Laser-driven cylindrical implosion experiments enable direct measurements of hydrodynamic instability growth in convergent geometries, providing a wealth of validation data in the high-energy-density regime. These experiments are designed to be nearly axially invariant, allowing for modeling with complementary two-dimensional slices of the cylinder. Two distinct hydrodynamics codes are employed to model a subset of these experiments, and the results are shown to be in very good agreement with each other and the available experimental data. While this 2D modeling approach adequately captures most of the physics of the implosion and ensuing instability growth, there are crucial aspects from the three-dimensional nature of the experiments that are missed in 2D. The first fully 3D simulations of these experiments are presented, and small but significant differences are found to arise from both the axial and azimuthal non-uniformity in the laser drive. Recent experimental results confirming the drive asymmetry are discussed.
Cylindrical implosions offer many advantages for studying properties of systems in the high-energy-density (HED) plasma environment relevant to inertial confinement fusion1,2 (ICF), as they include some amount of compression and heating due to the convergence of the system (albeit less than spherical systems) while also retaining direct diagnostic access by viewing down the cylinder axis. Many laser-driven cylindrical implosion experiments have been fielded3–9 that focused on studying hydrodynamic instability growth where convergence alters the behavior beyond the simple planar picture. More recent work has begun studying how magnetic fields impact these dynamics in experiments10,11 and simulations,12,13 and there is a natural overlap between these magnetized laser-driven cylinders and recent magnetically driven cylindrical experiments.14,15 Other extensions include adding a second cylinder inside of the first, allowing for studies of material mixing driven by the rebounding shock16 or investigations of instability growth in a classically unstable system17 where the effects of ablative stabilization are absent.
For the laser-driven experiments that we will focus on here, the cylinder is driven around the midsection of the target, and it implodes in the shape of an hourglass. The goal is to keep the highly driven neck region of the hourglass axially uniform over a desired extent, resulting in a quasi-2D cylindrical implosion with little variation along the cylinder axis. Often, a more opaque material is included in the cylinder midsection where the behavior is axially uniform. This diagnostic “marker” layer preferentially absorbs backlighter x-rays, providing radiographic contrast and allowing direct measurements at the converging interfaces. As noted previously, cylinders are much more amenable to diagnosing the unstable interface compared to spherical systems. These experiments provide a wealth of high-quality data that can be leveraged to inform our choice of best practice modeling options for HED systems.
Modeling of these cylindrical implosion experiments is slightly more complicated than for spherical systems, due to the reduced symmetry inherent in cylinders. In 1D, spherical targets have a definite size based on their outer radius, but in cylindrical systems the axial extent factors into the total size (and the intensity on the target surface for laser-driven experiments). In 2D, spherical systems are modeled with assumed cylindrical symmetry about the pole of the sphere. However, for cylinders, there are two different orthogonal ways to slice the cylinder: A slice along the cylinder axis with assumed cylindrical symmetry properly models the target and includes all axial non-uniformities that arise from the finite extent of the cylinder and the laser drive; a slice perpendicular to the cylinder axis with assumed translational invariance in the third direction can incorporate azimuthal non-uniformities in the target and laser drive, but again the axial extent must be specified in some way to constrain the system. Until recently, our modeling efforts have been mainly limited to just two dimensions due to computational resources, and we have combined the results of the two orthogonal simulations to form a 3D representation of the imploding cylindrical targets. We introduced our approach in Refs. 18 and 19, and it was further refined in Ref. 20.
Here, we present our first fully 3D simulations of a set of cylindrical implosion experiments using two different computational tools: xRAGE21,22 and FLASH.23,24 We find that these 3D simulations are in very good agreement with our previous 2D modeling paradigm, providing additional credibility for our former approach. However, we also find that there are subtle 3D effects that were not predicted in our 2D modeling framework, with important ramifications for our understanding of the relevant physics. We leverage these 3D simulations to design a set of experiments to test whether this subtle effect could be diagnosed, and experimental results presented here confirm the predictions of the 3D simulations.
The remainder of this paper is organized as follows: In Sec. II, we describe the details of the targets being simulated as well as the two computational models used in this study. The computational models are compared and contrasted in 2D simulations in Sec. III, and we compare these results to the experimental data as well. We discuss the 3D modeling approaches used for both codes in Sec. IV, and we compare the results of these 3D simulations to previous 2D work and experimental data in Secs. IV A and IV B. Lastly, in Sec. IV C, we introduce a new experimental design and present a preliminary comparison with 3D simulations. Finally, we end with a discussion and conclusions in Sec. V.
A. Experimental details
We focus on a set of cylindrical implosion experiments fielded at the OMEGA25 Laser Facility. The experimental results are discussed more fully in Refs. 8, 9, and 26, but we briefly summarize them here. The targets consisted of a hollow annular plastic (CH, 1.25 g/cm3) tube, a 12 thick embedded marker layer (Al, 2.70 g/cm3) that is set nearly flush with the inner surface of the plastic ablator, and a central plastic foam (CH, 300 mg/cm3). There is a small air gap between the outer surface of the foam and the inner surfaces of the ablator and marker as a result of fabrication, and this is important for accurately modeling the instability growth.9 Target dimensions are summarized in Table I.
|Material .||Outer radius .||Length .||Density .|
|CH foam||422.5||2500||0.30 g/cm3|
|Al marker||442.0||500||2.70 g/cm3|
|CH ablator||493.0||2500||1.25 g/cm3|
|Material .||Outer radius .||Length .||Density .|
|CH foam||422.5||2500||0.30 g/cm3|
|Al marker||442.0||500||2.70 g/cm3|
|CH ablator||493.0||2500||1.25 g/cm3|
One set of targets had a completely smooth Al marker layer, while a second set had a sinusoidal m = 20 perturbation machined on the inner surface of the Al marker layer with a 4 initial amplitude (8 peak-to-valley). In the absence of any azimuthal asymmetries in the laser drive (which we will subsequently show is not the case), the first set of targets can be modeled entirely in 2D. The azimuthal perturbation on the second set provides a seed for subsequent instability growth, which is driven by a combination of Richtmyer–Meshkov27,28 (RM), Rayleigh–Taylor29,30 (RT), and Bell–Plesset31,32 (BP) effects of convergence. The relatively large density of the foam (300 mg/cm3) results in the inner Al interface experiencing considerable deceleration after the 1 ns square pulse laser drive turns off, and the growth is dominated by RT instability with a modest contribution from BP effects due to the low convergence ratio: The initial radius of the interface compared to the final radius of the interface is only 2.25. More details on the drive conditions may be found in Ref. 20. Changing the density of the foam provides a means of controlling both the deceleration profile and the final convergence of these targets.
