, with a net transfer to the magnetic mode. Indeed, at the final time of the computation, the kinetic energy Ev has decreased by 80%, whereas the magnetic energyEM has decreased by only 52%. The end result is an excess of magnetic energy with EM/Ev = 1.76 at t ~ Tac, a feature commonly observed in computations of incompressible MHD also. The spectra, as expected, develop in time with the formation of small scales. Density fluctuations develop as well at all scales. The density contrast is still sizeable at the final time. Figs 2a and 2b show stacked sequences of the kinetic energy and magnetic energy spectra. A unit offset in the vertical direction is put between successive spectra which are shown at equal intervals of time. The energy spectra appear to follow a —2 law for the compressible part of the velocity and for the magnetic field as well. As pointed out by Balsara, Crutcher and Pouquet (1997) this spectral index is an implicit validation of Larson's laws.
3. Formation of Pre-Protostellar Cores We have carried out several simulations of pre-protostellar core formation in a magnetized, self-gravitating patch of molecular cloud. The size of the simulation is arranged so that it has several Jeans lengths in it. As demonstrated in Crutcher et al (1993) there is a positive correlation between the density and the magnetic field. In Fig 3 we show the mean magnetic pressure as a function of density, denoted by " 0", from one of our simulations. When possible we also show the one sigma fluctuation bounds on the mean, denoted by "1" , as obtained from our simulations. The mean density is normalized to unity. It becomes apparent from Fig 1 that at lower densities the magnetic
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3. shows the mean magnetic pressure as a function of density, denoted by "0". When possible we also show the one sigmafluctuationbounds on the mean denoted by "1" .
FIGURE
pressure shows a very tight positive correlation with the density. The higher densities probe regions where cores form. They too show a positive correlation though they show a considerable amount of scatter. The very highest densities could be poorly sampled. However, the correlation of magnetic field with density seems to be inverted. On examining the data we found that several of the densest cores form in regions with magnetic null points. Those are regions with the least magnetic pressure support and would, therefore, be most prone to gravitational collapse. Thus a dynamically consistent reason is found that would explain this inverted correlation at the highest densities. Examination of the data has shown that the cores are often interconnected by filamentary structures in both the density and the magnetic field. Cuts through the data, presented in Balsara, Crutcher and Pouquet (1996) , also support this view. Extensive volumetric rendering has shown that that cores are often not threaded by a magnetic field with a single polarity. This would explain why the cores often have multiple magnetic and density filaments emerging from them. A single polarity of magnetic field threading a core would imply that each core has just two filaments emanating from it. Furthermore, it would imply that the core assumes an hourglass morphology. The cores that form in our simulations, by and large, do not display such an idealized morphology. The complexity of the morphologies of the cores and the filaments that connect to them is consistent with a more complicated magnetic field topology. We have, in fact, carried out a JCMT study of the morphologies of fifty cores and found that only one has the idealized hourglass morphology! In Figs 4a - 4f we show simulated channel maps from a quadrant of one of our simulations. These have been done assuming optically thin radiative transfer and should correlate with isotopic lines. Fig 4a corresponds to line center. We invite the reader to focus on the most prominent core in the simulation which is in the center and towards the top in Fig 4a. Fig 4b shows that on shifting away from line center the maximal intensity comes from a location around the core but not on the core. Fig 4c shows a filament going off to the right as also an intensity peak going off to the north-west of
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4. shows a mosaic of simulated channel maps with velocity shifts from the mean shown on the top of each plot.
that core. Fig 4d shows that that trend continues. Fig 4e shows the original right-going filament bending and forming an arc. The fact that there is a variation in velocity along filaments is very interesting because it provides evidence for accretion onto the cores. Remember that these filaments are also regions of strong magnetic field. The magnetic field constrains the gas to flow along the filament and onto the core that it joins. Thus the simulations have provided a new insight into the nature of accretion onto cores and show that filaments of strong magnetic field play an important role in modulating the accretion. In Balsara et al (1988) we have made intercomparisons of this process with actual channel maps for S106 and found that the observational data shows the same trend as the simulated data thus lending support to our claim that magnetized filaments modulate accretion onto the cores.
4. Conclusions We have analyzed the spectra for MHD turbulence and found that the kinetic and magnetic energies follow a power law with a spectral index of — 2 . Diagnostics have
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been developed to show that a true inertial range has been achieved in our simulations. We also show that we obtain a self-similar decay. We have also shown the positive correlation between density and magnetic pressure in self-gravitating turbulence simulations. We show that the morphology consists of density and magnetic field filaments interconnecting the cores. Using simulated channel maps we show that the strongly magnetized filaments serve to channel the accretion onto the cores. Thus a new mechanism is found for accretion onto cores. It has been compared with actual observations for SI06.
REFERENCES D. S. 1998a, ApJS, 116, in press D. S. 1998b, ApJS, 116, in press D. S., CRUTCHER, R.M. & POUQUET, A. 1996, in Star Formation Near and Far, ed. S. Holt & L. G. Mundy, 89
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Direct Numerical Simulations of Compressible Magnetohydrodynamical Turbulence By JAMES M. STONE Department of Astronomy, The University of Maryland, College Park, MD 20742-2421 [email protected] We report the results of three-dimensional, direct numerical simulations of compressible MHD turbulence relevant to the internal dynamics of molecular clouds. Models of both driven and decaying turbulence are considered. The decay rate of driven supersonic MHD turbulence is found to be large, of order of one eddy turnover time at the driving scale. Non-ideal MHD effects can increase this decay rate by a factor of about two. In models where the magnetic field is strong (strong enough that the velocity dispersion in the saturated state is less than the Alfven speed), the power spectrum of the turbulence is remarkably similar to the expectations of the theory of incompressible MHD turbulence.
