The Astrophysical Journal, 662:379 Y388, 2007 June 10 # 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EXPERIMENT ON THE MASS STRIPPING OF AN INTERSTELLAR CLOUD FOLLOWING SHOCK PASSAGE J. F. Hansen, H. F. Robey, R. I. Klein,1 and A. R. Miles Lawrence Livermore National Laboratory, Livermore, CA 94550;
[email protected] Received 2006 May 16; accepted 2007 February 14
ABSTRACT The interaction of supernova shocks and interstellar clouds is an important astrophysical phenomenon which can lead to mass stripping (transfer of material from cloud to surrounding flow, ‘‘mass loading’’ the flow) and possibly increase the compression in the cloud to high enough densities to trigger star formation. Our experiments attempt to simulate and quantify the mass stripping as it occurs when a shock passes through interstellar clouds. We drive a strong shock (and blast wave) using 5 kJ of the 30 kJ Omega laser into a cylinder filled with low-density foam with an embedded 120 m Al sphere simulating an interstellar cloud. The density ratio between Al and foam is 9. Timeresolved X-ray radiographs show the cloud getting compressed by the shock (t 5 ns), undergoing a classical Kelvin-Helmholtz roll-up (12 ns) followed by a Widnall instability (30 ns), an inherently three-dimensional effect that breaks the two-dimensional symmetry of the experiment. Material is continuously being stripped from the cloud at a rate which is shown to be considerably larger than what is predicted by laminar models for mass stripping; the cloud is fully stripped by 80Y100 ns, 10 times faster than the laminar model. We present a new model for turbulent mass stripping that agrees with the observed rate and that should scale to astrophysical conditions, which occur at even higher Reynolds numbers than the current experiment. Subject headingg s: hydrodynamics — instabilities — ISM: clouds — ISM: kinematics and dynamics — methods: laboratory — shock waves — turbulence 1. INTRODUCTION
ing through interstellar clouds can trigger star formation or increase the star formation rate but also limit the time over which star formation is possible ( Hartmann et al. 2001). Recent analyses ( Palla & Stahler 2000) suggest that clouds are relatively short-lived (average age a few million years). The interaction with shocks may seed turbulence in the clouds, believed to play an important role in the fragmentation of the clouds ( Padoan et al. 2001; Clark & Bonnell 2005) and in the star formation process (Larson 1981). Shock-cloud interactions have been seen in several astrophysical objects, e.g., the Cygnus loop ( Bedogni & Woodward 1990; Patnaude & Fesen 2005), the Vela supernova remnant (SNR; Miceli et al. 2005), and SNR HB 21 (Byun et al. 2006), just to mention a few, and may play an important role in the evolution of clouds and in star formation everywhere from the solar neighborhood ( Hartmann et al. 2001) to intergalactic clouds ( Fragile et al. 2004). ( We also note that the situation with a cloud being stripped of its mass through a turbulent boundary layer has some similarity to a meteorite falling through the Earth’s atmosphere or the solar wind ablating a comet; although in these cases, the core of the ablated object remains solid.) Klein et al. (1994, 2000) performed a comprehensive study of the shock-cloud interaction using high-resolution adaptive-meshrefinement two-dimensional hydrodynamics, developed the theory for each stage of the interaction with detailed comparisons with their simulations, and also performed the first scaled laser experiments investigating the interaction of a supernova shock with an interstellar cloud. The experiments were initially performed at the Nova laser facility (Klein et al. 2000) and later at the Omega facility (Robey et al. 2002; Klein et al. 2003). The results of these experiments were compared in detail to the numerical simulations and the theoretical work and showed excellent agreement with the simulations for most stages of the interaction. The experiments were among the first to demonstrate the long-term behavior of compressible clouds immersed in supersonic flow. Recently, shock-cloud interaction evolved to a late time was observed and identified by Hwang et al. (2005) in SNR Puppis A by comparing morphological features in observational and experimental images.
Observations of astronomical media show that condensations embedded in more diffuse plasma are common phenomena. The flows present in the surrounding diffuse material can be initiated and enhanced by the injection of energy and momentum from supernovae, stellar winds, mass loss from active galactic nuclei, and possible mass loss from galaxy clusters. Mass loading is thought to be important in the Wolf-Rayet wind-blown bubble RCW 58 (Smith et al. 1984), for example, as well as the winds from T TauriYtype stars and other young stars and occurs from the ablation of molecular clumps of gas by these winds, creating high-velocity flows. The flow surrounding condensations results in pressure gradients across their surfaces, dissipation, and acceleration to the flow velocity. Mass loading, the transport of matter from the condensation into the flow, can indeed alter the structure of the flow itself. Although the process of mass loading is an important occurrence in nearly all astrophysical flows, the experimental study of such flows is still in an early stage of development. Early theoretical work on mass exchange concentrated on conduction fronts between cold material and more diffuse static plasma (Cowie et al. 1981). These descriptions are not strictly applicable to finite Mach number flows in turbulent media. Theoretical work on mass-loaded flows due in particular to mass-loaded winds has been done by Hartquist et al. (1986). Flows driven by mass loading may occur on all astronomical scales ( Hartquist et al. 1986) and may have important effects on scales ranging from galactic winds interacting with inhomogeneities in the early universe to winds from clusters of galaxies or starburst galaxies ablating very large-scale perturbations ( Marcolini et al. 2005). Woodward (1976) and Nittmann et al. (1982) have studied the interaction of a supernova shock with an initially pressure-confined cloud. Heathcote & Brand (1983) described the main features of this interaction after the passage of the shock. Shocks pass1
Also at: Department of Astronomy, University of California, Berkeley, CA 94720.
