A. Y. Poludnenko, A. Frank, and E. G. Blackman- Hydrodynamic Interaction of Strong Shocks with...

8/3/2019 A. Y. Poludnenko, A. Frank, and E. G. Blackman- Hydrodynamic Interaction of Strong Shocks with Inhomogeneous

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HYDRODYNAMIC INTERACTION OF STRONG SHOCKS WITHINHOMOGENEOUS MEDIA. I. ADIABATIC CASE

A. Y. Poludnenko, A. Frank, and E. G. BlackmanDepartment of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171;

[email protected], [email protected], [email protected]

Received 2001 September12; accepted 2002 May20

ABSTRACT

Many astrophysical flows occur in inhomogeneous (clumpy) media. We present results of a numericalstudy of steady, planar shocks interacting with a system of embedded cylindrical clouds. Our study uses atwo-dimensional geometry. Our numerical code uses an adaptive mesh refinement, allowing us to achieve suf-ficiently high resolution both at the largest and the smallest scales. We neglect any radiative losses, heat con-duction, and gravitational forces. Detailed analysis of the simulations shows that interaction of embeddedinhomogeneities with the shock/postshock wind depends primarily on the thickness of the cloud layer andarrangement of the clouds in the layer. The total cloud mass and the total number of individual clouds is nota significant factor. We define two classes of cloud distributions: thin and thick layers. We define the criticalcloud separation along the direction of the flow and perpendicular to it, distinguishing between the interact-ing and noninteracting regimes of cloud evolution. Finally, we discuss mass loading and mixing in suchsystems.

Subject headings: hydrodynamics ISM: clouds planetary nebulae: general shock waves stars: mass loss

1. INTRODUCTION

Mass outflows play a critical role in many astrophysicalsystems, ranging from stars to the most distant active gal-axies. Virtually all studies of mass outflows to date havefocused on flows in homogeneous media. However, the typi-cal astrophysical medium is inhomogeneous, with the clumps or clouds arising on a variety of scales. Theseinhomogeneities may arise because of initial fluctuations ofthe ambient mass distribution, the action of instabilities,variations in the flow source, etc. Whatever the origin of theclumps, their effect can be dramatic. The presence of inho-mogeneities can introduce not only quantitative but alsoqualitative changes to the overall dynamics of the flow.

A number of studies have attempted to understand therole of embedded inhomogeneities via (primarily) analyticalmethods (Hartquist et al. 1986; Hartquist & Dyson 1988;Dyson & Hartquist 1992, 1994; Redman, Williams, &Dyson 1998). In these pioneering works it was suggestedthat interactions of the global flow with inhomogeneitiesmay cause significant changes in the physical, dynamical,and even chemical state of the system. Two major conse-quences of the presence of clumps are mass loading (i.e.,seeding of material, ablated from the surface of inhomoge-neities, into the global flow) and transition of the global flowinto a transonic regime irrespective of the initial conditions.The papers cited above considered the potential effects ofmass loading on the global properties of a number of objectsin which inhomogeneities can be resolved. Such objectsinclude planetary nebulae, e.g., NGC 2392 (ODell, Weiner,& Chu 1990; Phillips & Cuesta 1999) and NGC 7293 (Bur-kert & ODell 1998), Wolf-Rayet stars, primarily RCW 58,which is believed to be mass-loadingdominated (Hartquistet al. 1986; Arthur, Henney, & Dyson 1996), and others.

A number of numerical studies of single-clump interac-tions have been performed (Klein, McKee, & Colella 1994,hereafter KMC94; Anderson et al. 1994; Jones, Ryu, & Tre-

gillis 1996; Gregori et al. 1999, 2000; Jun & Jones 1999;Miniati, Jones, & Ryu 1999; Lim & Raga 1999). In thesepapers, the basic hydrodynamics or MHD of wind-clumpand shock-clump physics have been detailed (often withmicrophysical processes included). A few studies of shockwaves overrunning over multiple clumps exist in the litera-ture (e.g., Jun, Jones, & Norman 1996). A detailed study ofmultiple clumps, however, where an attempt is made toarticulate basic physical processes and differentiate variousparameter regimes, has not yet been carried out. In thispaper (and those that follow) we address the problem ofclumpy flows, providing a description of the dynamics ofmultiple dense clouds interacting with a strong, steadyplanar shock.

The large parameter space and complexity of the problemrequire significant computational effort. To provide the nec-essary resolution of the flow, we have used an adaptive meshrefinement method. This is a relatively new computationaltechnology, and because of this we have chosen to investi-gate so-called adiabatic flows, in which radiative cooling isnot considered. In this regard our approach is similar to thatdescribed by Klein et al. (1994) for single clumps, and weutilize their results in understanding our multiclumpsimulations. We note that preliminary results, appropriateto active galactic nuclei (AGNs), were presented inPoludnenko, Frank, & Blackman (2002).

The plan of the paper is as follows. In x 2 we describe thenumerical experiments, the code used, and the formulationof the problem. In x 3.1 we consider the general propertiesof the shock-cloud interaction in the context of the multi-cloud systems; primarily we focus on the four major phasesof the interaction process. In x 3.2 we discuss the role ofcloud distribution in determining the dynamics of thesystem evolution. In x 3.3 we define several key parametersthat allow us to distinguish between various regimes ofshock-cloud interaction. Finally, in x 3.4 we address theissue of mass loading in such systems.

The Astrophysical Journal, 576:832848, 2002September 10

# 2002.The American Astronomical Society.All rights reserved. Printedin U.S.A.

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2. NUMERICAL EXPERIMENTS

2.1. Description of the Code Used

The code we used for this project is the AMRCLAWpackage, which implements an adaptive mesh refinementalgorithm for the equations of gasdynamics (Berger & Le-Veque 1998; Berger & Jameson 1985; Berger & Colella1989; Berger & Oliger 1984). In the AMRCLAW approach,the computational domain is associated with a logically rec-

tangular grid that represents the lowest level of refinement(level 1) and that embeds the nested sequence of logicallyrectangular meshes with finer resolution (levels 2, 3, . . .).The temporal and spatial steps of all grids at a level L arerefined with respect to the level L 1 grids by the same fac-tor, typically 4 in our calculations. The mesh ratios Dt=Dxand Dt=Dy are then the same on all grids, ensuring stabilitywith explicit difference schemes.

The solution on each grid is advanced via a second-orderaccurate Godunov-type finite-volume method inwhich second-order accuracy is achieved via flux-limitingand proper consideration of transverse wave propagation.The multidimensional wave-propagation algorithm is basedon the traditional dimensional splitting with the Riemann

problem solved in each dimension by means of a Roe-approximate Riemann solver (LeVeque 1997). It should benoted that our implementation of the Riemann solver,based on the Roe linearization, does not use any additionalprocedures to ensure satisfaction of the entropy condition,as usually employed for this type of Riemann solver. Ouranalysis shows that the numerical diffusion present in thesystem is sufficient to prevent entropy-violating waves frompropagating in the system.

The hydrodynamic equations we solve are appropriate toa single-fluid system, although a passive tracer is introducedin order to track advection and mixing of the cloud material.This was implemented as an additional wave family in theRoe solver.

