Three-dimensional modeling and hydrodynamic scaling of National Ignition Facility implosions

Inertial confinement fusion^1,2 experiments on the National Ignition Facility (NIF)^3–5 have yet to reach their namesake goal of ignition. Neutron yields in recent experiments using High Density Carbon (HDC) ablators^6–10 have exceeded 50 kJ; however, those yields remain well below the ∼1 MJ yields of unambiguous ignition.¹¹ Nevertheless, there is a growing understanding of the degradation sources that appear to have thwarted ignition attempts so far. And given that understanding, and the unchanged goal of reaching ignition and high yield,¹² increasing attention is being focused on the question of what implosion conditions will be required to finally achieve ignition on the NIF—or possibly on some future, upgraded facility.

Predicting what implosion conditions will be necessary for ignition obviously requires extrapolation beyond the database of current NIF experiments. In turn, accurately making such extrapolations requires a reliable model that can reasonably explain the current database, and so make those extrapolations plausible. While it remains unrealistic for any model to reproduce within error bars every datum returned from an NIF experiment, the gross features seen in the experiment should obviously be captured. Moreover, the model should be applicable to more than a single experiment. Ideally, the trends of a number of experiments that varied different implosion parameters (as might be anticipated in extrapolating to ignition) should be captured, and so lend confidence to extrapolations based on that model.

Crucial in making reliable extrapolations is that the model be “right for the right reasons.” That is, while it is often possible to match various aspects of NIF implosion data using artificial initial or boundary conditions, such as exaggerated surface perturbation amplitudes or heavily degraded x-ray drives, such models are not trustworthy for extrapolation. Models with these modified initial or boundary conditions may match certain experimental signatures exquisitely well, but there is little reason to expect that they will correctly predict performance trends as implosion parameters are changed. Put simply, if agreement is artificially contrived at one point in parameter space, there is no particular reason to believe that that the artifice should apply elsewhere. In this respect, it is important that models used for performance extrapolation adhere as closely as possible to all of the known conditions of a given experiment, including target dimensions and compositions, surface characterization data, x-ray drive conditions, etc., and that these known features be modeled as faithfully as possible. Obviously, not all initial and boundary conditions are perfectly known, and some level of uncertainty is inevitably introduced as a consequence. However, to the degree possible, all of the known experimental conditions should be equally respected, and thereby minimize any added uncertainty with extrapolation.

Perhaps the most obvious feature of NIF experiments that must be accounted for in any reliable model is the three-dimensionality (3-D) of current implosions. Not only is the initial target configuration at shot time three-dimensional (the presence of the capsule fill tube breaks the azimuthal symmetry of the hohlraum), but the drive conditions on the capsule are known to be three-dimensional due to the finite number of laser spots on the hohlraum wall and due to the beam-to-beam power variations in the delivered laser power. Randomly distributed defects or debris on the capsule surface will additionally introduce 3-D perturbations as these defects grow from Richtmyer-Meshkov^13,14 and Rayleigh-Taylor^15,16 instabilities. And most significantly, imaging data from NIF experiments clearly show the presence of 3-D structures at stagnation. At the same time, it has been known for some time that 3-D perturbations can degrade the yield significantly more than their 2-D equivalents.^17–19 Any reliable model, especially one applied to extrapolations, must therefore account for this important and obvious three-dimensionality. Indeed, as shown below, extrapolations based on simplified 2-D models differ markedly in their predicted ignition threshold compared to more rigorous 3-D results.

Equally important as the three-dimensionality of current implosions is the recognition that current NIF implosions are degraded by a number of perturbation sources and that no single effect explains the observed performance. The now well-known perturbations that appear to account for much of the observed degradation in NIF implosions are the capsule fill tube, the capsule support tent, asymmetries in the driving x-ray flux, and possibly also short-wavelength mixing between the deuterium-tritium (DT) fuel and ablator. Again, all of these perturbations must be reasonably accounted for in any reliable simulation model, especially since their relative importance could shift as an implosion is extrapolated to the threshold of ignition. A model that omitted one or more of the known perturbations would give very spurious predictions if the omitted effect proved to be the controlling factor at the threshold of ignition.

This prospectus article describes work that has been ongoing for several years to develop a modeling capability reliable enough to guide NIF experiments to ignition.^19–24 This modeling effort is founded in direct simulations of NIF implosions that account for all of the known degradation sources and avoid ad hoc modifications beyond the known experimental conditions. Given the need to respect the three-dimensionality of NIF implosions and the range of perturbation sources (and hence range of length scales) that are involved, simulations of this type are computationally very costly. Fortunately, continuing developments in computing power and modeling technique have recently made these simulations possible. While the modeling results as described here have improved significantly in their fidelity in recent years, and now reflect many of the features seen in the experiment, the comparison is still imperfect. Nevertheless, the simulation results are now sufficiently close to experiment to justify applying the current model to the question of extrapolation, and a preliminary assessment of the scaling of a recent NIF implosion to ignition is described below.

