Interstellar Cloud Formation through Aggregation of Cold Blobs in a ...

THE ASTROPHYSICAL JOURNAL, 480 : 694È700, 1997 May 10 ( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.

INTERSTELLAR CLOUD FORMATION THROUGH AGGREGATION OF COLD BLOBS IN A TWO-PHASE GAS MIXTURE HIDEYUKI KAMAYA Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan ; kamaya=kusastro.kyoto-u.ac.jp Received 1996 January 26 ; accepted 1996 December 9

ABSTRACT We propose a new formation scenario for interstellar clouds through the aggregation of dense cold blobs (phase II [PII]), which drift in a di†use warm medium (phase I [PI]). We examine how important it is that there exist numerous PII blobs when the properties of such a two-phase Ñow are studied. First, we solve a one-dimensional shock-tube problem and Ðnd that the shock wave in the mixture is considerably damped because of the drag force between the two phases. This is because the PII blobs are left behind the shock front, since their inertia is larger than that of PI, thus suppressing large spatial variations of PI gas via the drag force. The PII blobs thus play the role of anchors. Therefore, mass aggregation by shocks may be ine†ective in a two-phase medium. However, the PII blobs can still aggregate through a kind of Ñuid dynamical instability. We next suppose that the PI gas is accelerated upward by shocks against downward gravity, while the PII blobs are at rest because of balance between the drag force due to PI and gravity. If we put a positive perturbation in the number density of PII blobs, the upward PI Ñow above the perturbation is decelerated by the enhanced drag force, and the velocity di†erence between PI and PII is thereby reduced. Then the PII blobs above the perturbation are accelerated downward, since the gravity on PII now dominates the reduced drag force. As a result, the blobs will fall onto this perturbed region, and this region becomes denser and denser. This is the mechanism of the instability. Therefore, we expect efficient cloud formation by this instability in spiral arms, even when galactic shocks are extremely damped. Subject headings : hydrodynamics È ISM : clouds È ISM : general È planetary nebulae : general È shock waves È stars : formation 1.

INTRODUCTION

SpeciÐcally, the stability of the Ñow with the phase transitions has been Ðrst discussed in the context of interstellar physics (Shchekinov 1996). A preliminary study was made by Shu et al. (1972) in the context of star and cloud formation in spiral arms. However, although they treated phase transitions, the dynamics of the two-phase mixture was oversimpliÐed. Moreover, time evolution was not discussed. Thus, in this paper, we examine the dynamics of a twophase gas mixture by means of a nonlinear one-dimensional numerical analysis, although phase transitions are neglected. The outline of this paper is as follows : we introduce the basic idea of the multiphase Ñuid dynamics (MPFD) by Drew (1983) and explain our assumptions and basic equations in ° 2. We then give the numerical procedure and the results of a shock-tube problem for the gas (dense)Ègas (di†use) system in ° 3. In ° 4, we show that the aggregation of small blobs really can occur through a Ñuid dynamical instability, performing a numerical simulation. The Ðnal section is devoted to a summary and discussion.

It is known that recent star and cloud formation occurs in the arms of spiral galaxies. The processes of this formation are fundamental for understanding the evolution of galaxies. In some formation scenarios of interstellar clouds, for example, it is believed that the compression of the interstellar medium (ISM) by a shock wave plays an important role. Such a formation scenario is proposed by Tomisaka (1987) in the context of giant molecular cloud (GMC) formation. He succeeded in deriving the typical separation among GMCs in spiral galaxies, whose separation is known to be about 1000 pc from observations. However, he did not treat the dynamics of interstellar magnetic Ðelds. Note that this separation length also matches the length derived through one of the magnetohydrodynamical instabilities, the so-called Parker instability (Parker 1966). For an extensive review, for example, see Elmegreen (1993). It is known that interstellar space is Ðlled with interstellar gas, which is not perfectly uniform (see e.g., chap. 11 in Spitzer 1978). When we describe such a system, we adopt the so-called three-phase model (McKee & Ostriker 1977). These phases are the hot tenuous, the warm di†use, and the dense cold phases, respectively, and are usually treated independently at the level of simple theoretical considerations. Some models, however, consider the e†ects of the phase transitions between the di†erent phases (e.g., Habe, Ikeuchi, & Tanaka 1981 ; Tainaka, Fukazawa, & Mineshige 1993 ; Parravano 1996). Recently, Kamaya (1996, 1997) has proposed a description of the dynamics of such an ISM by the notion of multiphase Ñuid dynamics (e.g., Drew 1983). Notice that this method can not only calculate the dynamical evolution, but also describe the phase transitions between di†erent phases.