Our numerical experiments were performed on a coarsegrid with the resolution of 50 100 cells and with the maxi-mum number of refinement levels equal to 3 (meaning thatthe coarse grid associated with the computational domainembeds not more than two nested higher resolution levels).Each higher level has a temporal and spatial step refined bythe factor of 4 in comparison with the next lowest level, andwe kept this refinement ratio constant for all levels. Such asetup provides the equivalent resolution1 of 800 1600cells. In order to facilitate comparison of our numericalexperiments with those of KMC94, we describe the resolu-tion not in terms of the equivalent resolution but in terms ofthe number of cells that fit in the original maximal cloudradius a0, following the convention of KMC94. Then all ofthe runs described here in our paper have 32 cells per cloudradius.

KMC94 suggested that a minimum resolution of 120 cellsper cloud radius is necessary. We have performed the simu-lations of the cloud-shock interaction with the resolution of120, 75, and 55 cells per cloud radius. Although we do notdescribe the details of those runs in this paper, the principaldifference between the cases with maximum and minimum

resolution, i.e., 120 and 32 cells per cloud radius, is the rateof instability formation at the boundary layers.2 This doesnot seem to have any significant effect on the global proper-ties of the interaction or the averaged characteristics of theindividual cloud ablation processes. Therefore, we find theresolution of 30 cells per cloud radius and above to repre-sent accurately the global properties of the interaction proc-ess under consideration. Moreover, 30 cells per cloud radiusis a reasonable compromise between maximizing the size ofthe computational domain and capturing as many small-scale features of the interaction process as possible. Weemphasize that our problem requires a compromise betweenthe resolution needed for simulating details of individualcloud structures and capturing the global flow pattern.

Finally, another aspect of this problem is the connectionbetween the spatial resolution (which naturally sets thesmallest scale resolvable in the simulations) and the diffu-sion and thermal conduction length scales. As we show inx 3.1.4, viscous diffusion and thermal conduction in a realphysical system operate at length scales comparable to thesize of a computational cell at the highest refinement levelused in our simulations. Therefore, in a real system, featuressmaller than the ones that can be resolved with our resolu-tion could not survive over the dynamical timescales rele-vant to the problem. We will address this in greater detailwhen we discuss the mixing phase of cloud evolution. Ofcourse, numerical diffusion must also be considered but, asdiscussed above, our compromise resolution appears tosatisfy the need to capture both small- and large-scalebehavior.

2.2. Formulation of the Problem

We set up a two-dimensional computational volume,associated with the initial condition ofNdifferent clouds ofradius ai and density i embedded in the ambient medium ofdensity a, and an incident shock wave. Since all of the

experiments were performed in the Cartesian geometry, theclouds are actually cross sections of the infinitely long cylin-ders. We will address the importance of the cloud shape inmore detail in subsequent work, where we will consider thefully three-dimensional case of the shock interaction withspherical clouds. Denoting the maximum cloud radiuspresent in the system as amax, our computational domain is25amax 50amax. This allows us to track the dynamical evo-lution of the system over greater temporal and spatial inter-vals compared to the 6amax 16amax domain considered byKMC94.

All our calculations were performed in a fixed referenceframe in which both the clouds and the ambient medium arestationary at time t 0. In this reference frame the horizon-tal axis is taken to be the x-axis, and the vertical axis the y-axis. Initially, both the clouds and the surrounding inter-cloud medium are assumed to be in pressure equilibriumand have pressure P0. Typically, the extent of the regionoccupied by the cloud distribution at time t 0 is taken tobe not more than 30%35% of the horizontal extent of thecomputational domain with XL offset by 5% from the leftboundary of the computational domain and XR offset by35%40%. Table 1 below, describing the numerical experi-ments discussed in this paper, provides the details of the

1 By equivalent resolution hereafterwe mean theresolution of a uniformgrid covering all of the computational domain and possessing the temporalandspatialstep of thehighestrefinement level.

2 For the case of lower resolution, the lower rate of instability formationmaybe somewhat compensated by theuse of thecompressive flux limiters.

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cloud distribution in each simulation. Figure 1 illustratesthe setup of the computational domain at t 0.

In the most general case, we assume each cloud to havethe same nonuniform density profile. The clouds have con-stant density up to a smoothing transition region at thecloud edge, which is achieved through a linear or tanhrfunction. We typically set the extent of the transition regionto the outer 20% of a cloud radius ai and use the tanhr-type smoothing function. Therefore, the cloud density pro-file isof the form

ir i

const ; 0

r

ri ;

a i2

a i2

tanhr ai ri=2tanhai ri=2 ri r ai ;

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clouds, and amax maxai for cloud distributions of vary-ing size clouds.

Because of the scale-invariance of our simulations, onecan, using specific values for the shock velocity and the sizeof the inhomogeneities, easily convert the time units used inour discussion into physical ones. The parameter tSC is par-ticularly useful in characterizing the problem, since it hasclear physical meaning and does not depend on a specificdensity contrast, which is important in the case of systemscontaining clouds of different density.

Note that except for the very short period of time when acloud interacts with the shock front, the former finds itselfimmersed in a postshock flow or wind, the pressure anddensity of which vary only by several percent over the largerange of Mach numbers. Since KMC94 showed that the ini-tial interaction with the shock front does not alter the evolu-tion of the system for the varying Mach number, the detailsof the evolution should not change after the shock front haspassed the cloud. Therefore, conclusions about Mach scal-ing should be valid both for the durations of cloud-windinteractions discussed by KMC94 and for the much longerdurations in our experiments.

One final remark should be made concerning the boun-dary conditions used in our experiments. In all runs weimposed a constant inflow at the left boundary, describedby the postshock conditions, which is determined using theRankine-Hugoniot relations, and open boundary condi-tions at the right, top, and bottom boundaries. Those out-flow boundary conditions were implemented via zero-orderextrapolation.

2.3. Description of the Runs

All of the runs discussed in this paper contain a Mach 10shock wave as a part of the initial conditions and embeddedclouds with the density contrast of 500. Table 1 presents a

summary of our numerical experiments.In addition to the dependence on the shock Mach numberand the cloud density contrast, there are other degrees offreedom present even in the simplest adiabatic case. We con-sidered how the dynamical evolution, e.g., rate of momen-tum transfer from the shock wave and shock deceleration,

mass loading, mixing of cloud material, etc. of the systemdepend on (1) the number of clouds present in the system,(2) the total cloud mass, (3) the spatial arrangement ofclouds, and (4) the individual cloud sizes and masses.

In most of the runs we constrained ourselves to the caseof identical clouds, varying only their number and arrange-ment. Radii of the clouds in all runs except M14r is 2% ofthe horizontal extent of the computational domain. In orderto simplify consideration of the dependence on a specificcloud arrangement, most runs have a regular cloud distribu-tion, in which the clouds are placed in the vertices of themesh formed by the centers of the clouds in the run M14. Inaddition, we considered a more general case of a randomcloud distribution with random cloud spatial positions andradii.

All of our numerical experiments were run for about 100tSC.

5 By this time, each individual cloud has almost com-pletely lost its identity and gained a velocity comparable tothe velocity of the global flow. Mixing of cloud materialwith the global ambient flow is nearly completed by 100 tSCas well.

In order to facilitate our analysis, we track temporal evo-lution of the global averages and one-dimensional spatialdistributions of two quantities, namely (1) the kinetic energyfraction, kin Ekin=Etot, and (2) the volume filling factor,.

Because of the adiabatic nature of our simulations, thekinetic energy fraction also allows us to track the comple-mentary quantity, the thermal energy fraction term Eterm=Etot 1 kin.