Broadly, in extrapolating current NIF experiments to the threshold of ignition, there are two paths to consider. In the first of these, the quality of current implosions can be improved at a fixed drive energy. In this case, improvements in target fabrication, energy coupling to the implosion, or drive symmetry are hypothesized that enable faster, more stable, more symmetric, and/or higher compression implosions with consequently higher yield at fixed driver energy. Of course, these types of quality improvements have been sought since the beginning of NIF experiments in pursuit of higher yields.

In the second path, current implosions can simply be hydrodynamically scaled to higher drive energies in anticipation of a larger, upgraded driver. In this case, target dimensions and drive energies are scaled up, while the perturbations that degrade performance are assumed fixed, hence leading to higher yields. For the implosion performance demonstrated to-date, this brute force approach to ignition obviously reaches beyond NIF's current capabilities but is always a possibility for some future facility. Some combination of the two routes is of course also possible.

NIF implosions to-date have been exploring both of these paths in certain respects. As noted, quality improvements have been ongoing since the beginning of NIF experiments, but recent experiments comparing implosions with different sized fill tubes provide an archetypical example of a specific improvement in implosion quality. Until 2017, NIF implosions were fielded with 10 μm diameter fill tubes. In 2017, 5 μm diameter fill tubes became available and made possible the first NIF experiment with a yield greater than 10¹⁶ neutrons, shot N170601.²⁵ This implosion was then repeated with the previous 10 μm diameter fill tube, shot N170821, and the yield dropped roughly 40% to 9 × 10¹⁵ neutrons. From the modeling perspective, comparing these two shots affords an excellent opportunity to validate simulation capabilities for this specific improvement in the implosion quality. Moreover, N170601 was repeated a second time but now with the main laser drive prolonged by 200 ps to deliver NIF's full 1.8 MJ of laser energy. In the process, this experiment, N170827, assessed a scaling of implosion performance with increasing velocity. The shots N170601, N170821, and N170827 thus provide a convenient trio for validating modeling capabilities along two axes of quality improvements. Simulations for these three implosions are compared in Sec. III.

With respect to hydrodynamic scaling, some energy scaling tests have been performed on NIF, in this case going from “subscale” implosions driven with ∼1 MJ of laser energy to “full-scale” implosions at the limit of NIF's current power and energy. However, these scalings have so far covered only a very limited range of scales, and often with other target parameters also changed, so that the impact of the scaling is unclear. For this reason, direct tests of hydrodynamic scaling are planned in the near future, but there are no convenient pairs of experiments to use for model validation at this time, and the hydrodynamic scaling results presented in Sec. IV await further NIF data for validation. Note that these scaling results are anchored on the modeling of the recent implosion N170601. This implosion was a subignition design using 1.56 MJ and 450 TW of laser drive with a 1-D yield of ∼6 × 10¹⁶. That is, it is a design optimized for robust performance within NIF's current power and energy envelop, but not for high yield. Designs optimized for higher yield would scale differently.

Before proceeding, it must be emphasized that extrapolation to the threshold of a nonlinear phenomenon like ignition is inevitably highly uncertain. Though an effort of many years has been applied to understand current implosions, it remains difficult to predict the threshold at which α-particle self-heating will bootstrap the neutron yield from the current <100 kJ to ∼1 MJ. Small imperfections in the hot spot integrity or confinement that are difficult to predict, or could arise from currently unidentified perturbations, could shift the ignition threshold compared to current understanding. The results of the current, best-effort modeling are described below, but the caveat that the uncertainties with extrapolation are large should always be borne in mind.

This paper is organized as follows. Section II describes the methodology and perturbation sources used in the modeling and extrapolation studies. Section III shows the results of applying that model to recent, high-performance NIF implosions, in particular N170601, N170821, and N170827. Section IV then describes the results of applying the same model to hydrodynamic scaling of NIF shot N170601 and the implications for the threshold of ignition. Section V offers some perspectives on future directions and prospects for further model improvement. Finally, Sec. VI summarizes and concludes.