2.

DESCRIPTION OF BASIC EQUATIONS

2.1. Multiphase Fluid Dynamics (MPFD) Our motivation is to discuss the dynamics of a two-phase gas (di†use)Ègas (dense) system. For simplicity, we assume that numerous dense clouds (phase II [PII]) are embedded in a sea of di†use warm medium (phase I [PI]). This is the same as that postulated in a simple two-phase model of the ISM (Field, Goldsmith, & Habing 1969). For an analysis of the dynamics of the two-phase model, it is useful to describe the basic equations in the same way as those of ordinary Ñuid dynamics, because the same techniques can be used. Hence, we introduce MPFD. 694

INTERSTELLAR CLOUD FORMATION 2.2. Instability in a Fluidized Bed We consider the situation that the PI component Ñows upward against a downward external gravity. Inside this stream, PII blobs are suspended via the balance between the drag force due to the PI Ñow and the downward gravitational force. However, it is well known that there exists an instability in such a Ñuidized bed (see Jackson 1963 or the introduction in Batchelor 1988). We Ðnd the instability when a uniform distribution of blobs is perturbed. The spatial distribution of blobs is expressed by a volume fraction and denoted by 1 [ a. The a value is deÐned as the volume fraction of PI gas in a control volume, which is called a multiphase Ñuid element (MPFE) in this paper. Then the volume fraction of PII is expressed as 1 [ a. The physical mechanism of the instability is as follows : Suppose the spatial distribution of blobs is perturbed ; then 1 [ a also changes in space. Recall that the Ñow of gas has an opposite direction to that of the gravity and that the blobs su†er an upward drag force due to the PI Ñow and a downward gravitational force. Thus, not only does the upward drag force on the blobs Ñuctuate, but also the gas stream is decelerated in the regions with low a (or high 1 [ a). This means that the relative velocity between the gas Ñow and the blobs becomes small above the regions with low a. That is to say, the upward drag force on the blobs in the regions with higher a is reduced. Then the blobs are accelerated downward, since the gravity on the blobs dominates the drag force in the regions with high a. Therefore, the blobs can fall onto the region with ““ low ÏÏ a from the region with ““ high ÏÏ a. As a result, a becomes lower and lower in the low-a region. As the wave propagates, the contrast of the spatial pattern (i.e., the spatial distribution of a) grows. In other words, the void wave, which represents the propagation of the spatial pattern in terms of the MPFD, becomes unstable against inÐnitesimal disturbances. It is also known that this instability can occur even though the direction of gas Ñow is the same as that of gravity (Morioka 1989, in Japanese). Recently, Kamaya (1997) has proposed the basic equations for a compressible Ñuid with simple assumptions, neglecting the e†ect of drag between the two phases (PI and PII in this paper). His representative property of the basic set of equations is partly summarized in ° 2.6 in this paper. 2.3. Multiphase Fluid Element (MPFE) The basic concept of the underlying equations follows the philosophy of Drew (1983). In his paper, the basic equations for MPFD are derived on the basis of a simple law, the method of averages. Now, let q be any physical quantity. This q is averaged in a control volume and is expressed as