In order to obtain those quantities from the complex datastructure of the adaptive mesh simulations, we project thevalues of the state vector from each grid of the AMR gridhierarchy onto a uniform grid with the resolution of thehighest refinement level and that is associated with the com-putational domain. Such a projection does not cause loss ofdata or its precision. When this projection is done, we define

TABLE 1

Summary of the RunsDiscussed

RunNumber of

Cloudsa DistributionNumberof Rows x-Spacing y-Spacingc

M1 ................. 1 regular 1 . . . . . .M2 ................. 2 regular 1 . . . 4A2.................. 2 regular 1 . . . 12

M3 ................. 3 regular 1 . . . 4A5.................. 5 regular 1 . . . 4M14 ............... 14 regular 3 7b 4

M14r .............. 14 random . . . 6.34d 6.34d

a Total numberof cloudspresent in thesystem.b Spacing between the centers of clouds in two different rows, projected onto the x-axis, in

units of themaximumcloud radius amax.c Spacing between the centers of cloudsin the same row, projectedonto they-axis,in units

of themaximumcloud radius amax, exceptfor the run M14r.d Maximum absolute spacing between the cloud centers in any direction in the units of

the average cloudradius for the distribution.

5 For comparison, the experiments considered in KMC94, that havecomparable initial cloudambient medium density contrast, were run forabout 25 tSC.

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the global averages of the first quantity above as

hkini2D PNi

i1PNj

j1 kin;ijNiNj

%Rxmax

xmin

Rymaxymin

kinx;y dxdyxmax xminymax ymin ; 4

where Ni and Nj are the numbers of cells of the projected

grid in the x- and y-directions, respectively. Such averagingallows us to follow momentum transfer from the shockwave to the system of clouds, in the case ofhkini2D, and viathe complementary quantity of the global thermal energyfraction htermi2D, heating of the cloud system and inter-cloud material.

We also define the one-dimensional spatial average ofthat quantity as

hkini1Dx PNj

j1 kin;ijNj

%Rymax

yminkinx;ydy

ymax ymin : 5

Our code follows advection of a passive tracer markingcloud material. In order to follow mixing of the cloud mate-

rial with the global flow, we define the global average of thevolume filling factor hi2D as the ratio of the total numberof cells containing cloud material to the total number ofcells in the computational domain. We also define the one-dimensional spatially averaged volume filling factor hi1Das the variation with the coordinate x of the ratio of thenumber of cells containing cloud material in each verticalrow of the computational grid to the total number of cells inthe vertical dimension.

3. RESULTS

3.1. General Properties of the Shock-Cloud Interaction

Figures 25 show the time evolution of a shock waveinteracting with a single cloud (run M1), three clouds (runM3), 14 identical clouds in the regular distribution (runM14), and 14 clouds of random size in a random distribu-tion (run M14r). Shown are the synthetic Schlieren imagesof the system at four different times for all four sequences.Each image is obtained by calculating the density gradientat each point,6 plotted on a gray scale with the white denot-ing zero and black the maximum density gradient. Everyimage in each sequence roughly illustrates transitionsbetween the evolutionary phases discussed below.

3.1.1. Initial Compression Phase

After the initial contact, an external shock transmits an

internal forward shock into a cloud. This causes cloud com-pression and heating. At the same time, a bow shock formsaround the cloud. KMC94 subdivide this phase into twostages: initial transient and shock compression. Our numeri-cal experiments show that, in general, their description isapplicable for all cloud distributions except for the caseswhen individual clouds are almost in contact at time t 0.The cloud interior is dominated by the forward shock wave.In addition, it undergoes further compression because ofthe ram pressure from the global upstream postshock flow.

A reverse shock forms at the downstream surface of thecloud as a result of the backflow, caused by the global shockconvergence behind the cloud. This reverse shock in oursimulations never detaches from the downstream surface ofthe cloud, but instead propagates some small distanceupstream (toward the internal forward shock) together withthe cloud downstream surface. Moreover, typically thereverse shock is fairly weak, with shock Mach numbers notexceeding 1.21.3. This leads to lower maximum densities inthe cloud interior reached during this compression phasecompared to KMC94; we typically see i;maxd6i;0 asopposed to i;maxd10i;0 quoted by KMC94. The lattervalue was obtained analytically by KMC94 using theassumption of the collision of two strong shocks, asopposed to the collision of a strong and a weak shock in

6 To be more precise, the calculated quantity is the gradient of the den-sity logarithm. This makes the images clearer and easier to understand.

Fig. 2.Run M1. Time evolution of a system containing a single cloudand interacting with a MS 10 shock wave. Shown are the syntheticSchlieren images of the system at times 22:47tSC, 35:23tSC, 50:54tSC, and68:40tSC.

836 POLUDNENKO, FRANK, & BLACKMAN Vol. 576


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reality. Figure 6 illustrates the major flow structures presentin the system during the initial compression phase.

Propagation of the forward shock in the cloud allows usto define another important timescale governing the evolu-tion of the system and defining the duration of the compres-

sion phase, the cloud-crushing time, tCC. This is the timenecessary for the internal forward shock to cross the cloudand reach its downstream surface,7

tCC 2amaxvCS

: 6

In the above expression vCS is the internal forward shock

velocity and amax is again defined as a0 in the cases of clouddistributions with identical clouds, and as max

ai

in the

cases of cloud distributions with clouds of varying size. Fol-lowing KMC94, the velocity of the internal forward shockcan be written as

vCS vS1=2

Fc1Fst

1=2; 7

where vS is the velocity of the external shock. The factor Fstrelates the external postshock pressure far upstream withthe stagnation pressure at the cloud stagnation point andhas the form (Klein et al. 1994)

Fst 1 2:161 6:551=2 : 8

7 This was the principal timescale in the study of KMC94, although theydefined it as the time necessary for the internal forward shock to cross thecloud radius. We have changed the definition in our work, since the defini-tion of KMC94did notactually correspond to thedurationof thecompres-sion phase. Therefore, tCC in our work is about twice the tCC defined byKMC94.

Fig. 3.Run M3. Time evolution of a system containing three identicalclouds and interacting with a MS 10 shock wave. Shown arethe syntheticSchlieren images of the system at times 22:47tSC, 35:23tSC, 50:54tSC, and68:40tSC.

Fig. 4.Run M14. Time evolution of a system containing 14 identicalcloudsin a regular distributionand interacting with a MS 10 shock wave.Shown are the synthetic Schlieren images of the system at times 22 :47tSC,35:23tSC, 50:54tSC,and69:09tSC.



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3.1.2. Re-expansion Phase

This phase is initiated after the cloud internal forwardshock reaches the back of the cloud. Two major processesthen occur: lateral expansion of the cloud and the onset ofinstabilities at its upstream surface. At this stage Rayleigh-Taylor type instabilities dominate at the cloud/ambientflow interface. These are driven in part by the cloud expan-sion and incipient large-scale fragmentation. The flowdownstream with respect to the clouds is dominated byKelvin-Helmholtz instabilities operating in the growing tur-bulent region. The combined action of the lateral expansionand the instabilities causes the clouds to take the umbrella-type shape and eventually break up.

In the context of those two processes, the initial cloudseparation becomes of key importance in defining thesubsequent behavior of the whole system. We showbelow that it can be used to distinguish between the tworegimes of cloud evolution, interacting and noninteract-ing, and can serve as the basis for classification of clouddistributions. In x 3.3 we give a more rigorous discussionof the role of cloud separation. For now we give a quali-tative illustration.