The methodology applied here is the product of a long and continuing evolution in NIF postshot modeling.^19–24 As described previously, the simulations presented below are run with the radiation hydrodynamics code HYDRA²⁶ and include multigroup diffusive radiation transport, thermonuclear burn, Monte Carlo burn product transport, and tabular equation of state (EOS)^27–29 and opacity data.³⁰ In these capsule-only simulations, the hohlraum is treated as a spherical boundary surface where a prescribed inward x-ray flux is applied. This x-ray flux is independently adjusted to match tuning data from dedicated NIF shots that measure early-time shock propagation through surrogate targets using the Velocity Interferometer System for Any Reflector (VISAR) diagnostic,³¹ inflight shell implosion velocity and low-mode shell shape using the 1-D and 2-D ConA platforms,^32,33 hard x-ray fraction in the x-ray flux (“M-band”) from the View Factor platform,³⁴ and finally the measured bang time for the DT shot in question. Long-wavelength asymmetries in the x-ray flux are also included based on independent 3-D hohlraum simulations, also run using HYDRA, and account for the as-delivered beam-to-beam power imbalance for the particular shot in question.

A new feature in the 3-D simulations described here is that these simulations include physical viscosity and nonlocal thermodynamic equilibrium (NLTE) effects. The latter is important for the high-Z ablator dopants, namely tungsten, used with HDC ablators that can be injected into the hot spot due to the fill tube or other perturbations. For early silicon-doped plastic ablator implosions on NIF,³⁵ NLTE effects on hot spot emissivities have not been found to be significant. However, for such a high-Z element as tungsten injected into a hot spot with a background temperature greater than 1 keV, NLTE effects have been found to change the simulation results^36,37 according to the detailed configuration accounting (DCA) model³⁸ available in HYDRA. Similarly, including physical viscosity in the simulations damps many of the short-scale hot spot flows^39,40 and alters the simulated hot spot temperatures.²⁰ This effect is less significant than including NLTE effects but does slightly alter the results. Note that including both of these effects substantially increases run time and memory requirements in the simulations and has only recently become feasible due to improvements in computing resources and simulation technique.

The perturbation sources accounted for in the current model include the capsule support tent,^41,42 the capsule fill tube,^43–45 interface roughness on all capsule interfaces (inner DT ice surface, DT fuel-ablator interface, internal dopant interfaces, and ablator outer surface),⁴⁶ and the 3-D x-ray flux asymmetries described above. As described previoiusly, for the first two perturbation sources (45 nm thick tents and a 5 μm diameter fill tube), the very small spatial extents of these perturbations necessitate running highly resolved 2-D simulations to model both of these effects individually. Lower-resolution surrogate perturbations can then be developed that closely match the results of these resolved simulations but at reduced computational cost suitable for 3-D simulations. Even so, 2-D convergence tests have shown that it is necessary to run simulations with resolution up to Legendre mode numbers of ℓ ∼ 100 to adequately capture all of these perturbation sources. For a 3-D full-sphere simulation, this requires ∼500 × 10⁶ computational zones, and run times of up to a few weeks on ∼8000 compute cores.

For interface surface roughnesses, measured power spectra are used for the inner DT ice surface⁴⁷ and outer ablator surface.^48,49 Nominal values are used for the interior interfaces since these surfaces can only be characterized destructively. Note that the effect of the shell microstructure is not included in the current simulations. Recent 2-D VISAR experiments^50,51 and modeling⁵² have identified microstructure in HDC ablators as a potentially significant additional perturbation source, particularly at very short wavelengths. As the significance—or not—of this perturbation seed is clarified, it will be included in the modeling but has not been implemented so far.

In this respect, an effect that has received renewed attention in recent experiments is the potential for very short wavelength mixing between the DT fuel and HDC ablator inflight. This type of mixing has been known for some time^44,46,53 but did not appear to be a significant effect in earlier plastic ablator implosions.^21,22 HDC implosions with potentially more unstable fuel-ablator interfaces and possibly larger short wavelength perturbations appear to be impacted by this effect, however.⁵⁴ 

This effect is illustrated specifically in the 3-D simulation shown in Fig. 1. The figure shows a high-resolution (ℓ ∼ 1000) 3-D simulation of N170601 at the time of peak implosion velocity with interface roughness as the only perturbation seed. The inner DT interface and outer surface have been initialized with the measured surface roughness specific to this shot, while the fuel-ablator interface has been initialized with the inner surface features measured from a shard of a prototype HDC shell. Together, these roughness seeds have led to the significant, short-scale mixing between the HDC and DT at this time. In particular, narrow, nonlinear fingers of the ablator material can be seen penetrating deeply into the DT fuel. The spanwise density modulations due to this mixing are shown by the gray shaded region in the right panel of the figure, while the red curve shows the spanwise average. Comparing to the 1-D simulated density shown by the blue curve, the DT fuel has decompressed due to the presence of the 3-D mixing. In effect, the mixing of the higher temperature ablator material into the colder DT has raised its average temperature and so left it less compressed for the same drive pressure compared to its 1-D counterpart. The consequence of this mix-driven heating and decompression is to lower the ultimate compression of the fuel at stagnation, and so reduce the effective hot spot confinement.