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Fig. 1.— Experimental setup. A beryllium shock tube is filled with a low-density foam and is open at both ends. A small aluminum sphere is embedded in the foam (inside the shock tube and not visible in this drawing). The experiment starts when 10 laser beams launch a shock in the foam from one end of the shock tube. The shock develops into a blast wave and propagates through the shock tube and interacts with the embedded sphere. The evolution of the sphere is recorded by an X-ray camera (far outside the field of view of this drawing). X-rays for the camera are generated by additional laser beams striking a titanium foil. The X-rays can be focused either by a pinhole in the camera or by a pinhole in a tantalum substrate that is placed immediately adjacent to the Ti foil. A gold grid mounted on the outside surface of the shock tube provides a spatial reference in the X-ray images.
Beyond morphology, experiments may also give some quantitative measure of turbulence, and in this paper we present the results of a continued set of laser experiments designed to study the interaction between a supernova shock and an interstellar cloud, including a new model for turbulent mass stripping (turbulent transfer of material from the cloud to surrounding flow, mass loading the flow) that is in good agreement with the experimentally observed rate and that should scale to astrophysical conditions. The experiments were carried out at the Omega laser at the Laboratory of Laser Energetics in Rochester, New York ( Boehly et al. 1997). 2. EXPERIMENTAL SETUP AND DIAGNOSTICS The strong shock (and blast wave) of a supernova explosion is simulated by experiments at Omega in the following manner; a small beryllium shock tube (2.25 mm long, 0.8 mm inner diameter, and 1.1 mm outer diameter) is filled with a low-density (300 mg cm3) carbonized resorcinol formaldehyde foam (CRF). The CRF at one end of the shock tube is then ablated by laser beams, causing the ejection of ablated material in one direction to launch a planar shock in the opposite direction. A total of 10 beams were superimposed to launch the shock, each with a super-Gaussian beam profile created by a phase plate in the focusing optics; the super-Gaussian is exp½ðr/d Þn , with n ¼ 4:7 and d ¼ 430 m; the beam angles of incidence were 10.3 (one beam), 31.7 (three beams), 42.2 (four beams), and 50.5 (two beams). Each laser beam has an energy of 500 J with a pulse duration of 1.0 ns. Note that after the laser beams are turned off, a rarefaction follows the shock into shock tube, and thus the shock soon develops into a blast wave. Figure 1 shows the experimental setup.
The scaled interstellar medium (ISM ) cloud is simulated by an aluminum sphere (radius R0 ¼ 60 m) embedded in the CRF a short distance into the shock tube (on the shock tube axis, 500 m from the ablated CRF surface). The density ratio between the Al (density 20 ¼ 2:7 g cm3) and the surrounding CRF is chosen to match the density ratio 20 /10 for an actual ISM cloud (20 /10 10Y100 [Klein et al. 1994]) and other experimental parameters are also scaled to preserve the physics regime of the astrophysical case (see below). The cloud is imaged using a gated X-rayYframing camera ( Budil et al. 1996). X-rays for the image are generated by a second set of time-delayed laser beams (backlighter beams) pointed at a metal foil, typically titanium, located on the opposite side of the shock tube from the camera. These X-rays (mainly He radiation at 4.7 keV ) pass through the shock tube and are imaged by a 10 m pinhole at the front end of the camera. (This is called ‘‘area radiography’’; over the course of our experiments we also developed and used a more advanced imaging technique, ‘‘point-projection radiography,’’ in which the pinhole is located right next to the Ti foil and has a diameter of 20 m.) The imaging element of the camera is either a microchannel plate (MCP) and film or MCP and a charge coupled device (CCD); in both cases it has a size of 35 mm. The distance from shock tube to Ti foil is 4.0 mm (for area radiography, 6.5 mm for point-projection radiography), and the imaging element is located to give a magnification such that the shock tube image roughly fills the imaging element. The time delay for the backlighter beams is chosen to obtain an image at a desired time t after the initial ablative laser pulse has started the shock in the shock tube. The camera MCP is triggered to coincide with the backlighter beams. The high-voltage pulse for the MCP was set
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Fig. 2.— HYADES calculation (without a cloud present) of the free-stream velocity Uf (t), the Mach number M ðtÞ, the compression 1 ðtÞ/10 (where the initial density 10 ¼ 300 mg cm3), and the temperature T ðtÞ at a location 500 m from the ablated surface of the shock tube. Note that due to experimental limitations, a rarefaction reverses the direction of the flow just before t ¼ 40 ns.