Clouds located far enough from each other are notgreatly influenced by their neighbors, and their interactionwith the flow proceeds independently, as described byKMC94. This case is illustrated in Figure 7. Compared tothe evolution of a single cloud system, shown in Figure 2,the two clouds evolve up to the point of their destructionvery similarly to the single-cloud case. However, cloud sepa-rations can be small enough for the mutual interaction tomanifest early during the re-expansion phase, as in Figure 8.This mutual interaction causes changes primarily in the flowbetween clouds. As a result, the lateral expansion andgrowth of the Rayleigh-Taylor instabilities in the cloudmaterial is affected. The tails behind the clouds are alsodeformed outward (see, e.g., also Fig. 3).

The unperturbed supersonic flow that forms behind theexternal shock wave undergoes a transition from a super-sonic to a subsonic regime as it passes through a cloud bowshock. As a consequence, it suffers a significant velocitydrop whose magnitude is larger for smaller cloud separa-tions, because of the larger volume of the stagnation zone infront of the clouds. Clouds, acting as de Lavalle nozzles,then cause the flow material to reaccelerate. The flowreaches a sonic point next to a cloud core for the regions ofthe flow adjacent to a cloud, and farther downstream for theregions of the flow located farther from the clouds. It isimportant to note that this reacceleration results in rarefac-tion of the flow and a gradual decrease of both thermody-namic and dynamical pressure. Eventually, as a result ofacceleration in the intercloud region, the flow becomeshighly supersonic and finally shocks down through a sta-tionary shock formed downstream of the clouds to theregime close to the unperturbed flow behind the externalshock (see Figs. 78).

From the above discussion it is clear that the lateralexpansion velocity depends critically on the cloud separa-tion. For sufficiently low flow speeds, the cloud material willexpand at the cloud internal sound speed. With increasingglobal flow velocities (or, equivalently, with increasingvelocities of the external shock front), the lateral expansionvelocity will increase as well. This velocity is limited, in prin-ciple, by the terminal expansion velocity into vacuum.

For a fixed unperturbed upstream flow, the flow velocitynear a cloud lateral surface (facing the space in between theclouds) will be highest in the case of a single cloud or a cloudlocated far from the neighboring ones. With decreasing

cloud separation this velocity will decrease as well, causinghigher dynamical pressure on the lateral surface, and there-fore lower lateral expansion velocities. This occurs becausethe velocity drop across a bow shock in the cases of smallcloud separations is much larger, because of a stronger stag-nation effect in between the bow shock and the clouds.Therefore, flow adjacent to the cloud does not reach veloc-ities as high as in cases of large cloud separations.9 Anotherway to look at this process is as follows. The flow adjacent

Fig. 7.Run A2. Illustration of the noninteracting regime of cloud evo-lution: interaction of a MS 10 shock wave with a system of two identicalcloudswith thecloud centerseparation of 12:0a0 % 2:86dcrit. Shown are thesynthetic Schlieren images of the system at times 22:47tSC, 35:23tSC,50:54tSC,and68:40tSC.

9 It should be noted that eventually the velocities reached by the flowdownstream after passing the region between clouds are much higher, andconsequently the strength of the stationary shock downstream is muchlargerin thecase of small cloud separations.



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to the cloud surface passes through a sonic point, but in thecases of small cloud separations, densities in the stagnationregion are much higher. Thus, flow densities at the sonicpoint near the cloud lateral surface are much higher. This

leads to lower sound speeds, and therefore lower flowspeeds.Following KMC94, the effective lateral expansion veloc-

ity vexp can be defined as the internal cloud sound speed,

vexp CC vCS 2 1 1=2

1 ; 11

where vCS is the velocity of the cloud internal forward shock(eq. [7]). Our numerical experiments prove this to be a verygood approximation during almost all of the re-expansionphase. The expansion velocity exceeds this value by the endof the re-expansion phase because of stagnation pressure inthe regions, formed by the Rayleigh-Taylor instability.

We are now in a position to articulate the temporal evolu-tion of a cloud radius in the direction perpendicular to theupstream flow, a?t. From the moment of their initial con-tact with the external shock to the moment of their destruc-tion, the clouds first undergo slight compression in thedirection perpendicular to the flow and subsequently re-expand. KMC94s analytic model did not explicitly includecloud compression but instead tried to account for its effectvia a reduced monotonic expansion rate from t

0. Since

a?t is intimately related to the drag exerted on a cloud bythe global flow, the theoretical rate of momentum pickup bya cloud (or the rate of cloud deceleration in the referenceframe used by KMC94) differed from the numerical result.Namely, in Figure 12b of the paper by KMC94 numericaland theoretical results are practically the same up to thetime %2.0tCC, when the rate of cloud deceleration suddenlyincreases and the numerical and theoretical results drasti-cally diverge. This moment of time corresponds to thebeginning of the re-expansion phase, when the cloud crosssection starts to increase, causing an increase of the rate ofthe momentum transfer from the flow to the cloud. To avoidthis problem and simplify an expression for a?t, we usethe following form for evolution of a cloud radius normal tothe flow:

a?t a0 t tCC ;a0 CCt tCC tCC t tCD :

&12

Here CC is given by equation (11), and tCD is the clouddestruction time, defined below in equation (13).

3.1.3. Cloud Destruction Phase

Depending on the cloud separation, via the process of re-expansion clouds may come into contact and merge into asingle coherent structure. This subsequently interacts withthe flow as a whole and eventually breaks up. Thus, for the

case of small cloud separations we can define the moment ofcloud merging as the onset of the cloud destruction phase.For large cloud separations in which individual clouds getdestroyed before ever merging, it is difficult to define theprecise onset of the destruction phase, as it may be effec-tively viewed as a part of the re-expansion phase.

We define the end of the cloud destruction phase as thetime at which the largest cloud fragment contains less than50% of the initial cloud mass. For single-cloud systems orsystems of weakly interacting clouds, we define the totaltime from t 0 until the end of the cloud destruction phaseas the cloud destruction time, tCD,

tCD tCC 0tSC : 13

Typically % 2:0, consistent with KMC94, and using equa-tion (10) we find 0 % 24 in our simulations.

In addition to tCD there is also a cloud system destructiontime, tSD, which we define as the time when the largest frag-ment of a cloud located farthest downstream contains lessthan 50% of its initial mass. For thick-layer systems (to bedescribed later), including strongly interacting cloud distri-butions, tCD becomes less relevant as a description of thesystem than tSD, because tCD < tSD.

3.1.4. Mixing Phase

After the end of the destruction phase, the cloud materialvelocity is still only a small fraction of the global flow veloc-

Fig. 8.Run M2. Illustration of the interacting regime of cloud evolu-tion: interaction of a MS 10 shock wave with a system of two identicalclouds with the cloud center separation of 4 :0a0 % 0:95dcrit. Shown are thesynthetic Schlieren images of the system at times 22:47tSC, 35:23tSC,50:54tSC, and 68:40tSC.



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ity (see eq. [36] below). The velocity difference promotesKelvin-Helmholtz instabilities at the cloud materialglobalflow interfaces, and therefore the transition of the system toa turbulent regime. Typically, by the beginning of this phaseeach cloud has lost its identity as a result of merging withneighboring clouds. As the individual fragments becomesmaller and the velocity of the global flow relative to thecloud material decreases, Kelvin-Helmholtz instabilitiesgrow faster than the Rayleigh-Taylortype ones. This even-tually results in complete domination by the former of thesmall-scale fragmentation and causes mixing of cloud mate-rial with the flow (Klein et al. 1994).

In our numerical experiments, as can be seen in Figures25, turbulent mixing produces a two-phase filamentarysystem. The appearance of such a two-phase system resultsbecause our code does not include viscous diffusion or ther-mal conduction, restricting all dissipative effects to numeri-cal diffusion only. The latter acts at the length scalescomparable to a cell size at the highest refinement level.

