A Laboratory Astrophysical Jet Validation Test of the Radiation Hydrodynamics Capabilities of the FLASH Code
AA Laboratory Astrophysical Jet Validation Test of the Radiation HydrodynamicsCapabilities of the FLASH Code
Chris Orban , Milad Fatenejad , and Don Q. Lamb , (1) Department of Physics,The Ohio State University, Columbus, OH 43210(2) ASC Flash Center for Computational Science,University of Chicago, Chicago, IL(3) 6157 N Kenmore, Chicago, IL 60660(4) Department of Astronomy,University of Chicago, Chicago, IL ∗ (Dated: June 19, 2020)The potential for laser-produced plasmas to yield fundamental insights into high energy densityphysics (HEDP) and deliver other useful applications can sometimes be frustrated by uncertaintiesin modeling the properties and behavior of these plasmas using radiation-hydrodynamics codes.In an effort to overcome this and to corroborate the accuracy of the HEDP capabilities thathave been added to the publicly available FLASH radiation-hydrodynamics code, we presentdetailed code-to-code comparisons between FLASH and the HYDRA code developed at LawrenceLivermore National Laboratory using previously published HYDRA simulations from Grava etal. 2008. That study describes a laser experiment that produced a jet-like feature that theauthors compare to astrophysical jets. Importantly, the Grava et al. 2008 experiment includeddetailed x-ray interferometric measurements of electron number densities. Despite radicallydifferent methods for treating the computational mesh, and different equation of state and opacitymodels, the FLASH results greatly resemble the results from HYDRA and, most importantly, theexperimental measurements of electron density. Having validated the FLASH code in this way, weuse the code to further investigate and understand the formation of the jet seen in the Grava etal. (2008) experiment and discuss its relation to the Wan et al. (1997) experiment at the NOVA laser. PACS numbers:
I. INTRODUCTION
The potential for laser experiments to yield fundamen-tal insights into High-Energy-Density Physics (HEDP)is in many ways limited by the sophistication and ac-curacy of current-generation “three-temperature” (3T) radiation-hydrodynamics codes that simulate the heat-ing, conduction and radiation of laser-irradiated fluids.In deconstructing the results from ultra-high intensity,short-pulse laser experiments, for example, Particle-In-Cell (PIC) simulations of the ultra-intense pulse interac-tion with the target may depend sensitively on radiation-hydrodynamics simulations of the heating and ionizingeffect of stray “pre-pulse” laser energy in the nanosecondsbefore the arrival of the main pulse. It is not always pos-sible to use interferometric instruments to measure elec-tron number densities in the “pre-plasma” created by thispre-pulse, as the target geometry may not permit probe ∗ Electronic address: [email protected] We use the term “three-temperature” (or “3T”) to denote theapproximation that electrons and ions move together as a singlefluid but with two different temperatures, and that this fluid canemit or absorb radiation. In the 3T simulations presented in thispaper, each cell has an electron temperature, an ion temperature,and radiation energy densities in a number of photon energy bins. beams to access the pre-plasma, e.g., in cone targets as inRef.[1], so the pre-plasma properties must be predictedusing a radiation-hydrodynamics code. The uncertaintiesin these simulations may frustrate efforts to gain a betterunderstanding of ion acceleration or electron transportthat could prove to be valuable for a variety of applica-tions, such as radiation therapy, x-ray generation, or theactivation and detection of fissile materials.Another important use of these codes is in modelinginertial confinement fusion experiments at laser facili-ties like Omega and the National Ignition Facility (NIF)[2–4]. Rosen et al. [5] describe some of the subtletiesencountered in understanding indirect-drive experimentsand a 2012 panel report by Lamb & Marinak et al. [6]outlines a number of remaining uncertainties in simulat-ing ignition-relevant experiments at NIF. Lamb & Mari-nak et al. [6] emphasize the need for code-to-code com-parisons and validation in a wider effort to reproducethe diagnostics of NIF implosions. Although there havebeen some recent investigations with other codes [7, 8],the HYDRA code [9–11], developed at Lawrence Liver-more National Laboratory (LLNL), is frequently used forradiation-hydrodynamics modeling of these experiments.Uncertainties and inaccuracies in radiation-hydrodynamics modeling can also frustrate the designand interpretation of experiments to investigate fun-damental plasma properties (e.g., opacities, equation a r X i v : . [ phy s i c s . p l a s m - ph ] J un of state, hydrodynamic instabilities) at HEDP-relevantdensities and temperatures [12–14]. Both Omega andNIF, among other facilities, have completed a number ofexperiments in this category, and will continue to do soin the future [15, 16].With these concerns in mind, and in an effort to con-firm the accuracy of the HEDP capabilities of the FLASHradiation-hydrodynamics code [17–19], we compare thepredictions of FLASH to previously-published resultsfrom nanosecond laser irradiation of an Aluminum tar-get and previously published modeling of this experimentusing the HYDRA code [20]. FLASH is a finite-volumeEulerian code that operates on a block-structured meshusing Adaptive Mesh Refinement (AMR) [21], whereasthe HYDRA code uses an Arbitrary-Lagrangian-Eulerian(ALE) scheme to determine the computational grid [22–24], which can deform and stretch in response to themovement and heating of the fluid. In other respectsthe codes are very similar in that they use a tabulatedEquation of State (EOS) and make many of the same as-sumptions regarding laser propagation and absorption aswell as various aspects of the hydrodynamics. The simu-lations in this paper