to 500 ps, resulting in a gain duration of 300 ps in a trade-off between maximizing X-ray exposure on the MCP while minimizing motion blurring (e.g., when the plasma moves 20 km s1, the motion blurring is 6 m, comparable to the pinhole diameter). The experiment is repeated with different time delays to generate an image sequence. Physical quantities in the CRF (without a cloud present) can be evaluated using the hydrodynamic code HYADES. (This is a one-dimensional Lagrangian hydrodynamics and energy transport code in which electron and ion components are treated separately in a fluid approximation, each in local thermodynamic equilibrium [ LTE] and described by Maxwell-Boltzmann statistics, but loosely coupled to each other. Radiation is coupled only to the electron fluid and is assumed in our calculations to be Planckian and is treated as a single group [gray approximation]. All energy transport is modeled using the diffusion approximation. For further information on the HYADES code, refer to Larsen & Lane [1994].) We have used HYADES to calculate the free-stream velocity Uf ðt Þ, the temperature T (t), the Mach number M ðt Þ, and the density 1 ðt Þ for the CRF, and these quantities are plotted in Figure 2 at a location 500 m from the ablated surface. This is the approximate location of the cloud in the experiment, and so the calculated quantities represent the physical environment of the cloud in the experiment, but calculated without a cloud present. Figure 2 shows that the cloud environment is first shocked and compressed by the passing blast wave, then subject to a high-Mach-number flow with a gradually decreasing velocity (which reverses direction as the rarefaction follows the blast wave through the CRF). The velocity Uf ðt Þ plays the most important role in the outcome of the experiment, while the temperature and density only have second-order effects, e.g., the temperature dropping from 24 to 4 eV alters the viscosities. The simulation used an in-line quotidian equation-of-state model
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(QEOS; an equation-of-state model based on Thomas-Fermi physics [ More et al. 1988]) with a bulk modulus of 3 ; 109 Pa and 112 zones to represent the 2.25 mm long CRF, with the first 46 zones feathered for ablation with a zone-to-zone scaling ratio of 1.15, the final 46 zones feathered for shock release with a scaling ratio of 0.87. The laser energy in the simulation was tuned to match shock velocities with experimental data. This tuning of the simulation compensates for loss of energy in the experiments due to lateral expansion of shock tube walls, laser-plasma interaction, etc. (Note that the CRF may be more compressible than what the QEOS allows for, meaning at a given energy a shock in the experiment would move slower than in the numerical simulation. Then when we reduce the simulation energy to match the shock position with the real shock position, we end up with a simulation with a slower than real flow speed. However, this change is fairly small, likely smaller than that of the 6 times less dense CRF [50 mg cm3] used in Miles et al. [2004], and is assumed negligible.) The experimental parameters were scaled to preserve the physics regime of the astrophysical case. In particular, a flow described by the Euler equations will remain governed by the Euler equations after a parameter scaling if the following four criteria ( Ryutov et al. 1999) are satisfied. (1) Particles are localized on spatial scales smaller than the characteristic length scale D0 of the system, e.g., the collisional mean free path kc TD0 . (2) Convective (hydrodynamic) transport of heat must dominate conductive transport of heat, e.g., the Peclet number Pe31 so that the travel time at the speed of sound over a characteristic length scale is shorter than the thermal or radiative conduction times. (3) Energy fluxes carried by the flow ( hydrodynamics) must be large compared to radiative energy fluxes. (4) Viscous dissipation needs to be small compared to inertial forces, i.e., the Reynolds number Re 31. ( In addition to these four criteria, the flow is assumed to be nonmagnetic, and we also note that while the blast wave width likely has some impact on the evolution of the cloud, it is unlikely to affect the final conclusions of this manuscript simply because the blast wave width is no larger than the cloud dimension [we estimate it at 1, since in this case D0 3 kR . Using the values above, Bo 25.) 4. The Reynolds number Re ¼ D0 U / , where the kinematic viscosity , strictly speaking, must include both photon viscosity and particle viscosity (although we expect the particle viscosity to dominate). The photon viscosities are easily calculated (Ryutov et al. 1999) from pho ¼ ð1:7 ; 1025 cm4 g 2 s1 K4 ÞAT 4 /Z 2 , and for T 10 eV we get pho ¼ 6:8 ; 108 m2 s1 for the CRF and pho ¼ 8:8 ; 1010 m2 s1 for the Al cloud. The particle viscosity is more complicated, as the plasma-coupling parameter > 1, i.e., the Braginskii viscosity for weakly coupled plasma
ions, is invalid. We must use the more extensive viscosity model by Cle´rouin et al. (1998) and Robey (2004), part ¼ ð6:55 ; 1010 cm9=2 s1 g1=2 ÞZmi1=2 n5=6 ½k I1 þ ð1 þ kI2 Þ 2/kI3 , where mi is the ionic mass and the parameters k, I1, I2, andP I3 are calculated from the plasma-coupling parameter ¼ Z 1=3 j xj Zj qe2 ð4 / 1=3 3nÞ /4 "0 kB T , where xj is the mass fraction of the jth ionic species with atomic number Zj. The parameters are k ¼ 4 (3) 3=2/3, = I1 ¼ (180 3=2 )1 , I2 ¼ (0:49 2:231 3 )/60 2 , and I3 ¼ =9 3=2 1 2:41 /10 . We then calculate part ¼ 5:4 ; 107 m2 s1 for the CRF and part ¼ 1:5 ; 105 m2 s1 for the Al cloud. The Reynolds number is then clearly much greater than unity (for reference, the flow Reynolds number is Re 2 ; 105 ), and consequently, all four criteria for a properly scaled experiment are satisfied, meaning the overall morphology of the cloud is properly scaled. 3. RESULTS Results from the experiment can be seen in Figure 3. As the shock runs over the cloud (t ¼ 5 ns), its speed inside the cloud is greatly reduced. The velocity difference between surrounding flow and cloud material eventually leads to a Kelvin-Helmholtz instability and its characteristic roll-up (t ¼ 12 ns). Soon thereafter, a Widnall-type instability (Widnall et al. 1974) occurs, creating a low mode number azimuthal perturbation of order 5 when viewed from a point on the extended shock tube axis (Robey et al. 2002; Klein et al. 2003). Here we see the Widnall instability as four ‘‘fingers’’ at the trailing edge of the cloud at t ¼ 30 ns, indicating a mode number of 4Y8 (depending on if each finger is or is not overlapping another finger along the line of sight). Material is constantly being stripped away from the Al plasma cloud and is visible in the images as a cone of diffuse material behind the cloud (t 19 ns) extending all the way to the shock. By t ¼ 40 ns this cone extends outside our diagnostic field of view.