Real dissipative effects also constrain the overall stabilityof cold dense plasma embedded in a tenuous hottermedium. KMC94 considered the overall effect of thermalconduction on the stability of such two-phase media againstevaporation. They concluded that the cloud ablation timedue to evaporation, expressed in terms of the shock-crossingtime tSC, defined in equation (3), has the form

tab 9F00

2

1

1=2Fc1Fst

1=2tSC ; 14

where F00 is typically of order unity (Klein et al. 1994).Therefore, for the case of our simulations, the typical abla-tion time is tab $ 30tSC, or comparable to the cloud destruc-tion time.

One can also estimate an effective depth over which diffu-sion and thermal conduction will disrupt the boundary layer

between the two phases over a dynamical timescale, tSC.This can be estimated as follows (see Kuncic, Blackman, &Rees 1996 and references therein). For viscous diffusion

ddiff $ DdifftSC1=2

2ffiffiffi

p a0MSn0p

1=2: 15

Here Ddiff vT;p is the coefficient of viscous diffusion, isthe effective path length between collisions in the cloudmaterial, vT;p kTp=mp1=2 is the proton thermal velocityin the ambient postshock gas, tSC is the shock-crossing time,defined by equation (3), n0 is the initial cloud number den-sity, p is the proton collisional cross section, and MS is theglobal shock Mach number. If we assume n0

$1000 cm3,

then for the cases presented in our simulations, ddiff is about1% of the initial cloud radius or, equivalently, is about 13 of acell of the computational domain at the highest refinementlevel. For thermal conduction, the effective depth is

dterm $ DtermtSC1=2 2ffiffiffi

p a0MSn0p

mp

me

1=2" #1=2

ddiff

mp

me

1=4; 16

where Dterm vT;e is the thermal diffusion coefficient, isagain the effective path length between collisions in the

cloud material, vT;e kTe=me1=2 is the thermal electronvelocity in the ambient postshock gas,10 me is the electronmass, and other quantities have the same meaning as inequation (15). Then dterm is about 6% ofa0, or equivalently,about twice the size of a computational cell at the highestrefinement level. Therefore, should we have included realdissipative effects, they would destroy the smallest resolv-able structures over the dynamically relevant timescales.Consequently, any further increase in resolution withoutproviding for appropriate mechanisms able to significantlyinhibit diffusion and thermal conduction would not provideadditional insights into the real physical evolution of asystem.

The importance of the dissipative effects is therefore two-fold: on the one hand, for the dynamical stability of small-scale structures and, on the other, for the overall stability ofthe system as a whole. Consider the stability of the initialsystem against destruction due to thermal conduction anddiffusion. From the arguments given above, dissipativeeffects prevent survival of the system for any dynamicallysignificant amount of time. As a solution to this problem,KMC94 suggested that weak magnetic fields inhibit thermalconduction and diffusion. Indeed, as shown by Mac Low etal. (1994), evolution of weakly magnetized clouds duringthe compression and re-expansion phases does not differ sig-nificantly from the purely hydrodynamic description. How-ever, the presence of magnetic fields would raise otherissues. During the mixing phase the system undergoes tran-sition to turbulence, which may amplify the initially dynam-ically insignificant magnetic fields. Turbulence can lowervalues of the plasma parameter Pg=PB, where Pg is theideal gas pressure and PB B2=8 is the magnetic pressure,to 1 or even smaller values. This may alter the evolution ofthe system during the later periods of the mixing phase. Inthis respect only, a fully magnetohydrodynamic study of theevolution of a system of clouds interacting with a strongshock is fully self-consistent (for a series of single cloudMHD studies, see Mac Low et al. 1994; Gregori et al. 1999,2000; Jones et al. 1996; Miniati et al. 1999; Jun & Jones1999).

3.2. Role of Cloud Distribution

In order to characterize the global properties of theshock/cloud system interaction, we plotted the time evolu-tion of the global quantities defined in x 2.3 for the runs M1,M2, M3, A5, M14, and M14r. Those plots are presented inFigures 910.

The important feature of those plots is the striking simi-larity of the behavior of systems containing similar clouddistributions. The systems containing from one to fiveclouds arranged in a single layer exhibit exactly the samerate of momentum transfer from the global flow. This ismanifested by the linear rates of fractional kinetic energy,hkini2D, increase from t 0 up to t 24tSC (see Fig. 9).The value of the slope for those five cases is 0:193 1:6%.The thermal energy, htermi2D, behaves complementarily.Such behavior of single-layer systems contrasts with that ofthe multiple-layer systems, namely, the runs M14 and M14r,which we now discuss.

The two 14 cloud runs have different cloud distributions

10 We assume that electron temperature and proton temperature areequal, Te Tp.



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(regular as opposed to random), different total cloud mass,and different cloud sizes. Nevertheless, the evolution of theirfractional energies is similar. The rate of the kinetic energy

increase during compression and re-expansion is the samefor both M14 and M14r, and yet is different from that forsingle-layer cases. The slope in the multilayer cases is alsopractically constant throughout the two phases, with valuesof 0:146 4:5%.

Note that for all cases the kinetic (thermal) energy reachesits maximum (minimum) at the time t 24tSC, or the timedefined above as the cloud destruction time tCD, eventhough for the 14 cloud runs the cloud system destructiontime tSD, defined in x 3.1.3, is greater than tCD. The fact thatthe kinetic energy peaks at t 24tSC signifies that the clouddestruction time is the characteristic timescale for the onsetof the fully developed turbulence in the system, which limitsthe ultimate growth of the kinetic energy fraction. We dis-

cuss this in greater detail in x 3.3.2.After passing through its maximum, the kinetic energyfraction begins to decrease as a result of the transition toturbulence, and consequently turbulent energy dissipation.It is difficult to define a value of the slope for the mixingphase because of the complex nature of the turbulent flow,but the average rate of kinetic energy dissipation in the sys-

tem is %0:013 27% for the single-layer systems and%0:015 25% for the multiple-layer ones. The proximity ofthese two values (within the standard error) is evidence thatthe systems have lost any unique details of the initial clouddistribution and developed turbulence that depends primar-ily on the rate of energy input at the largest scale, i.e., on therelative velocity of the global flow with respect to the cloudmaterial.

The time evolution of the volume filling factor,hi

2D, isanother example of the similarity in behavior of single-versus multiple-layer systems. As can be seen in Figure 10,the rate of cloud material mixing into the global flow is simi-lar for the single-layer systems but is different from that inmultiple-layer ones. The higher mixing rate in the case ofmultiple-layer distributions results because upstream cloudspick up momentum faster than the downstream ones.Upstream clouds promote destruction of the downstreamones and consequently the overall mixing of the system. Themost important feature is that the slopes of the two 14clump runs, which have different cloud distributions anddifferent total clump mass, from time t 0 to roughlyt 35tSC, when they both reach their maximum, is thesame, with d

hi2D

=dt%

0:196

3:9%. In cases of single-layer runs the differences are somewhat greater(dhi2D=dt % 0:067 25:9%). However, this is primarilydue to the fact that small numbers of clumps exhibit statisti-cal behavior to a lesser extent, and details of individualclump evolution are more important. With increasing num-ber of clumps, even though the distribution does notchange, the slopes become more similar: for the runs M3and A5, the slope value is dhi2D=dt % 0:078 7:1%. Notethat all of the single-layer runs undergo a slight breakaround t 25tSC, which is associated with clumpbreakup.11

These results lead us to conclude that cloud distributionplays a more important role than the number of clouds orthe total cloud mass. We use this conclusion in the next sec-tion as the foundation for classifying possible cloud distri-butions and defining the general type of the cloud systemevolution in each category.