P NP

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example, a pressure scale height in the presence of a constant external gravity. Thus, we deÐne an L3 cube as a new Ñuid element and call an L3 cube an MPFE. Note that the MPFE corresponds to the control volume of Pistinner & Shaviv (1993). It must also be noted that L has been introduced only for the purpose of constructing the basic equations. In this sense, L is only a hypothetical length (see also Drew 1983). The di†erence between ordinary Ñuid elements and the MPFE is that the former are homogeneous, while the MPFE is a mixture of PI elements and PII elements. Here it must be noted that the previous studies (e.g., Sweet 1963 ; Kato 1973 ; Ikeuchi, Nakamura, & Takahara 1974 ; Jog & Solomon 1984) assumed that the two kinds of Ñuid elements have a uniform distribution inside an MPFE for both phases. In other words, a void fraction, a, was assumed to be spatially constant in an MPFE. It is thus impossible to describe aggregation e†ects of blobs by the previous methods. In the present formulation, however, we introduce a volume fraction, a, which can vary as time goes on in each MPFE. The deÐnition of a is the volume fraction of PI (di†use phase) in an MPFE. Thus the time evolution of spatial patterns can be described hydrodynamically in terms of the time and space variations of a. This provides a new tool for describing the features of two-phase gas systems (e.g., van Wijngaarden & Kapteyn 1990 ; Pistinner & Shaviv 1993 ; Kamaya 1996 ; Harris 1996). 2.4. Construction L aw of Basic Equations We next explain the derivation of the basic equations through a function X (x, t). If a position x is occupied by k we let X (x, t) \ 1. Otherwise, we phase k at time t, then k may write assign X (x, t) \ 0. With X (x, t), we k k a \ SX (x, t)T \ k k

P

NP

X dv dv , (2.3) k L3 L3 where a means the volume fraction of the k phase. That is, k a \ 1 [ a in this paper. Using this construction a \ a and I II the following relations for physical paramlaw, we deÐne eters, q8 \ SX qT/a , (2.4) k k k qü \ SX oqT/a o8 . (2.5) k k k k If we assign q \ o(x, t), for example, equation (2.4) gives a mean density of k-phase gas, o8(x, t) . If we put q \ v(x, t) or k q \ v(x, t), we obtain mass-weighted mean velocities or mass-weighted speciÐc energy via equation (2.5). These quantities are described as ¿ and vü , respectively. From now on, however, the notationsk q8 andkqü are expressed as q k k k for simplicity. Drew (1983) also adopts the following general rules for integration : S f ] gT \ S f T ] SgT ,

(2.6)

V V Here, dv is a volume element in a volume that is expressed as L3. The length scale, L , represents the size of an MPFE, which is discussed below, and should satisfy the following relation ;

SS f TgT \ S f TSgT ,

(2.7)

ScT \ c ,

(2.8)

l>L >H ,

S+f T \ +S f T .

SqT \

q dv

dv .

(2.1)

(2.2)

where l is the mean free path among the gas elements in the system, and H is the scale length of the system, which is, for

TU

Lf L \ SfT , Lt Lt

(2.9) (2.10)

Here f (x, t) and g(x, t) are functions of position and time, but c is a constant. The Ðrst three relations are called Rey-

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noldsÏs rules, while the fourth is called LeibnizÏs rule, and the Ðfth is called GaussÏs rule. 2.5. Basic Equations For simplicity, we neglect (1) the e†ect of deformations of PII, (2) surface tension forces, (3) phase transitions, (4) radiative cooling, and (5) thermal conduction. Although these processes are very important, we are more concerned with the dynamical properties of a two-phase gas. For completeness, we Ðrst write down the basic equations including these terms, following Drew (1983). For mass conservation, Lao I ] $ Æ (ao ¿ ) \ ! , I I Lt

(2.11)

L(1 [ a)o II ] $ Æ [(1 [ a)o ¿ ] \ [! , II II Lt

(2.12)

for PI and PII, where the subscripts I and II represent quantities of PI and PII, respectively. Here ! is the rate of phase transitions between PI and PII and is expressed as follows : ! \ S[o(¿ [ ¿*)]K Æ $X T . (2.13) I I Here ¿* represents the Euler velocity of the position of the surfaceI of the PI in an MPFE. The quantity [o(¿ [ ¿*)]K denotes a jump quantity across the interface betweenI PI and PII. For momentum equations, we have L (ao ¿ ) ] $ Æ (ao ¿ ¿ ] ap ) I I I I I I Lt \ [D Æ (¿ [ ¿ ) [ ao g ] M , I II I