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By t ¼ 60 ns so much material has been stripped away that the remaining cloud is quite diffuse (we are showing the 60 ns image at a higher contrast than the earlier point projection radiography images in Fig. 3). By t ¼ 100 ns the cloud has been completely stripped away and can no longer be identified in the point projection radiography images. We also obtained an image at 80 ns in which the cloud is completely gone, but this target unfortunately had its gold spatial reference grid (see Fig. 1) mounted outside the view of the X-rayYframing camera and was therefore not included in Figure 3. Note that in the images up to t ¼ 60 ns the cloud remains (roughly) at its starting location, 500 m from the ablated CRF surface, and that the flow speed Uf is low enough that the cloud cannot have translated out of view in the later images. To quantify how far out in time our experiment goes, allowing comparison to other experiments and numerical simulations, we = estimate the cloud-crushing time tcc ¼ ð 20 /10 Þ1 2 R0 /Ub , where 10 ¼ 300 mg cm3 is the initial CRF density, 20 ¼ 2:7 g cm3 the initial Al density, and the shock speed when the shock runs over the cloud can be simply estimated as Ub 500 m/5 ns ¼ 100 km s1. For the current experiment this means tcc 1:8 ns. In the Nova experiments presented by Klein et al. (2000, 2003), the much longer tcc ¼ 8:3 ns gives about 5:3tcc for their latest image. The same dimensionless time is reached by 15 ns in our experiment, and we continue to take data to well over 30tcc. Because the point projection radiography technique illuminates the MCP in a very uniform fashion, we can use the point projection radiography images to estimate the cloud mass. Mass attenuation along the line of sight in an X-ray image can be expressed as I ¼ I0 expð1 m/AÞ, where I is the measured pixel intensity, I0 is the X-rayYsource intensity (i.e., the intensity we would have expected to measure had there not been mass attenuation), m is the integrated line of sight mass, is the X-ray attenuation coefficient in units of area per mass, and A is the pixel area in the image. With a background intensity level Ib and multiple X-ray frequencies passing through multiple materials i the general formula becomes Y X I0; ei ð Þmi =A þ Ib : ð1Þ I¼
i¼1; 2; 3; : : :
To estimate the cloud mass, the only two materials of interest in equation (1) are the Al cloud material m2 and the CRF material of the surrounding flow m1. Other materials (e.g., blast shields in the camera protecting the detector from target debris) do not move, and their contributions to X-ray attenuation are constant, so equation (1) can be expressed as X I0; e1 ð Þm1 =A e2 ð Þm2 =A þ Ib ; ð2Þ I¼ where the new coefficients I0; include attenuation from m3 , m4 , . . . , etc. To be precise, m1 ¼ m1 (m2 ), but because the opacity of the cloud material is much larger than that of the flow material, i.e., 2 ð Þ 3 1 ð Þ for all frequencies , m1 can be assumed constant, giving X I ðm2 Þ ¼ I1; e2 ð Þm2 =A þ Ib ; ð3Þ
where I1; ¼ I0; expð1 ð Þm1 /AÞ. The relative (but not absolute) strength of the nonattenuated intensities I0; can be measured using an X-ray spectrometer, and this was done for Ti backlighters (for a type very similar to ours)
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in the energy range 4Y7 keV by Glendinning et al. (2000). (Ti plasma may emit X-rays at energies outside this energy range, but this has little impact on our images; see below.) The absolute intensity I0; þ Ib can be measured in a portion of each image that is not occupied by the shock tube (or alternatively n uI1; þ Ib can be measured in unshocked material m1 ). The background intensity Ib includes all sources of nondirectional exposure, such as film fogging, nondirectional X-rays, energetic particles, etc., and can be measured if the image includes an object that is opaque to all contributions under the frequency summation; in our case this is accomplished using the Au spatial reference grid. With all the intensity coefficients known, equation (3) could be solved numerically for m2. However, we avoid a numerical iteration by making a further simplification taking advantage of our backlighter X-rays being near-monochomatic; 80% are He at 4.7 keV (with He at 5.6 keV dominating the remaining 20%). By approximating all 2 ð Þ with the same constant value 2, equation (3) simplifies to I ðm2 Þ ¼ I1 e2 m2 =A þ Ib ; which can be rearranged for an expression of the mass, A I Ib ln m2 ¼ ; 2 I1 but we have then introduced an error I in I ðm2 Þ, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi X dI ð Þ I ¼ d2 ð Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi X maxðm2 Þ 2 ð Þ maxðm2 Þ=A e I1; 2 ð Þ ; A
ð4Þ
ð5Þ
ð6Þ
where 2 ð Þ ¼ j2 ð Þ 2 j and maxðm2 Þ is the maximum cloud mass we would expect in any pixel (which in our case is safely estimated by a [hypothetical] pixel containing the original center point of the Al sphere). This error is added to a measurement error I1 of the mean pixel value in an area of the CRF flow surrounding the cloud (an area that excludes the cone of stripped material). There is also a measurement error Ib of the mean pixel value of the Au grid. We take both to be standard deviations, and the total error in the measured mass (in any given pixel with intensity I ) then becomes s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 @m @m2 2 2 2 2 I1 þ I þ Ib m2 ¼ @I1 @Ib sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 A I1 þ I Ib2 þ ¼ : ð7Þ 2 2 I1 ð I Ib Þ 2 The total error in the measured mass of the entire cloud, consisting of N pixels, then becomes vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N X I 2 þ I2 A u 1 2 tN 1 : ð8Þ þ I mtot ¼ b 2 2 2 I1 p¼1 ð I Ib Þ Using an X-ray attenuation coefficient for our Al cloud material of 2 ¼ 191:2 cm2 g1 and a pixel area A ¼ 3:9 ; 109 cm2, we then calculate the cloud mass to be 0:67 0:11 g at t ¼ 30 ns
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Fig. 4.— Normalized plasma height h (dimension along shock tube axis) as a function of normalized time for this work and compared to a recent experiment by Ranjan et al. (2005).