3.3. Critical Density Parameter

We have seen that the cloud distribution plays the defin-ing role in determining the evolution of a shock-cloud sys-tem. We now quantify this statement and define criteria fordetermining the behavior of a given system.

We define a set of all possible cloud distributions for agiven number of clouds N. We consider only the clouds ofequal or comparable size and density contrast. We defineeach set of cloud distributions N for any given number of

clouds N to be a set of all possible N pairs of cloud centercoordinates satisfying two conditions: (1) each pair ofclouds is separated by some minimum distance rmin, and (2)clouds are confined to a layer extending from the positionXL to the position XR (see Fig. 1):

8N ! 1 : N nxi;yi; 1 i N : rij

hxi xj2

yi yj2i1=2

! rmin % 2a0; xi 2 XL;XRo: 17

Fig. 9. Time evolution of the global average of the kinetic energyfraction hkini2D for the runsM1,M2, M3, A5, M14,and M14r.

Fig. 10.Time evolution of the global average of the volume fillingfactor hi2D for the runsM1,M2, M3, A5, M14,andM14r.

11 The presence of the maximum values in hi2D in Figure 10 for all runsis due to the eventual loss of the cloud material through the outflow boun-daries.



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Next, we consider the complete set of all possible cloud dis-tributions for all possible cloud numbers defined as

[N!1

fNg : 18

We define two subsets within this set : a subset of thin-layer cloud distributions I, and a subset of thick-layer cloud distributions M, sothat

I [ M and I \ M ; : 19In our numerical experiments those two subsets are associ-ated with the single-row and multiple-row distributions.

In order to give a precise definition of those two funda-mental classes of cloud distributions, we need to introduceseveral auxiliary quantities.

3.3.1. Cloud Velocity and Displacement

We now estimate the distance that the cloud material willtravel before the cloud breakup, i.e., within the time tCD.

The equation of motion of a cloud in the stationary refer-ence frame of the unshocked ambient medium takes the

form

mCdvC

dt 1

2CDPS vPS vC 2ACt ; 20

where mC is the mass of the cloud, vC is the cloud velocity inthe stationary reference frame, CD is the cloud drag coeffi-cient, PS is the undisturbed postshock flow density, andACt is the cloud cross section area normal to the flow. Itshould be noted that this equation is valid only until thecloud destruction is complete, i.e., until t % tCD. From thispoint on we assume that the drag coefficient CD % 1, whichis a rather good approximation for a cylindrical bodyembedded in a supersonic flow ofMPS 1:31 (see Klein etal. 1994; Bedogni & Di Fazio 1998; Miller & Bailey 1979).

Let us assume for a moment that the clouds have finiteextent in the z-direction, z0. Note that then

mC 0a20z0 ; 21where 0 is the cloud density and a0 and z0 are the clouddimensions at time t 0. Moreover, the cross section area is

ACt 2a?tz0 ; 22where a?t is the cloud radius in the direction normal to theflow. Substituting equations (21) and (22) into equation (20)and using equation (12) for a?t, we get the modified equa-tion of motion12

dvCdt

PS0a20

vPS vC 2a?t : 23

This equation describes motion of the cloud as a result of itsinteraction with the postshock wind. However, we also needto account for the velocity that the cloud material acquiresafter its initial contact with the external shock front. Thisvelocity may be comparable to the velocity acquired duringthe compression and re-expansion phases, and thereforemust be carefully taken into consideration.

Recall that the initial contact of the incident shock frontdrives an internal forward shock into the cloud with velocityvCS. Cloud material behind the internal shock front gains avelocity vC;PS that can be determined from the Rankine-Hugoniot relations in the usual manner,

vC;PS 2vCS 1

1 1

M2CS

: 24

Here MCS is the Mach number of the cloud internal forwardshock, which can be expressed in terms of the external shockMach number as

MCS vCSCC;0

MS

Fc1Fst1=2

; 25

where CC;0 denotes the sound speed in the unshocked cloudmaterial, and we have used equation (7) for vCS. For thesimulations discussed in this paper ( 500 and 5=3),the internal cloud shock Mach number isMCS 1:86MS 18:6.

Substituting equation (7) for vCS and equation (25) forMCS into equation (24), and expressing the external shockvelocity vS in terms of the unperturbed upstream postshockvelocity vPS, we obtain the following expression for thevelocity of the cloud material due to the cloud interactionwith the external shock front:

vC;PS vPS

Fc1Fst1=2

M2S

Fc1Fst1h i

1=2

M2S 1

8>>>>>>>>>>>>>>>>>>:

33

In the first case ofMS 10, the coefficients a1, a2, and b1have the values

a1 1:79 103 ; a2 1:09; b1 8:35 105 : 34Substituting these into equation (33), we find that at the endof the compression phase, i.e., at the time t 12tSC, thecloud velocity is 10% of the postshock velocity vPS and 7.5%of the shock velocity vS. Onthe other hand,at theendof there-expansion phase, i.e., at the time t 24tSC, the cloudvelocity is 12.66% ofvPS and 9.4% ofvS.

For the case MS 1, the above coefficients have thevalues13

a1 1:83 103 ; a2 1:09 ; b1 8:51 105 : 35Substitution into equation (33) gives us the maximum val-ues of the velocity that a cloud can reach in the case of aninfinitely strong shock:

vC;max 10:1 102

vPS 7:55 102

vSfor the compression phase ; 36

vC;max 12:8 102vPS 9:57 102vSfor the re-expansion phase :

Similarly, we can determine the values of cloud displace-ment for the two cases considered above. Equation (31) forLC can be rewritten as

LCt

a0c1t

tSC 1

a1ln

t

tSC a1

a2 1

!& '0 t tCC ;

a0c1

t

tSC

c2 tan1 t=tSC 12t=tSCc4 c5

c3

!tCC t tCD ;

8>>>>>>>>>>>>>:

37

where for the case MS 10 the coefficients a1 and a2 havethe values defined in equation (34), and for MS 1 thevalues defined in equation (35).

In the case MS 10 the coefficients c1, c2, c3, c4, and c5have the values

c1 1:49 ; c2 104:17 ; c3 10:89 ;c4

9:34

102 ; c5

113:72 :

Substitution into equation (37) gives us the displacementthat the cloud material undergoes by the end of the com-pression and re-expansion phases: 1:6a0 and 3:5a0, respec-tively.

In the limiting case MS 1, the values of the coeffi-cients c1, c2, c3, c4,and c5 are

c1 1:5 ; c2 103:22 ; c3 10:9 ;c4 9:43 102 ; c5 112:56 :

Substituting these coefficients into equation (37) we findthat by the end of the compression phase the cloud is dis-placed by the distance of 1:65a0, whereas by the end of the

re-expansion phase the displacement is 3:53a0.It is clear from the results obtained above that both thevelocity and cloud displacement values in the case MS 10are practically identical to the maximum values achieved inthe limiting case of MS 1. Therefore, our resultsobtained for the case of a Mach 10 shock can be consideredas the limiting ones for the cases of strong shocks.

These results, derived for single clouds or systems withlarge separation, are in good agreement with numerical

13 Note that theassumption here is thesame as in thediscussion of Machscaling, namely, while increasing the shock Mach number, we keep theshockfront velocity in the stationary referenceframe constant.



