Mach reflection in a warm dense plasmaJ. M. Foster, P. A. Rosen, B. H. Wilde, P. Hartigan, and T. S. Perry

Citation: Physics of Plasmas 17, 112704 (2010); doi: 10.1063/1.3499690View online: http://dx.doi.org/10.1063/1.3499690View Table of Contents: http://aip.scitation.org/toc/php/17/11Published by the American Institute of Physics

Mach reflection in a warm dense plasmaJ. M. Foster,1 P. A. Rosen,1 B. H. Wilde,2 P. Hartigan,3 and T. S. Perry4

1Atomic Weapons Establishment, Aldermaston, Reading RG7 4PR, United Kingdom2Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA3Department of Physics and Astronomy, Rice University, 6100 South Main, Houston,Texas 77521-1892, USA4Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550, USA

�Received 2 August 2010; accepted 13 September 2010; published online 8 November 2010�

The phenomenon of irregular shock-wave reflection is of importance in high-temperature gasdynamics, astrophysics, inertial-confinement fusion, and related fields of high-energy-densityscience. However, most experimental studies of irregular reflection have used supersonic windtunnels or shock tubes, and few or no data are available for Mach reflection phenomena in theplasma regime. Similarly, analytic studies have often been confined to calorically perfect gases. Wereport the first direct observation, and numerical modeling, of Mach stem formation for a warm,dense plasma. Two ablatively driven aluminum disks launch oppositely directed, near-sphericalshock waves into a cylindrical plastic block. The interaction of these shocks results in the formationof a Mach-ring shock that is diagnosed by x-ray backlighting. The data are modeled using radiationhydrocodes developed by AWE and LANL. The experiments were carried out at the University ofRochester’s Omega laser �J. M. Soures, R. L. McCrory, C. P. Verdon et al., Phys. Plasmas 3, 2108�1996�� and were inspired by modeling �A. M. Khokhlov, P. A. Höflich, E. S. Oran et al., AstrophysJ. 524, L107 �1999�� of core-collapse supernovae that suggest that in asymmetric supernovaexplosion significant mass may be ejected in a Mach-ring formation launched by bipolar jets.�doi:10.1063/1.3499690�

I. INTRODUCTION

After their discovery by Ernst Mach in the late 19thcentury, irregular shock reflection phenomena �so-calledMach reflections consisting of a particular configuration ofincident, reflected and Mach stem shock waves, which, to-gether with a slip discontinuity, meet at a triple point� havebeen intensively studied following the pioneering work ofvon Neumann.1,2 An understanding of such interactions ofstrong shock waves is central to many fields of high-energy-density science, including explosively driven blast waves,high-temperature gas dynamics, astrophysics, and inertial-confinement fusion �ICF�. The Mach reflection configurationis important in height-of-burst studies,3 is apparent in theflow field from supersonic jets,4,5 and has been postulated asa mechanism for the structure observed in astrophysicalHerbig–Haro objects.6,7 Mach reflection of two sphericalshocks has been proposed to explain x-ray emission from theVela supernova remnant8 and a Mach stem formed followingthe interaction of bipolar jets may explain some observa-tional features of core-collapse supernovae.9 In ICF,Richtmyer–Meshkov instability and the interaction of shockwaves that traverse capsule defects may lead to the formationof jets of material that initiate mixing and thus degradeyield.10,11

Most experimental studies of irregular shock reflectionhave used supersonic wind tunnels or shock tubes and havestudied the pseudosteady reflection of a shock wave at aninclined wedge, or the mutual interaction of two shocks�launched from opposing wedges�. In spite of its relevancefor astrophysics and laboratory high-energy-density science,few or no data are available for Mach reflection phenomena

in the plasma regime. In this paper, we report the first directobservation, and numerical modeling, of Mach stem forma-tion for a warm dense plasma �typically, 5–10 eV tempera-ture, 1–2 g cm−3 density�.

Mach reflection phenomena have been reviewed in detailby Hornung,12 Glass,13 and Ben-Dor14,15 who note that thegeneral Mach reflection �MR� configuration may be furthersubdivided into specific wave structures, encompassingsingle Mach reflection �SMR�, transitional Mach reflection�TMR�, and double Mach reflection �DMR�. In the self-similar, pseudosteady flow regime, criteria for transition be-tween regular reflection �RR�, SMR, TMR, and DMR struc-tures may be derived analytically and are frequentlyrepresented16,17 as transition lines in the �M ,�w� plane, whereM is the flow Mach number ahead of the incident shock, and�w is the angle of the shock-forming wedge. In the case ofnon-self-similar, unsteady shock reflections �such as the re-flection of a planar shock at a concave or convex cylindricalsurface, or the reflection of a spherical shock at a plane sur-face� simple transition criteria do not exist, and recoursemust be made to numerical computation. Further complica-tion arises from hysteresis14 that may arise in transition be-tween the different types of reflection. Such unsteady shockreflection phenomena do frequently occur in many cases ofcurrent interest �astrophysics, high-energy-density laboratoryplasma physics, etc.�, and in a review of future researchneeds for shock-wave reflection, Ben-Dor15 lists transitioncriteria for unsteady shocks and Mach stem curvature,among other topics, as deserving further research. The pur-pose of the present work is to investigate the regular-to-Mach-reflection transition in the case of interacting, near-

PHYSICS OF PLASMAS 17, 112704 �2010�

1070-664X/2010/17�11�/112704/9/$30.00 17, 112704-1

spherical shocks in a dense plasma for the purpose of testingnumerical radiation-hydrocode simulations.