and 0:54 0:11 g at t ¼ 40 ns. These values can be compared to the original sphere mass of 2.44 g. Note that in the calculation above, only 5% of the 0.11 g total error comes from using a single attenuation coefficient 2, justifying our simplification of disregarding other X-ray frequencies in the 4Y7 keV energy range. Outside this energy range, lower energy X-rays emitted by the Ti plasma have energies less than 0.5 keV, so these can be ignored, as X-ray attenuation coefficients at such energies are 3 orders of magnitude larger than the coefficient used in the calculation. It is also conceivable that the Ti plasma could produce emission at higher energies from thermal bremsstrahlung. However, an analysis of the 5 ns image of Figure 3 reveals no significant contribution from higher energy X-rays (other than that higher energy X-rays may possibly be one additional contributor to the background intensity Ib ). In this early image, the Al cloud is still contained within its original spherical volume, and no stripped mass is present in the surrounding CRF. Accounting for the nonuniform intensity of the area backlighter, we measure X-ray intensities in this image of 26 at the cloud center and 128 in the unshocked CRF (these are average pixel intensities on a scale from 0 to 255 over sampling areas of 30 ; 30 m). From these two values we can calculate Ib and a single I0 in equation (1), assuming a monochomatic source, and then use these in turn to calculate an expected X-ray intensity in the shocked part of the CRF. Over a finite area of 30 ; 30 m just behind the shock, the shocked CRF has a density of 1.2 g cm3, according to numerical simulations done using the Lawrence Livermore CALE code (an arbitrary Lagrangian-Eulerian code; Barton 1985), giving an expected X-ray intensity of 78. The actual measured X-ray intensity was 80. While there are some uncertainties in these numbers, it is clear that contributions from X-rays more energetic than those measured by Glendinning et al. (2000) are not significantly affecting our mass measurements. Figures 4 and 5 show the physical dimensions, height and width, respectively, of the Al plasma cloud at different times. The quantities in these plots are normalized for comparison to recent data by Ranjan et al. (2005) from another shock-cloud interaction experiment (Ranjan et al. 2005; Niederhaus et al. 2005),
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Fig. 5.— Normalized plasma width w (dimension perpendicular to the shock tube axis) as a function of normalized time for this work and compared to a recent experiment by Ranjan et al. (2005).
in which 5 cm soap bubbles filled with Ar are placed in a N2 environment and shocked by planar shocks at a Mach number M ¼ 2:88. In our experiment the dimensionless time ¼ ðt 5 nsÞ max Uf /D0 . ( The 5 ns offset comes from our definition of the experimental time t starting when the laser pulse is fired, while Ranjan et al. defines from when the shock hits the cloud.) The normalized height h is the size of the cloud measured along the shock tube axis divided by D0, and the width w is the size perpendicular to the shock tube axis divided by D0. The qualitative response of the cloud is the same in both experiments; the height is initially compressed, and the width increases rapidly. At around 5 the height returns to unity, while the width is noticeably reduced. This lasts until 9, when both height and width begin a gradual, roughly linear growth with time. Quantitatively, we see a larger width but smaller height than Ranjan et al. at comparable times. The issue of preheat must be taken into consideration in laser experiments, and measurements of the Al sphere in the 5 ns image of Figure 3 allow quantitative estimates of preheat. The laser radiation ablating the CRF surface generates preheat of two types: radiative preheat and hot electrons. Radiative preheat by X-rays produced in the ablation front will heat the CRF volumetrically but will tend to heat only the surface of the Al sphere. The front side of the sphere (the hemisphere facing the ablation front) will be heated more than the back side of the sphere; the ‘‘equator’’ between the two hemispheres is heated to an intermediate value. Hot electrons will heat both the CRF and the Al sphere volumetrically, but likely to much lower temperatures than the radiative preheat; radiative preheat tends to mask preheat due to electrons in ultraviolet ( UV ) laser experiments ( Yaakobi et al. 2000). Since the Al sphere has a higher density than the surrounding CRF, and since it also tends to absorb more preheat and thus become hotter, the sphere will expand due to preheat, creating a density gradient of Al plasma. In the 5 ns image of Figure 3, we measure a density gradient at the sphere equator extending over 4 m (after adjusting for the expected transmission through the sphere and more importantly the finite size of our pinhole). At the back side of the sphere, at the point furthest from the ablation front, we
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measure a density gradient extending over 1 m. We have checked this expansion using the CALE code (Barton 1985). CALE’s laser package is used to deposit energy into the problem. This is done according to the time-dependent energy profile and super-Gaussian radial intensity profile characteristic of the experiment. Simulations were run with CALE’s radiation diffusion package in order to assess the effect of radiative preheat, but CALE does not include the effect of hot electrons. The radiation diffusion allows for energy transfer from the laser deposition region to the sphere ahead of the incident shock. The radiation field is assumed to be Planckian, and Rosseland and Planck mean opacities are used to calculate the rates of radiation diffusion and matter-radiation coupling, respectively. The radiation transport in the code assumes a uniform medium and will likely differ somewhat from that actually produced in the CRF, as foam consists of