(2.14)

L [(1 [ a)o ¿ ] ] $ Æ [(1 [ a)o ¿ ¿ ] (1 [ a)p ] II II II II II II Lt \ [D(¿ [ ¿ ) [ (1 [ a)o g [ M , (2.15) II I II for both phases, where M is related to the surface tension force on the boundaries between PI and PII, and g is a constant gravitational acceleration. The momentum exchange between PI and PII is also carried through M, which is explicitly expressed as M \ S[o¿(¿ [ ¿*) [ T]K Æ $X T , (2.16) I I where T is the stress tensor. The term D is a drag coefficient and is expressed as (2.17) D \ 3 Ca(1 [ a)o o ¿ [ ¿ o/SR T , I II I c 8 where SR T is typical radius of clouds, and C is the dimenc coefficient adopted in Shu et al. (1972). In this sionless drag paper, both ! and M are set to be zero and 3C/8SR T is c assumed to be unity for simplicity, since we are concerned with the dynamical properties of the a. More rigorous discussion of our basic equations is given in Drew (1983) and also in Kamaya (1997). We employ the following energy equations :

A

B

C A

BD

¿2 ¿2 L ao v ] I ] $ Æ ao v ] I ¿ I I 2 2 I Lt I I

\ $ Æ aT ¿ [ D(¿ [ ¿ ) Æ ¿ [ ao g¿ ] E , I I I II I I I

(2.18)

and

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A

B

C

A

BD

L ¿2 ¿2 (1 [ a)o v ] II ] $ Æ (1 [ a)o v ] II ¿ II II II II Lt 2 2 II

\ $ Æ aT ¿ [ D(¿ [ ¿ ) Æ ¿ [ (1 [ a)o g¿ [ E , II II II I II II II (2.19) respectively. Here, v is a speciÐc energy and E \ S[o(v ] ¿2/2)(¿ [ ¿*) [ T¿]K Æ +X T , (2.20) I I I but E is also selected as 0, since thermal conduction between the phases is neglected. For each phase, the ideal gas conditions are employed : p \ o RT /k , (2.21) I I I I p \ o RT /k , (2.22) II II II II where R is the gas constant. To close these equations, we assume pressure equilibrium in all space and time between PI and PII for the calculations that follow in ° 3, that is, p \p . (2.23) I II There follow some remarks concerning the description of physical quantities. Since o and o are the densities of I phases I and II, we may expect o /oII \ 0.01 for a simple I ¿II and v are the velocitwo-phase model. We also note that ties and speciÐc energy at the center of mass in each MPFE cell for each phase, respectively. Pressure is given by p for both phases if pressure balance is assumed. However, ap has an unclear physical meaning. In this paper, p is considered to be related to internal energy for both phases. Then ap corresponds to an energy fraction or a fraction of total pressure in each MPFE. Finally, we remark that our physical model for ISM may be correct for long timescale phenomena, since the details of the phenomena occurring on short timescales are smoothed out in an MPFE. Thus, if we selected a typical scale, a \ 0.01 pc, where a is the hypothetical size of blobs in an MPFE, the phenomenon that can be described by our procedure should have longer timescales than a sound crossing timescale over a : 0.01 pc/C D 104 yr. Here, C is the sound speed inside a PII blobsIIand is approximatelysII105 cm s~1. Adding to this idea about the timescales of phenomena, we also note that the wavelength for phenomena must be much longer than the intercloud separation in an MPFE. 2.6. V oid W ave (s-Mode) Here we summarize the properties of the s-mode, which can be described by the expression in Kamaya (1997), although the drag terms are neglected. This s-mode emerges through the repulsive e†ect of ambient PI gas that can be compressed by the PII blobs. Inversely, if the compressibility of the ambient gas is neglected, the s-mode never appears. Its dispersion relation is derived from equations (2.11), (2.12), (2.14), and (2.15) and is written as (u2 [ C2 k2)(u2 [ C2 k2) \ 0 (2.24) sI sII (see eq. [2.24] in Kamaya 1996 or full derivation in Kamaya 1997). Here u is wave frequency, k is wavenumber, C is the sI sound speed of the ambient PI gas, isothermal conditions are assumed, and any phase transitions are neglected. C is sII phase velocity but is estimated as sound speed in the cold blob. The part (u2 [ C2 k2) denotes the s-mode. Then we sII