II. EXPERIMENTAL DETAILS AND GENERALCONSIDERATIONS

Figures 1 and 2 show our experimental arrangement. Wegenerate counter-propagating, convex-curved shock frontsusing a laser-target assembly consisting of two ablativelydriven aluminum foil “pushers” at opposite ends of a cylin-

drical hydrocarbon foam cylinder. Two laser-heated hohl-raum targets provide x-ray ablation of the outer faces of thealuminum disks, leading to the propagation of 50 Mbar �peakpressure� shocks through the aluminum foils, and subse-quently into the hydrocarbon foam. The aluminum disks are220 �m diameter and are set �flush with both faces� within50 �m thickness, 2000 �m diameter gold washers. Thecentral, cylindrical component has 800 �m diameter,450 �m length, resorcinol formaldehyde aerogel foam�C15H12O4� at 0.51 g cm−3 density, and pore size �0.1 �m.The two hohlraum targets have 1600 �m diameter,1200 �m length, and 1200 �m diameter laser entranceholes. The experimental package is illuminated through a800 �m diameter hole in the hohlraum wall, opposite thelaser entrance hole. This axisymmetric configuration enablesus to model the entire �both hohlraums plus experimentalpackage� assembly using two dimensional radiation hydro-codes. Each hohlraum is heated using 12 beams of theOmega18 laser with a total energy of 6 kJ in a 1 ns duration,constant-power laser pulse of 0.35 �m wavelength. The re-sulting peak radiation drive temperature lies in the range190–200 eV. The propagation and interaction of shockswithin the target assembly is measured using x-ray backlight-ing radiography along two �mutually orthogonal� lines ofsight, perpendicular to the axis of symmetry of the experi-mental package. Two x-ray backlighting targets are used �onefor each line of sight�, positioned 4 mm from the axis ofsymmetry. Each target consists of a 3 mm square, 25 �mthickness foil of titanium illuminated with six laser beams�400–600 ps pulse duration, 0.35 �m wavelength�, incidentfrom both sides of the target in a 800 �m diameter spot. Thebacklighter spectrum is dominated by the 4.75 keV reso-nance line emission of He-like titanium. X-ray backlit im-ages are recorded on Kodak DEF x-ray film using two cam-eras, each incorporating a 16-hole array of 10 �m diameterpinholes. The combination of geometric spatial resolutionand motion blurring results in a net spatial resolution of ap-proximately 15 �m. The experimental arrangement isclosely similar to that used for our complementary study ofsupersonic jet and shock interactions.19 Figure 3 shows ex-perimental, x-ray backlit, images of the motion of thealuminum-pusher-driven shocks in the hydrocarbon foamand the formation of a curved, radially propagating Mach-ring shock. Figure 4 identifies the principal features of theMach reflection structure.

Before turning to the detailed hydrocode modeling ofour experiment, it is interesting to infer approximately thephysical conditions in the shocked material and ask if themeasured critical angle is in line with expectations from ana-lytic shock reflection theory. In the present experiment, theincident and Mach shock velocities, us and uM, lie in therange 20–30 �m ns−1. Although it is difficult to assess withany certainty the preheated temperature of the undisturbedhydrocarbon foam ahead of the shock, a simple calculationdemonstrates that our experiment lies in the strong shockregime, as follows. We calculate the sound speed �cs

= ��P /��1/2, where the adiabatic index, �=Cp /Cv� in thefoam ahead of the shock, and thus the shock Mach number�M =u /cs�. SESAME equation-of-state tables20 for polysty-

X-ray backlight

Dante

12 Omegalaser beams,6 kJ, 1 ns

FIG. 1. Experimental arrangement for the Mach reflection experiment. Twoopposing laser-heated hohlraums are used to ablate aluminum pushers thatin turn drive counter-propagating shocks into a resorcinol-formaldehydefoam experimental package. The shock positions are recorded by x-raybacklighting radiography, using an area backlighting source and time-gatedx-ray camera. The “Dante” x-ray diode array is used to monitor the hohl-raum temperature.

FIG. 2. The experimental package for the Mach reflection experiment, inwhich the incident �i�, reflected �r�, and Mach �m� shocks are shown sche-matically and �bottom� typical experimental data. Dimensions are inmicrons.