1. Introduction Numerical tools are likely to play an important role in the investigation of MHD turbulence in cold molecular clouds if for no other reason than because the observed linewidths are highly supersonic, and as of yet there does not exist a comprehensive analytic theory of compressible MHD turbulence. Our group (C. Gammie, E. Ostriker, and myself) has begun a project to study systematically the internal dynamics of magnetized, self-gravitating molecular clouds in two- and three-dimensions. Our motivations are two-fold: not only do we wish to understand the dynamics of compressible MHD turbulence as a well-defined physics problem, but also we would like to use the dynamical models as a basis with which to interpret the enormous collection of astronomical observations of molecular clouds that have been collected over the past several decades. The issues being addressed by this project, along with some results from a campaign of two-dimensional simulations, are summarized by Ostriker (this volume) and (Ostriker, Gammie, & Stone 1998). In this paper, I briefly summarize some of the results of three-dimensional calculations. The simulations all use a version of the ZEUS compressible MHD code (Stone &; Norman 1992). The initial conditions consist of a cubic box of size L which contains a plasma with uniform density p0 threaded by a uniform magnetic field in the a;-direction, B = (Bx, 0,0). The sound speed in the fluid Cs is constant, so that the relevant timescales for the dynamics are the sound crossing time ts = L/Cs, and the Alfven crossing time tA = L\Z4irpo/Bx. The dynamics is computed using an isothermal equation of state. We use periodic boundary conditions in each direction, and grid resolutions which vary between 323 and 5123. Most of the simulations assume the magnetic field is perfectly coupled to the fluid (the ideal MHD approximation), but in some models we include ambipolar diffusion in the strong coupling limit; this is described more fully below. In some simulations we include self-gravity computed using Fourier transform techniques. Additionally, in all models we evolve a passive contaminant which initially fills a cylindrical volume in the center of the grid orientated with the symmetry axis parallel to B and with diameter and axial length equal to L/2. Studying the distribution of this contaminant at later times not only allows us to follow field line tangling (at least for 267
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the ideal MHD simulations), but also allows study of the mixing rate of passive chemical species in compressible MHD turbulence. Following (Gammie & Ostriker 1996), we study three kinds of turbulence models: (1) randomly driven turbulence, (2) decaying turbulence from saturated initial conditions, and (3) decaying turbulence with self-gravity. We present results from the first kind of models in the next section, and from the second kind in section 3. Results from the third kind of models are given by Ostriker (this volume) and Ostriker, Gammie & Stone (1998) for 2-D simulations, and by Gammie, Stone, & Ostriker (1998) for 3-D simulations.
2. Results from Driven Turbulence Models To drive turbulence in our simulations, we add random velocity perturbations 5v at discrete time intervals At with At/ts = 0.001. The velocity perturbations are chosen to satisfy several constraints. First, we set the power spectrum of the fluctuations so that \6vl\ oc k6 exp(-8/c/fcpfc). This form gives a steeply rising spectrum at small wavenumbers with an exponential cut-off near k — kpk- We set kpk = 8. Second, the perturbations are drawn from a Gaussian random field and normalized so that the kinetic energy input rate is a constant value E. Third, we constrain the fluctuations so that no net momentum is added to the box. Finally, the fluctuations are chosen to be incompressible, i.e. k-<W = 0. There are three free parameters which describe our driven turbulence models. First we have the energy injection rate E. We study models with E/poL3Cg = 1000, which corresponds to about 10L© for typical cloud parameters (Gammie & Ostriker 1996). We find this value gives a saturated energy comparable to observed values. Second, we specify a magnetic field strength by choosing the ratio of the square of the sound to Alfven speeds /? = C^/V^. We study values of @ — 0.01,1.0 and oo (corresponding to pure hydrodynamics). Finally, the driving spectrum chosen above must be considered as a free parameter; we study how variation of both the form of the spectrum, and the wavenumber at which the spectrum is peaked, affect the saturated turbulence in our simulations. One motivation for this study is to measure the decay rate of saturated MHD turbulence. We define the wave energy Ew as the sum of the kinetic and magnetic energy associated with velocity and magnetic field fluctuations respectively. Studying how the wave energy varies with time reveals when and at what amplitude saturation occurs: at saturation the decay rate is then equal to the driving rate E. Figure 1 (left panel) plots Ew{t) for three simulations using different magnetic field strengths. Each model is computed at a resolution of 2563. The plot reveals that saturation occurs quickly, within only a few hundredths of a sound crossing time. Saturated amplitudes for Ew are around 15 - 20 in units of p(,L3Cg. The RMS sonic Mach number is about 5 in all models, whereas the RMS Alfvenic Mach number is roughly 0.5 in the /3 = 0.01 model, but is about 3 in the /3 = 1.0 model. Note this implies the time for the turbulence to saturate is roughly one eddy turnover time at the scale at which the turbulence is being driven. Through a large number of simulations which surveyed parameter space, (Gammie & Ostriker 1996) found an empirical saturation predictor for driven MHD turbulence in 1-2/3 D. Using this result (see eq. 19 in Gammie & Ostriker 1996), we would expect the amplitude of Ew at saturation to be 50 - 100 in our models, which is 3 - 5 times larger than actually observed in Figure 1 (left panel). Equivalently, this indicates the decay rate of compressible MHD turbulence is 5 - 10 times larger in three-dimensions than in one-dimension. We find the dissipation time in units of the sound crossing time
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FIGURE 1. left Wave energy versus time for the strong field 0 = 0.01 model (solid line), the weak field 0 = 1.0 model (dotted line), and the pure hydrodynamic 0 = oo model (dashed line). right Power spectrum for the 0 — 0.01 model (solid line), and the 0 = 1.0 model (dashed line). Also shown is the spectrum of velocity perturbations with which the turbulence is driven.
is tdiss/ts ~ Ew/E = 0.02. However, in units of the eddy turnover time at the driving wavenumber kpk, the decay time is of order unity. Several interesting results are revealed by studying the power spectrum of the velocity and magnetic field fluctuations in the saturated state at different magnetic field strengths. Figure 1 (right panel) shows the three-dimensional power spectrum (averaged over spherical shells) for f3 = 0.01 and 1.0 at a numerical resolution of 2563, as well as the spectrum with which the turbulence is driven. At high k, the spectra are clearly affected by numerical dissipation, evident as an increase in slope above k — 50 - 60 (corresponding to length scales of about 4 A i ) . At low k the spectra are flat, which is remarkable given that virtually no energy is fed into these wavenumbers by the forcing. Instead, the energy at large scales (small k) must come through an inverse cascade from smaller scales. At intermediate scales, we identify an inertial range. Although this inertial range does not span a large domain of wavenumbers, it is large enough to measure a slope of the power spectrum in each model. Interestingly, the best-fit slope measured from the weakly magnetized turbulence model is steeper compared to that measured from the strongly magnetized case. In fact, we find a slope for the /? = 1.0 model which is roughly —4, consistent with the expectation that strongly supersonic turbulence is dominated by shocks. On the other hand, the slope of the /? = 0.01 model is roughly -11/3, consistent with a Kolmogorov spectrum. In fact, theories of strong incompressible MHD turbulence predict the slope of the power spectrum should be nearly Kolmogorov (Goldreich & Sridhar 1995). It would therefore seem that compressibility effects are reduced if the velocity dispersion is comparable to or less than the Alfven speed, even though it may greatly exceed the sound speed. Figure 2 shows two-dimensional power spectra for the strongly and weakly magnetized turbulence models plotted as functions of wavenumber perpendicular and parallel to the mean field (k± and fc|| respectively). The contours are clearly circular in the weak field case, indicating the turbulence is isotropic. However, in the strong field case, the contours are elliptical; elongated perpendicular to the mean field. In fact, anisotropic turbulence with a cascade perpendicular to the field has emerged as one of the most
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characteristic properties of incompressible MHD turbulence. Thus, we find important similarities between our compressible MHD simulations and the theory of incompressible MHD turbulence (e.g., Goldreich k Sridhar 1995).