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INTERSTELLAR CLOUD FORMATION

Ðnd that the s-mode has a slower phase velocity than the ordinary Ñuid (in this case, the other mode has phase velocity C ). The restoring force of the s-mode is the repulsive sI force that originates from the ambient PI medium compressed by blob motions. We note that this restoring force is expressed as p+a in the momentum equations. 3.

SHOCK-TUBE PROBLEM

For a numerical analysis, we adopted the cubic interpolated pseudoparticle (CIP) method (e.g., Takewaki & Yabe 1987), since this code has a simple algorithm but is nevertheless stable and accurate for shock problems. This point is important because the number of free parameters treated in the MPFD is about twice that of ordinary hydrodynamics. More speciÐcally, the following algorithm is adopted. First, physical quantities belonging to PI are calculated by means of a one-component CIP solver. Next, the quantities of PII are also calculated, separately. Finally, we can Ðnd the distribution of a from the following equation : ao /(1 [ a)o \ av /(1 [ a)v , (3.1) I II II I through the assumption of pressure equilibrium. Adiabatic exponents are chosen to be the same for both phases (c \ 1.4), and it must be noted that ao , (1 [ a)o , v , v , v , I II I II I and v are directly calculated in this paper. II Our calculations are limited to one dimension and perform a shock-tube problem, assuming free boundaries at x \ [200 and x \ X \ 200. The basic equations are ' a CIP scheme with an artiÐcial solved numerically by using viscosity. The mesh spacings are *x \ 1.0 (for example, ““ pc ÏÏ). The total number of mesh points is N \ 400. The x (t \ 100 in total elapsed time of all the runs is 100(105 yr) dimensionless time), if the spatial dimension is a parsec. For more details of our numerical procedure and tests of this code, see Yabe & Takei (1988). Initially, we set the medium to have a uniform void fraction (a \ 0.9). We set o \ 1.0, o \ 100.0, and p \ p \ p \ 1.0 at x \ 0, and o I\ 0.125, IIo \ 12.5, and pI \ pII \ II initially set toI be zero. II p \ 0.1 at x º 0. All the Ivelocities are In such condition, a shock wave travels to the right, and a rarefaction wave travels to the left only if a single gas phase were considered. In Figure 1, the results of the calculations are displayed. The top (a) and the middle (b) panels correspond to the shock-tube problem of a two-phase gas mixture, including the e†ect of drag force. Figure 1a shows the spatial distribution of a. In Figure 1b, the solid line corresponds to the velocity distribution of PI, and the dotted line to that of PII. For comparison, the result of the ordinary one-component (a \ 1) shock-tube problem is depicted in Figure 1c as a velocity distribution. The numerical results of Figure 1c coincide with the analytic solution to within 1%. As the di†erence between the solid lines in Figures 1b and 1c suggests, the shock wave is extremely damped through the drag force. On the other hand, the PII blobs are accelerated by this drag force. The velocity distribution for the PII blobs is depicted in Figure 1b by the dotted line. As a result, the spatial inhomogeneity of the void fraction is clearly seen in panel Figure 1a. The cause of the inhomogeneous distribution of a is that the inertia of the PII blobs is much larger than that of PI. That is to say, the blobs are left behind the propagating shock wave. The signiÐcance of this inhomogeneous distribution of a is discussed in the next section. In any cases, the method of MPFD can conÐrm the

697

(a) 1

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(b) PI :solid line PII:dotted line

.6 .4 .2 0 -200

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(c)

.8 .6 .4 .2 0 -200

FIG. 1.ÈSet of solutions of the shock-tube problem. (a, b) These panels show the result for a two-phase gas mixture (a \ 0.9). (c) This panel 0 problem (a \ 1.0). corresponds to the one-component (PI) shock-tube 0 of PI Each panel shows (a) the void fraction, (b) the velocity distributions (solid line) and PII (dotted line), and (c) the velocity distribution of PI. Comparing the solid line in (b) with (c), we Ðnd that the drag e†ect suppresses the shock. The spatial distribution of the void fraction in (a) shows the global picture of a postshock two-phase gas mixture. The number of meshes is 400, and the elapsed time is 100 sound crossing times.

properties of a shock wave in a two-phase medium with drag. 4.