112704-2 Foster et al. Phys. Plasmas 17, 112704 �2010�

rene �a material similar to that used in the present experi-ment� indicate a pressure of 0.5 GPa at 0.1 eV temperatureand 0.5 g cm−3 density, and 20 GPa at 1 eV temperature and0.5 g cm−3 density. With the assumption, for now, that�=5 /3, the corresponding sound speeds are 1.3 and8 �m ns−1 suggesting M �3–20, and so M �1 even at theimplausibly high preheat temperature of 1 eV. We proceedfurther by noting from the strong-shock relations for agamma-law gas21 that the specific internal energy, �, of theshocked foam is given by �=0.5�2 / ��+1��2u2. Let us as-sume that the foam is shocked to peak theoretical density of���+1� / ��−1���0=2 g cm−3. With reference to theSESAME equation-of-state for polystyrene, we find that at adensity of 2 g cm−3, the specific internal energy is relatedapproximately to temperature by �=45T1.125, where � and Tare in units of MJ kg−1 and eV, respectively. Inspection of theexperimental data �Fig. 3� shows that the radial velocity ofthe Mach shock is approximately 30 �m ns−1, and we thusinfer a temperature of approximately 5 eV for the materialbehind the Mach stem. The Mach and incident shock veloci-ties are related by uM =us /sin , and we infer an incidentshock speed of 20 �m ns−1 and a corresponding incidentshock temperature of approximately 3 eV. It remains to dem-onstrate that our assumption of an ideal polytropic gas isapproximately reasonable. The ideal gas equation-of-state isP= ��−1���. Again from the SESAME equation-of-state, wefind, for example, that at T=5 eV and �=2 g cm−3,�=246 MJ kg−1, and P=344 GPa. By substitution, we inferan effective ��1.7, close to our assumption of �=5 /3.More detailed inspection shows that for material at some-

what lower temperature �representative of the incidentshock�, we find an effective � somewhat less than 5/3, be-cause of the “loss” of energy to dissociation and ionization.

Now, the critical angle for Mach reflection depends onthe Mach number, M, of the shock and the adiabatic index,�, of the material in which the shock is propagating. A fre-quently quoted expression due to Courant and Friedrichs22

for the critical angle in the limit of infinitely strong shocksand using the “detachment criterion” of von Neumann is

cr = arcsin� 1

� ,

and for a �=5 /3 ideal gas we infer cr=37°. De Rosa et al.23

discuss the limitations of this expression and provide thefollowing �better� approximation:

FIG. 3. Experimental images �obtained using a 4.75 keV x-ray backlightingsource� showing formation of the Mach stem as a function of time. Dimen-sions are in microns. Note the changes of offset on the left-hand axis, whichwere chosen to centralize approximately the radially propagating Mach stemin each image.

FIG. 4. �Top� Experimental data from the Mach stem experiment att=11 ns, in which the incident shock �I�, reflected shock �R�, Mach stem�M�, and slip surface discontinuity �S� may all be identified. �Bottom� No-tation for analysis of the experimental data. The angles 1 and 2 are de-fined by the tangent to the incident shock front at the position of the triplepoint �T�. LM is the length of the Mach stem. The angle is the deviationfrom planarity of the Mach stem at the triple point.

112704-3 Mach reflection in a warm dense plasma Phys. Plasmas 17, 112704 �2010�

cr = arctan 1

� − 1� ��2 − 1

� − ��2 − 11/2

� �1 − ��2 − ��� − 1�1/2�� .

The Courant and Friedrichs and De Rosa treatments showlittle difference of critical angle for �=5 /3, but diverge forsmaller �, with a difference of nearly 10° at �=1.2. Analyti-cal calculations by Glass and Ben-Dor16,17 identify the tran-sition lines between RR and DMR, TMR and SMR as afunction of Mach number, M, and wedge angle, �w �note that�w+=90°�, although for a monatomic gas and all M �4, �w

is close to 55° for the transition from RR �corresponding tocr=35°�. Of course, the analytic treatment of Ref. 17 maynot be strictly appropriate to our own case of two counter-propagating, curved shock fronts. Rather, our own case maybe similar to that of the interaction of two explosively drivenblast waves,24,25 or the reflection of an explosively drivenblast wave at the ground plane.3 In these cases a curvedMach stem is typically observed, and the experimentallymeasured26 critical angle for Mach reflection may differ by10° �smaller or greater� from the analytically calculated tran-sition line from RR to MR. Further consideration is given tothis point in discussion of our numerical hydrocode simula-tions.