B. Modeling paradigm
We model these cylindrical implosion experiments using two distinct Eulerian radiation-hydrodynamics codes, xRAGE21,22 and FLASH.23,24 Each utilizes adaptive mesh refinement (AMR), which enables selectively refining regions of interest (shocks and material interfaces) while reducing computational resolution elsewhere (i.e., far from the marker layer). The differences in each code's AMR approach are discussed in Appendix A 1. Here, both xRAGE and FLASH use a Cartesian mesh with cubic cells. While FLASH is capable of employing a cylindrical polar mesh, this is not currently used (though there are plans to explore this in the future).
Both codes solve the three-temperature (3T: electrons, ions, and radiation) hydrodynamics equations, though in this work we will ignore the radiation field completely. For these low-convergence foam-filled targets, radiation has only a small impact on the simulated trajectories, and neglecting it is important for keeping the computational costs of our 3D simulations reasonable. Each code employs flux-limited Spitzer and Härm33 thermal conduction with a flux limiter of . This value was chosen as it more closely matches the coronal plasma conditions in a calculation using a more sophisticated non-local electron thermal conduction model.34,35
We use tabular equations of state (EOS) as inputs for both codes. The plastic foam and ablator are modeled with SESAME36 7592, and the aluminum marker layer is modeled using SESAME 3720. (A reader has recently been written that allows the use of SESAME data in the FLASH code.37) The background “vacuum” material is modeled as helium in both codes: In xRAGE, we use SESAME 5762, while in FLASH, we use data from IONMIX38 for helium. However, the results are insensitive to the helium EOS that is used. Previous attempts at comparing the two codes found that the results are rather sensitive to the choices of EOS for the other target materials.
Previous xRAGE simulations18,20 used an ad hoc laser power multiplier in an attempt to account for cross-beam energy transfer39 (CBET) and other laser plasma instabilities that were not being modeled. There, we found that a constant power multiplier of applied to the nominal 18 kJ, 1 ns square pulse drive (total simulation drive energy of 14.4 kJ) provided a good match to the experimental results. We use the same laser power multiplier in both the xRAGE and FLASH simulations presented here.
Although the laser ray tracing in 2D xRAGE simulations is performed in a full 3D geometry, the results are azimuthally uniform; the target is cylindrically symmetric from the view of the laser package, and only the m = 0 Fourier component of the deposition is returned to advance the hydrodynamics. Additional details on the implementation of the laser package in xRAGE may be found in Ref. 40. In 2D FLASH simulations, the ray trace is also performed in a full 3D geometry. However, the target is represented as a thin wedge, and the laser deposition is averaged over the azimuthal area. Additional details on the FLASH laser package may be found in the user manual.
III. 2D SIMULATION RESULTS
A. Code to code comparison
We begin by comparing the two codes in 2D simulations in order to demonstrate the level of agreement between them and their utility for modeling these types of experiments. Here, we model a slice of the target aligned with the cylinder axis, previously referred to as r–z computations based on the two dimensions that are modeled. Cylindrical symmetry about r = 0 is assumed. More details on the domain sizes and AMR settings are given in Appendix A 2.
The early-time behavior for both codes is compared in Fig. 1. The laser power deposition per unit volume is shown in the colored contours, and the logarithm of the mass density is shown in grayscale to highlight the imploding target. Contours of the electron temperature are also shown: blue = 400 eV, green = 800 eV, yellow = 1200 eV, and red = 1600 eV. The xRAGE results are shown in the top subplots, while the FLASH results are contrasted in the bottom subplots.
Qualitatively, the two codes appear to be in reasonable agreement. The axial extent of the laser deposition is similar, and the radial extent of the deposition region evolves similarly in both. At late time, ray effects appear in the extended deposition region, and these differ slightly as a result of the different laser packages in both codes. There are also some noticeable differences in the electron temperature contours evident both early and late into the laser pulse, but the electron temperature gradients near the target surface agree well.
These small differences do not significantly impact the drive on the target, which is in remarkably good agreement between the two codes. This can be seen in Fig. 2 that shows filled contours of the logarithm of the mass density at several intermediate points during the implosion. Again, the xRAGE and FLASH results are shown in the top and bottom subplots, respectively. The laser drive sends a shock into the target, which is clearly evident in the density jump in the central CH foam, and the axial structure is remarkably similar for both codes over the entirety of the implosion. Both codes show similar small levels of axial non-uniformity in the aluminum layer and a modest amount of marker rollup at the axial ends of the marker driven by the impedance mismatch in materials during shock transit.
More detailed comparisons between the two codes are made by considering lineouts along the axial midplane, z = 0. In particular, we compare density, electron and ion temperature, and laser power deposition per unit volume at several different times for the two codes in Fig. 3. Here, the xRAGE results are shown in the colored lines with symbols (see legend), while the FLASH results are plotted on top in the black dots. The two vertical lines indicate the inner and outer surface of the aluminum marker layer, with the solid vertical lines showing xRAGE results and the dashed vertical lines showing FLASH results. The marker layer from the xRAGE result is also highlighted with gray shading.