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FIGURE 2. Two-dimensional power spectra of driven models plotted along wavenumbers parallel fc|| and perpendicular k± to the mean magnetic field, left Result for weakly magnetized turbulence, right Result for strongly magnetized turbulence. Note the spectrum is anisotropic in this case.
By plotting the distribution of the passive contaminant along directions parallel and perpendicular to the field, it is also possible to determine mixing rates. Our preliminary analysis indicates that the mixing rate is isotropic in the weak field case, but 10 times larger along field lines than across them in the strong field case. Finally, we have studied the effect of ambipolar diffusion in the strong coupling limit on the decay rate of driven turbulence. In this limit, the inertia of the ions is neglected, allowing one to write the induction equation as
^
+ V x (B x un) = V x ( — ? — x [B x (V x B)]l,
(2.1)
where un is the velocity of the neutrals, pi and pn are the density of the ions and neutrals respectively, and 7 is the ion-neutral collision coupling constant. The ion density can be eliminated from the expression if ionization equilibrium is assumed, so that p, = Cpn where C is a constant. The importance of ambipolar diffusion to the dynamics can be measured in terms of an effective "Reynolds" number expressed as the ratio of the Alfven crossing time to the ion-neutral collision time. In real clouds, we expect this ratio to be (Re) A D =
tA
45
B
-1
(2.2)
We have computed driven MHD turbulence models including ambipolar diffusion in the strong coupling limit (i.e. we solve equation 2.1) using effective Reynolds numbers of 25 and 100. We find that the decay rate of driven turbulence in these models can be increased by a factor of about two above the ideal MHD case. Preliminary investigation of the power spectra in the models including ambipolar diffusion indicate no additional power at large k in the magnetic field, which might be expected if sharp current sheets were being enhanced by the ambipolar diffusion (Brandenburg & Zweibel 1994).
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3. Results from Other Models We have studied the properties of decaying supersonic MHD turbulence using the saturated state computed in the driven models as an initial condition. At some late time t0 we simply turn off the driving mechanism, and let the turbulence decay. Because these decay models begin from a self-consistent saturated initial condition (at least for the forcing scheme used here); this technique avoids any questions about what initial power spectrum to choose to describe the initial state. We find the wave energy in decaying turbulence models asymptotically falls as a power law in time with a slope near -0.85, in agreement with previous results (e.g. see MacLow et al, this volume). The slope shows only small variation with the strength of the mean magnetic field in the model. Because the wave energy decreases roughly as t~l, this means the dissipation rate, which is proportional to Ew/E\v, also is proportional to t~l. Therefore, the decay rate in these models is not well defined as it depends on the initial amplitude of the turbulence, and the time in the problem. 4. Summary We have used direct numerical simulations of driven compressible MHD turbulence to study questions of direct relevance to the dynamics of the cold ISM. We find the decay rate of supersonic turbulence is large, roughly one turnover time at the scale at which the turbulence is driven. The power spectrum of velocity and magnetic field fluctuations at saturation shows several interesting features, including evidence for an inverse cascade at large scales. Strongly magnetized turbulence (defined as a velocity dispersion in the saturated state less than the Alfven speed) shows an anisotropic power spectrum with more power perpendicular to the field than parallel to it. Moreover, the 3-D spectrum appears to have a slope close to Kolmogorov, whereas weakly magnetized and hydrodynamic compressible turbulence have a slightly steeper spectrum consistent with a flow dominated by shocks. These latter results are in accordance with the expectations of studies of strong incompressible MHD turbulence (see Sridhar, this volume, and Goldreich & Sridhar 1995). Much work has yet to be done in further studying the gas dynamics revealed by these simulations, and in directly comparing the properties of the turbulence to observations of molecular clouds. Support for this work is provided by NASA under contract number NAG53840. Computations were performed on the Cray/SGI Origin 2000 system at the National Center for Supercomputing Applications. I thank my colleagues Charles Gammie and Eve Ostriker who contributed significantly to the work presented here.
REFERENCES BRANDENBURG, A., & ZWEIBEL, E.G. GAMMIE, C.F., GAMMIE,
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GOLDREICH, P., & SRIDHAR, S. 1995, ApJ, 438, OSTRIKER, E . C , STONE,
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ApJ
Fragmentation in Molecular Clouds: The Formation of a Stellar Cluster By RALF KLESSEN AND ANDREAS BURKERT Max-Planck-Institut fur Astronomie, Konigstuhl 17, 69117 Heidelberg, Germany The isothermal gravitational collapse and fragmentation of a molecular cloud region and the subsequent formation of a protostellar cluster is investigated numerically. The clump mass spectrum which forms during the fragmentation phase can be well approximated by a power law distribution dN/dM <x M~15. In contrast, the mass spectrum of protostellar cores that form in the centers of Jeans unstable clumps and evolve through accretion and iV-body interaction is best described by a log-normal distribution. Assuming a star formation efficiency of ~10%, it is in excellent agreement with the IMF of multiple stellar systems.