III. HYDRODYNAMIC MODELING

We have modeled the experiment using radiation hydro-codes employing both fixed-mesh and adaptive-mesh-refinement �AMR� Eulerian schemes, implemented in theAWE hydrocode PETRA and the LANL hydrocode RAGE. Inboth cases, the simulation includes a representation of boththe hohlraum and the experimental package, although notreatment of laser light propagation and absorption is in-cluded. Instead, radiation drive is simulated by imposing atemperature in those cells of the calculational mesh that areinitially within the internal volume of the hohlraum and filledwith low-density �10−4 g cm−3� material. Our simulation ofthe hohlraum driver for this experiment �using the AWE La-grangian radiation hydrocode NYM� is explained in detail inRef. 19, together with a fuller discussion of our prescriptionfor the radiation drive and further details of Eulerian model-ing using the PETRA and RAGE hydrocodes.

PETRA is a two-dimensional Eulerian hydrocode devel-oped by AWE.27,28 Radiation transport is treated by single-group �gray� diffusion, and ion, electron, and radiation tem-peratures are assumed equal �one-temperature model�.Equation-of-state �EOS� and opacity data are input in tabularform, provided by off-line calculations. In simulation of thepresent experiment, EOS was obtained from tables that de-rive from the combination and smooth interpolation of ex-perimental data �where these are available� and Thomas–Fermi calculations �elsewhere�. Radiative opacities wereobtained from off-line simulation using the IMP �Ref. 29�opacity code. Specifically, standard EOS data were used forthe aluminum and gold components of the experiment, andthe resorcinol-formaldehyde foam was represented by the

tabular EOS for polystyrene. The experimental package waszoned in the following way: a cell size of 1.25 �m per cellwas used in the axial direction and 2 �m per cell in theradial direction. Outside of this region, coarser zoning wasused to enable the calculation to run in a time-efficient man-ner. A region of low-density CH was placed outside the hohl-raum, immediately adjacent to the laser-entry holes, to over-come time-step problems in the simulation. A startingtemperature of 0.1 eV was used throughout the entire mesh.

Radiation Adaptive Grid Eulerian �RAGE� is a multidi-mensional, multimaterial Eulerian radiation-hydrodynamicscode developed by Los Alamos National Laboratory and Sci-ence Applications International.30

RAGE uses a continuous �intime and space� adaptive-mesh-refinement �CAMR� algo-rithm to follow interfaces, shocks, and gradients of physicalquantities such as material densities and temperatures. Ateach cycle, the code automatically determines whether tosubdivide or recombine Eulerian cells. The user also has theoption to dezone �that is, reduce the resolution of the mesh�as a function of time, space, or material. Adjacent squarecells may differ by only one level of resolution, that is, by afactor of 2 in cell size. Interface tracking was not used in thepresent simulations �although several different interface pre-servers are now available� and the simulation easily followscontact discontinuities with fine enough zoning at the mate-rial interfaces �1.5 �m minimum resolution was used in thepresent case�. The CAMR method speeds calculations by asmuch as two orders of magnitude over straight Eulerianmethods. RAGE uses a second-order-accurate Godunov hy-drodynamics scheme similar to the Eulerian scheme ofColella.31 Mixed cells are assumed to be in pressure andtemperature equilibrium, with separate material and radiationtemperatures. Radiation transport is approximated with im-plicit gray flux-limited diffusion. Opacity and EOS data aregenerated off-line and input in tabular form. The SESAME�Ref. 20� data tables were used for the equation of state, andthe EOS of the resorcinol-formaldehyde foam was based onthat of carbon �SESAME material no. 7831�. The RAGE codehas been validated with analytical test problems and appliedto many shock tube, laser and pulsed-power experiments.32,33

In the case of both PETRA and RAGE, the hydrocodesimulation is postprocessed to generate synthetic x-ray back-lit images for comparison with the experimental data.

In the following, we illustrate our modeling of the hy-drodynamics by considering simulations obtained using thePETRA code, although the results of RAGE simulations are notsignificantly different. Figure 5, taken from a PETRA simula-tion of the experiment, shows the material interfaces, mate-rial density, and synthetic x-ray backlit image at 11 ns afterthe origin of the laser drive. The locations of the incident andMach stem shocks are clearly identifiable in the density dis-tribution and synthetic radiograph, as is the twice-shockedmaterial between the slip discontinuity and the reflectedshock. The region of gold that is evident in the materialdistribution �and synthetic radiograph� arises from the axialstagnation and subsequent jetting of the hohlraum wall and isprobably overestimated because of the two-dimensionalsymmetry of the simulation.34 Figure 6, also taken from thePETRA simulation at 11 ns, shows temperature and velocity

112704-4 Foster et al. Phys. Plasmas 17, 112704 �2010�

distributions. The higher temperature �in comparison withthe incident shock� of material within the Mach stem andreflected shock is apparent. Two points of interest are evidentin the velocity distribution: the velocity shear at the slip sur-

FIG. 6. �Color� PETRA simulation �at t=11 ns� of the Mach stem experimentshowing temperature and velocity distribution. Note the temperature differ-ence between the incident and Mach shocks, the velocity shear at the slipdiscontinuity �evident in the axial component of velocity�, and the jetting atthe symmetry plane �evident in the radial component of velocity, just insidethe Mach shock�. Dimensions are in microns.