During the laser pulse [Figs. 3(a) and 3(b)], we see small differences in the laser power deposition (yellow inverted triangles) between xRAGE and FLASH, while the subtle differences in the electron temperature contours noted previously are too small to be observable on the logarithmic scales used here. There is clear separation between the electron and ion temperatures in the low-density coronal plasma, but both codes predict nearly identical behavior. The shock speed through the CH ablator is the same for both codes, and the shock breaks out of the inner surface of the marker layer just as the laser drive is turning off.
At later times, the electron and ion temperatures equilibrate in both codes, as can be seen in Figs. 3(c)–3(f). The marker density drops slowly following shock passage, but a very similar behavior is observed in xRAGE and FLASH. The shock speed in the central CH foam is also consistent, with the shock reaching r = 0 near t = 6.5 ns and rebounding to hit the incoming marker at t = 9 ns. Interestingly, the aluminum marker material appears to extend to slightly larger radius in the xRAGE computation compared to the FLASH computation, though there is no noticeable impact on the density. The vertical lines here represent the points where the aluminum marker concentration drops to 5%. The fact that the FLASH computation drops below 5% at a smaller radius than the xRAGE computation indicates that FLASH may be slightly more diffusive—it is losing more aluminum than xRAGE.
B. Comparison to data
In order to compare with the experimental data, we create synthetic radiographs from the results of our simulations. Rays are traced through the computational domain, and the transmission loss is integrated incrementally as the rays pass through the different materials. The absorption coefficients are calculated based on each cell's density and temperature using the OPLIB database41 and assembled using the TOPS code.42 Here, the rays are projected through a pinhole that is perfectly aligned with the cylinder axis, eliminating the effects of parallax in our synthetic images to simplify the analysis. Previous work finds that the observed asymmetries in the experimental images are nearly entirely explained by parallax,26 and this can be removed in the analysis of the experimental radiographs. The integrated transmission through the simulated target is convolved with a Gaussian kernel to approximate the effects of the finite size pinholes (15 ) used in the experiments. Motion blurring is neglected here, as the targets are moving very slowly, slowing from /ns down to /ns, compared with the integration time of the framing camera. The experiments used an 80 ps full-width-at-half-maximum Gaussian voltage pulse, and the pulse takes 200 ps to cross one strip (four pinholes per strip) on the microchannel plate.
Sample synthetic radiographs from the xRAGE and FLASH simulations are shown on the right and left halves of Figs. 4(a)–4(d), respectively. [An experimental radiograph from these experiments can be seen in Fig. 7(a) of Ref. 9 for comparison.] The images are framed by a circular aperture formed by a tungsten washer placed at one end of the cylindrical target, which is used to prevent crosstalk on the microchannel plate between neighboring pinholes. The aluminum marker layer is readily apparent as the very dark ring in the images. The leading edge of the shock is also visible as it compresses and then recompresses the central CH foam after rebounding from the cylinder axis.
We make quantitative comparisons by extracting the trajectories of the inner and outer surfaces of the aluminum marker along with the shock front. This is done by taking radial line-outs of the transmission at several discrete azimuthal angles and averaging the results over all angles. (Although the simulations themselves are angle-invariant, the synthetic radiographs are converted to a Cartesian grid to include the effects of pinhole smearing.) The key features are identified based on extrema in the radial gradients of transmission, as is done in the experimental analysis. The same framework is used for both simulations, and these trajectories are compared in Fig. 4(e). The xRAGE (solid black lines) and FLASH (dashed purple lines) trajectories are in remarkably good agreement with each other. The shock positions differ by at most 10 , which occurs when the shock is rebounding from the axis and identifying the location is more difficult. The average difference in the shock position over the entire time window (4–8 ns) is only . The maximum discrepancies for the outer and inner surfaces are 4.8 and 1.6 , respectively, though the average differences are smaller: 3.8 for the outer surface, and 0.8 for the inner surface.
The experimental data points for four smooth targets (OMEGA shots 93062, 93063, 93066, and 93067) are also shown in Fig. 4(e). The experimental points here represent the azimuthally averaged value of the feature (outer surface, inner surface, or shock front), identified as a local extrema in the radial gradient of image intensity, and the vertical error bars show the standard deviation. The deviations between the xRAGE and FLASH trajectories are within 1 standard deviation of the experimentally inferred values, which are estimated at , and for the outer, inner, and shock positions, respectively. (See Refs. 9 and 26 for more details.)
C. Testing an alternative drive
As noted previously, the xRAGE code currently lacks the ability to run the laser package in 3D simulations. In order to facilitate comparisons to the 3D FLASH simulations, we devised an axisymmetric internal energy source that drives the target in approximately the same way as the laser package in our 2D calculations. A similar approach was previously used for ICF calculations.43 The details of the internal energy source drive used here are provided in Appendix B.
In our 2D xRAGE simulations, the internal energy source is able to reproduce many of the characteristics of the laser drive, as can be seen in Fig. 5. The laser-driven case is shown in the top half of each subplot, and the energy-source-driven case is shown in the bottom half of each subplot. Although there are some differences between the two, most notably additional instability growth on the incoming CH ablator near the dezone block boundary at , the aluminum marker layer dynamics are very similar. Since we are primarily concerned with getting the hydrodynamics of the marker layer correct, this is believed to be an acceptable drive for our 3D runs.
IV. 3D SIMULATION RESULTS
As discussed at the outset, these experiments were initially designed entirely with 2D xRAGE simulations. For targets with no azimuthal asymmetries, either from the target itself or the laser drive, the 2D simulations examined in Sec. III do a remarkably good job of reproducing the experimental observables. However, when azimuthal asymmetries are included in the target (or expected from the laser drive), these 2D simulations alone are unable to correctly model this behavior. Instead, we have previously combined these 2D simulations with results from separate 2D simulations modeling an orthogonal slice of the cylinder. This modeling approach is extensively discussed in Refs. 18 and 20, and we find that we are able to capture the behavior of targets with machined azimuthal perturbations reasonably well using this framework. However, in spite of the successes of our 2D modeling paradigm, the question remained as to whether 3D effects were important to these cylindrical implosion experiments. We are now able to address this concern.