AGGREGATION OF COLD BLOBS

4.1. Dispersion Relation of the Instability of Fluidized Bed As seen above, there is the possibility that shocks themselves are not e†ective in causing aggregation of clouds. There exists, however, an alternative mechanism to form clumps after the propagation of the shock wave. First, in this section, we derive the one-dimensional dispersion relation of the Ñuidized bed instability, for the purpose of showing the existence of the instability. For simplicity, both phases are assumed to be incompressible and isothermal, and the drag coefficient (D ) is chosen to be a constant. Pressures are also neglected1 in this subsection. Then we can Ðnd a closed set of equations La Lav I\0 , ] Lx Lt

(4.1)

L(1 [ a) L(1 [ a)v II \ 0 , ] Lx Lt

(4.2)

Lv Lv II \ [D (v [ v ) [ g , II ] v II 1 II I Lx Lt

(4.3)

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KAMAYA

from equations (2.11), (2.12), and (2.15), respectively. Here D has di†erent dimensions from equation (2.17). The above set1 of equations has a steady solution given by g (4.4) v \ 4v ; v \0 . I0 II I D 1 If we set a ] a ] a@, . . . , we obtain the following linearized 0 equation : a\a ; 0

La@ La@ L2a@ ]D ] D (1 [ a )v \0 . a 1 Lt 1 0 I0 Lx 0 Lt2

1 u \^ r 2a 0

G

] [

(4.6)

H

1 1 0.5 D2 ] [D4 ] 16a2 D2v2 k2(1 [ a )2]0.5 , 0 1 I0 0 2 1 2 1

and 1 u\ i 2a 0

E G

space, a \ 0.95, o \ 1, o \ 100, v \ 10.5, v \ 0, p \ 1, I II I II I and p \ 1. We comment again that the PI is not in an II equilibrium condition. It must be noted that the velocity dependence in equation (2.17) is neglected, and the term o v [ v o in (2.17) is set to unity in this simulation. We then I II add a perturbation, a@ \ (0.95È0.1) ] 0.95 sin

n(x [ 101) for 101 \ x \ 151 . (151 [ 101) (4.8)

(4.5)

Assuming that a P exp [[i(kx [ ut)], we Ðnd the following dispersion relation : [a u2 ] iD u [ iD (1 [ a )v k \ 0 . 0 1 1 0 I0 This equation has the following solutions :

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H F

The resultant evolution is shown in Figure 2. The top panel depicts the evolution of the void fraction. The elapsed times are 0, 10, 20, 30, 40, 50, 60.0, 70, 80, and 90.0 from the right dishes to left valleys. Obviously, the Ñuidized bed instability emerges, since the void fraction shows a larger dip as time goes on. To check whether that instability really occurs, the velocity distributions of both phases are depicted in the middle and bottom panels for PI and PII, respectively. According to the summary of the instability in ° 2.2, the Ñow of the PI above the dip of the void fraction must be decelerated because the Ñow su†ers the drag forces by blobs of PII. We Ðnd that this deceleration appears in the middle panel. On the other hand, the blobs must be accelerated downward in the region where the Ñow of PI is decelerated, since the gravity on the PII blobs dominates the drag force exerted by the PI Ñow. Then, in the bottom panel, we also

1 0.5 1 ] D ^ D2 ] [D4 ] 16a2 D2v2 k2(1 [ a )2]0.5 , 1 0 1 I0 0 2 1 2 1

1

(a)