were performed with FLASH 4-beta[25] with an added Lee-More conductivity and thermalequilibration model [26]. The Lee-More model becamepart of the publicly available version of the code startingwith FLASH 4.2.2 [27].We show results for an experiment in which a targetconsisting of an Al slab with a mm-long triangular grooveis irradiated by a rectangular laser beam. The results ofthis experiment, which is translationally invariant alongthe groove and so is a test of plasma expansion in 2DCartesian geometry, were modeled with HYDRA simu-lations in Grava et al. 2008 [20], hereafter referred toas GRAVA. They investigated this problem for its re-semblance to astrophysical jets where radiative coolingplays an important dynamical role, and as a miniatureversion of similarly-motivated experiments at the Novalaser carried out by Wan et al. [28] (hereafter WAN)and [29]. The experiment was performed at ColoradoState University and, importantly, GRAVA present x-rayinterferometric measurements of electron number den-sity from 1-20 ns after the target begins to be irradi-ated that afford a powerful validation test. We thereforecompare the results of FLASH simulations to both theHYDRA simulations they present and the experimentaldata GRAVA present. Using a commercially-available[30] EOS and opacity model with FLASH, we find over-all good agreement comparing FLASH with experimen-tal measurements and HYDRA simulations presented inGRAVA.Section II presents comparisons of FLASH predictionsto the experiments and modeling in GRAVA. Section IIIuses FLASH simulations to investigate and understandthe formation of the jet in the GRAVA experiment. Wesummarize and conclude in Sec. IV. II. COMPARISON TO GRAVA ET AL. 2008
In this section, we describe code-to-code comparisonsbetween FLASH and HYDRA for an experiment thatwas carried out at Colorado State University and mod-eled using HYDRA (GRAVA). The GRAVA study isunique in both the quality of the experimental data thatwas collected and the sophistication of the radiation-hydrodynamics modeling that was done. In the experi-ment, an Al target with a V-shaped groove was irradiatedby a rectangular laser beam striking the target perpen-dicular to its face. The intensity of the laser beam had aGaussian cross section with a FWHM of 360 µ m in thenarrow direction and was highly uniform in the wide di-rection. The peak of the Gaussian was aligned with thecenter-line of the V-shaped groove and the peak inten-sity of the Gaussian was ∼ W/cm . The energy ofthe laser was 0.8 J and the duration of the laser pulsewas 120 ps. The data from the experiment was used asa validation test for HYDRA. Here we will use the dataas a validation test for FLASH.The geometry of the groove-shaped target, which isreminiscent of a well-known validation experiment doneat the NOVA laser by WAN and [29], allows interferomet-ric measurements of the electron density in the blowoffplasma. GRAVA conducted these measurements with afew-ns cadence using soft x-rays with a wavelength of46.9 nm. This implies a critical density of 5 × cm − ;however, taking into account instrumental resolution andother details, the largest measurable electron density isreported to be 5 × cm − .GRAVA pursued the experiment as a scaled version ofastrophysical radiative shocks, explaining that the radia-tive energy loss timescale in the problem, τ rad , is com-parable to hydrodynamic expansion timescale, τ hydro .They were also motivated by the fact that similar, ear-lier NOVA experiments produced puzzling results, rais-ing the question whether radiation-hydrodynamics codesmight be inadequate to model the experiment and col-lisionless Particle-In-Cell (PIC) codes might be neededinstead(WAN). GRAVA and later work by the same col-laboration [31, 32] showed that radiation-hydrodynamiccodes are able to model this kind of experiment, and fora variety of different target elements. A. Non-Radiative Results:Electron Number Density
GRAVA presents the results of HYDRA simulationswith and without multi-group radiation diffusion todemonstrate the importance of radiation on their sim-ulation results. We use GRAVA’s non-radiative HY-DRA simulations as the starting point for our compar-ison, since it removes any dependence on the opacitymodel. We performed FLASH simulations using thePROPACEOS EOS model [30] which included physicsfrom the so-called QEOS model [33] for near-solid-density −200 0 200−200 0 2003002001000−100−200−300
FLASH (PROPACEOS)
HYDRA (no rad.) E l e c t r on N u m be r D en s i t y ( c m − ) Distance ( µ m) D i s t an c e ( µ m ) FIG. 1: Electron number densities at various times from aHYDRA simulation of the irradiation of a V-shaped groove without including radiation diffusion (left panels) and resultsfrom the same setup with FLASH using the PROPACEOSequation of state, also without radiation transport (right pan-els). HYDRA panels are adapted with permission from Fig. 9in GRAVA [20] (copyrighted by the American Physical Soci-ety). interactions. Figure 1 compares the results of the FLASHsimulation and the HYDRA simulation, which also usedQEOS in its tabulated EOS. The comparison showsthat the results of the two simulations agree qualita-tively. Specifically, there are no important features inthe FLASH simulations that are not in the HYDRA sim-ulations, and vice versa; and many of the contours fromthe simulations bear a remarkable resemblance to eachother. −200 0 200 −200 0 200−200 0 2003002001000−100−200−300
FLASH
HYDRA
Experiment E l e c t r on N u m be r D en s i t y ( c m − ) D i s t an c e ( µ m ) Distance ( µ m) FIG. 2: Comparing inferred electron number density fromsoft x-ray interferometry (left column) to HYDRA (centercolumn) and FLASH (right column) simulations, both includ-ing multi-group radiation diffusion. The FLASH simulationuses PROPACEOS opacity and EOS data. Left and centercolumns are adapted with permission from Fig. 9 in GRAVA(copyrighted by the American Physical Society).