FIG. 5. �Color� PETRA simulation �at t=11 ns� of the Mach stem experimentshowing material interfaces and density distribution and synthetic 4.75 keVradiograph. Dimensions are in microns.

112704-5 Mach reflection in a warm dense plasma Phys. Plasmas 17, 112704 �2010�

face discontinuity �evident in the axial velocity distribution�and the jetting phenomenon at the symmetry plane �evidentin the radial velocity distribution, inside the Mach stemshock�. A similar jetting phenomenon is discussed in detailby Henderson et al.35 It is not clearly discernable in theraw-data late time experimental radiographs, although it is atthe limit of spatial resolution of the present work. Figure 7shows synthetic radiographs from the PETRA simulation, attimes chosen for direct comparison with the experimentaldata of Fig. 3.

IV. DISCUSSION

We identify shock positions in the experimental and syn-thetic radiographs by use of the Canny edge-detectionalgorithm.36,37 Figure 8 shows a comparison of shock posi-tions identified by edge-detection in the radiographs shownin Fig. 3 �experimental images� and in the PETRA and RAGE

simulations �Fig. 7 shows synthetic images from PETRA�.Figure 9 compares the radial positions of the Mach stemshock at the plane of symmetry.

We attempt to identify the critical angle for Mach reflec-tion from our experimental data and simulations. We con-struct tangents to the incoming shocks �at the position of thetriple points� and measure the angles of incidence �1 and2, in the notation of Fig. 4� relative to the plane of symme-try and the length of the Mach stem �LM�. Figure 10 showsthe measured length of the Mach stem as a function of themean angle of shock incidence, �1+2� /2, at the triple

point in the experiment and in the PETRA and RAGE simula-tions.

In the case of steady-flow Mach reflections, the experi-mental data of Hornung and Robinson38 suggest a nearlylinear dependence of Mach stem length on angle of shockincidence. Although we have no reason, a priori, to assumethat such a linear dependence extends to our own nonstation-ary flow experiment, the experimental data nevertheless sug-

FIG. 7. Synthetic radiographs from PETRA simulation of the Mach reflec-tion experiment. The Mach stem and slip discontinuity are evident at 10 nsand later, and the jetting phenomenon at the plane of symmetry is evident at13 ns. Dimensions are in microns. Note the changes of offset on the left-hand axis, which were chosen to centralize approximately the radiallypropagating Mach stem in each image.

FIG. 8. �Color� Location of shock positions in the experiment �black�, andthe PETRA �red� and RAGE �blue� synthetic radiographs. The shocks are iden-tified by using the Canny edge-detection algorithm. In the case of the PETRA

simulations, the data have been offset by a maximum of 20 �m �comparewith Fig. 9, which shows absolute positions� to aid comparison of shape.The amplitude of noise in the experimental data is dependent on the degreeof spatial smoothing employed in the edge-detection process, although thisdoes not affect the shape of the detected shock front.

FIG. 9. �Color online� Radial position of the Mach shock at the plane ofsymmetry in experiment and simulation.

112704-6 Foster et al. Phys. Plasmas 17, 112704 �2010�

gest that this may be a reasonable approach for the purposeof identifying the critical angle. We do not intend to imply afunctional relationship between and LM for our nonsteadyshock, but instead simply use Fig. 10 to identify the criticalangle at which the Mach stem first appears. On this basis,backward extrapolation suggests a critical angle, cr, forMach reflection of close to 48° for our present experiment�different by approximately 10° from the analytically calcu-lated angle�. Now, the analytic derivation of critical angleassumes that the Mach stem is perpendicular to the wedgesurface �or plane of symmetry, in our own case�. Glass39

discusses the effect of Mach stem curvature on critical angleand notes that the critical angle may differ from calculationby an amount equal to the angle of curvature of the Machstem �the angle , defined in Fig. 4� at the triple point. It isdifficult to measure this angle accurately from the experi-mental data, although we note that �15°, and that this may,in part, explain the difference between our measured criticalangle and analytic estimates. Hysteresis14 may also delay theonset of Mach stem formation in a case such as the presentexperiment, where the angle increases progressively to thatrequired for Mach reflection.

Our experiments differ from other work using shocktubes and supersonic wind tunnels, and aside from the higherincident shock Mach number that can be addressed byplasma experiments, it is interesting to consider the potentialconsequences of the high temperature and small spatial scalethat is accessed in the present experiment.