1. Introduction Understanding the processes leading to the formation of stars is one of the fundamental challenges in astronomy and astrophysics. However, theoretical models considerably lag behind the recent observational progress. The analytical description of the star formation process is restricted to the collapse of isolated, idealized objects (Whitworth & Summers 1985). Much the same applies to numerical studies (e.g. Boss 1997; Burkert et al. 1997 and reference therein). Previous numerical models that treated cloud fragmentation on scales larger than single, isolated clumps were strongly constrained by numerical resolution. Larson (1978), for example, used just 150 particles in an SPH-like simulation. Whitworth et al. (1995) were the first who addressed star formation in an entire cloud region using high-resolution numerical models. However, they studied a different problem: fragmentation and star formation in the shocked interface of colliding molecular clumps. While clump-clump interactions are expected to be abundant in molecular clouds, the rapid formation of a whole star cluster requires gravitational collapse on a size scale which contains many clumps and dense filaments. Here, we present a high-resolution numerical model describing the dynamical evolution of an entire region embedded in the interior of a molecular cloud. We follow the fragmentation into dense protostellar cores which form a hierarchically structured cluster. 2. Numerical Technique and Initial Condition To follow the time evolution of the system, we use smoothed particle hydrodynamics (SPH: for a review see Monaghan 1992) which is intrinsically Lagrangian and can resolve very high density contrasts. We adopt a standard description of artificial viscosity (Monaghan &; Gingold 1983) with the parameters av = 1 and j3v = 2. The smoothing lengths are variable in space and time such that the number of neighbors for each particle remains at approximately fifty. The system is integrated in time using a second order Runge-Kutta-Fehlberg scheme, allowing individual timesteps for each particle. Once a highly condensed object has formed in the center of a collapsing cloud fragment and has passed beyond a certain density, we substitute it by a 'sink' particle which then continues to accrete material from its infalling gaseous envelope (Bate et al. 1995). By doing so we prevent the code time stepping from becoming prohibitively small. This procedure implies that we cannot describe the evolution of gas inside such a sink particle. For a 272
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detailed description of the physical processes inside a protostellar core, i.e. its further collapse and fragmentation, a new simulation just concentrating on this single object with the appropriate initial conditions taken from the larger scale simulation would be necessary (Burkert et al. 1998). To achieve high computational speed, we have combined SPH with the special purpose hardware device GRAPE (Ebisuzaki et al. 1993), following the implementation described in detail by Steinmetz (1996). Since we wish to describe a region in the interior of a molecular cloud, we have to prevent global collapse. Therefore, we use periodic boundaries, applying the Ewald method in an PM-like scheme (Klessen 1997). The structure of molecular clouds is very complex, consisting of a hierarchy of clumps and filaments on all scales (e.g. Blitz 1993). Many attempts have been made to identify the clump structure and derive its properties (Stutzki & Giisten 1990, Williams et al. 1994). We choose as starting conditions Gaussian random density fluctuations with a power spectrum P(k) oc l/kN and 0 < N < 3. Thefieldsare generated by applying the Zel'dovich (1970) approximation to an originally homogeneous gas distribution: we compute a hypothetical field of density fluctuations in Fourier space and solve Poisson's equation to obtain the corresponding self-consistent velocity field. These velocities are then used to advance the particles in one single big timestep St. We present simulations with 50 000 and 500 000 SPH particles, respectively. 3. A Case Study As a case study, we present the time evolution of a region in the interior of a molecular cloud with P(k) oc \/k2 and containing a total mass of 222 Jeans masses determined from the temperature and mean density of the gas. Figure 1 depicts snapshots of the system initially, and when 10, 30 and 60 per cent of the gas mass has been accreted onto the protostellar cores. Note that the cube has to be seen periodically replicated in all directions. At the beginning, pressure smears out small scale features, whereas large scale fluctuations start to collapse into filaments and knots. After t^O.3, the first highlycondensed cores form in the centers of the most massive and densest Jeans unstable gas clumps and are replaced by sink particles. Soon clumps of lower mass and density follow, altogether creating a hierarchically-structured cluster of accreting protostellar cores. For a realistic timing estimate, the Zel'dovich shift interval St = 2.0 has to be taken into account. In dimension-less time units, the free-fall time of the isolated cube is r^ = 1.4. 3.1. Scaling Properties The gas is isothermal. Hence, the calculations are scale free, depending only on one parameter: the dimensionless temperature T = E-mt/\Epot\, which is defined as the ratio between the internal and gravitational energy of the gas. The model can thus be applied to star-forming regions with different physical properties. In the case of a dark cloud with mean density n(H2) ^ 100 cm" 3 and a temperature T ~ 10 K like Taurus-Auriga, the computation corresponds to a cube of length 10 pc and a total mass of 6 3 0 0 M Q . The dimensionless time unit corresponds to 2.2 x 106yrs. For a high-mass star-forming region like Orion with n{H2) ^ 105cm~3 and T ~ 30 K these values scale to 0.5 pc and IOOOM0, respectively. The time scale is 6.9 x 104yrs. 3.2. The Importance of Dynamical Interaction and Competitive Accretion The location and the time at which protostellar cores form, is determined by the dynamical evolution of their parental gas clouds. Besides collapsing individually, clumps stream towards a common center of attraction where they merge with each other or undergo
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t = 1.3 FIGURE 1. Time evolution and fragmentation of a region of 222 Jeans masses with initial Gaussian densityfluctuationswith power law P(k) oc 1/fc2. Collapse sets in and soon forms a cluster of highly-condensed cores, which continue to accrete from the surrounding gas reservoir. At t = 0.7 about 10% of all the gas mass is converted into "protostellar" cores (denoted by black dots). At t = 1.3 and t — 2.0 these values are 30% and 60%, respectively. Initially the cube contains 50000 SPH particles.
further fragmentation. The formation of dense cores in the centers of clumps depends strongly on the relation between the timescales for individual collapse, merging and subfragmentation. Individual clumps may become Jeans unstable and start to collapse to form a condensed core in their centers. When clumps merge, the larger new clump continues to collapse, but contains now a multiple system of cores in its center. Now sharing a common environment, these cores compete for the limited reservoir of gas in their surrounding (see e.g. Price & Podsiadlowski 1995, Bonnell et al. 1997). Furthermore, the protostellar cores interact gravitationally with each other. As in dense stellar clusters, close encounters lead to the formation of unstable triple or higher order systems and alter the orbital parameters of the cluster members. As a result, a considerable fraction of "protostellar" cores get expelled from their parental clump. Suddenly bereft of the massive gas inflow from their collapsing surrounding, they effectively stop accreting and their final mass is determined. Ejected objects can travel quite far and resemble the weak
Klessen & Burkert: Fragmentation in Molecular Clouds 2.0 1.5
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1.0 ]
0.5 0.0 -0.5
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'iri/1.
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-1.5
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FIGURE 2. a) - d) Mass distribution of gas clumps (thin lines) and of protostellar cores (thick lines) at times t = 0.0, 0.7, 1.3 and 2.0 when 0%, 10%, 30% and 60% of the total gas mass is condensed in cores, respectively. The vertical lines indicate the resolution limit of the simulation with 500000 particles (Bate & Burkert 1997), and the dashed lines illustrate the observed clump mass spectrum with dN/dM oc M~ (Blitz 1993). e) Comparison of the final core mass spectrum (thick line) with different observationally based models for the IMF. The thick dashed line denotes the log-normal form for the IMF, uncorrected for binary stars as proposed by Kroupa et al. (1990). In order for the peaks of both distributions to overlap, a core star formation efficiency of 10% has to be assumed. The agreement in width is remarkable. The multiple power-law IMF, corrected for binary stars (Kroupa et al. 1993) is shown by the thin solid line. As reference, the thin dashed line denotes the Salpeter (1955) IMF. Both are scaled to fit at the high-mass end of the spectrum. All masses are normalized to the overall Jeans mass in the system. line T Tauri stars found via X-ray observation in the vicinities of star-forming molecular clouds (e.g. Neuhauser et al. 1995).