cells essentially free of matter surrounded by higher density walls. Even so, we find that the observed expansion in Figure 3 is consistent with our CALE simulation, which has a 3.7 m density gradient on the equator and 1.5 m on the back side. Since CALE does not include hot electrons, we therefore conclude that hot electrons, as expected from the findings of Yaakobi et al. (2000), play a much smaller role than radiative preheat in our experiment. There are two ways that the radiative preheat could pose a problem to our experiment. First, the expansion of the sphere could change the starting conditions for the mass-stripping process. However, the amount of sphere mass affected by preheat is relatively small (at most a shell 4 m thick), so even if this mass is stripped faster than what otherwise would have been the case, it would not change the conclusions of our mass-stripping analysis. Second, the CRF could be volumetrically heated, thus changing post-shock quantities of the flow. However, post-shock quantities tend to go as 1 þ 1/M 3 2 (Drake 2006), so as long as the Mach number M 3 1, the preheat is insignificant for the interpretations of this experiment. We can use the measured density gradient to estimate the sound speed a2 of the Al sphere surface and also set a limit on the CRF sound speed a1 . When the surface of the sphere is heated, expansion will occur at a speed no higher than the adiabatic expansion speed 2/ð 1Þa2 (the speed may be less than this as the CRF slows the expanding Al plasma), where is the adiabatic index. A heat wave is at the same time moving toward the interior of the sphere at the sound speed a2 , so at a given time t we have a2 t < p < ½2/ð 1Þ þ 1a2 t, where p is the thickness of Al plasma that has a density gradient due to radiative preheat. To be precise, the limiting quantities should really be integrals, as a2 is likely to vary slightly over time, but for our purposes of roughly estimating preheat sound speeds, a constant a2 will do. Furthermore, we assume a worst-case scenario and simply set p ¼ a2 t. For our measured p ¼ 4 m over t ¼ 5 ns, we then obtain a2 ¼ 0:8 km s1. If we assume that the CRF is heated to the same temperature as the surface of the sphere (even though the CRF is likely heated to a lower temperature) then the sound speed in the CRF would be a1 ¼ 1:8 km s1, and since the shock is moving at Ub 100 km s1, the Mach number would still be M > 50 3 1. Consequently, radiative preheat will not change the final conclusions of this manuscript. 4. ANALYSIS We present in this section a new mathematical model that describes mass stripping from a cloud under turbulent, high Reynolds number conditions. We compare this model to our experimental data and to an existing model ( Taylor 1963; Ranger & Nicholls 1969) for laminar mass stripping. Our model combines four separate concepts of fluid mechanics: (1) the integral momentum
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equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere (we use potential flow with good result; for further refinement this could be changed, e.g., to Fage’s equation [Fage 1936], albeit with the penalty of more complicated algebra), (3) Spalding’s law of the wall for turbulent boundary layers (Spalding 1961), and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. We begin with the integral momentum equations for a stationary, viscous boundary layer, @ @x
Z
1
u1 ðU u1 Þ dy1 þ
dU dx
Z
1
ðU u1 Þ dy1
Z 1 dr 1 @u1
þ u1 ðU u1 Þ dy1 ¼ 1 ; ð9Þ r dx 0 @y1 y1 ¼0
Z 2 Z @ 1 dr 2 2 1 dp @u2
2 ¼ 2 u22 dy2 þ u2 dy2 þ ; @x 0 r dx 0 2 dx @y2 y 2 ¼0 0
0
ð10Þ
@u1
1 1 @y
1 y1 ¼0
@u2
¼ 2 2 @y
ð11Þ
;
2 y 2 ¼0
where x is a coordinate along the surface of the cloud [we approximate the cloud with a sphere at all times so that x ¼ 0 at the flow stagnation point and x ¼ ð /2ÞR at the equator], y is a coordinate perpendicular to the cloud surface, r is the distance from the cloud surface to the cloud axis of symmetry, U ¼ U ð xÞ is the free-stream flow velocity immediately outside the boundary layer, u ¼ uð x; yÞ is the flow velocity inside the boundary layer, is the kinematic viscosity, is the density, p ¼ pð xÞ is the pressure, and ¼ ð xÞ is the thickness of the boundary layer. The flow properties U, u, , , p, and are also functions of time t, but are changing relatively slowly (the flow is changing over timescales which are longer than the transit time past the cloud), and so we have chosen to use the equations for a stationary boundary layer to make the problem more tractable. Subscript ‘‘2’’ denotes plasma in the cloud, and subscript ‘‘1’’ denotes the surrounding flow, e.g., 2 is the boundary layer thickness inside the cloud. The geometry is sketched out in Figure 6. Taylor (1963) developed a theory for wind stripping the surface layer of a fluid, and subsequently, Ranger & Nicholls (1969) extended Taylor’s model to estimate how much mass is stripped from falling raindrops. Taylor’s theory is laminar, and while it works beautifully for raindrops, i.e., for low Mach and Reynolds numbers, we see below that it does not accurately predict mass stripping under turbulent, high Reynolds number conditions. However, we follow Taylor’s lead in relating the flow velocity on the surface of the cloud to the free-stream flow velocity through the expression AU, where A is a constant, as indicated in Figure 6. Inside the boundary layer, away from the surface in either direction, our expressions become more complicated than Taylor’s, who assumes ad hoc relations for the flow velocities inside the boundary layers, u1/U ¼ 1 ð1 AÞ expðy1 /1 x p Þ and u2 /U ¼ A exp (y2 /2 x p ), where Taylor sets p ¼ 12 and solves the system of equations for the coefficients 1, 2, and A. Instead of these arbitrary expressions for the velocities u1 and u2 , we use Spalding’s law of the wall for turbulent boundary layers (Spalding 1961), " þ
þ
y ¼u þe
B
2
e
uþ
ðuþ Þ ðuþ Þ 1 u 2 6 þ
3
# ;
ð12Þ
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HANSEN ET AL.