Scaling from the laboratory plasma regime to an entirelydifferent spatial scale requires that dissipative processes areunimportant, as discussed by Ryutov et al.40,41 The condi-tions of temperature, density, typical scale size, and fluidvelocity of the present experiment are sufficiently close tothose of our earlier jet-shock interaction experiments for theresults of Ref. 19 to apply also in the present case. In briefsummary we note that the warm, dense plasma of the presentexperiment is well described by a hydrodynamic treatment.The Reynolds number is high �of order of 106� and viscous

effects are negligible. With reference to Figs. 5 and 6, wechoose two aspects of the experiment for more detailed, il-lustrative discussion: thermal conduction from the shock-heated material to its surroundings and hydrodynamic insta-bility at the shear surfaces within the Mach stem.

The thermal diffusivity is given42 by

�cm2 s−1� = 3.30 � 10−3 T�eV�5/2A

Z�Z + 1�ln ���g cm−3�,

and for a �by means of example� carbon plasma shocked tofour times initial density and a temperature of approximately10 eV �a typical upper limit for our present work�, we find athermal diffusivity in the order of 1 cm2 s−1. For a typicalvelocity �u� of 20 �m ns−1 and spatial resolution distancescale �r� of 10 �m, the corresponding Peclet number �Pe=ur / � is of the order of 2000, and thermal conduction isevidently negligible. RAGE simulations with and withoutthermal conduction included confirm this point.

Axial fluid velocity on either side of the slip surface�PETRA simulations shown in Fig. 6� is typically�5 �m ns−1. For the radially propagating jet that forms atthe plane of symmetry �Fig. 6�, radial velocity is typically30 �m ns−1 within the surrounding fluid’s radial flow of ve-locity of 20 �m ns−1. In each case, the velocity shear is10 �m ns−1, over some distance of separation of the flows.The Kelvin–Helmholtz �KH� growth rate at these interfacesis given by �=ku��1�2 / ��1+�2�2�0.5, where k is the wave-number of the perturbation and u is the relative fluid velocity.For a wavenumber of 0.3 �m−1 characteristic of the spatialresolution of our experiment, densities ��1 ,�2� of 1.5 and3 g cm−3 on either side of the slip surface, and fluid velocityof 10 �m ns−1, we find a KH growth rate of the order of1.5 ns−1. During the time of the Mach stem’s formation �8ns� and our latest measurement �13 ns�, approximately eighte-foldings of growth occur for wavelengths that we might beable to resolve. Shorter spatial wavelengths would rapidlysaturate at the resolution limit of our experiment. We notethat the slip surface and any jetting phenomenon at the planeof symmetry are not clearly observed in Fig. 3, but that thismay be because of the limited spatial resolution �and becauseof statistical “shot noise”� in the experimental radiographs,and not necessarily because of instability growth at positionsof velocity shear. In this context, a further plasma experimentat larger spatial scale and with longer duration of observationwould be interesting. An experimental signature of velocityshear instability may appear as break-up of the Mach stem atlate time. As far as the present experiment is concerned, theconsequence of working in the dense plasma regime may beevident in departure of the EOS from the ideal gas approxi-mation.

With reference to Fig. 10, we note that the PETRA andRAGE simulations are in close agreement �within the limits ofaccuracy of measurement of critical angle in the syntheticradiographs� and that both imply a critical angle close to thatobserved in experiment. To identify the effect of departuresfrom ideal gas EOS in the simulations, we have repeated thePETRA simulations with an ideal gas EOS in place of thetabular EOS, and for two choices of � �1.67 and, arbitrarily,

FIG. 10. �Color� Observed Mach stem length, LM, and mean angle of inci-dence, �1+2� /2, of the incoming shock at the position of the triple point,for the experimental data and the PETRA and RAGE simulations.

112704-7 Mach reflection in a warm dense plasma Phys. Plasmas 17, 112704 �2010�

1.33�. These data are also shown in Fig. 10. The ideal gassimulations show the expected trend of critical angle withadiabatic index, and we infer that the departure of criticalangle from that expected from analytic treatments of Courantand Friedrichs, and De Rosa, is a consequence of the non-steady, curved incident shock, and the curved Mach stem, inour experiment. The experimental data tend to depart furthestfrom simulation at late time, although the single data pointfor a 65° angle of incidence may be affected by asymmetryin the experimental data that is evident in the 13 ns imageshown in Fig. 3. As explained previously, the precise two-dimensional symmetry of the simulation may overestimatethe late time jetting of inward expanding gold from the hohl-raum wall, and this may contribute to the late time differenceof experiment and simulation. By contrast, detailed inspec-tion of the simulations suggests that this late time jettingphenomenon does not contribute significantly to the hydro-dynamics at early time and for the determination of the criti-cal angle.