Similar to the tent and fill tube perturbations, lower resolution surrogates must be used to account for this high-mode mixing in full-sphere simulations of these implosions. Ideally, subgrid or Reynolds average Navier-Stokes (RANS)-type mix models^55,56 could be tuned to reproduce the mixing seen in the high-mode, resolved 3-D simulations, and those lower-resolution mix models could then be included in the 3-D global simulation. Unfortunately, the mix models currently available in HYDRA are not compatible with the 3-D full-sphere simulations described here. As a consequence, motivated by the effective heating that the mix adds to the DT fuel, a preheat model has been used to mimic the effect of the high-mode mixing in the following simulations. Specifically, 60 J of preheat is added adiabatically to the DT fuel over the course of the acceleration of the shell to gradually decompress the fuel similar to what is seen in the resolved 3-D simulation. The green curve in Fig. 1 shows the effect of adding this preheat source to the 1-D simulation. The decompression of the bulk of the fuel mass is quite well matched by this preheat source; however, the inner part of the fuel is not as well matched. Further refinements of the preheat source, or ideally including better subgrid or RANS models in HYDRA could improve this agreement, and this is a subject of ongoing work. It should be emphasized, however, that this preheat model is exclusively a numerical expediency. There are no known physical preheat sources that have this effect in current NIF implosions. Here, the preheat is merely a surrogate for the impact of the unresolved high-mode mixing.

Finally, similar to the results summarized in Ref. 22, numerical convergence of the simulation model has been confirmed with respect to zoning, the number of energy groups used for radiation transport, and the number of Monte Carlo particles used to model fusion products. With respect to the complex hydrodynamics seeded by features such as the fill tube, verification test have also been run comparing HYDRA to the high-order Eulerian code Miranda⁵⁷ and to the adaptive mesh refinement (AMR) Eulerian code xRAGE.^58,59 The results of these tests have shown quite favorable agreement through most of the implosion; however, differences inevitably emerge at late times for such nonlinear features as the fill tube jet.⁶⁰ Work is ongoing to understand the origin of these differences and their consequences. Independent 2-D simulation work with xRAGE⁵⁹ has also shown very similar results to 2-D HYDRA when modeling the same NIF implosion with the same simulation inputs. This lends confidence to the independent hydrodynamic schemes in both codes. Finally, it is beyond the scope of this article, but extensive validation work has also been done to compare HYDRA predictions to dedicated experiments aimed at isolating the effects of particular perturbation sources.^{42,53,61–67}

The combined effect of the multiple perturbations described in Sec. II is shown by the 3-D simulation of N170601 in Fig. 2. The figure shows four times close to the time of peak neutron production (bang time) with the color scale of the left cutaway showing the ion temperature and the right cutaway showing the mass density. The outer surface shows the ablation front location at each time colored by the electron temperature. At the first time (t = 7.50 ns), the perturbation seeded by the fill tube can be seen penetrating the shell at the equator while the imprint of the tent perturbation is barely visible as the ringlike perturbations on the ablation front at roughly 45° and 135° in polar angle. The random modulations of the ablation front are due to broadband surface roughness. At the subsequent times, the fill tube jet penetrates the center of the hot spot but is also deflected upward by the up-down asymmetry in the x-ray drive on this implosion, itself due to the as-delivered laser power imbalance for this shot. At bang time (t = 8.18 ns), this flux asymmetry has led to an accumulation of mass in the lower hemisphere and a corresponding thinning of the confining shell in the upper hemisphere. At the last time shown (t = 8.25 ns), the hot spot can be seen to be venting through the combined weak spot caused by the flux asymmetry and the defect from the tent perturbation. As already emphasized and is made clear in this figure, multiple effects combine to determine the morphology of the hot spot at bang time, and the correct superposition of these effects is only possible in a realistic 3-D geometry.