Vol. 662
and trivially, dr/dx ¼ 0. Carrying out the spatial derivative and setting x ¼ R/2, we then find that equation (9) becomes Z
1
þ 2 v10 uþ 1 ð1 2AÞU 2v1 u1 dy1 ¼ v1 ;
ð14Þ
0
where v10 ¼ dv1 ð xÞ/dx and U ¼ 3Uf /2. Equation (10) becomes Z 0
Fig. 6.— Flow geometry around the cloud is modeled with the potential flow around a sphere with the boundary layer flow calculated using a local Cartesian coordinate system where the coordinate x is along the flow (i.e., along the surface of the sphere) and y is the distance into the boundary layer from the sphere surface. The velocity in the boundary layer is u1 outside the sphere radius R and u2 inside the sphere radius. At the outside edge of the boundary layer y1 ¼ 1, the velocity u1 ¼ U, i.e., matches the potential flow velocity. At the sphere radius u1 ¼ u2 ¼ AU, where A is a constant. At the inside edge of the boundary layer y2 ¼ 2 , the velocity u2 ¼ 0. The cylindrical coordinate r is the distance from the axis of symmetry to the sphere surface.
an empirical relation that has been shown to hold in numerous fluid dynamics experiments and a multitude of regimes. Here, the dimensionless coordinate yþ yv / , and the dimensionless velocity uþ u/v, or in the appropriate coordinate frame v2 uþ of reference u1 ¼ v1 uþ 1 þ AU and u2 ¼ AU 2 . The wall 2 friction velocity v is defined through v ¼ ðdu/dyÞjy¼0 , and Coles (1955) gives the coefficients ¼ 0:41 and B ¼ 5:0 (using the values ¼ 0:4 and B ¼ 5:5 given by Nikuradse [1930] does not produce significantly different results). Next, we proceed by carrying out the spatial derivatives with respect to x as far as we can in equations (9) and (10). To reduce the algebra, we assume that the free-stream velocity around the cloud follows the potential flow of a sphere, 3 x ; ð13Þ U ð xÞ ¼ Uf sin 2 R
2
þ 2 2v20 uþ 2 v2 u2 AU dy2 ¼ v2 :
Next, we carry out the integration with respect to y. This is done by substituting duþ for dy using Spalding’s law of the wall (eq. [12]) and noting that when y ¼ then uþ 1 ¼ ð1 AÞU /v1 in þ the CRF and u2 ¼ AU /v2 in the Al cloud. Integrating equation (14), we then obtain " 3 1 1 2 U 0 B ð1 þ 2AÞð1 AÞ þ 2 eB v1 1 1 e 6 v1 6 4 5 U 1 3 B U e ð3 þ 10AÞð1 AÞ 4 ; ð1 þ AÞð1 AÞ 3 þ v1 40 v1
U ð1AÞ 1 U U þ e B ð1 2AÞ e v1 ð1 A Þ 1 þ 1 v1 v1 ( 2 U ð1AÞ 1 U 2 2 eB e v1 ð1 AÞ v1 ) !# U 2 2 ð1 AÞþ 2 2 ¼ v1 : ð16Þ v1 Integrating equation (15) gives AU 3 1 1 B 2 AU 4 B e 2v 2 1 e þ 6 v2 12 v2 1 B 3 AU 5 1 B AU vAU AU 2 þ e e e 1 þ1 40 v2 v2 v2 # )! ( " AU 1 AU 2 AU 2 þ 2 e B e v2 2 þ2 2 ¼ v2 : v2 v2 0
ð17Þ This is as far as we can go without saying something about the wall-friction velocity v, or equivalently, the skin-friction coefficient Cf , as the two are related through 2
where Uf is the flow velocity far from the sphere, and we assume Uf to be uniform over the dimension of the cloud (this is very nearly true in the experiment). One could certainly use a nonuniform flow field, e.g., Uf ð xÞ ¼ U ð1 þ 1 cosð x/RÞÞ, where the numerical constant 1 is negative for a decelerating flow field, or a different expression for the free-stream velocity, e.g., the Fage (1936) equation 3 5 7 U x x x x ¼ 1:5 0:4371 þ 0:1481 0:0423 ; Uf R R R R but the potential flow expression has the advantage that dU /dx ¼ 0 at the equator x ¼ R/2, which will simplify the algebra substantially. In addition, at the equator dp/dx ¼ g UdU /dx ¼ 0,
ð15Þ
v ¼ 12 Cf ð xÞU 2 ð xÞ:
ð18Þ
For calculations of skin-friction drag, many researchers, beginning with Dryden & Kuethe (1930) and Millikan (1932), have used velocity distributions for a flat plate in nonflat geometries and found that the results do not differ seriously from measured values (Goldstein 1965). We do the same and use the skin friction coefficient for a turbulent boundary layer on a flat plate ( White 1974), ; Cf ð xÞ 0:0592Re1=5 x
ð19Þ
where the Reynold’s number Rex ¼ Ux/ , but with a modification; if we use equation (19) as is, the problem is overdetermined.