3.3. Mass Spectrum - Implications for the IMF Figures 2a - d describe the mass distribution of identified gas clumps (thin lines) and of protostellar cores (thick lines) that formed within unstable clumps in a simulation analogous to Fig. 1, but with 10 times higher resolution. To identify individual clumps we have developed an algorithm similar to the method described by Williams et al. (1994), but based on the framework of SPH. As reference, we also plot the observed canonical form for the clump mass spectrum, dN/dM oc M~1-5 (Blitz 1993), which has a slope of —0.5 when plotting N versus M. Note that our initial condition does not exhibit a clear power law clump spectrum. The Zel'dovich approximation generates an overabundance of small scale fluctuations. However, in the subsequent evolution, these small clumps are immediately damped by pressure forces and non-linear gravitational collapse begins to create a power-law like mass spectrum. A common feature in all our simulations is the broad mass spectrum of protostellar cores which peaks slightly above the overall Jeans mass of the system. This is somewhat surprising, given the fact that the evolution of individual cores is highly influenced by complex dynamical processes. In a statistical sense, the system retains 'knowledge' of its (initial) average properties. The present simulations cannot resolve sub-fragmentation in condensed cores. Since detailed simulations show that perturbed cores tend to break up into multiple systems (e.g. Burkert et al. 1997), we can only determine the mass function of multiple systems. Our simulations predict an initial mass function with a log-normal functional form. Figure 2e compares the results of our calculations with the observed IMF for mul-
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tiple systems from Kroupa et al. (1990). Assuming a typical Jeans mass Mj ss 1 M 0 and a star formation efficiency of individual cores of 10%, the agreement between the numerically-calculated mass function and the observed IMF for multiple systems (thick dashed line; from Kroupa et al. 1990) is excellent. For comparison, also the IMF corrected for binary stars (Kroupa et a. 1993) is indicated as thin solid line, together with the mass function from Salpeter (1955) as thin dashed line.
4. Discussion Large-scale collapse and fragmentation in molecular clouds leads to a hierarchical cluster of condensed objects whose further dynamical evolution is extremely complex. The agreement between the numerically-calculated mass function and the observations strongly suggests that gravitational fragmentation and accretion processes dominate the origin of stellar masses. The final mass distribution of protostellar cores in isothermal models is a consequence of the chaotic kinematical evolution during the accretion phase. Our simulations give evidence, that the star formation process can best be understood in the frame work of a probabilistic theory. A sequence of statistical events may naturally lead to a log-normal IMF (see e.g. Zinnecker 1984; Adams & Fatuzzo 1996; also Price & Podsiadlowski 1995; Murray & Lin 1996; Elmegreen 1997).
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MNRAS, 277, 727
Accretion Disk Turbulence By CHARLES F. GAMMIE1-2 1 2
Isaac Newton Institute, 20 Clarkson Rd., Cambridge, CB3 OEH, UK
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, 02138, USA
I review recent developments in the theory of turbulence in centrifugally supported astrophysical disks. Turbulence in disks is astrophysically important because it can transport angular momentum through shear stresses and thus allow disks to evolve and accrete. Turbulence can be initiated by magnetic, gravitational, or purely hydrodynamic instabilities; I give an abbreviated review of the linear and nonlinear theory of each of these possibilities, and conclude with a list of problems.
1. Introduction Spiral galaxies, quasars, active galactic nuclei, X-ray binaries, cataclysmic variables, and young stars: these are a few of the astronomical objects that contain disks. Disks are common in astrophysics because it is usually difficult to change the specific angular momentum of gas, but easy to radiate away its thermal energy. Gas injected into in a spherically symmetric potential thus naturally shocks, radiates, and settles down into a plane normal to its mean angular momentum. Because they are so common, disks occupy a lot of the astronomical community's time and energy (that would otherwise be entirely dissipated in attempting to measure f2o). Although there are enormous differences between individual disk systems in global structure and observational appearance, there are a number of fluid dynamical processes common to all disks. These processes are worth understanding in detail. The most fundamental process in disks, analogous to nuclear reactions in stars, is angular momentum transport. The disk cannot evolve unless gas in the disk can be persuaded to give up some of its angular momentum and spiral down the gravitational potential. Accretion disks (e.g. CV disks, but not spiral galaxies) are mainly heated by frictional processes associated with this gradual inflow; we would not see them at all absent some process for redistributing angular momentum. The central role of angular momentum transport is evident if we write down an equation for the evolution of the disk surface density E, obtained directly from the angular momentum and continuity equation in the limit that the disk is thin: r, t)
1
0 / 1 0 ,
2__,
,
here Cl = orbital frequency ~ r~q and t,w is the mass lost in a wind. Angular momentum is either redistributed (diffused) through the disk by the height-integrated and azimuthally averaged shear stress Wrip = J dzd<j>wr(f,/(2TV) (using cylindrical coordinates centered on the disk) or else removed directly from the disk by an external torque r per unit area, provided perhaps by a magnetohydrodynamic (MHD) wind. Without angular momentum transport (or a wind) the disk does not evolve. The shear stress wr
+ ^Q^9r9
(1-2)
278
Gammie: Disk Turbulence
Here 5 implies departures from the mean value and g is the gravitational field. One of the main goals of accretion disk theory is to calculate wr^,. Disk turbulence is interesting because of its importance to accretion disk evolution, but it is also interesting from a fluid-dynamical viewpoint as a model, self-sustained turbulent system. In this review I will discuss turbulence initiated by magnetohydrodynamical instabilities (Section 2), gravitational instabilities (Section 3), and purely hydrodynamical instabilities (Section 4). In conclusion (Section 5) I discuss directions for future research. Finally, there are a number of other recent reviews that focus on similar issues: Balbus & Hawley (1998) gives a complete discussion of the literature; Gammie (1998a) focuses on recent numerical experiments; Stone et al. (1998) discuss transport processes in protostellar disks; Brandenburg (1998) gives a different perspective on the numerical experiments with an emphasis on the connection to dynamo theory.