A. Targets with no azimuthal perturbation
As a first test of our 3D simulation capabilities, we compare results from targets with no azimuthal perturbation to the previous 2D results. As we have seen previously for these smooth targets, there is a small amount of axial non-uniformity in the marker layer, which arises from the axial variation in the laser drive. This effect is well captured in our 2D calculations, and so it provides a simple comparison for our 3D calculations. In the absence of any azimuthal asymmetries, these 2D and 3D calculations should match very well. The details of the 3D modeling choices are given in Appendix A 3.
We see very reasonable agreement between the smooth target calculations in 2D and 3D with the two different codes, as can be seen in the synthetic radiographs shown in Fig. 6. By virtue of the assumed rotational symmetry, the 2D computations show perfectly circular signatures in the radiographs. However, there is a bit more structure in the 3D computations. The 3D xRAGE simulation shows a minor diamond shape on the marker, a hallmark of grid imprint when using square cells to simulate a circular cross section. Similar behavior is observed for the 3D FLASH computation, though this also shows additional higher mode features on the inner surface of the ablator. Some of this is likely a result of the finite number of laser rays used to drive this computation, resulting in some modes feeding through to the marker layer during the ablative acceleration phase of the target.
Key features are extracted from these synthetic radiographs following the previous procedure, and trajectories are compared to each other and to the experimental data in Fig. 7. Recall that the experimental data points represent the average feature position over the azimuthal angle, and the error bars show the standard deviation. To better accentuate the differences between the different simulations, we compare trajectories from each simulation to the original 2D laser-driven xRAGE results. The outer surface, shown in Fig. 7(a), is generally very similar for all of the simulations. The 2D FLASH computation shows the most disagreement with the outer surface position from the 2D laser-driven xRAGE computation, but this is only on the order of about 4 at most. At late times, all of the simulations tend to underpredict the outer surface position relative to the experimental data for reasons that are not yet clear.
The agreement for the inner surface, shown in Fig. 7(b), is quite reasonable for all of the simulations. Deviations are very small, with the largest outlier being the 3D laser-driven FLASH run, which shows the inner surface slightly advanced compared to the others. This is likely due to the small-scale numerically seeded modes that grow on the inner surface, pushing the average surface to smaller radius. As the AMR resolution in the FLASH computations is increased, this difference is expected to diminish. Despite the noticeable features forming in the 3D xRAGE calculation along the diagonals, the average inner surface position is nearly indistinguishable to the comparable 2D calculations.
We see the largest deviations between computations in the shock trajectories, shown in Fig. 7(c). The xRAGE computations are generally in very good agreement with each other, but there appears to be a bit more deviation in the FLASH runs. We speculate that this may be due to the lower AMR resolution that is allowed for the central CH foam in these cases (details in Appendix A 3), and we expect this discrepancy to get smaller as the AMR resolution there is increased.
An unexpected result of the 3D FLASH computation is shown in Fig. 8, which shows slices along several planes through the imploding cylinder. There is a faint signature of an m = 5 sinusoidal variation on the aluminum marker layer, most evident at the axial ends of the marker, shown in the slices at in Figs. 8(b) and 8(d). The phase of the m = 5 perturbation varies at the two axial ends, and there is a cancelation at the axial midplane, z = 0, shown in Fig. 8(c). The presence of an m = 5 signature arising from the laser beam geometry at OMEGA was previously noted in self-emission images looking at one axial end of a driven cylindrical target for the mini-MagLIF campaign.44 However, this effect is washed out when viewing down the entire axial extent of the cylinder, as in our experiments. Most importantly, this effect is not able to be captured in our previous 2D modeling paradigm at all. While the magnitude of the m = 5 variation is small here, it can contribute to an apparent thickening of the marker when viewing down the axis: Essentially one is looking down through two pentagonal “washers” of aluminum material that are exactly 180° out of phase with each other.
Although the pentagonal shape is visible to the eye, the m = 5 amplitude is quite small. We analyze the inner and outer surfaces identified by taking the extrema in radial gradients of the density along various lineouts, in a similar manner to how the synthetic radiographs were investigated. We find that the mode 5 component of the Fourier decomposition is between 3 and 4 on both the inner and outer surfaces, and there is a 180° phase change between the slice at compared to the slice at .
B. Targets with an m = 20 azimuthal perturbation
Having confirmed that our 3D calculations are behaving as expected, we now turn to consider the targets with a pre-machined azimuthal perturbation on the inner surface of the aluminum marker layer. We expect these targets to show the most benefit from our ability to model them in fully 3D simulations, as compared to the 2D framework that was used previously. We visualize the dynamics of the implosion using ParaView,45 a 3D visualization tool. We show the density along slices of the x, y, and z midplanes in Fig. 9, and we also extrude the aluminum marker material, colored by the density, to highlight the primary region of interest. The results from the laser-driven 3D FLASH simulation are shown in the left column, and the results from the energy-source-driven 3D xRAGE simulation are shown in the right column.
Examining the first images at t = 4 ns in Fig. 9(a), we see the shock propagating into the central CH foam, and the perturbation on the inner surface of the aluminum marker is beginning to grow. Perturbation growth continues, and the shock reflects off of the central cylinder axis just after 6 ns, as can be seen in Fig. 9(b). Finally, the rebounding shock strikes the incoming spike tips on the aluminum marker just after 8 ns, as shown in Fig. 9(c). Qualitatively, the perturbation growth on the aluminum marker appears to be quite similar for the two computations. Although the perturbation growth appears to be fairly axially uniform, there are clear end effects at the edges of the marker. This can be seen by examining the extruded marker material: The upper left extrusion shows the entire axial extent of the marker, while the lower right extrusion is clipped at . Aluminum material wraps around into the bubble pockets as a result of the impedance mismatch during shock passage, as was previously observed in our 2D simulations. As we will see, the amount of aluminum swept around the axial ends is insufficient to significantly alter the transmission down the cylinder axis. This 3D effect has little impact on the dominant instability growth and does not negatively affect our ability to measure these systems.