B. Results including Radiation:Electron Number Density
Radiating plasmas can be compressed to higher den-sities than non-radiating plasmas because the loss of en-ergy cools the plasma, lowering the pressure. This is evi-dent in comparing Figs. 1 and 2. Fig. 1 shows the spatialdistribution of the electron number density at four differ-ent times for HYDRA and FLASH simulations withoutradiation transport and Fig. 2 shows the same informa-tion for HYDRA and FLASH simulations with radiationtransport. Also shown in Fig. 2 is the spatial distribu-tion of the electron number density from experimentalmeasurements (left panel). In both figures the ablatingplasma is colliding with itself at 1.1 ns, creating a rela-tively thin jet of high density, high temperature Al ex-tending from center of the groove in the target. In Fig. 1the jet expands due to the high pressure, thus creatingthe double horn feature seen there at later times. How-ever in Fig. 2 the jet stays compressed for longer so that time (ns) J e t w i d t h ( µ m ) FLASHHYDRAInterferometry
FIG. 3: A quantitative comparison of the jet width versustime as inferred from the electron number density (Fig. 2) ona line perpendicular to the laser axis that is centered at (0,0).HYDRA results (green points) and interferometric measure-ments (black points) from GRAVA are presented at 1.1, 2.6,4.6, 5.9 and 9.7 ns. Measurements from the FLASH simu-lation (solid blue line) are more finely spaced in time. Forbrevity, the comparison at 5.9 ns is not shown in Fig. 2. at 4.6 ns the width of the jet is only modestly larger.Only by 9.7 ns has ablation from the diagonal walls ofthe target lessened enough that the pressure can broadenthe jet and produce the double horn structure in the den-sity similar to what is seen in Fig. 1. For a more extremeexample of the effect of radiation in this problem see theCu or Mo results in [32].Despite the possibility of important differences arisingfrom the implementation of flux-limited, multi-group dif-fusion and/or the opacity models used, the FLASH andHYDRA radiative results shown in Fig. 2 agree about aswell as do the non-radiative results shown in Fig. 1. Im-portantly, both codes agree well with the experimentaldata.To provide a quantitative comparison, we compare inFig. 3 the width of the jet as a function of time as mea-sured in the experiment and given by the FLASH andHYDRA simulations.As the width of the jet, we take thedistance between the most steeply rising features in theelectron density along a line perpendicular to the laseraxis centered at (0,0). This definition is convenient formeasuring the jet width from the figures in GRAVA, sincethe point of steepest rise is simply where the electronnumber density contours are closest together. We assignto the HYDRA results in Fig. 3 an error bar of width ± µ m, which is the typical distance between the twoclosest contours. We measure the width of the jet derivedfrom the interferometric measurements in the same way.As our estimate of the error bars for the interferometricjet width, we take roughly half the distance between thefringes ( ± . µ m), except at 1.1 ns when the jet is stillforming. The true resolution of the interferometric data may be slightly finer than indicated in Fig. 3, although,in principle, a slight misalignment of the laser on targetwould be an additional source of uncertainty [32]. Thewidth of the jet in the FLASH simulations can be inferredin a precise way using the above definition and at a largenumber of times.Fig. 3 shows that the width of the jet as a functionof time that is given by the HYDRA simulation is some-what closer to the experimental width derived from theinterferometry than that given by the FLASH simula-tion, which slightly under predicts the width of the jet at4.6 ns and after. The reason for this is unclear. Interest-ingly, FLASH and HYDRA both over predict the widthof the jet at the first measurement at 1.1 ns. C. Results including Radiation: ElectronTemperature and Mean Ionization State
Fig. 4 compares the electron temperatures in FLASHand HYDRA at the same times reported in Figs. 1 & 2.In comparing these results, it is important to note thatthe interface between the expanding Al plasma and theHe is visible at 1.1 ns and 2.6 ns. The ¯ Z contours in Fig. 4show that at this boundary the plasma goes from a re-gion where Al is significantly ionized to a region where¯ Z can at most be equal to two. As a result, the nar-row region of closely spaced contours corresponding tothis transition moves steadily away from the target; by4.6 ns the transition has left the grid. Fig. 4 indicatesthat at 1.1 ns the He temperatures rightward of the inter-face are somewhat higher in the FLASH simulation thanin the HYDRA simulation. This difference could stemfrom a difference in how key physical processes are han-dled in this very low density region, including the non-equilibrium (i.e. T ele (cid:54) = T ion ) nature of the shock. Moreprosaically, the He density assumed but not reported inGRAVA may simply be higher than what we assumed forthe FLASH simulation (which was ρ = 5 · − g/cm ).In the expanding Al plasma leftward of this Al/Heinterface, the results of the FLASH and HYDRA sim-ulations are again qualitatively similar, with T ele beingslightly higher at 1.1 ns in the FLASH simulation than inthe HYDRA simulation. The plasma is slightly more ion-ized in the FLASH simulation than in the HYDRA sim-ulation, due to some combination of this slightly higher T ele and possible differences between the EOS models.The ¯ Z contours in Fig. 4 are consistent with an overalldifference of ∆ ¯ Z ∼ Z ∼
11. BothFLASH and HYDRA simulations agree that the meanionization state never reaches ¯ Z ∼
12, which would re-quire much higher temperatures. GRAVA states thatthe highest mean ionization state in their HYDRA sim-ulation is ¯ Z ∼
10, and that this result is confirmed bythe absence of signatures of more-highly-ionized charge −200 0 200−200 0 2002001000−100−200
FLASH E l e c t r on T e m pe r a t u r e ( e V ) D i s t an c e ( µ m ) Distance ( µ m) FIG. 4: Comparing electron temperatures in radiation-hydrodynamics simulations from HYDRA (left column) toresults from FLASH (right column); the FLASH simulationuses PROPACEOS opacity and EOS data. Contours for themean ionization state, ¯ Z , are over plotted. HYDRA panelsare adapted with permission from Fig. 6 in GRAVA with per-mission (copyrighted by the American Physical Society). states in extreme UV spectroscopy.The contours of ¯ Z can be used as another measureof the width of the jet versus time to quantify the levelof agreement in Fig. 4. Aluminum ablating from thewalls expands and collides with plasma flowing outwardon axis, creating peaks in T ele and ¯ Z just above and be-low the axis of the laser beam instead of on axis. Thisproduces the double-horned features in T ele and ¯ Z seenin Fig. 4 prior to 5 ns. The distance between the peaksin ¯ Z can be measured on a line perpendicular to the laseraxis centered at (0,0) for the FLASH and HYDRA sim-ulations, giving a different determination of the width ofthe jet. Fig. 5 compares this measurement for the HY-DRA simulation to a very precise measurement for theFLASH simulation at many times. Before 4.6 ns, this time (ns) J e t w i d t h ( µ m ) FLASHHYDRA