Figure 3 compares some of the imaging data recorded on shot N170601 against the corresponding synthetic images from the 3-D simulation. Experimental images are shown in the top row with the synthetic images in the bottom row. The comparison includes the equatorial and polar time-integrated x-ray images attenuated through a vanadium filter, the primary neutron⁶⁸ image, and the flange nuclear activation detector (fNAD)⁶⁹ “sky map.” For each image, the synthetic results are generated using the same filter response and pinhole blurring as in the experiment. Prominent in each of the x-ray images is the bright spot due to the fill tube jet superimposed on the background hot spot self-emission. In the equatorial image, the fill tube enters from the left, while from the polar line of sight the fill tube enters from the right. The high-Z material entrained by the jet is the cause of the bright x-ray emission seen in both the experiment and simulation. The primary neutron image is effectively the inverse of the x-ray image: Where the fill tube penetrates and radiatively cools the hot spot (from the right on this line of sight), the neutron emissivity is correspondingly low, while on the left where the hot spot remains hot and unperturbed, the emissivity is high.

The synthetic x-ray and neutron images qualitatively reproduce the features seen in experiment. In both of the x-ray images, there are bright spots of approximately the correct size and shape corresponding to the simulated fill tube jet penetrating the hot spot. The background hot spot emission also has approximately the correct shape and size. Likewise, the synthetic neutron image is dim on the right side where the fill tube enters the hot spot, and shows the correct bright spot on the left where the hot spot is unperturbed. However, note that, in both of the x-ray images, the relative brightness of the fill tube jet compared to the background hot spot appears lower in simulation than in experiment. Similarly, though the neutron image has roughly the correct size compared to experiment, the simulated image is more elongated than seen in experiment. These differences are being further quantified, and identifying their origins is the subject of ongoing work.

Finally, the right most images in the figure compare the fNAD sky maps for this shot. Again, a good qualitative agreement is found. In both simulation and experiment, the thinning of the shell in the upper hemisphere leads to higher fNAD activation in that hemisphere (red in the color scale), while the mass accumulation in the lower hemisphere leads to lower activation there (blue in the color scale). The simulation also qualitatively reproduces the lower activation region in the very middle of this view due to a mass accumulation to the left of the fill tube location. Again, the agreement between the simulation and experiment is not perfect, and residual differences are being investigated, but the approximate mass distribution around the hot spot is encouragingly close in this simulation.

Interestingly, in the simulation, it is also possible to determine that all of the modulations seen in the synthetic fNAD sky map are due to the x-ray flux asymmetries on the capsule which in turn are driven by the as-shot laser power imbalance in this experiment. That is, for this case, the low activation region in the center of the sky map is not due to the fill tube, as has been hypothesized, but appears to be entirely due to the as-delivered laser imbalance.

Another class of diagnostic signatures to compare with the simulation is the neutron spectra recorded along separate lines of sights around the NIF target chamber.^70,71 These directional spectra may be characterized by a primary neutron peak width, or effective burn-weighted ion temperature along that direction, and a neutron down scatter ratio (DSR), the ratio of down scattered neutrons in the 10–12 MeV energy band to the primary 13–15 MeV energy band. The first quantity encodes information about the state of flow in the hot spot relative to the thermal motion of the reacting ions,^72,73 while the second records information about the areal density along a given line of sight.

Simulated DSR and burn-weighted ion temperatures along four different lines of sight are compared to the experimental results for N170601 in Fig. 4. The simulated DSR results are within the experimental error bars on two of the directions and only just outside of the error bars on the other two. Overall, the global variation of DSR, and hence areal density, is fairly well captured in the simulation consistent with the qualitative agreement also shown from the fNAD sky map. The agreement is less close on the simulated ion temperature, however, with only one of the directions within the error bars of the measurement. On average, the simulation overpredicts the temperature, but only one of the directions is significantly outside of the error bar.

Note that these results are in contrast with previous simulations of NIF implosions where the experimental ion temperatures have typically been underpredicted. This is due both to improvements in modeling and to the reduced long-wavelength perturbations in more recent experiments. In particular, these simulations use updated DT thermal conductivity tables that show less hot spot conductive heat loss⁷⁴ and hence slightly higher hot spot temperatures than in previous simulation studies. Additionally, recent implosion experiments are more symmetrically driven and consequently have smaller residual hot spot flows, especially compared to the high foot experimental database. These implosions have proven easier to model with the resulting improvement in agreement between measured and simulated ion temperatures.

Lastly, Table I compares several of the scalar observables between simulation and experiment for N170601. As can be expected, the 3-D simulation shows a closer agreement with experiment than the 2-D equivalent and matches all of the observables listed within their error bars with the exception of the burn width and the neutron yield. The simulated burn width is shorter than measured, but only just outside of the experimental error bar, and is consistent with the slightly lower than measured yield. That is, the ∼10% lower simulated yield is consistent with a burn width that is ∼20% shorter than the experiment. Note that the 2-D simulation is significantly shorter in burn width but overpredicts the yield, DSR, and neutron image size while underpredicting the ion temperature.