V. CONCLUSIONS

The phenomenon of irregular shock-wave reflection�Mach reflection� is important throughout the fields of high-temperature gas dynamics, astrophysics, inertial-confinementfusion, and related areas of high-energy-density science.Aside from the terrestrial context of, for example, height-of-burst studies and the flow fields of supersonic jets, shockwave interactions may play a key role in shaping the struc-ture of astronomical objects on both large and small scales.Understanding how Mach stems form in an experimental set-ting has important astrophysical applications. Most astro-physical flows are unsteady, and the preshock medium can behighly inhomogeneous. For example, in stellar jets these twofactors lead to multiple curved bow shocks at some workingsurfaces where wings of individual bows intersect one an-other at oblique angles. Were a Mach stem to form at theseintersection points it would significantly affect the dynamicsof the region, because incident gas at this location wouldnow experience a normal shock instead of a highly obliqueone. As a result, a hot spot would form at that location andgive rise to much higher-excitation line emission than wouldotherwise occur. In addition, a nearly normal shock like theMach stem accelerates preshock material to the flow speedrather than deflecting it like an oblique shock would, leadingto markedly differing behavior in the dynamics of the post-shock flow. The enhanced luminosity recorded in multiple-epoch observation by one of the authors �Hartigan� at theintersection of large sheetlike bow shocks in astrophysicaljets �for example, the Herbig–Haro object HH34� may itselfresult from heated material at a Mach stem shock.

Almost all experimental data for Mach reflection phe-nomena result from shock tube work and do not extend intothe plasma regime. In the present paper, we have describedan experiment that extends the study of Mach reflection phe-nomena into the dense plasma regime and reported compari-son of these data with simulations using two-dimensionalradiation hydrocodes. The critical angle for Mach stem for-mation is reproduced by the hydrodynamic modeling and

differs from that expected on the basis of analytic treatmentsfor an ideal gas. This difference arises in part from the tran-sient nature of the experiment in which curved shock frontsinteract, which may also be due in part to deviation of theequation of state of the shocked material from ideal gas�where it is coldest and densest�. These initial experimentssuggest obvious extensions to the further study of denseplasma Mach reflection phenomena in other geometries�such as the interaction of shock with a conical wedge ofvariable angle, differences of curvature of the interactingshocks, and alternative choices of the material�s� throughwhich the shocks propagate�.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge the guidance provided byPaul Drake and Alexei Khokhlov in the design and interpre-tation of this experiment. The authors also gratefully ac-knowledge the technical expertise and assistance of VernRikow and Sharon Alvarez, the staff and operations team ofthe Omega laser facility, and the LLNL and AWE target-fabrication groups. This work was supported by the Los Ala-mos National Laboratory under the auspices of the UnitedStates Department of Energy under Contract No. DE-AC52-06NA25396.

1J. von Neumann, Oblique Reflection of Shocks, Explos. Res. Rep. Vol. 12�Navy Department, Bureau of Ordnance, Washington, D.C., 1943�.

2J. von Neumann, Refraction, Intersection and Reflection of Shock Waves,NAVORD Rep. Vol. 203-45 �Navy Department, Bureau of Ordnance,Washington, D.C., 1943�.

3T. C. J. Hu and I. I. Glass, AIAA J. 24, 607 �1986�.4L. L. Smarr, Physica �Amsterdam� 12D, 83 �1984�.5H. Li and G. Ben-Dor, AIAA J. 36, 488 �1998�.6M. L. Norman, L. Smarr, K.-H. A. Winkler, and M. D. Smith, Astron.Astrophys. 113, 285 �1982�.

7P. Hartigan, Astrophys. J. 339, 987 �1989�.8V. V. Gvaramadze, Astron. Lett. 24, 144 �1998�.9A. M. Khokhlov, P. A. Höflich, E. S. Oran, J. C. Wheeler, L. Wang, and A.Yu. Chtchelkanova, Astrophys. J. 524, L107 �1999�.

10S. R. Goldman, S. E. Caldwell, M. D. Wilke, D. C. Wilson, C. W. Barnes,W. W. Hsing, N. D. Delameter, G. T. Schappert, J. W. Groves, E. L.Lindman, J. M. Wallace, R. P. Weaver, A. M. Dunne, M. J. Edwards, P.Graham, and B. R. Thomas, Phys. Plasmas 6, 3327 �1999�.

11S. R. Goldman, C. W. Barnes, S. E. Caldwell, D. C. Wilson, S. H. Batha,J. W. Grove, M. L. Gittings, W. W. Hsing, R. J. Kares, K. A. Klare, G. A.Kyrala, R. W. Margevicious, R. P. Weaver, M. D. Wilke, A. M. Dunne, M.J. Edwards, P. Graham, and B. R. Thomas, Phys. Plasmas 7, 2007 �2000�.

12H. G. Hornung, Annu. Rev. Fluid Mech. 18, 33 �1986�.13I. I. Glass, AIAA J. 25, 214 �1987�.14G. Ben-Dor, Shock Wave Reflection Phenomena �Springer-Varlag, New

York, 1991�.15G. Ben-Dor and K. Takayama, Shock Waves 2, 211 �1992�.16G. Ben-Dor and I. I. Glass, J. Fluid Mech. 92, 459 �1979�.17G. Ben-Dor and I. I. Glass, J. Fluid Mech. 96, 735 �1980�.18J. M. Soures, R. L. McCrory, C. P. Verdon, A. Babushkin, R. E. Bahr, T.