Turning our attention to other parts of the cylinder, we see some differences in the shock propagation near the axial ends of the cylinder. In particular, the FLASH computation shows a much coarser shock, and the large AMR cells are very evident. In contrast, the xRAGE computation shows a more gradual transition to coarser behavior, having stepped down the AMR resolution less abruptly (see details in Appendix A 3). We can also see numerical instabilities arising from the boundaries between the energy source regions, evident near and . This was also observed in our 2D runs, but as it is far from the marker region, this was deemed to be less important.
The synthetic radiographs constructed from the 3D xRAGE and 3D FLASH computations are in very good agreement with the experimental data, as can be seen in Fig. 10(a). The experimental radiograph is taken from OMEGA shot 93068, and it has been rotated slightly here for more direct comparison with the simulation data. The 3D xRAGE and FLASH simulations both include the modest amount of axial non-uniformity, which results in an apparent thickening of the marker layer in the radiographs. In contrast, the 2D xRAGE radiograph, produced following the procedure of Ref. 20, shows a slightly thinner marker compared to the experimental data. The 2D framework for producing the synthetic radiographs for targets with an azimuthal asymmetry does not incorporate any of the small axial non-uniformity that is predicted by the 2D r–z computations (see Figs. 2 and 5).
Note that the 3D FLASH computations also show more variation in the apparent spike tip positions. We attribute some of this behavior to the influence of the m = 5 perturbation arising from the laser drive. Despite the small 3–4 amplitude of the m = 5 perturbation, the asymmetry has the opposite phase on the north pole end of the cylinder compared to the south pole end. This gives rise to an m = 10 pattern when viewing down the axis and manifests here as alternating radial depths for the m = 20 spike tips.
The spike tips in the experimental data appear to be much less well defined compared with both the 2D and 3D simulations. Previous work46 identified morphological changes in the instability growth depending upon both dimensionality and the inclusion of low-amplitude “noise” in addition to the dominant mode. In particular, they find that a 3D simulation of a single mode feature with very small amplitude noise added evolves to an intact mushroom tip surrounded by a diffuse mixed region, rather than a pronounced mushroom tip as seen in 2D, and this is in much better qualitative agreement with their experimental data. A similar effect may be happening in our cylindrical implosion experiments, where the low-amplitude noise may be seeded by small-scale structures in the CH foam and/or native roughness from the machining of the aluminum marker layer. Although the current simulations necessarily include a small amount of “surface roughness” that is introduced by the misalignment between the circular cross section of the target and the Cartesian simulation grid, this noise spectrum is axially invariant. Simulations that include fully 3D noise are planned for future work.
The thickening of the marker due to the 3D axial non-uniformity is quantified for these perturbed targets in Fig. 11, which shows the radius vs time plots of various features and compared to the experimental data (taken from OMEGA shots 93068, 93069, 93070, and 93071). The experimental data are divided up into 20 different arcs, and the location of key features is identified by the local extrema in the radial gradient of transmission. The data points here represent the average value over the 20 different segments, and the error bars represent the standard deviation. Here, we track the outer and inner surfaces of the marker at the bubble pockets, the spike tip positions, and the shock front. As before, we compare these all against the same metric, trajectories from a 2D laser-driven xRAGE computation, to better accentuate the differences in these plots. The width, determined by the difference in the outer surface vs bubble pocket, is on the order of 3–5 thicker in the 3D calculations compared to the 2D calculations. Similarly, the spike tips in the 3D results are generally about 3–5 further in, compared to the 2D results, as seen in Fig. 11(c). Both of these differences are attributed to the small axial non-uniformity present in the aluminum marker layer, which is absent in our 2D modeling framework used for these perturbed targets.
The shock trajectories show a bit more disagreement between the two sets of 3D simulations, as can be seen in Fig. 11(d), similar to what was observed previously for the unperturbed targets [see Fig. 7(c)]. The 3D FLASH computation shows a slightly faster shock compared to the 3D xRAGE computation, which we attribute to the differences in AMR resolution between the two codes (see discussion in Appendix A 3).
Note that the largest disagreement between our simulations and the experimental data occurs for the bubble pocket, Fig. 11(b), which is consistently found to be about 10–15 further in compared to the simulation results. This is well outside the uncertainty estimates for our experimental data, though the identification of the bubble pocket in experiment is very challenging. The synthetic radiographs show a very defined gradient in transmission at the bubble pocket position, as can be seen in the radial lineouts shown in Fig. 10(b). (These lineouts are taken through a single bubble pocket at ; the values plotted in Fig. 11 are averaged over all 20 sectors.) In contrast, the experimental data show a very gradual intensity gradient through the bubble pocket, making a clear identification much more difficult.
Previous attempts to explain this discrepancy in the bubble pocket position in our 2D framework found that we could push the 2D simulations toward better agreement by invoking a mix model or adding a small amount of energy to the aluminum layer to mock up preheat. This was generally unsatisfying, however, as it was never clear that we were getting the right answer for the right reasons. Now, with the advent of our 3D simulations, we can conclusively state that at least 4 of this discrepancy can be attributed solely to 3D effects: namely, the small amount of axial non-uniformity present in the implosion. The takeaway message here is that care must be taken before invoking additional physics in a 2D modeling framework in order to justify results that could otherwise be explained by a higher fidelity 3D model. A few additional comments on the comparisons with the experimental data are given in Appendix C.