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experiments. Typically, the maximum difference betweennumerical and analytical values of cloud velocity and posi-tion never exceeds 10%. The analytical results are usually anoverestimate of the numerical ones. This is due to a slightoverestimation of the initial velocity gain after the contactwith the external shock front and because we assumed thecloud cross section to be constant during the compressionphase, whereas it undergoes a small decrease in the experi-ments.

Therefore, the maximum distance a cloud with a densitycontrast of 500 can travel before its destruction after the ini-tial interaction with a strong shock is

LCD;max % 3:5a0 : 38Finally, it should be noted that the values of the maxi-

mum cloud velocity at the moment of breakup, vC;max, givenin equation (36), and the maximum cloud destructionlength, LCD;max, given in equation (38), are still functions ofthe cloudambient medium density contrast . Therefore,for a different value of the values of vC;max and LCD;maxshould be determined by direct evaluation of equations (27)and (31).14 Although the change in vC;max can be quite signif-

icant, we find that LCD;max does not change so prominentlywith varying . For example, for 40:0, which is morethan an order of magnitude less than the value 500:0used above, vC;max 36:5 102vPS, which is almost 3times the value given in equation (36). However, the corre-sponding value of LCD;max % 3:54a0, which is only a 1%increase compared to the value in equation (38). This is dueto the fact that clouds with lower density accelerate faster,although their destruction time is shorter. Therefore, eventhough the value of the cloud velocity at the time of breakupshould be evaluated for each particular value of, the clouddestruction length value given in equation (38) can be usedwith a sufficiently high accuracy over the full range of den-sity contrasts considered in this work, i.e.,

10 1000.

3.3.2. Critical Cloud Separation

We first define the average cloud separation between aclump and its nearest neighbor, projected on to the directionof the flow, hDxNi, and perpendicular to it, hDyNi, for agiven cloud distribution,

hDxNi 1N

XNi1

minj2 1;N

fjxi xjjg ; 39

hDyNi 1N

XNi1

minj2 1;N

fjyi yjjg : 40

We can also define a maximum cloud separation projected onto the direction of the flow, or the cloud layerthickness,

DxNmax maxi;j2 1;N

fjxi xjjg : 41

Now we are in a position to give a precise definition of the thin-layer and thick-layer systems. We define a distri-bution of clouds to belong to the subset I if its maximumcloud separation DxNmax does not exceed the cloud

destruction length, LCD. The distribution belongs to thesubset M in all other cases:

I fN : DxNmax LCDg ;M fN : DxNmax > LCDg :

42

A more intuitive way to look at this classification is as fol-lows. As we have seen, a cloud interacting with the post-shock flow re-expands and breaks up before it proceeds into

the mixing phase. The above criterion tells us if any cloud ora row of clouds will complete its destruction phase prior toencountering any other clouds located downstream. Thedefinition (42) appears to rather accurately draw the linebetween cloud systems of two types.

In practice, the maximum cloud separation DxNmax (eq.[41]) is simply the thickness of the layer of inhomogeneitiesin a real system and should be compared to the clouddestruction length. This thickness can be obtained from theobservations of a particular object, or it can be found ana-lytically, e.g., via consideration of instabilities at the inter-face between two flows.

Having defined the two classes, or subsets, of cloud distri-butions, we now consider the behavior of the clouds in each

class. First we consider I, the single-row distributions.On average, by the time the clouds are displaced by the dis-tance LCD, all of them will be destroyed and will proceed tothe mixing phase. Thus, the time of the destruction shouldbe approximately tCD tSD.

The question arises whether clouds will interact duringthe process of re-expansion and destruction. We can give aformal criterion for this. Consider two clouds with separa-tion hDyNi d and DxNmax LCD. Both clouds willexpand laterally at the velocity vexp defined by equation(11). Consequently, the time for the clouds to come intocontact is

tmerge %

d

2a0

2vexp:

43

Such re-expansion starts after the cloud compression phase,i.e., after the time tCC, and cannot proceed beyond the clouddestruction time tCD. Therefore, setting tmerge tCD tCC,we find the following critical cloud separation transverse tothe global flow:

dcrit 2 a0 vexptCD tCC

: 44Substituting equation (11) explicitly for the expansionvelocity and equation (13) for the cloud destruction timeinto the equation (44), we obtain

dcrit 2a0( tCD tCCtSC

Fc1Fst

1=23 1 1

!1=21) :45

In other words clouds whose separation transverse to theflow is less than dcrit will come into contact and merge beforetheir destruction is completed. Therefore, their evolutionduring the destruction phase (and for the most part of there-expansion phase) cannot be considered as the evolutionof two independent clouds.

The critical separation does not depend on the globalshock Mach number, which is consistent with the Machscaling, discussed above. Therefore, this parameter is

14 Note that dependence on also comes via the cloud destruction timetCD, whichshould be substitutedinto eqs. (27) and (31).



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universal for all strong shocks and for all possible distribu-tions from the subset I. For the case 5=3 and 500,we find the critical cloud separation to be approximately

dcrit % 4:2a0 : 46For cloud distributions from the subset I that have an

average separation hDyNi4dcrit, the evolution of the systemwill proceed in the noninteracting regime. On the other

hand, for the distributions for which hDyNiddcrit, thecloud-cloud interactions are important throughout the re-expansion and destruction phases, placing them in an inter-acting regime.

It is more difficult to formulate a unified criterion forthe behavior of the systems in the class M. WhenhDxNi > LCD, such systems can be considered as a set ofthin layers with an average separation greater than LCD, i.e.,each row can be considered as a system from the subset I.Consider, for example, the run M14, presented in Figure 4.From Table 1, the average separation hDxNi for this run isequal to 7, i.e., hDxNi > LCD. Indeed, the evolution of theleftmost row of clouds proceeds as a simple single-row case,and its destruction is completed by the time tCD. This results

in the fractional kinetic energy reaching a maximum at thetime tCD % 24tSC (see Fig. 9). However, it is clear from Fig-ure 4 that the evolution of the downstream rows is alteredby the destruction of the leftmost one. Therefore, whenhDxNi > LCD one must account for the fact that the destruc-tion of an upstream layer of clouds will change the proper-ties of the global flow for the next, downstream layer. Thenew averaged values of the velocity, density, and pressure inthe global flow should then be used as an input for theanalysis of the downstream cloud layer.

3.4. Mass Loading

One of the principal questions concerning the effects ofshock/cloud-system interactions is the role of mass loading(Hartquist & Dyson 1988). Mass loading is defined as thefeeding of material into the global flow by nearly stationaryclouds. Analytical studies have predicted a number ofimportant changes when mass loading occurs. The mostimportant of these is the transition of the flow to a tran-sonic regime (Hartquist et al. 1986; Hartquist & Dyson1988; Dyson & Hartquist 1992, 1994). In our numericalexperiments we consider whether mass loading indeed isprominent.

Mass loading can occur only from time t 0 up to themoment of cloud destruction at time t tCD. In our experi-ments the cloud destruction time is fairly short compared tothe total age of most relevant astrophysical objects. Indeed,cloud destruction is practically completed by the time theshock wave reaches the right boundary of the computa-tional domain, i.e., by the time the shock wave travels thedistance of about 2030 cloud sizes. This could, for exam-ple, be compared with clump systems in planetary nebulae(PNs). Assuming typical size for PNs clouds to be about 100AU (which is the size of cometary knots in NGC 7293;Burkert & ODell 1998), a density contrast of 500, and ashock wave velocity of 100 km s1, we find that clouds getcompletely destroyed within approximately 100150 yr.This ismuch less than the typical age of the PN(104105 yr).