Given the relatively close agreement seen between the 3-D simulation and experiment for N170601, it is interesting to query this model for the relative importance of each separate perturbation source. The impact of the individual perturbations on the simulated yield is summarized in Fig. 5. The 1-D, unperturbed yield in this implosion model is 5.6 × 10¹⁶. The presence of the high-mode fuel-ablator mix, again modeled by a time-dependent fuel preheat source tuned to the high-resolution 3-D simulation, reduces the yield by a factor of 1.6 to 3.4 × 10¹⁶. The fill tube alone results in a larger yield degradation by a factor of 2.2 to 2.5 × 10¹⁶, and the 3-D x-ray flux asymmetries result in a degradation of 3.1 to 1.8 × 10¹⁶. For comparison, all effects combined leads to the simulated yield of 1.3 × 10¹⁶, or a reduction by a factor of 4.2, while the experimental yield is a factor of 3.9 less than 1-D. Note that, according to these results, even if the fill tube were totally removed from the implosion or the x-ray drive on the capsule were completely symmetrized, the yield would only increase to ∼2 × 10¹⁶. That is, due to the presence of the multiple large perturbation sources, more than one of these perturbations must be significantly mitigated in order to substantially improve the performance.

As a further validation test of the methodology, the same model as used to simulate N170601 has been applied to the companion shots N170821 (10 μm fill tube) and N170827 (200 ps longer laser pulse). The fNAD maps for these three shots are compared in Fig. 6. As with N170601, the agreement between experiment (upper row) and simulation (lower row) is imperfect, but similar trends are seen. The larger equatorial blue spot seen in N170821 compared to N170601 is reproduced in the simulation, and it is rotated slightly to the right consistent with the experiment. The peak activation in the simulation is somewhat higher than observed experimentally, however. Likewise, the equatorial blue spot in N170827 is diminished compared to N170601, and this is captured in the simulation, although the simulated peak activation in the upper hemisphere is higher than seen in the experiment, and the minimum activation in the lower hemisphere is lower than the experiment. Note that, as with N170601, the features seen in the simulated fNAD sky maps are entirely determined by the 3-D flux asymmetries that in turn result from the as-delivered laser power imbalance. According to the simulations, the perturbation caused by the fill tube does not contribute to the modulations seen in the fNAD maps.

The same scalar observables as listed in Table I for N170601 are given for N170821 and N170827 in Table II. As with the fNAD maps, the agreement between the simulation and experiment is less close for the two later shots than it is for N170601. Encouragingly, the yield trend is captured in the simulations as the laser drive and fill tube size are varied. The results for the DSR and ion temperatures are more variable, however. The burn width and bang time for N170827 are also not well matched in the simulation. Although the comparison to the experiment for these two shots is imperfect, that the model captures some of the scaling trends lends confidence to applying the model to extrapolations of NIF performance beyond the currently tested regimes.

The model described in Sec. II and compared to the experiment in Sec. III has been applied to hydrodynamic scaling starting from the results for shot N170601. Note that pure hydrodynamic scaling,⁷⁵ where all of the capsule dimensions and the timing of the x-ray drive on the capsule are scaled by a similar factor (but all other implosion features are unchanged), does not apply for x-ray driven implosions such as those discussed above. This is due to the nonscaling features of indirect drive implosions, namely the physical optical depth associated with the x-ray radiation that drives the implosion. If the capsule is naïvely scaled up, the penetration depth of the x-ray flux into the ablator becomes proportionately smaller with increasing scale and other implosion characteristics (interface Atwood numbers, implosion velocities, etc.) that should remain invariant with hydrodynamic scaling in fact change with the scale. To compensate for this, it is necessary to adjust the net opacity of the ablator in order to preserve the implosion velocity, Atwood numbers, etc. with increasing scale. Empirically, in the simulaitons, it is found that reducing the ablator dopant concentration by the reciprocal of the scale factor and also thinning the ablator shell thickness by 5 μm with every 20% increase in scale very closely conserves the relevant implosion characteristics.²⁴ This recipe was applied in the simulations described here.

Note also that, in the simulations described here, the x-ray flux on the capsule was assumed to scale ideally. That is, only the timing of the x-ray flux on the capsule was changed with the scale, and there were no changes in the net flux or the spectral character of the x-ray radiation incident on the capsule. Effectively, this is to assume that the hohlraum performance remains identical with increasing scale and that there are no changes in the x-ray conversion efficiency, hard x-ray fraction, or in hohlraum laser-plasma interactions with scale. These assumptions will remain approximately valid for sufficiently small scale changes but will clearly be violated at some scale. Quantifying the detailed changes in hohlraum performance with scale is an ongoing area of research,⁷⁶ and any findings of nonideal scaling features of hohlraum performance will be incorporated in future scaling studies.