FIG. 5: A quantitative comparison of jet width versus timeas measured from the distance between the peaks in the meanionization state ( ¯ Z ) on a line perpendicular to the laser axisthat is centered on (0,0). The HYDRA measurements (greenpoints) come from the ¯ Z contours in Fig. 4, which are derivedfrom GRAVA. Measurements from the FLASH simulation(solid blue line) are presented at finely spaced intervals intime. definition of the jet width gives results similar to thosein Fig. 3. However, after 5 ns the double-horned fea-ture in ¯ Z is much less pronounced because the Al plasmaarriving along the axis of the laser beam is no longerplasma that was directly heated by the laser (c.f. Fig. 10in GRAVA). The speed and momenta of the plasma thatis ablating from the walls and arriving along the axisof the laser beam has decreased significantly, causing atransition in which the double-horned feature nearly dis-appears and the width of the jet suddenly decreases. Theabrupt decrease in the width of the jet seen in Fig. 5 ataround 5 ns is due to this transition.The results of the FLASH and HYDRA simulationsagree well on the width of the jet as measured by thedistance between the two peaks in ¯ Z , despite the over-all difference of ∆ ¯ Z ∼ D. Results including Radiation: Total Pressure
Fig. 6 compares the total plasma pressure at earlytimes in the FLASH and HYDRA simulations with ra-diation. While FLASH and HYDRA use EOS modelsthat are related to or are the same as the QEOS modelconstructed by More et al. [33], the implementations areclearly somewhat different. In HYDRA the total pressureat solid density, i.e. well into the target, is vanishinglysmall, as one expects for a cold solid. FLASH uses the −200 0 200−200 0 2002001000−100−200
800 ps
FLASH
300 psHYDRA P r e ss u r e ( k B a r s ) D i s t an c e ( µ m ) Distance ( µ m) FIG. 6: Comparing total plasma pressure at early times fromradiation-hydrodynamics simulations from HYDRA (left col-umn) and FLASH (right column). The FLASH simulationwas performed using PROPACEOS opacity and EOS data.HYDRA panels are adapted from Fig. 8 in GRAVA with per-mission (copyrighted by the American Physical Society). (QEOS-based) PROPACEOS EOS which achieves van-ishingly small total pressures for cold solids, in part, byallowing the electron pressure to be negative. FLASHcurrently requires positive electron and ion pressures insolving the momentum equation for the fluid and in in-terpolating EOS data. We therefore set the few negativeelectron pressures that exist in the PROPACEOS tableto a very small, positive value. The ∼ ion pressure reported fromPROPACEOS. Bearing in mind, from a hydrodynamicpoint of view, only pressure differences matter and thatthe laser-heated Al quickly enters a regime where the to-tal pressure is well above zero, it is understandable thatthis difference seems not to have significantly affected theagreement between the two codes. III. FORMATION AND PROPERTIES OF THEJET IN THE GRAVA EXPERIMENT
The GRAVA experiment had two objectives: (1) toobtain data that would make possible an important ad-ditional validation test of radiative hydrodynamics codesin the wake of the apparent failure of LASNEX [34] sim-ulations to reproduce a similar experiment done usingthe Nova laser WAN; and (2) to create a jet analogousto astrophysics jets, following Stone et al. [29]. Havingvalidated FLASH for the GRAVA experiment, we nowuse FLASH simulations to better understand the forma-tion and properties of the jet. We focus on early times( ≤ ± µ m section ofthe target removed. This latter configuration resemblesan earlier experiment done by WAN in which two slabswith a gap between them were oriented perpendicular toeach other and irradiated by beams of the NOVA laser.GRAVA cites this experiment as an important motiva-tion for their work.A number of questions arise in drawing parallels be-tween the self-colliding, ablating plasma in GRAVA andastrophysical jets. While in both the astrophysical andlaboratory context the radiative cooling timescale maybe similar in magnitude to the hydrodynamic expansiontimescale (and therefore the adiabatic cooling timescale),how similar are these situations in other respects? Herewe address three specific questions about the formationand properties of the jets seen in the laser experimentswhose answers enable us to compare them with astro-physical jets:1. What determines the physical conditions (e.g., thedensity, temperature, and velocity) in the core ofthe jet?2. Is the collimating effect of the plasma ablating fromthe angular sides of the groove due to thermalpressure, i.e., the internal energy of the ablatingplasma, or ram pressure, i.e., the component of themomentum of the ablating plasma perpendicular tothe mid-plane of the experiment?3. Does the collimation of the flow by the plasma ab-lating from the angular sides of the groove and theentrainment of this plasma in the resulting jet in-crease or decrease the velocity of the jet? z ( µ m) v z ( k m / s ) −200 −100 0 100 2000200400600800
100 ps200 ps300 ps500 ps800 ps1.1 ns
FIG. 7: Lineouts along the laser axis of the component of the velocity parallel to the laser axis at various times. The thicksolid lines show results from the FLASH simulation of a V-shaped groove target described in Sec. § II. These thick solid linesbecome dotted lines at the point where the cell material is mostly very low density He instead of Al. Thin solid lines (whichbecome dashed lines at the Al/He transition) show the same measurements from a simulation where a flat target of the samematerial is irradiated with the GRAVA laser pulse. z ( µ m) v z ( k m / s ) −200 −100 0 100 2000200400600800
100 ps200 ps300 ps500 ps800 ps1.1 ns
FIG. 8: Lineouts along the laser axis of the component of the velocity parallel to the laser axis at various times. The thicksolid lines show measurements from the FLASH simulation of a V-shaped groove target described in Sec. II. These thick solidlines become dotted lines at the Al/He transition. Thin solid lines (which likewise become dashed lines at the Al/He transition)show the same measurements from a simulation where, similar to the target geometry of WAN, a large gap of material ismissing from the center of the V-shaped groove. This target is irradiated by the GRAVA laser pulse.