2. Magnetohydrodynamic turbulence 2.1. Balbus-Hawley instability: Linear Theory Of all the unstable modes discovered to date one stands out as having the largest growth rates in the largest part of disk "phase space." It is a local, linear, MED instability (here local means that it does not depend on the global radial or vertical structure of the disk) first understood in the context of accretion disks by Balbus & Hawley in 1991 (although it was discovered earlier in the context of magnetized Couette flow by Velikhov 1959). The BH instability is most easily explained using a mechanical analogy first developed by Balbus & Hawley (1992). Consider two equal masses in coplanar orbit in a Keplerian potential. The masses are close together in the sense that |<5r|/|r| -C 1, and they are connected by a spring with natural frequency 7. The masses represent "fluid elements"; the spring represents the magnetic field if j 2 = (k • VA) 2 , where VA = B/y/Anp is the Alfven velocity and k the wavevector of the perturbation. This analogy is exact in the case of a purely vertical magnetic field and a vertical wavevector in ideal MHD. In a frame coorbiting with the center of mass, the governing equations are <Jr = -2SI x6r + 3fi2<5r? - 72<5r,
(2.3)
where the terms on the right hand side are the Coriolis, tidal, and spring accelerations, respectively. These equations are already linear and so, taking 5r ~ est, we find that In the limit of a weak spring (-y2 -C Q2), s ~ ±%/3~7 or s ~ ±iU. The unstable member of the first pair is the BH mode; the second pair are just the usual epicyclic oscillations of the particles. One can recover several interesting properties of the BH instability from the dispersion relation: its maximum growth rate is 3Q/4; this maximum is achieved at 7 = \/l5fi/4; the instability vanishes if 7 > \/3fi. Returning to the analogous magnetic problem using 7 —> |k • VA|, we see the most remarkable property of the BH instability: no matter what the magnetic field strength, the maximum growth rate is always dynamical! As the field strength decreases, the fastest growing mode is simply pushed to smaller and smaller scales. The local, linear analysis of a magnetized disk for a general field is somewhat involved (Balbus & Hawley 1992). The main complication is differential rotation, but this can be handled by the shearing plane wave formalism invented by Goldreich & Lynden-Bell (1965). Because of differential rotation there are no nonaxisymmetric local modes, but this does not mean the BH mechanism is absent. Instead it takes the form of transient,
Gammie: Disk Turbulence
279
finite amplification of nonaxisymmetric, shearing plane waves, but the amplification factor can be made as large as required by adjusting the initial wavevector. It turns out, then, that the BH instability is present for any magnetic field orientation and a broad range of field strengths. Because the BH instability is so powerful and is present under such general conditions, it is an important matter to discover when the instability is not present. Two potentially relevant effects can shut off the BH instability. (1) The disk is poorly ionized. Then the instability can be damped by either ordinary resistivity (see Balbus & Hawley 1998; Jin 1996; Papaloizou & Terquem 1997) or ambipolar diffusion (Blaes & Balbus 1994). Protostellar disks (Hayashi 1981; Gammie 1996) and CV disks in quiescence (Gammie & Menou 1998) may fall within this regime. (2) The magnetic field is too strong. In this case one must examine the vertical or radial structure of a specific disk model to determine precise stability conditions (see Section 2.4 below). A convenient rule of thumb, however, is that a strong field can only shut off the instability if the smallest-scale unstable mode has wavelength comparable to the disk scale height, so that the unstable modes no longer "fit" within the disk. This happens when VA <; cs. Weaker fields are certainly unstable. It is unclear if any realistic disk model is stabilized by a strong magnetic field; one might expect that a strong field would be difficult to anchor within the disk, i.e. that the system would be unstable to some other global or quasi-global instability. Nevertheless strong fields have been proposed as one way of turning off turbulence in quiescent CV disks (Armitage et al. 1996).
2.2. Balbus-Hawley Instability: Nonlinear Outcome Studies of the nonlinear outcome of the BH instability have mainly used a local model of the disk. The local model is a first order expansion of the equations of motion in H(R)/R (H = disk scale height) in a frame comoving with some fiducial point in the disk. Together with a set of boundary conditions called the shearing box (see Hawley et al. 1995), this model allows one to study the nonlinear development of the BH instability in a practical fashion. Experiments typically evolve the equations of compressible MHD, sometimes including the vertical structure of the disk and sometimes not. They have used a variety of vertical boundary conditions. I have reviewed the results of these numerical experiments in detail elsewhere (Gammie 1998a). To give a brief summary, these experiments show that: (1) the instability leads to fully developed MHD turbulence; (2) this turbulence transports angular momentum outward; (3) in the absence of a mean field, this turbulence is self-sustained even in the presence of explicit or numerical dissipation. It can therefore be described as a dynamo; (4) the mean shear stress in the outcome, a = 2(iyr^)/(3pCg), depends on the mean magnetic field:
_ o.oi + 4 K ^
+
IK^>i,
(2.5)
where (VA) is the mean Alfven velocity; (5) Zero-mean field experiments with finite explicit resistivity saturate at a level that depends on the resistivity. At magnetic Reynolds numbers below about 103 turbulence and transport die away completely (Hawley et al. 1996); (6) the slope of the power spectrum of the turbulence is consistent with Kolmogorov, but the turbulence is anisotropic.
280
Gammie: Disk Turbulence
2.3. Quasi-Global MHD Instabilities Magnetized disks are also subject to quasi-global instabilities, by which I mean instabilities that depend on the local vertical structure of the disk, but not on its radial structure. The Parker and interchange modes are one example of a linear, quasi-global MHD instability. The original linear analysis is due to Tserkovnikov (1960) for the interchange mode and Newcomb(1961) for the Parker mode. The stability criterion for Parker modes in a stratified atmosphere with uniform gravity g, adiabatic index 7, and a horizontal magnetic field, is _ ^ > 5_£. (STABLE) (2.6) dz IV This is equivalent to the Schwarzschild criterion written as if the magnetic field were absent (it is, however, implicitly present in the equilibrium). I will only quote the stability criterion for the Parker mode, since it always goes unstable before the interchange mode. Notice that a large radiative diffusivity, as in accretion disks, can erase the stabilizing effects of stable stratification (Acheson 1978, Hughes 1985), driving the Parker mode stability criterion back to that for an adiabatic atmosphere: dlnB/dz > 0. Brandenburg et al. (1995) and Stone et al. (1996) have studied the nonlinear outcome of the Balbus-Hawley instability in a stratified disk, which is then potentially subject to magnetic Rayleigh-Taylor instabilities. In the experiments of Stone et al. (1996) there was no significant vertical Poynting flux, suggesting that magnetic buoyancy was not dynamically important. The vertical run of magnetic pressure was consistent with marginal stability to the Parker mode. Radiation-dominated disks, such as might be found in disks around neutron stars and black holes accreting near the Eddington limit, are subject to yet another type of quasi-global instability called "photon bubbles" (Arons 1992, Gammie 1998b). The instability occurs in radiation dominated regions where the magnetic pressure exceeds the gas pressure. The nonlinear outcome of the instability is not yet known, although it has been investigated in the context of neutron star polar cap accretion by Hsu et al. (1997), where it greatly enhances the vertical transport of energy. 