The results of hydrodynamic scaling from the 3-D simulation of N170601 are summarized in Fig. 7. Here, the purple points represent a subset of the database of HDC ablator experiments conducted on NIF. As illustrated by the figure, the experimental database does cover some range in energy scaling; however, many other features other than simply the implosion scale and energy were changed in these experiments such that no pair represents a clear hydrodynamic scaling test. N170601 is represented by the gold point in the figure, and as discussed above, the 3-D simulation of this experiment, represented by the gold square, is quite close to the experimental result. The result of hydrodynamically scaling this 3-D simulation, accounting for the nonscaling aspects just discussed, is shown by the gold curve. Appropriate for these capsule-only simulations, the scaling is shown vs capsule absorbed energy calculated for each scale. According to the simulation model, an absorbed energy of 500 kJ is required to reach neutron yields greater than 1 MJ for this class of targets. Note again that N170601 is a subignition design with a 1-D yield of ∼6 × 10¹⁶; ignition designs would scale differently.

For comparison, the result of performing the same scaling exercise but using 2-D simulations is shown by the blue curve. Notably, the anchor point of this scaling curve, the 2-D simulation of N170601, appears fairly close to the experimental result. However, the projected ignition threshold occurs at a significantly lower drive energy. This difference underscores the importance of properly modeling NIF implosions in their realistic 3-D configuration. Even though the 2-D simulation results can be fairly close to the observed yields at the current energy scale, the scaling properties of 2-D and 3-D simulations are quite different.

These 2-D-3-D differences can be ascribed to the fundamental difference in the stagnation dynamics between 2-D axisymmetric flows and realistic 3-D flows. In 2-D flows, the material reaching the symmetry axis is by construction met by an exactly opposing flow. This perfect momentum balance maximizes the conversion of shell kinetic energy to hot spot internal energy, and hence maximizes the compression and heating of the hot spot material and in turn enhances the fusion reactivity. In realistic 3-D flows, this perfect momentum balancing of course does not occur. The conversion of kinetic to internal energy is then less efficient with more energy left in unuseful, residual hot spot flows. The well-known higher growth rate of nonlinear Rayleigh-Taylor instabilities in 3-D vs 2-D geometry⁷⁷ also contributes to 2-D vs 3-D differences. Specifically, the development of weak spots that can rupture the confining shell and cause a loss of confinement can happen more rapidly in realistic 3-D geometry, leading to effectively shorter burn durations and lower hot spot pressures. Precisely this type of 3-D rupture is seen in the last snapshot in Fig. 2.

In addition, it is interesting to note that at the threshold of ignition (∼1 MJ in yield), the relative burn duration in these simulations shortens compared to the time-scaled burn duration of the nonburning, anchor case. That is, the burn duration scales roughly linearly with the spatial scale factor as the implosion is scaled up until the yield exceeds ∼1 MJ. At that point, the burn duration drops from ∼150 ps to ∼120 ps, a value less than the starting, unscaled burn duration. Similarly, the burn-averaged ion temperature increases almost linearly with scale for yields less than 1 MJ but increases abruptly when the yield exceeds ∼1 MJ, nearly doubling from ∼6.5 keV to ∼12.3 keV. Beyond the obvious signature of higher neutron yield, this shortening of the burn duration and abrupt increase in the burn-averaged temperature could serve as additional experimental signatures that the threshold of ignition is being approached.

Finally, in addition to hydrodynamic scaling, the 3-D simulations have been rerun to test some near-term improvements in implosion quality. The first test was to assess the effect of zeroing all of the extrinsic x-ray flux asymmetries in the 3-D simulation. That is, all of the odd Legendre modes and all of the nonaxisymmetric modes in the x-ray flux asymmetries were removed, so that only the intrinsic hohlraum asymmetries represented by Legendre modes ℓ = 2, 4, 6, and 8 are retained. Physically, this corresponds to arranging perfect beam-to-beam power balance from the NIF laser and also perfect radiation smoothing in the azimuthal direction around the hohlraum. Eliminating the extrinsic asymmetries alone results in only a modest improvement in yield from 1.31 to 1.47 × 10¹⁶ neutrons. A second near term improvement is to replace the 5 μm diameter fill tube with a 2 μm diameter tube. Similarly, this single improvement results in only a slight yield improvement to 1.62 × 10¹⁶. Combining these two improvements in the simulation leads to a slightly larger yield improvement to 1.86 × 10¹⁶.