T=0

T=50 .6

T=30

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(b)

velocity of phase I

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(c) T=10

velocity of phase II

4.2. Numerical Simulation of the Instability in a Compressible Fluidized Bed In this subsection, we perform a one-dimensional MPFD numerical simulation. To show the evolution of the Ñuidized bed instability clearly, we assume g \ 0.1, isothermal conditions for both phases, and incompressibility for PII. For PI, however, we consider the compressibility. Thus, we estimate the spatial distribution of a from the numerical results of (1 [ a)o , since pressure balance between both II phases is now not assumed. Then we adopt basic equations (2.11), (2.12), (2.14), and (2.15), assuming (1), (2), (3), (4), and (5) in ° 2.5. The numerical code is the same as that described in ° 3. As an initial condition, we set all the quantities constant in

void fraction

.8

where u and u are the real and imaginary parts (u \ i that u can take negative values. This u ] iu ).r We Ðnd r i i result indicates that there exists the instability introduced in ° 2.2 of this paper. We must pay attention to the fact that the PI component is not steady under real conditions. If we adopt the basic equations derived in ° 2, then the PI is always accelerated by gravity everywhere. This is because the drag term in equation (2.14) does not cancel out the gravity even if the blobs are in a steady state. All the above points give us a hint as to how we can select the initial condition for numerical analysis. That is to say, we can select the nonÈsteady state as an initial condition when numerical simulations are performed. We thus perform a numerical simulation to examine the growth of the Ñuidized bed instability in a more realistic situation, adopting equation (2.14), although the momentum equation of PI is not solved in this linear analysis.

-2

T=30

-4

T=50

-6

T=70

-8

T=90

-10 -200

FIG. 2.ÈEvolution of the Ñuidized bed instability is depicted in (a). The displayed times are 0, 10, 20, 30, 40, 50, 60, 70, 80, and 90.0 from right dishes to left valleys. The perturbed void fraction decreases as time goes on. The velocity distributions of PI and PII are also shown in (b) and (c), respectively. The elapsed times are 30, 50, 70, and 90.0 from the top line to the bottom line in (b) for PI (di†use component). In (c), however, the times are 10, 30, 50, 70, and 90.0 from the top line to the bottom line for PII (dense component). We have chosen v \ 10.5 and v \ 0.0 as the initial I II conditions.

No. 2, 1997

INTERSTELLAR CLOUD FORMATION

see that the velocity of the PII above the minimum of the void fraction becomes lower than that of the blobs below the minimum. We comment here that the overall spatial distribution of PII is also decreased by gravity, since the background phase (PI) is always accelerated downward by the gravity, everywhere. Then the dense phase blobs are also accelerated downward through the frictional term in our basic equations. Thus, the cold blobs can aggregate through the Ñuidized bed instability. 5.

SUMMARY AND DISCUSSIONS

We examined two mechanisms for the aggregation of small clouds, namely, a shock and an instability. Consequently, we performed two sets of numerical simulations in this paper. First, we carried out the shock-tube problem of a two-phase gas, including the e†ect of drag force. Then we conÐrmed that the shock wave is extremely damped because the kinetic energy of the PI component is transferred to the motion of PII through the drag e†ect. Note that the inertia of PII is greater than that of PI. This means that the motion of the blobs is delayed behind the shock. The shock can thus cause mass segregation or spatial inhomogeneities in the a distribution because of the di†erence in inertia between the two phases. MPFD can describe the evolution not only of the shock, but also of the postshock gas, whose properties were discussed in ° 7 of Shu et al. (1972). Next, we Ðnd that an inhomogeneous distribution of a really develops through the Ñuidized bed instability (see ° 2.2). The instability will be enhanced, since the drag e†ect becomes greater if the (1 [ a) is increased. The number density of blobs is believed to become large through thermal instabilities in the arm regions of spiral galaxies (Field 1965). Then interstellar clouds may easily be formed through the Ñuid dynamic instability even when the galactic shock (Roberts 1969) is damped. As a Ðnal process, we may expect that the blobs stick together by collisions (e.g., Zhang, Wang, & Davis 1993), since the mean free path in an MPFE becomes very short. The present study di†ers from Shu et al. (1972) in the following two points : (1) in their approximation, they assumed ¿ \ ¿ , while we allow ¿ D ¿ , so that we can II I drag II ; (2) next, they discuss theI dynamical e†ect of the assumed the a distribution to be spatially constant. Since we consider the dynamical properties of a, we can discuss the Ñuidized bed instability. For describing the more realistic ISM conditions, however, we must construct an MPFD numerical code including the e†ect of phase transitions, which were discussed in Shu et al. (1972). We may expect that our model would be able to extend or conÐrm the work by Shu et al. (1972) if phase transitions were treated, although the present work seems to contradict the conclusions in their paper. This is because the steepening of the void fraction is regarded as a kind of shock (Fanucci, Ness, & Yen 1979). If the e†ect of gravity is included in an explicit code, it is known that a numerical instability inevitably appears in the MPFD solver. This is because the instability in ° 4.1 corresponds to a negative di†usion for PII components. It is well known that the simple explicit numerical procedure is unstable against negative di†usion. Thus, even in recent work (e.g., Anderson, Sundaresan, & Jackson 1995 ; Glasser, Kevrekidis, & Sundaresan 1996), the approach is too complex to carry out MPFD simply. The complexity