It should be noted that, aside from the small axial non-uniformity in the 3D simulations, we see no significant differences between the predicted instability growth of the dominant m = 20 mode in our 3D and 2D simulations. This dominant mode continues to grow as a nearly axially invariant 2D feature. We see no evidence of secondary instabilities with axial variation causing breakup of the fin-like structures here. In the future, we plan to explore the feasibility of machining targets with both axial and azimuthal components to the perturbation, enabling an assessment of 2D–3D instability growth and possible differences in instability saturation.
C. Testing the drive asymmetry
Although the m = 5 asymmetry from the laser drive was previously observed experimentally by others,44 we had not seen any evidence for it in our cylindrical implosion experiments. This is consistent with our 3D FLASH simulation results, which show that the opposite phase of the m = 5 perturbation at either pole of the cylinder ends up obscuring the overall m = 5 perturbation. However, a question remained: Was this prediction from our 3D FLASH simulation physical? In order to test this effect, we designed a set of experiments using FLASH and leveraging some advances in target fabrication technology. While the data analysis is still ongoing, we will present preliminary results here that confirm that this m = 5 asymmetry is indeed a real effect.
The target fabrication advance here replaces the existing 2.7 g/cm3 aluminum marker layer with a 1.5 at. % iodine-doped CH (CHI) marker that is density-matched to the surrounding epoxy ablator at 1.25 g/cm3. Here, we model the CHI in an identical manner to the surrounding epoxy (CH) ablator. While this is an oversimplification, the exclusion of radiation or any other source of selective preheat of the iodine-doped marker in our simulations makes this a reasonable first approximation. The density match to the ablator results in no impedance mismatch as the shock is traveling past the edges of the marker layer, and the vortical rollup features that developed for our aluminum marker layers (see Figs. 2 and 5) are predicted to vanish for CHI markers. This allows us to selectively dope different axial extents of the cylinder, enabling us to isolate asymmetries introduced by the driver in different regions of the target. A significant uncertainty here is whether the CHI will experience any differential heating or expansion compared to the epoxy ablator, which would confound the experimental measurements.
The previous fabrication technique (outlined in Ref. 9) coated the aluminum onto a copper mandrel, then coated the mandrel with epoxy, and then drilled and leached out the copper, leaving a hollow cylinder with the marker flush with the inner surface of the epoxy. Due to difficulties in coating thin layers of CHI onto a copper mandrel, these targets required a different approach. A solid CHI rod was machined down to leave a protrusion of a desired axial extent near the center of the rod. This was then coated with epoxy and allowed to cure. Following the curing process, the entire CHI/epoxy rod was bored out, and the bore diameter was greater than the thin portion of the solid CHI rod. This left an epoxy tube with the desired length CHI marker embedded flush with the inner surface of the epoxy, similar to the aluminum markers in our previous targets. The biggest issue was concentricity of the CHI marker inside of the epoxy tube, and many targets displayed somewhat significant offsets. However, if the CHI responds in a hydrodynamically similar fashion to the epoxy (CH), then this will not perturb the implosion in any way as long as the total wall thickness (CHI + epoxy) is angle-invariant.
Examining our previous FLASH simulation results, we observed that the m = 5 amplitude and phase are approximately constant between and . The m = 5 drive asymmetry vanishes between and where the beams on each pole overlapped. The m = 5 amplitude is again non-zero between and , and the phase is constant and opposite to the south-pole end of the cylinder. Based on these findings, we opted for “north-pole” targets with the CHI selectively placed between and and “south-pole” targets with the CHI selectively placed between and .
In order to accentuate the m = 5 asymmetry, we also pushed these targets to higher convergence ratio by lowering the foam density to 40 mg/cm3 (down from the 300 mg/cm3 foams discussed previously). This increases the convergence ratio to nearly 5 by the time the rebounding shock strikes the incoming marker layer. Our pre-shot FLASH calculations of a south-pole target demonstrate this increased m = 5 amplitude, as can be seen in Fig. 12. Here, the m = 5 amplitude is nearly 4.6 at the z = −200 slice, compared to roughly 3 in the previous design. Note, however, that the 3D FLASH computation does predict that the CHI marker behaves in a slightly dissimilar fashion as the CH ablator, despite the fact that we have neglected radiation diffusion or any other sort of preheat in this case. Some of this may be an artifact of the step-down in the AMR resolution going from marker material (1 minimum AMR cell size) to the ablator material (2 minimum AMR cell size), and this is being investigated further.
These experiments were fielded at the OMEGA laser facility in August 2022, and the m = 5 pattern is confirmed in axial radiographs from a south-pole target, as can be seen in Fig. 13(a). There is remarkable qualitative agreement with the synthetic radiograph produced from the FLASH pre-shot simulation, shown in Fig. 13(b). Similar behavior, albeit with the opposite phase, is observed for the north-pole targets at late times. More detailed analysis of this experimental data is under way, and we plan to make more quantitative comparisons between simulations and data in the future. For now, it suffices to note that the m = 5 asymmetry arising from the laser drive that is predicted in our 3D FLASH simulations is confirmed in experimental data.
We have presented updated simulation results of recent cylindrical implosion experiments using the xRAGE and FLASH radiation-hydrodynamics codes. We find that these experiments behave in an approximately 2D manner with only limited variation in drive conditions along the axial extent of the diagnostic marker layer, in agreement with previous results. For targets with no machined azimuthal asymmetry, the dynamics are well captured in both 2D xRAGE and 2D FLASH simulations. When considering targets with a prescribed single-mode sinusoidal perturbation, designed to act as a seed for subsequent hydrodynamic instability growth, we find that the previous framework combining two separate 2D computations to approximately represent the imploding cylinder is in reasonable agreement with the experimental data. We can capture the spike tip and shock front trajectories quite well in 2D simulations, though the marker appears slightly thinner in the simulated radiographs.