A. What determines the physical conditions in thecore of the jet?
Figures 3 and 5 show that the width of the jet at earlytimes ( ≤ µ m whichimplies that most of the laser energy is deposited within ± µ m of the laser axis. Accordingly, the intensity ofthe laser beam will be much greater on the relatively flatportion in the middle of the V-shaped groove than on thesides of the groove. In this way, the interaction near thecenter of the groove resembles the simple case of an Al slab (i.e. flat) target. If the physical mechanism produc-ing the jet is the same in for both the V-shaped groveand slab target then one would expect that the velocityat various locations along the core of the jet should besimilar in the two cases.To investigate this possibility, we compared lineouts ofthe velocity along the mid-plane for the two problems atfive relatively early times; i.e., ≤ v z (cid:38)
350 km/s) ve-locities of very low density gas and focusing on v z (cid:46)
350 km/s, the velocity profiles for the two problems agreeclosely at all five times, supporting our conjecture that themechanism producing them is the same . The differencesseen between the two simulations at very high velocities( v z (cid:38)
350 km/s) are at very low densities and near orapproaching the transition from cells that are mostly Alto cells that are mostly very low-density He. This tran-sition is marked by a change in line type from thick solidto dotted lines for the V-shaped target or from thin solidto dashed lines for the flat target. Because the He servesonly as an approximation to vacuum conditions, the re-sults near this very low-density interface are not relevantto the questions we are concerned with in this section.We also compared the lineouts of the velocity along themid-plane for the Al target with a V-shaped groove to atarget consisting of two Al slabs oriented perpendicularto each other with a gap between them, similar to theWAN experiment. This comparison allows us to contrastthe properties of the jet produced by a V-shaped groovethat has a relatively flat portion near the mid-plane andone that does not. Figure 8 shows the results of the sim-ulation for the V-shaped groove target and that for theWAN-like target. As in Fig. 7 the change from a solid lineto a dotted or dashed line indicates the transition frommostly Al to mostly He. Focusing on the thin and thicksolid lines that correspond to Al target material, the twovelocity profiles differ greatly. The absence of a relativelyflat portion of the target near the mid-plane means thatformation of the jet is delayed until the plasma ablatingfrom the sloping sides of the target has had time to meetat the mid-plane. Furthermore, the velocity profile of thejet is much shallower and its maximum velocity is muchsmaller. These results provide further support for the hy-pothesis that the physical mechanism producing the jetin GRAVA is the same as in the slab problem.Figure 9 compares lineouts of the density along themid-plane for the V-shaped groove target (thick solidlines) and a flat Al slab target (thin solid lines), whileFig. 10 compares the same measurement from the V-shaped groove target simulation (thick solid lines) to re-sults from a target consisting of two Al slabs orientedperpendicular to each other with a gap between them(thin solid lines), similar to the WAN experiment. As inFigs. 7 and 8 these lines become dotted or dashed whenthe cells are mostly He instead of Al.Clearly, ablation from the sloping sides of the V-shaped groove confines the flow, as discussed by WANand GRAVA, greatly increasing the density in the jetrelative to the density in the case of the flat Al slab tar-get, where the flow can expand laterally as well as awayfrom the surface of the target. Collimation of the flowby the plasma ablating from the sloping sides of the V-shaped groove in GRAVA raises the question of whetherthe collimation is due primarily to thermal pressure or toram pressure. We now address this question.
B. Is the collimating effect of the plasma ablatingfrom the angular sides of the groove due to thermalpressure or ram pressure?