Thus, clouds with low density contrast,i % 10 100, can-not provide significant mass loading because of the easewith which they are advected and destroyed by the global

flow. On the other hand, clouds with higher density con-trasts, i > 100, retain their low velocities with respect tothe global flow for much longer periods of time, and there-fore may potentially be efficient mass-loading sources.However, it should be noted that this higher relative velocityof a cloud increases the efficiency of the instability forma-tion, thereby promoting cloud destruction and its mixingwith the flow.

We can also consider the amount of mass seeded into theflow, i.e., stripped off from the clouds and assimilated intothe global flow, before cloud destruction. Typically, in ourexperiments the amount of seeded cloud material does notexceed a few percent of the total cloud mass, which isunlikely to be enough to switch the flow into a mass-loadedregime. Figure 11 shows the distribution of cloud materialalong the direction of the flow or, to be more precise, the dis-tribution of the parameter hi1Dx for the three-cloud runM3. There the clouds have the separation hDyNi % 0:95dcrit.The first graph corresponds to the end of the compressionphase, while the second corresponds to the end of thedestruction phase. The graphs show that cloud materialremains localized in the vicinity of the cloud cores until themoment of cloud destruction, and the system does notexhibit any significant mass loading. Moreover, the graphs3 and 4 of Figure 11, showing cloud material distributionearly in the mixing phase, indicate that even after destruc-tion cloud material remains localized within the region ofabout 8 cloud radii and retains almost the same averagevelocity with respect to the global flow. Only further on inthe mixing phase does cloud material spread significantly.

In conclusion, we can say that for the cloud density con-trast values in the range i % 10 1000 and practically allvalues of the global shock wave Mach number, the flows arenot likely to be subject to mass loading. This is due to thefact that such relatively low density clumps on one handaccelerate rather rapidly, and on the other fairly quicklybecome destroyed by instabilities. These flows will be domi-nated by the mixing of cloud material with the global flowthat occurs after cloud destruction. Systems with very denseclouds, i41000, may provide sites suitable for mass load-ing. Future numerical studies are required to verify this.

4. CONCLUSIONS

We have numerically investigated the interaction of astrong, planar shock wave with a system of inhomogene-ities. These clumps are considered to be infinitely longcylinders embedded in a tenuous, cold ambient medium. Wehave assumed constant conditions in the global postshockflow, thereby constraining the maximum size of the cloudsonly by the condition of the shock front planarity. Ourresults are applicable to strong global shocks with Machnumbers 3dMSd1000. The range of the applicable cloud/ambient density contrast values is 101000.

We considered four major phases of the cloud evolutiondue to the interaction of the global shock and postshockflow with a system of clouds. These are: initial compressionphase, re-expansion phase, destruction phase, and mixingphase. We describe a simple model for the cloud accelera-tion during the first three phases, i.e., prior to its destruc-tion, and derive expressions for the cloud velocity anddisplacement. The results of that model are in excellentagreement with the numerical experiments. The differencein the values of cloud velocity and displacement between



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analytical and numerical results is d10%. The maximumcloud displacement due to its interaction with a strongshock (prior to its destruction) does not exceed 3.5 initialmaximum cloud radii. The maximum cloud velocity is notmore than 10% of the global shock velocity.

The principal conclusion of the present work is that theset of all possible cloud distributions can be subdivided

into two large subsets, I and M. The first subset is I,thin-layer systems. This subset is defined by the conditionthat the maximum cloud separation along the direction offlow, or the cloud layer thickness, is not greater than thecloud destruction length, DxNmax LCD. The thick-layersystems, M, are defined by the condition DxNmax > LCD.The evolution of cloud distributions within each subsetexhibit striking similarity in behavior. We conclude that theevolution of a system of clouds interacting with a strongshock depends primarily on the total thickness of the cloudlayer and the cloud distribution in it, as opposed to the totalnumber of clouds or the total cloud mass present in the sys-tem. The key parameters determining the type of cloud sys-tem evolution are therefore the critical cloud separationtransverse to the flow, dcrit (this is also the critical linearcloud density in the layer), and the cloud destruction length,LCD.

For a given astrophysical situation, our results indicatethat one might determine, either from observations or fromtheoretical analysis, the thickness of the cloud layerDxNmax. This will then determine the class of the givencloud distribution, I or M. For cloud distributions fromthe set I with average cloud separation

hDyN

i> dcrit, evo-

lution of the clouds during the compression, re-expansion,and destruction phases will proceed in the noninteractingregime, and the formalism for a single-cloud interactionwith a shock wave (e.g., Klein et al. 1994; Jones et al. 1996;Mac Low et al. 1994; Lim & Raga 1999) can be used todescribe the system. On the other hand, if the cloud separa-tion is less than the critical distance, the clouds in the layerwill merge into a single structure before their destruction iscompleted. Although throughout the compression phasethey can still be considered independently of each other,their evolution during the re-expansion and destructionphases clearly proceeds in the interacting regime.

When the distribution belongs to the subset M, it is nec-essary to determine the average cloud separation projectedonto the direction of the flow hDxNi, defined by equation(39), and compare it to LCD: ifhDxNi > LCD, evolution ofthe cloud system can be roughly approximated as the evolu-tion of a set of distributions from the subset I, and theabove thin-layer case analysis applies. If, on the otherhand, hDxNi LCD (especially ifhDyNi < dcrit), the systemevolution is dominated by cloud interactions, and a thin-layer formalism is inappropriate.

Finally, we have considered the role of mass loading.Here our principal conclusion is that the mass loading is notsignificant in the cases of strong shocks interacting with asystem of inhomogeneities for density contrasts in the range101000. In part this is due to short survival times of cloudsunder such conditions, and in part it is due to the very lowmass-loss rates of the clouds even during the times prior totheir destruction. Mass loading may well be important inhigher density clouds (Dyson & Hartquist 1994).

The major limitation of our current work is the purelyhydrodynamic nature of our analysis, which does notinclude any consideration of magnetic fields. As discussed inx 3.1.4, cold dense inhomogeneities (clouds) embedded intenuous hotter medium are inherently unstable against thedissipative action of diffusion and thermal conduction. Thisevaporates the clouds on the timescales comparable to, orshorter than, the timescales of the dynamical evolution ofthe system. It was suggested that the magnetic fields mayplay a stabilizing role against the action of the dissipative

Fig. 11.Distribution of cloud material along the horizontal dimensionof the computational domain for the run M3. Shown are the one-dimensional spatial averages of the volume filling factor hi1D at times12:26tSC, 25:02tSC, 37:78tSC,and50:54tSC.



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mechanisms. Although weak magnetic fields that aredynamically insignificant up to the moment of clouddestruction can inhibit thermal conduction and diffusion,those magnetic fields may become dynamically important asa result of turbulent amplification during the mixing phase.A fully magnetohydrodynamic description of the interac-tion ofa strong shock with a system ofclouds will need to becarried forward in future works.

This work was supported in part by the NSF grant AST97-02484 and the Laboratory for Laser Energetics underDOE sponsorship. The most recent results and animationsof the numerical experiments, described above and not men-tioned in the current paper, can be found at http://www.pas.rochester.edu/~wma.

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848 POLUDNENKO, FRANK, & BLACKMAN

A. Y. Poludnenko, A. Frank, and E. G. Blackman- Hydrodynamic Interaction of Strong Shocks with...

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