R. Boehly, R. Boni, D. K. Bradley, D. L. Brown, R. S. Craxton, J. A.Delettrez, W. R. Donaldson, R. Epstein, P. A. Jaanimagi, S. D. Jacobs, K.Kearney, R. L. Keck, J. H. Kelly, T. J. Kessler, R. L. Kremens, J. P.Knauer, S. A. Kumpan, S. A. Letzring, D. J. Lonobile, S. J. Loucks, L. D.Lund, F. J. Marshall, P. W. McKenty, D. D. Meyerhofer, S. F. B. Morse, A.O. Kishev, S. Papernov, G. Pien, W. Seka, R. Short, M. J. Shoup III, M.Skeldon, S. Skupsky, A. W. Schmid, D. J. Smith, S. Swales, M. Wittman,and B. Yaakobi, Phys. Plasmas 3, 2108 �1996�.

19J. M. Foster, B. H. Wilde, P. A. Rosen, T. S. Perry, M. Fell, M. J. Edwards,B. F. Lasinski, R. E. Turner, and M. L. Gittings, Phys. Plasmas 9, 2251�2002�.

20K. S. Holian, “T-4 handbook of material properties data bases. Vol Ic:

112704-8 Foster et al. Phys. Plasmas 17, 112704 �2010�

Equations of state,” Los Alamos National Laboratory Report No. LA-10160-MS, 1984.

21Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena �Academic, New York, 1966�,Vol. 1, pp. 50–55.

22R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves�Springer-Verlag, New York, 1986�, p. 327.

23M. De Rosa, F. Fama, V. Palleschi, D. P. Singh, and M. Vaselli, Phys. Rev.A 45, 6130 �1992�.

24J. M. Dewey, D. J. McMillin, and D. F. Classen, J. Fluid Mech. 81, 701�1977�.

25M. De Rosa, F. Fama, M. A. Harath, V. Palleschi, A. Salvetti, D. P. Singh,M. Vaselli, E. M. Barkudarov, M. O. Mdivnishvili, I. V. Sokolov, and M.I. Taktakishvili, Phys. Lett. A 156, 89 �1991�.

26J. Wisotski, quoted in G. Ben-Dor, Phys. Lett. A 156, 265 �1991�.27D. L. Youngs, in Numerical Methods for Fluid Mechanics, edited by K. W.

Morton and M. J. Baines �Academic, London, 1982�.28D. L. Youngs, Physica �Amsterdam� 12D, 32 �1984�.29S. J. Rose, J. Phys. B 25, 1667 �1992�.30M. L. Gittings, R. Weaver, M. Clover, T. Betlach, N. Byrne, R. Coker, E.

Dendy, R. Hueckstaedt, K. New, W. R. Oakes, D. Ranta, and R. Stefan,Comput. Sci. Disc. 1, 015005 �2008�.

31P. Colella, SIAM J. Comput. 6, 104 �1985�.32R. M. Baltrusaitis, M. L. Gittings, R. P. Weaver, R. F. Benjamin, and J. M.

Budzinski, Phys. Fluids 8, 2471 �1996�.33R. L. Holmes, G. Dimonte, B. Fryxell, M. L. Gittings, J. W. Grove, M.

Schneider, D. Sharp, A. L. Velikovich, R. P. Weaver, and Q. Zhang, J.Fluid Mech. 389, 55 �1999�.

34B. H. Wilde, M. L. Gittings, T. S. Perry, J. M. Foster, P. A. Rosen, M. Fell,and J. A. Cobble, Bull. Am. Phys. Soc. 44, 39 �1999�.

35L. F. Henderson, E. I. Vasilev, G. Ben-Dor, and T. Elperin, J. Fluid Mech.479, 259 �2003�.

36W. K. Pratt, Digital Image Processing �Wiley, New York, 1991�.37J. Canny, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679 �1986�.38H. G. Hornung and M. L. Robinson, J. Fluid Mech. 123, 155 �1982�.39I. I. Glass, American Institute of Aeronautics and Astronautics Report No.

AIAA-86–0306, 1986.40D. D. Ryutov, R. P. Drake, J. Kane, E. Liang, B. A. Remington, and W. M.

Wood-Vasey, Astrophys. J. 518, 821 �1999�.41D. D. Ryutov, B. A. Remington, H. F. Robey, and R. P. Drake, Phys.

Plasmas 8, 1804 �2001�.42L. Spitzer, Jr., The Physics of Fully Ionized Gases, 2nd ed. �Wiley Inter-

science, New York, 1962�, pp. 143–147.

112704-9 Mach reflection in a warm dense plasma Phys. Plasmas 17, 112704 �2010