To address this question, we calculate the ratio of thespecific kinetic energy, e kin = 12 | (cid:126)v | (1)to the total specific internal energy, e int = e ele + e ion . (2)If e kin /e int (cid:29)
1, the kinetic energy of the ablation flowis dominant. On the other hand, if e kin /e int (cid:46)
1, theinternal energy due to the temperature of the plasma isdominant.Figure 11 shows the ratio e kin /e int throughout the com-putational domain at six different times for the V-shapedgroove target, while Fig. 12 shows the same quantity atthe same times for the target comprised of two slabs ori-ented perpendicularly to each other with a gap in be-tween, similar to the WAN experiment. Figure 13 com-pares the these measurements at 1.1 ns to measurementsfrom the simulation of a flat Al slab target, In Figs. 11 -13, we see a similar behavior: even at very early times,in the plasma very near the target, the internal energy ofthe plasma dominates the kinetic energy of the bulk flowaway from the target, but further out the kinetic energyof the bulk flow dominates the internal energy. Once thelaser turns off, the region where the kinetic energy of thebulk flow dominates the gas internal energy begins togrow substantially larger.Examining the properties of the ablation flow as it ap-proaches the mid-plane, we see that the internal energyof the plasma dominates at distances < µ m from thetarget, but the kinetic energy of the bulk flow dominatesat all larger distances. This indicates that the collima-tion of the jet is due primarily to ram pressure exceptvery near the target, where it is due primarily to thermalpressure of the hot plasma.The plasma in the jet is heated when the plasma ab-lating from the sloping sides of the target in both theGRAVA and the WAN-like experiments collides in thecenter, converting kinetic energy to thermal energy. Thisprocess is simplest in the [28] target and this collisionproduces a low ratio of e kin /e int along the center.The GRAVA target simulation shown in Fig. 11 has amore complex structure to e kin /e int than the other cases.Material from the target collides in the center, but thereis also material originating from the middle of the V-shaped groove that is moving rapidly away from the tar-get. The result is a complex lateral structure within thejet in which e kin /e int is large in the core of the jet andsmall at its edges, and large again in the ablating plasmaabove and below the jet. This is the origin of the double-horn structure evident at very late times in the electrondensity, which can be seen in Figs. 1-3 and in the ioniza-tion state, which can be seen in Figs. 4-5. z ( µ m) D en s i t y ( g / c m ) −200 −100 0 100 2000.0000.0010.0020.003
100 ps200 ps300 ps500 ps800 ps1.1 nsz ( µ m) D en s i t y ( − g / c m ) −200 −100 0 100 200012345 FIG. 9: Lineouts of the mass density along the laser axis at various times. The thick solid lines show measurements fromthe FLASH simulation of a V-shaped groove target described in Sec. II. Thin solid lines show the same measurements from asimulation where a flat target of the same material is irradiated with the GRAVA laser pulse. As in Fig. 7, the change from asolid line to a dotted or dashed line indicates the Al/He transition. z ( µ m) D en s i t y ( g / c m ) −200 −100 0 100 2000.0000.0010.0020.003
100 ps200 ps300 ps500 ps800 ps1.1 nsz ( µ m) D en s i t y ( − g / c m ) −200 −100 0 100 200012345 FIG. 10: Lineouts of the mass density along the laser axis at various times. The thick solid lines show measurements fromthe FLASH simulations of a V-shaped groove target described in Sec. II. Dashed lines show the same measurements from asimulation where, similar to the target geometry of WAN, a large gap of material is missing from the center of the V-shapedgroove. This target is irradiated by the GRAVA laser pulse. As in Fig. 8, the change from a solid line to a dotted or dashedline indicates the Al/He transition.
While the complex lateral structure of the jet in theseexperiments is comparable to astrophysical jets, it differsin that the ratio e kin /e int in astrophysical jets is expectedto be large in the core of the jet and progressively smallervalues further away from the jet axis with e kin /e int → C. Does the ablation from the angular sides of thegroove increase or decrease the velocity of the jet?
We are now in a position to address whether the ab-lating plasma from the angular sides of the groove in thetarget increases or decreases the velocity of the jet. Akey piece of information is that the velocity of the jetis much smaller in the WAN-like experiment in which the target is two Al slabs oriented perpendicular to eachother with a gap in between than in the GRAVA exper-iment in which the target is an Al slab with a V-shapedgroove in it. We can now understand the reason whyfrom the answer we obtained to the previous question.The energy density of the ablating plasma is dominatedby its bulk kinetic energy by the time it approaches themid-plane, except at very small distances ( < µ m) fromthe target. The component of the momentum of the ab-lating plasma that is perpendicular to the mid-plane willgo into heating the jet, while the component parallel tothe mid-plane will add to the velocity of the jet. However,because the laser intensity is much lower away from themid-plane, due both to the profile of the laser beam andthe slanted angle of the surface of the groove, the spe-cific internal energy (i.e., the internal energy per gram)0 −200 −100 0 100 200