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emerges since they consider terms representing ““ viscous dissipation.ÏÏ We note that such terms can correspond to positive di†usion terms and that the numerical code is stabilized by these terms. On the other hand, we found that the term p+a in the momentum equations (2.14) and (2.15) can also perform a similar role to the viscous dissipation terms. These spatial derivative terms of a also function as positive di†usion, since the terms denote that the PI compressed by PII blobs repulses the blobs (see also ° 2.6). This mode is known as the s-mode (Kamaya 1997). Thus, we can expect that the numerical code becomes stable if the spatial derivative of a is included in the basic equations. One might think that any mechanisms that succeeded in stabilizing the compressible Ñuidized bed would prevent the aggregation of the cold blobs. Here we note that the s-mode is damped through the friction among the components. Although the stabilization of our instability through the s-mode occurs only locally, the s-mode itself is damped on a larger scale. Thus, in ° 4.2 we Ðnd the steepening of the void fraction. We note that the stabilization of the Ñuid instability at small scales assures numerical stability. Such a steepening of the void fraction is properly discussed by Fanucci, Ness, & Yen (1979). They performed a characteristic method calculation and showed that the distribution of the void fraction must steepen when the viscous terms are neglected. Since our basic set of equations also includes no viscous terms, we comment on the di†erence between Fanucci et al. (1979) and the present work. We note that they solved their problem with three independent variables. That is to say, both phases in the two-phase Ñow were assumed to be incompressible. On the other hand, there are four independent variables in our numerical simulation in ° 4.2. We consider the compressibility of the ambient medium, and so one more variable (the density of the ambient phase) is added to Fanucci et al. (1979). If we consider the PI as an incompressible Ñuid in our basic set of equations, our set would correspond to that in Fanucci et al. (1979). However, we again stress that their set of equations is numerically unstable when an explicit numerical scheme is used, since no stabilizing e†ect is considered at smaller scales. Finally, we comment on other possible astrophysical applications. If we include the e†ects of interstellar magnetic Ðelds, then it becomes more realistic to study the magnetized interstellar two-phase gas in terms of MPFD (Mouschovias, Shu, & Woodward 1974). If we consider radiative processes, it may be interesting to compare the C-type shock discussed by Draine (1980) with our case. Recently, Miyahata & Ikeuchi (1995) has formulated a twophase protogalaxy. Since their model is a steady model, we can reformulate it, performing numerical MPFD. We may expect that the settling down of cold components in twophase protogalaxies leads to the formation of a disk inside the primordial galactic halos. Our approach may easily present the above dynamical phenomena, since the dynamic range in our numerical code is much smaller than in other Ñuid dynamical simulations. I am grateful to an anonymous referee for critical but extremely useful comments and English revisions. I am also thankful to S. Mineshige for reading the manuscript and his continuous encouragement and to T. Kudo and Yu. Shchekinov for useful discussions.

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