A more complete 3D xRAGE simulation, driven by an azimuthally symmetric energy source, reveals that there is a subtle increase in the thickness of the marker layer and the penetration depth of the Rayleigh–Taylor spikes. This occurs because the 3D simulation properly accounts for the small axial non-uniformity present in these targets that is neglected in the combined 2D modeling framework. This results in a small but significant increase in the apparent thickness of the marker layer, which might otherwise have been incorrectly attributed to preheat expansion of the layer, additional mixing processes, or other phenomena commonly used to explain away discrepancies between the experimental data and 2D simulation results.
Further, the 3D laser-driven FLASH simulations expose an additional small asymmetry arising from the laser drive, an effect which is not able to be reproduced in either the 2D framework or the 3D xRAGE simulations driven by the energy source. This m = 5 drive asymmetry is due to the laser beam geometry at the OMEGA laser facility, which has beams arranged in a pent-hex pattern. Although this was previously identified in self-emission images of cylindrical targets,44 our experimental radiographs imaging down the axis of the cylinder did not display any such signature. Our inability to observe this effect is due to the fact that the m = 5 asymmetry switches phase from one pole of the cylinder to the other, and imaging of a symmetric (about z = 0) layer washes out this feature. This effect is confirmed in recent cylindrical implosion experiments that selectively doped only a finite axial extent of the target, enabling a direct measurement of the drive conditions at different axial positions.
Although the 3D effects identified here are subtle, they are not insignificant. We confirm that the 2D framework can provide a reasonable first approximation when modeling these laser-driven cylindrical implosions, and this will remain a workhorse for experimental design going forward. However, detailed comparisons with the experimental data with the goal of validating simulation models will require complete 3D simulations, lest small changes arising from three-dimensional effects be inadvertently attributed to other physics. The 3D laser-driven FLASH simulations are already being leveraged to understand additional cylindrical implosion experiments, and these results will be discussed in future publications.
This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). Simulations were run on the LANL High Performance Computing clusters provided through the Advanced Simulation and Computing (ASC) Program. Much of this work was completed while J.P.S. was visiting the Flash Center for Computational Science at the University of Rochester, and J.P.S. is very grateful for their support and assistance. J.P.S. thanks A. Koskelo, E. Merritt, and K. Stalsberg of LANL for supporting this visit.
The Flash Center for Computational Science acknowledges support from the U.S. DOE NNSA under Award No. DE-NA0003856 to the Laboratory for Laser Energetics, Award No. DE-NA0003842 to the Center of Matter under Extreme Conditions, and Subcontract Nos. 536203 and 630138 with Los Alamos National Laboratory and B632670 with Lawrence Livermore National Laboratory. The software used in this work was developed in part by the U.S. DOE NNSA and U.S. DOE Office of Science-supported Flash Center for Computational Science at the University of Chicago and the University of Rochester.
Conflict of Interest
The authors have no conflicts to disclose.
Joshua Paul Sauppe: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (lead); Project administration (lead); Supervision (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Yingchao Lu: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Petros Tzeferacos: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (lead); Software (lead); Supervision (equal); Writing – review & editing (equal). Adam Reyes: Resources (equal); Software (lead); Validation (equal). Sasikumar Palaniyappan: Data curation (equal); Formal analysis (lead); Investigation (lead); Validation (equal). Kirk A. Flippo: Formal analysis (equal); Investigation (equal). Shengtai Li: Investigation (equal); Software (equal). John L. Kline: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Resources (lead); Supervision (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: ADDITIONAL CODE DETAILS
1. AMR approach and interface treatment
The xRAGE code uses cell-based AMR, meaning that every individual cell is selected for refinement independently of every other cell (with the exception that neighboring cells can differ by at most one level of refinement). In contrast, FLASH uses block-based AMR, where all cells within a given cell block are refined collectively if any of them require further refinement. Cell-based AMR allows for much more selective refinement at the cost of algorithmic complexity, while block-based AMR is conceptually simpler at the cost of reduced flexibility in refining regions of interest.
An “interface preserver” is available in the xRAGE code, which acts to reduce numerical diffusion of materials by steepening material gradients at their boundaries. Our previously published simulations use this approach. Recently, however, we have determined that disabling the interface preserver in xRAGE results in no noticeable differences in the primary points of comparison with the experimental data (synthetic radiographs), provided the AMR resolution is sufficiently refined. Because the FLASH code currently lacks a similar interface preserver, we have opted to disable the interface preserver in these xRAGE runs. However, for reasons discussed in Appendix B, a volume-of-fluid47 approach is used for several auxiliary materials in our 3D xRAGE simulations.
Often, FLASH computations will use a hyperbolic tangent function in the initial conditions to smoothly transition species' concentrations at interfaces between different materials. This relaxes the gradients and reduces some of the numerical diffusion that would otherwise occur. We do not use this here, and instead, we use a sharp transition between materials. This introduces a bit more numerical diffusion in the FLASH results, but it allows for a more direct comparison with the xRAGE simulations.
2. Computational domain and AMR choices in 2D
For the 2D xRAGE simulations, the level 1 AMR cell size is 64 , and the minimum AMR cell size allowed is 0.5 . Dezoning is used to limit the number of cells outside of the cylindrical target. The 2D xRAGE simulation spans and , chosen so that the target sits far from the domain edges and boundary conditions become unimportant.
In the 2D FLASH simulations, the level 1 AMR cell size is 4 , though we use 16 × 16 cells in a level 1 block. All blocks with marker material are allowed to refine to a minimum cell size of 0.5 , while blocks with just ablator or foam material are refined to 1 . The 2D FLASH simulation only extends to and , but it allows mass, momentum, and energy (heat) to flow out of the domain boundaries.
The differences in domain sizes and boundary conditions result in behavioral discrepancies at the axial ends of the cylinder, , as can be seen in Fig. 2. In particular, the xRAGE simulation shows a weak shock being driven into the target down the axis, arising from ablation of foam material at the axial ends. This ablation is driven by the heat front that wraps around to the endcaps, as can be seen most clearly in Figs. 1(b) and 1(c). In contrast, the FLASH computations show no such effect. We attribute this to the different computational domains used in both codes. In FLASH, the heat front escapes the computational domain through the upper and lower boundaries before wrapping around to the axial ends of the cylindrical target. However, because we are most interested in the behavior of the central marker layer, these differences at the ends of the cylinder are ignorable.