2.4. Global MHD Instabilities Disks are potentially subject to an enormous variety of global instabilities. Global or quasi-global linear analyses of model astrophysical disks may be found in Papaloizou & Szuszkiewicz (1992), Gammie & Balbus (1994), Ogilvie & Pringle (1996), Terquem & Papaloizou (1996), Curry & Pudritz (1996), and Ogilvie (1998), to name a few. There is also an extensive literature on global instabilities in contexts other than disks (e.g. Couette flows); see Balbus & Hawley (1998) and references therein. Global analyses can exhibit the effects of boundaries and background gradients in the flow, but they lack the generality of local analyses since they depend on particular choices of the equilibrium. The three-dimensional nonlinear outcome of Balbus-Hawley and other global MHD instabilities in disks has not yet been studied, except in an idealized Couette flow model (Armitage 1998). 3. Gravitational Instability In the outer parts of AGN disks and protostellar disks gravitational instability may compete with or completely dominate the BH instability. Local stability of the disk is determined by Toomre's Q = cs«;/(7rGE), where K = epicyclic frequency. For Q < 1 the
Gammie: Disk Turbulence
281
disk is axisymmetrically unstable.(the precise value depends on the vertical structure of the disk; 0.676 is the critical value for an isothermal disk finite thickness disk, 1 for a zero thickness disk). In an a disk, the instability criterion can be rewritten in the form r3
/
r3
\3
s (3.7) M > 3a-?- ~ 7 x 10~3a —r M 0 yr~\ (UNSTABLE) G \ lkms / a condition easily satisfied in the outer parts of AGN disks for a ~ 1, and in YSO disks for a « l . Disks that are grossly unstable do not exist in nature, so the nonlinear theory of such systems is a mathematical exercise. Instead it is likely that disks are driven unstable, either by cooling (lowering cs) or by mass-loading (raising S, possibly via infall), and that stability is partially recovered in the nonlinear outcome either by dissipation (raising cs, possibly by shock heating) or by mass-shedding (lowering E, in AGNs possibly by star formation). For an a disk, cooling gives d\nQ/dt ~ ad. If infall is to dominate this cooling, then it is easy to show that the infall accretion rate per logarithmic interval in radius must exceed the accretion rate within the disk by a factor of order (R/H)2. It thus seems likely that cooling is the main driver of gravitational instability in most circumstances. The nonlinear outcome of gravitational instability with cooling has been studied in the context of a thin, local model of a gaseous disk by Gammie (1998c). I find that the disk goes unstable due to cooling and that, if certain conditions are satisfied, it then shock heats and returns to marginal stability. In the outcome the disk contains fluctuating surface density variations of order unity, and the density correlation length is of order 2nQH. The density structure transport angular momentum through both Reynolds stress and through gravitational stresses. Finally, a note on linear theory: it is somewhat under appreciated that self-gravitating disks with constant dynamic viscosity are secularly unstable, a point first noticed by Lynden-Bell and Pringle (1974) and later discussed in the context of differentially rotating disks by Safronov (1991), Willerding (1992), and Gammie (1996). The instability grows on the viscous timescale ("viscosity" is here a proxy for smaller-scale turbulence; molecular viscosity is negligible). In the limit of weak viscosity, the growth rate s of an axisymmetric mode in a zero thickness disk is
For an a disk (uturb ~ acsH), in the limit that Q > 1 and a <§: 1 the maximum growth rate of the instability is 27afi/(16Q4). Why are self-gravitating disks secularly unstable? It is clearly energetically advantageous for disks to bunch up into long-wavelength rings, thereby increasing their gravitational binding energy at little expense in compressional heating. But in inviscid disks there is an obstacle to this: the conservation of potential vorticity £ = (V x v)/E. Once viscosity is introduced then £ can evolve and the rotational support of the disk at long wavelengths is compromised.
4. Hydrodynamic Instabilities Absent self-gravity and magnetic fields, we are left with purely hydrodynamic mechanisms for generating turbulence. The local linear stability criterion for rotating fluid flow is the Rayleigh criterion: d(r2Q)2/dr > 0, i.e. specific angular momentum should
282
Gammie: Disk Turbulence
increase outwards. Most disks, and in particular Keplerian disks, satisfy the Rayleigh criterion. It has been suggested that Keplerian disks are locally nonlinearly unstable because of their high Reynolds number (e.g. Shakura & Sunyaev 1973, Lynden-Bell & Pringle 1974). This idea has been developed in some detail by Dubrulle & Zahn (1991), Dubrulle (1993), and Kato & Yoshizawa (1997). Numerical experiments in the local model (Balbus et al. 1996), however, fail to find any evidence of nonlinear instability in Keplerian shear flows. Nonlinear instability is found in a narrow band near dlnil/dlnr — —2, i.e. in disks that are marginally stable by the Rayleigh criterion. While one can always ask whether the numerical experiments achieve sufficiently high Reynolds number, Balbus et al. (1996) present an argument based on moments of the momentum equations that suggests, but does not prove, that Keplerian disks are nonlinearly stable. Disks can also suffer quasi-global instabilities such as convection (see Ruden et al. 1988 for the axisymmetric linear theory). One point that is not generally appreciated is the degree to which ordinary convective instabilities are damped by radiative diffusion in disks (although there are other, inertial, oscillations that become overstable in the presence of radiative diffusion). Workers had long thought that convection might lead to enhanced turbulent transport of angular momentum in disks, the idea being that turbulence always implies transport. An early sign that this expectation might be incorrect was a quasi-linear calculation (Ryu & Goodman 1992) of the angular momentum flux associated with linear, nonaxisymmetric convective motions; the direction of the flux was found to be inwards rather than outwards. Subsequent numerical experiments (Stone & Balbus 1996; Cabot 1996) showed that in the nonlinear regime the angular momentum flux was small and inwards. This nonintuitive result is a nice illustration of the value of numerical experiments. Finally, disks are susceptible to a wide variety of global hydrodynamic instabilities. One example is the Papaloizou-Pringle (1984, 1985) instability, subsequently elucidated by Narayan et al. (1986); see Savonije & Heemskerk (1990) for a readable physical account of this and allied global instabilities. A different type of global instability has been discovered by Goodman (1993). It requires a tidal field capable of distorting the disk streamlines into an oval shape. The instability grows from the free energy available in this oval distortion, causing it to decay by parametric instability into small scale inertial oscillations.