No. 1, 2007
MASS STRIPPING OF CLOUD FOLLOWING SHOCK PASSAGE
We need to introduce the equivalent of Taylor’s 1 and 2. For very small uþ , we have y¼
v
(
"
þ 2
þ 3
u ðu Þ ðu Þ þ þ e B eu 1 uþ v 2 6
where ¼ K2 A2 :
#)
ð29Þ
ð20Þ where
Comparing this to Taylor’s
1 K3 ¼ 2
u2 y2 ¼ 2 x ln AU p
suggests that we should replace the coefficient 0.0592 with a coefficient that will be determined by our system of equations. We set Cf ð xÞ ¼
ð28Þ
Next, we relate 1 to 2 by rewriting equation (11) as 1 ¼ K3 2 ;
u 2 u : ¼ Cf ð xÞU 2 ð xÞ v 2
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2 ðUx= Þ1=5 ; 2
ð21Þ
which allows us to calculate the spatial derivative of v as v 0 ð xÞ ¼
1 v : 10 x
In the CRF at the equator x ¼ R/2, we then get 3 Uf R 1=10 U ¼ 1 ¼ K1 1 ; v1 4 1
ð22Þ
where we have defined a new coefficient K1 that depends only on known quantities, and we can rewrite equation (16) as
2 1200 2 e B K19 ð1 AÞ1 1 ¼ 120e ð1þ 2AÞ ð3 þ 2AÞ þ 4 3ð3 þ 10AÞ 5 20ð1 þ AÞ 4 þ 20 1 e B 1 ð1 þ 2AÞ 3 120ð1 2AÞ 480;
ð24Þ
where
¼ K1 ð1 AÞ1 :
ð25Þ
We expect A to be a fairly small quantity because of the difference in densities between cloud and surrounding flow (it will certainly be smaller than unity), so one might be tempted to linearize equation (24) with respect to A, but this only simplifies terms where A (or ) does not appear in the exponents and does not lead to an analytical solution for A, as it does in the laminar theory. Consequently, some form of simple numerical scheme must be employed to calculate A, and we have therefore chosen to not linearize equation (24) with respect to A, but to keep the exact form. Similarly, in the Al cloud, defining 3 Uf R 1=10 ð26Þ K2 ¼ 4 2 allows us to rewrite equation (17) as 5 4 600 2 e B K29 1 2 ¼ 120e ð 2Þ 3 10 1 B e 1 3 þ 120 þ 240; ð27Þ þ 20
ð30Þ
We eliminate 1 by substituting equation (29) into equations (24) and (25), leaving us with two equations, equations (24) and (27), for two unknown coefficients A and 2. This equation pair can easily be solved numerically, e.g., equation (24) can be solved for A by simple iteration as it converges rapidly, and a simple regula falsi (secant) method can be used for equation (27), but other numerical schemes will work too, and we used a globally convergent Newton’s method. With A and 2 at hand, one easily calculates the mass stripped from the cloud by integrating the cloud material flowing through the boundary layer at the equator ( Ranger & Nicholls 1969), dm ¼ 2 R 2 dt
ð23Þ
1=2 1=10 1 : 2
Z
2
u2 dy2 ¼ 2 R 2 2 ðÞ;
ð31Þ
0
where we have defined a mass-stripping coefficient ð Þ ¼
1 2 1 B 1 2 1 3 1 4 e þ 1 þ e : 2 2 2 6 24 ð32Þ
It should be noted thatdm/dt is not proportional linearly to R, 2, or 2, because ¼ R; Uf ; 1 ; 2 ; 1 ; 2 from the numerical solution above. Using the specific physical quantities for our experiment, we can now calculate the mass stripped as a function of time and compare the calculation to our experimental data. For the cloud radius Rðt Þ we use measured values from the experiment images, namely, half of the measured width in Figure 5, and interpolate to other times. For values of the free-stream flow velocity Uf ðt Þ, the density 1 ðt Þ, and the temperature T (t), we use values from HYADES. The density 2 ðt Þ is obtained by applying the same compression as for 1 ðt Þ. In addition, the peak compressions are independently verified from the experiment at t ¼ 5 ns, where the left side of the sphere is compressed to an ellipsoid shape with minor radius 30 m, corresponding to a compression of 4 (which is the strong shock limit for a polytropic gas with adiabatic index ¼ 5/3). With our given physical quantities, the coefficients K1 5, K2 3, and K3 14 at all times (varying slightly over time as the input parameters gradually change, e.g., K2 ¼ 3:5 initially, but gradually drops to 2.9 by t ¼ 35 ns, then increases again after the rarefaction has changed the direction of the flow). From solving equations (24), (27), and (29) we calculate the coefficients A 15, 1 6, and 2 27 at all times, and we find that the compound quantity varies between 6 and 8 (except very briefly when the rarefaction changes the direction of the flow, i.e., Uf < 1 km s1), so that the mass-stripping coefficient is in the range 4 ; 10 2 P P 4 ; 10 3 . The mass of the cloud as a function of time is plotted
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HANSEN ET AL. low to achieve the cloud being completely stripped by t 80 ns; if the mass stripping was done by laminar flow and continued under the same conditions past t ¼ 80 ns (ignoring experimental limitations), the laminar mass-stripping time would be 1 s. As a final note, to illustrate the nonlinearity between dm/dt and the various physical quantities, one can arbitrarily double, say, the value of the viscosity 2 and see that this leads to only a 12% increase in dm/dt. 5. CONCLUSIONS
Fig. 7.— Cloud mass remaining as a function of time calculated using a laminar model ( Taylor 1963; Ranger & Nicholls 1969; dashed line) and the turbulent model presented in this manuscript (solid line), compared to experimentally measured values of the cloud mass (two squares). The turbulent model agrees with the measured values and also predicts that the cloud is completely stripped by 90 ns, which compares well with the experimental observation of the cloud being stripped by 80Y100 ns. In the laminar model (assuming unchanged condition from 80 ns), the cloud is not stripped until 1 s.
in Figure 7 and reaches m ¼ 0 (fully stripped) by t 90 ns. This agrees well with the experiment where the cloud can no longer be observed by 80Y100 ns. By comparison, the equivalent massstripping coefficient ¼ ð2 Rl l Þ1 dm/dt in the laminar theory is P4 ; 10 2 for all times of interest in the experiment, which is too
We observe the rapid stripping of all mass from a simulated interstellar cloud in a laser experiment. We present a model that agrees very well with our experimental observations. The model combines (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding’s law of the wall for turbulent boundary layers, and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. By comparison, a laminar model overestimates the stripping time by an order of magnitude. This suggests that mass stripping in the experiment must be of a turbulent nature, and with its even higher Reynolds numbers, this must also hold in the astrophysical case; the relatively short lifetimes of interstellar clouds may be due to shock-cloud interactions, where mass is stripped from a boundary layer on the cloud surface by passing shocks and subsequent high Reynolds number flows.
We would like to thank C. F. McKee, Departments of Physics and Astronomy, University of California, Berkeley for his support of this project. This work was performed under the auspices of the US Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
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