300 ps −200 −100 0 100 2003002001000−100−200−300 0.01.02.03.04.05.0
800 ps200 ps
500 ps100 ps ( s pe c i f i c k i ne t i c ene r g y ) / ( s pe c i f i c i n t e r na l ene r g y ) D i s t an c e ( µ m ) Distance ( µ m)FLASH FIG. 11: Plotting the ratio of e kin /e int at various times forthe FLASH simulation of the V-shaped groove described inSec. II. The original target location is shown with a white line.The transition from mostly Al to mostly He cells is indicatedwith a solid black line in each panel. generated by the component of the momentum of theablating plasma when the ablating plasma collides withthe jet, and the specific component of the momentum ofthe accreting plasma parallel to the mid-plane (i.e., themomentum per unit mass) are both smaller than in thejet flow itself, which is generated by the most intensepart of the laser beam illuminating the nearly flat partof the groove near the mid-plane. This suggests that theentrainment in the jet of the plasma ablating from thesloping sides of the groove will decrease slightly the ve-locity of the jet compared to the velocity along the mid-plane of the freely expanding plasma in the case of a slabtarget. This expectation is consistent with the resultsshown in Figure 7. IV. CONCLUSIONS AND SUMMARY
We compared the results of FLASH hydrodynamicsimulations to previously-published experimental resultsand HYDRA simulations for the irradiation of a mm-longV-shaped groove cut into an Al target GRAVA. Impor- −200 −100 0 100 200
300 ps −200 −100 0 100 2003002001000−100−200−300 0.01.02.03.04.05.0
800 ps200 ps
500 ps100 ps ( s pe c i f i c k i ne t i c ene r g y ) / ( s pe c i f i c i n t e r na l ene r g y ) D i s t an c e ( µ m ) Distance ( µ m)FLASH FIG. 12: Plotting the ratio of e kin /e int at various times for aFLASH simulation of a target with a gap in the center (similarto WAN) that is irradiated by the GRAVA laser pulse. Theoriginal target location is shown with a thick white line. Thetransition from mostly Al to mostly He cells is indicated witha solid black line in each panel. tantly, these experiments, conducted at Colorado StateUniversity, included soft x-ray interferometric measure-ments of the electron density in the Al blowoff plasma asa powerful validation test. We performed these FLASHsimulations without the exact same EOS and opacitymodels that were used in the previously published HY-DRA simulations. Instead we used a commercially avail-able PROPACEOS EOS and opacity model [30] that in-cluded QEOS physics for near-solid-density fluids [33].In all cases the FLASH results greatly resemble theresults from HYDRA and, most importantly, the exper-imental measurements of electron density. This includesthe properties of the underdense Al blowoff plasma,which matters most for the use of these codes in calcu-lating pre-plasma properties as initial conditions for PICsimulations of ultra-intense, short-pulse laser-matter in-teractions (e.g. [1]). This result is encouraging for thewider HEDP community since FLASH is a“user” codethat is freely available to the academic community. Itis also encouraging because FLASH uses a finite-volumeAdaptive Mesh Refinement (AMR) scheme that makes it1 −200 −100 0 100 2003002001000−100−200−300 1.02.03.04.05.0 ( s pe c i f i c k i ne t i c ene r g y ) / ( s pe c i f i c i n t e r na l ene r g y ) D i s t an c e ( µ m ) Distance ( µ m) FIG. 13: Plotting the ratio of e kin /e int at 1.1 ns for threedifferent FLASH simulations: (Top) the V-shaped groove sim-ulation described in Sec. II, (middle) results from a V-shapedgroove with a gap as in WAN (bottom) results from a FLASHsimulation of a flat target that is likewise irradiated by theGRAVA laser pulse. In each plot the original target locationis shown with a thick white line. The transition from mostlyAl to mostly He cells is indicated with a solid black line ineach panel. straightforward for the user to configure a simulation tomaintain high resolution even in areas where the plasma is expanding rapidly due to laser heating.Having validated the FLASH simulations for the Altarget with a V-shaped groove, we used these and otherFLASH simulations to better understand the formationand properties of the jet in the experiment. We showthat the velocity of the jet is produced primarily by theheating of the target in the relatively flat region of theV-shaped groove at the mid-plane, as in standard slabtargets. We show that the jet is collimated primarilyby the ram pressure of the plasma that ablates from thesloping sides of the groove. Further, we find that the in-teraction of the plasma ablating from the sloping sidesof the groove with the jet produces the observed com-plex lateral structure in it, a structure that is comparableto astrophysical jets but differs significantly from them,quantitatively. Finally, we show that the entrainment inthe jet of the plasma ablating from the sloping sides ofthe groove slightly decreases the velocity in the jet com-pared to the velocity along the mid-plane of the freelyexpanding plasma in the case of a slab target.In this work we validated the FLASH code for a spe-cific, previously-published experiment on laser-irradiatedAl plasmas. Similarly high-quality interferometric dataalso exists for C, Cu and Mo [31, 32], which can be usedin future efforts to validate the HEDP capabilities inFLASH. Acknowledgements