5. Conclusions Great progress has been made in the last few years in understanding the origins and development of turbulence in accretion disks. We know that under a broad range of conditions the BH instability can initiate turbulence that transports angular momentum outwards. We also have strong numerical evidence that other types of turbulence in disks, such as convective turbulence, do not necessarily provide the angular momentum transport required for disk evolution. But there are still many interesting open questions about turbulence in disks; I will conclude with three of particular current interest. 1. Is angular momentum transport local? Numerical studies of the three dimensional nonlinear outcome of the BH instability have so far been restricted to regions of the disk of order H in size (but see Armitage 1998). It is always found that most of the energy, and angular momentum flux, is contained in structures that are as large as allowed in the experiments. Thus the outcome is limited by the experiment size. What will happen in more realistic, larger-scale experiments? One possibility is that largest scale structures will have a small fraction of the turbulent
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energy, with most the turbulent energy being concentrated at scales of order H. In this case angular momentum transport would be truly local. But another possibility is that most of the energy is always contained in the largest scale structures allowed. Then angular momentum transport would be mainly due to structure much larger than the disk scale height: it is nonlocal. This would be inconsistent with current approaches to modeling disk evolution embodied in the a model. Numerical experiments may be able to decide between these, and intermediate, alternatives in the near future. 2. Are unmagnetized Keplerian disks nonlinearly stable ? Numerical experiments have diligently sought nonlinear instability in Keplerian disks and not found it (Balbus et al. 1996). But there remains a pool of skeptics who point out that the numerical experiments do not reach astrophysical Reynolds numbers, and so there is still the possibility of nonlinear instability. Since astrophysical Reynolds numbers will never be computationally accessible, what is needed is either a proof of nonlinear stability- a mathematically challenging problem- or an explicit demonstration of nonlinear instability. But for now the bulk of the evidence seems to favor the nonlinear stability of Keplerian shear flows. 3. How do waves and turbulence interact in disks? It is common to model the effect of turbulence on waves as a viscosity. This is done in studies of the tidal interaction between planets and protostellar disks (e.g. Lin & Papaloizou 1993), and in studies of warped disks (e.g. Pringle 1996); in these examples the turbulent viscosity completely governs the evolution of the disk. But the viscous model is completely untested. It could be quite misleading if, for example, it amplifies certain modes, or couples together linear modes of the laminar disk, or even gives the disk gas elastic properties. Numerical experiments that are immediately practical could measure the effects of turbulence on large-scale waves and settle this issue. I am grateful to Jim Stone, Eve Ostriker, and Gordon Ogilvie for their comments and suggestions. This work was supported in part by NASA grant NAG 52837.
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LIST OF PARTICIPANTS
1. Alvarez, Cesar 2. Antnez Garcia, Joel 3. Arias, Lorena 4. Arthur, Jane 5. Avila, Remy 6. Balsara, Dinshaw 7. Ballesteros-Paredes, Javier 8. Barrera, Pablo 9. Benjamin, Robert 10. Bensch, Prank 11. Bhattacharjee, Amitava 12. Blackman, Eric 13. Blom, Jon J. 14. Braun, Robert 15. Bregman, Joel 16. Brunt, Christopher 17. Cardona, Octavio 18. Carraminana, Alberto 19. Carrasco, Luis 20. Carrasco, Esperanza 21. Castafieda, Lizbeth A. 22. Colombon, Laura 23. Cordes, James 24. Crutcher, Richard 25. Cuevas, Salvador 26. Chatterjee, Tapan 27. Chavez, Miguel 28. Chavira, Enrique 29. Dalessio, Paola 30. Desai, Ketan 31. Dottori, Horacio 32. Duschl, Wolfgang J. 33. Elmegreen, Bruce 34. Falgarone, Edith 35. Flores, Aaron 36. Franco, Jose 37. Gammie, Charles 38. Garcia, Nieves 39. Garcia, Jose 40. Garcia-Segura, Guillermo 41. Gazol, Adriana 42. Gehman, Curtis 43. Gibson, Carl H. 44. Goldreich, Peter 45. Gomez-Reyes, Gilberto 46. Gonzalez, Alejandro
285
[email protected] j ag@bufadora. astrosen. unam. mx [email protected] j ane@astroscu. unam. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] blackman@ast .cam .ac.uk [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] cordes@spacenet. tn. Cornell, edu [email protected] chavoc@astroscu. unam. mx mchavez@inaoep. mx [email protected] dalessio@astroscu. unam. mx [email protected] [email protected] [email protected] [email protected] edith. falgarone@ensapb. ens. fr [email protected] [email protected] cgammie@cfa. harvard. edu nieves@astroscu. unam. mx [email protected] ggs@bufadora. astrosen. unam. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
286 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96.
List of Participants Gredel, Roland Guichard, Jose Gulati, Ravi Kumar Gwinn, Carl Ross Heithausen, Andreas Heyer, Mark Jokipii, Randy Joncas, Gilles Kessel, Olaf Klessen, Ralf Stephan Korpi, Maarit Lacey, Christina LaRosa, Ted Lazarian, Alex Lazio, Joseph Liljestrom, Tarja Lis, Darek Liszt, Harvey Lucas, Robert Luna, Abraham Mac Low, Mordecai-Mark MacLeod, Gordon Magnani, Loris Maron, Jason Martinez-Bravo, Oscar Mario Mayya, Divakara McKee, Christopher Melnick, Jorge Mendoza-Torres, Jose-Eduardo Minter, Anthony Miville-Deschenes, Marc-Antoine Muders, Dirk Munch, Guido Myers, Phil NG, Chung-Sang Nordlund, Ake Oey, Sally Orlov, Valeri Ortega, Maria Luisa Ossenkopf, Volker Ostriker, Eve Owocki, Stan Padoan, Paolo Palacios, Maria Norma Pereyra, Antonio Perez, Enrique Piccineli, Gabriela Pichardo, Barbara Pineda, Leopoldo Pouquet, Annick
[email protected] [email protected] [email protected] [email protected] [email protected] heyer@fermat .phast. umass.edu [email protected] [email protected] [email protected]. de [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] omartin@inaoep. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] dmuders@as. arizona. edu [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] .uni-koeln.de [email protected] [email protected] [email protected] palacios@inaoep. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
List of Participants 97. Puerari, Ivanio 98. Rajagopalan, Ramachandran 99. Recillas, Elsa 100. Reyes-Ruiz, Mauricio 101. Romano, Emilio 102. Romano, Roberto 103. Rosa, Daniel 104. Roth, Miguel 105. Santillan, Alfredo 106. Segura, Juan 107. Serrano, Alfonso 108. Seshadri, Sridhar 109. Silantev, Nikolai 110. Spangler, Steven 111. Stone, James 112. Sunada, Kazuyoshi 113. Tedds, Jonathan A. 114. Terlevich, Elena 115. Terlevich, Roberto 116. Thilker, David 117. Tovmassian, Hrant 118. Trinidad, Miguel Angel 119. Tufte, Stephen L. 120. Valdes-Parra, Jose Ramon 121. Valdez, Margarita 122. Valenzuela, Octavio 123. Van Atta, Charles 124. Vazquez, Gerardo 125. Vazquez-Semadeni, Enrique 126. Walterbos, Rene 127. Wall, William 128. Williams, Jonathan 129. Yam, Joel Omar 130. Zweibel, Ellen
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[email protected] [email protected] [email protected] maurey @bufadora. astrosen. unam .mx eromano@inaoep. mx rromano@inaoep. mx danrosa@inaoep. mx [email protected] [email protected] j segura@inaoep. mx [email protected] [email protected] [email protected] srs@ vest a. physics. uiowa. edu [email protected] sunada@nro. nao. ac .j p [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] mago@inaoep. mx [email protected] c vanat ta@ames. ucsd. edu gerar@astroscu. unam. mx [email protected] rwalterb@nmsu .edu [email protected] [email protected] [email protected] [email protected]