3. Computational domain and AMR choices in 3D
The 3D xRAGE simulations presented here are limited to only one octant of the cylinder: , , and . Reflection symmetry is enforced about each axis. This is reasonable because the internal energy source is axisymmetric, and we confirmed that the same results are obtained when we selectively extended the domain to contain the full extent along one of the directions.
In the 3D xRAGE simulations, all materials except the energy source regions are allowed to refine down to a minimum AMR cell size of 0.5 , though extensive dezone regions are used away from the cylinder midsection to reduce the cell count. In a 2D convergence scan, we found changes in marker positions differing by at most 3 as we increased the resolution from 0.5 to 0.25 with the interface preserver disabled. The internal energy sourcing regions are zoned at 1 minimum AMR cell size, and a volume-of-fluid approach47 is applied to only these regions to reduce diffusion at this coarse scale.
These 3D xRAGE simulations are somewhat sensitive to the resolution used in the energy sourcing regions. However, because the drive here is only intended to be a lowest-order approximation to what the full laser drive would be, this is not a concern here. If we changed the energy-source region's resolution, we would simply need to “retune” the energy-source drive slightly. Finally, it should be noted that previous 3D xRAGE simulations48 find that a newer unsplit hydrodynamics advance produces more vorticity than the default split hydrodynamics advance used in our simulations. At present, however, the volume-of-fluid approach is not supported with the unsplit hydrodynamics, though we hope to test this feature in the future.
Our 3D FLASH simulations use a level 1 cell size of 32 , and the cells within a block give a level 1 block length of 512 . There are five level 1 blocks along the axial extent of the cylindrical target that spans , and the central level 1 block spanning contains the marker layer. In order to cut down on the total number of blocks in the 3D computations, the x and y domains are limited to , and outflow conditions are specified at all boundaries.
Blocks containing the marker material are allowed to be refined down to 1 cell size here, while blocks containing just ablator material can be refined to 2 cell size. In contrast to the 2D simulations, here we restrict blocks containing just the foam material to be refined down to 4 cells. In addition, regions outside of the central extent are only refined as far as is needed to ensure that neighboring blocks differ by just one level. While the minimum AMR cell sizes here are larger than those allowed in our 3D xRAGE simulations, these 3D FLASH computations are simulating the full cylinder, not just one octant. Increasing the refinement in our 3D FLASH simulations is an ongoing effort.
APPENDIX B: TUNING THE INTERNAL ENERGY SOURCE
We follow the prescription outlined in Ref. 43 to generate a suitable internal energy source to approximately replicate the conditions of the laser drive. Here, we place tracer particles near the edge of the ablator in a 2D r–z laser-driven simulation, and we track the laser energy deposition into these particles as a function of both time and position (axial and radial). Tracer particles are loaded every 0.2 radially between 492.9 and 489.1 , and they are spaced every 22.5 axially between . The laser power deposited per unit mass is tracked for each particle, and by integrating in time, we can find the total energy deposition per unit mass. The outermost tracer particles near the edge of the cylinder begin absorbing energy first, followed by those placed deeper into the ablator.
We group tracer particles based on their initial positions to define separate energy source regions, and we average the energy deposition per unit mass per time over each group of tracer particles to construct the energy sourcing function for each distinct region. Here, we utilize five different energy sourcing regions, defined in Table II. The axial and radial structure of these different regions, combined with the different energy source functions in each region, allows us to qualitatively reproduce the curvature of the shock front, as we see in Fig. 5.
|Region .||Radial domain .||Axial domain .|
|Region .||Radial domain .||Axial domain .|
These five regions along with their time-dependent energy sources are used as inputs for a second r–z simulation without the laser drive. Scalar multipliers are added to each region's energy sourcing function, and we tune these in order to approximately match the shock and marker to that of the calculation with the laser drive. This sourced 60% of the energy into the electrons and 40% of the energy into the ions. In the energy-source drive computation, the minimum AMR cell size for all materials is , but the thin energy-sourcing regions are zoned at just 1 (the entire radial extent of each region). Extensive dezoning is also used away from the marker layer. The AMR is stepped down one level outside of , two levels outside of , and three levels outside of . All cells outside of and are set to level 1 (here 64 ).
APPENDIX C: COMMENTS ON COMPARISONS TO EXPERIMENTS
With respect to the comparisons with the experimental data, it should be noted that the simulations presented here use the dimensions from a nominal target (see Table I) and the requested laser drive. In actuality, the as-built targets exhibited small variations in the key dimensions, and the as-fired laser pulses differed slightly from the requested drive. These variations were included in previous simulations [see Fig. 7(c) of Ref. 9], though we neglect them in the present work. Most of the target dimensions were within of the nominal design, though the ablator outer radius varied between and from nominal for some targets.
Within the four smooth target datasets considered in Fig. 4, the laser energy varied between –4% and from the average drive. Comparing the new 2D results presented here to our previous 2D simulations, we observe that using the as-built dimensions and as-fired laser energies can change the simulated trajectories by as much as ±10 , which is on the same order as the experimental uncertainty. Similar variations are seen for the four perturbed target datasets shown in Fig. 11.
Finally, although it appears that the 3D FLASH results for the shock front are nearly spot-on with the experimental data in Fig. 11, we caution against drawing very strong conclusions from this agreement. In particular, it should not be concluded from this alone that the 3D FLASH simulations are providing a better match with the experimental data than the 3D xRAGE simulations. Additional 3D FLASH simulations that increase the AMR refinement are needed, and it is also necessary to use the as-built dimensions and as-fired laser pulses for each shot.