Supercomputer allocations for this project includedtime from the Ohio Supercomputer Center. The FLASHcode used in this work was developed in part by theFlash Center for Computational Science at the Univer-sity of Chicago through funding by DOE NNSA ASC,DOE Office of Science OASCR, and NSF Physics. COwas supported by U.S. Department of Energy contractDE-FG02-05ER54834 (ACE). DL was supported in partat the University of Chicago by the U.S. Department ofEnergy NNSA ASC through the Argonne Institute forComputing in Science under field work proposal 57789. [1] K. U. Akli, C. Orban, D. Schumacher, M. Storm,M. Fatenejad, D. Lamb, and R. R. Freeman, Phys. Rev.E , 065402 (2012).[2] T. R. Boehly, D. L. Brown, R. S. Craxton, R. L. Keck,J. P. Knauer, J. H. Kelly, T. J. Kessler, S. A. Kumpan,S. J. Loucks, S. A. Letzring, et al., Optics Communica-tions , 495 (1997).[3] E. I. Moses, R. N. Boyd, B. A. Remington, C. J. Keane,and R. Al-Ayat, Physics of Plasmas , 041006 (2009).[4] J. D. Lindl, L. J. Atherton, P. A. Amednt, S. Batha,P. Bell, R. L. Berger, R. Betti, D. L. Bleuel, T. R. Boehly,D. K. Bradley, et al., Nuclear Fusion , 094024 (2011).[5] M. D. Rosen, H. A. Scott, D. E. Hinkel, E. A. Williams,D. A. Callahan, R. P. J. Town, L. Divol, P. A. Michel, W. L. Kruer, L. J. Suter, et al., High Energy DensityPhysics , 180 (2011).[6] D. Lamb and M. Marinak, Panel Report on IntegratedModeling, Workshop on Science of Fusion Ignition onNIF, LLNL-TR-570412 (2012), URL https://lasers.llnl.gov/workshops/science_of_ignition/ .[7] C. Bellei, P. A. Amendt, S. C. Wilks, M. G. Haines, D. T.Casey, C. K. Li, R. Petrasso, and D. R. Welch, Physicsof Plasmas , 012701 (2013).[8] V. A. Thomas and R. J. Kares, Phys. Rev. Lett. , 075004 (2012), URL http://link.aps.org/doi/10.1103/PhysRevLett.109.075004 .[9] M. M. Marinak, R. E. Tipton, O. L. Landen, T. J. Mur-phy, P. Amendt, S. W. Haan, S. P. Hatchett, C. J. Keane, R. McEachern, and R. Wallace, Physics of Plasmas ,2070 (1996).[10] M. M. Marinak, S. W. Haan, T. R. Dittrich, R. E. Tip-ton, and G. B. Zimmerman, Physics of Plasmas , 1125(1998).[11] M. M. Marinak, G. D. Kerbel, N. A. Gentile, O. Jones,D. Munro, S. Pollaine, T. R. Dittrich, and S. W. Haan,Physics of Plasmas , 2275 (2001).[12] E. C. Harding, J. F. Hansen, O. A. Hurricane, R. P.Drake, H. F. Robey, C. C. Kuranz, B. A. Remington,M. J. Bono, M. J. Grosskopf, and R. S. Gillespie, Phys.Rev. Lett. , 045005 (2009).[13] B. van der Holst, G. Tth, I. Sokolov, L. Daldorff, K. Pow-ell, and R. Drake, High Energy Density Physics , 161(2012), ISSN 1574-1818.[14] J. M. Taccetti, P. A. Keiter, N. Lanier, K. Mussack,K. Belle, and G. R. Magelssen, Review of Scientific In-struments , 023506 (2012).[15] DOE, Basic Research Directions for User Science at theNational Ignition Facility (2011).[16] P. A. Keiter, K. Mussack, and S. R. Klein, High EnergyDensity Physics , 319 (2013).[17] B. Fryxell, K. Olson, P. Ricker, F. X. Timmes, M. Zin-gale, D. Q. Lamb, P. MacNeice, R. Rosner, J. W. Truran,and H. Tufo, ApJS , 273 (2000).[18] A. Dubey, L. B. Reid, K. Weide, K. Antypas, M. K.Ganapathy, K. Riley, D. Sheeler, and A. Siegal, ArXive-prints (2009), 0903.4875.[19] P. Tzeferacos, M. Fatenejad, N. Flocke, C. Graziani,G. Gregori, D. Lamb, D. Lee, J. Meinecke, A. Sco-patz, and K. Weide, High Energy Density Physics , 24 (2015), ISSN 1574-1818, 10th Interna-tional Conference on High Energy Density Labora-tory Astrophysics, URL .[20] J. Grava, M. A. Purvis, J. Filevich, M. C. Marconi, J. J.Rocca, J. Dunn, S. J. Moon, and V. N. Shlyaptsev, Phys.Rev. E , 016403 (2008). [21] P. MacNeice, K. M. Olson, C. Mobarry, R. de Fainchtein,and C. Packer, Computer Physics Communications ,330 (2000).[22] C. W. Hirt, A. A. Amsden, and J. L. Cook, Journal ofComputational Physics , 227 (1974).[23] J. I. Castor, Radiation Hydrodynamics (Cambridge Uni-versity Press, 2004).[24] M. Kucharik, Ph. D. Thesis, Czech Technical Universityin Prague , 1273 (2006).[25] The Flash Center for Computational Science, User GuideVersion-4.0-beta (2012).[26] Y. T. Lee and R. M. More, Physics of Fluids , 1273(1984).[27] Flash code user guides , http://flash.uchicago.edu/site/flashcode/user_support/ .[28] A. S. Wan, T. W. Barbee, R. Cauble, P. Celliers,L. B. Da Silva, J. C. Moreno, P. W. Rambo, G. F.Stone, J. E. Trebes, and F. Weber, Phys. Rev. E ,6293 (1997), URL http://link.aps.org/doi/10.1103/PhysRevE.55.6293 .[29] J. M. Stone, N. Turner, K. Estabrook, B. Remington,D. Farley, S. G. Glendinning, and S. Glenzer, ApJS ,497 (2000).[30] J. J. Macfarlane, I. E. Golovkin, and P. R. Woodruff,JSQRT , 381 (2006).[31] J. Filevich, M. Purvis, J. Grava, D. P. Ryan, J. Dunn,S. J. Moon, V. N. Shlyaptsev, and J. J. Rocca, HighEnergy Density Physics , 276 (2009).[32] M. A. Purvis, J. Grava, J. Filevich, D. P. Ryan, S. J.Moon, J. Dunn, V. N. Shlyaptsev, and J. J. Rocca, Phys.Rev. E , 036408 (2010).[33] R. M. More, K. H. Warren, D. A. Young, and G. B.Zimmerman, Physics of Fluids , 3059 (1988).[34] G. B. Zimmerman and W. L. Kruer, Comments PlasmaPhys. Controlled Fusion , 51 (1975).[35] J. M. Stone and M. L. Norman, ApJ413