This example again illustrates the important point that several perturbations degrade current NIF implosions, and mitigating only one of these perturbation sources is likely to show only modest improvements in the performance. According to the simulations, significant progress must be made to address at least two perturbation sources to see a substantial improvement. Even so, when two perturbations are substantially mitigated, the model still predicts yields well below ∼1 MJ. This underscores a second point that, even after mitigating the current perturbation sources, implosions of the type currently being test on the NIF will likely require some amount of hydrodynamic scaling or an increase in convergence and compression to reach yields in the MJ range. Simulations are in progress to assess the hydrodynamic scaling of these improved versions of N170601, but it is clear that somewhat less than 500 kJ of absorbed energy will be required.

With respect to the extrapolations discussed in this section, it again bears emphasizing that the uncertainties in these extrapolations are large. The model discussed in Sec. III represents the current best model of NIF implosion performance, and the results of directly extrapolating that model to the ignition threshold have been presented. However, small discrepancies or as yet unidentified perturbation sources could easily swing the onset of ignition and hence significantly change the shape of the extrapolation curve. Further work will aim to quantify the uncertainties in the extrapolation curve in Fig. 7, as well as improve the accuracy in modeling current experiments and reduce the uncertainties inherent in extrapolation.

Sections II–IV have reviewed the current state of detailed modeling of NIF ignition implosions and the results of extrapolations based on those models. What are the largest uncertainties in those predictions, and what are the prospects for future model improvements? This section offers some perspectives on these questions.

The largest source of uncertainty in the current model, as in all predecessor models, is almost certainly the x-ray flux on the capsule. While this x-ray flux is heavily constrained by measurements of the implosion timing and trajectory, and considerable effort is invested in matching these data, the x-ray flux itself remains unmeasured, and therefore uncertain. At the same time, implosion performance, in particular, yield, is a very sensitive function of implosion velocity and also fuel adiabat, both of which depend directly on the x-ray drive on the capsule. Small errors in the x-ray flux on the capsule can therefore subtly compound into large differences in the predicted yields or other observables. Considerable effort is being made to improve hohlraum modeling,⁷⁶ and so develop a genuinely predictive capability for the x-ray flux on the capsule; however, substantial progress still needs to be made to meaningfully reduce the uncertainties in the x-ray drive.

A recent, very promising approach to characterizing the x-ray flux on the capsule is to treat the measurements constraining the implosion timing and trajectory as an inverse problem for the x-ray drive. This “Green's function” approach⁷⁸ to inferring the x-ray drive in a particular implosion exploits the remarkable linearity of the simulated response of NIF implosions to small perturbations in the x-ray drive. 1-D or 2-D simulations with a sequence of infinitesimal drive perturbations can be used to numerically compute the “response matrix” of a given implosion to 1-D and 2-D perturbations of the x-ray drive, that is, the effect on simulated observables from a small perturbation to the drive at a given point in the pulse. By inverting this response matrix, a particular set of implosion observables (VISAR shock timing traces, inflight 2-D ConA data, and hot spot x-ray self-emission shape data) can be matched very closely in a single operation. The effective implosion x-ray drive can then be inferred by working backward from the observables assuming a linear relationship between the x-ray drive and the subsequent observables. (Numerical experiments with actual data have shown that this linearity assumption is very accurate.) Furthermore, the technique can be applied equally well to both the 1-D x-ray drive on the capsule responsible for the implosion velocity and adiabat, and also to the low order Legendre asymmetries in the that drive that lead to subsequent hot spot distortions. More importantly, the same inversion procedure may be applied to the error bars associated with the experimental observables. By applying the “inverse response matrix” to the error bars, the uncertainty in the x-ray drive may then be directly computed. This technique therefore offers an important inroad into not only the nominal x-ray drive on the capsule but also quantifies the uncertainties in that drive.

Another leading modeling uncertainty is the role of ablator microstructure as a perturbation seed. As noted in Sec. II, recent 2-D VISAR measurements of shocked HDC ablator samples have shown shock front modulations that are larger than expected based on surface roughness alone.⁵¹ The grain structure in HDC is a likely candidate to explain this, and a modeling effort is under way to quantify the impact of the grain structure in these VISAR experiments and also to assess its potential impact in DT implosions.⁵² Current results suggest that it could enhance the mixing already seeded by surface roughness, such as shown in Fig. 1, but the exact amount remains to be determined. At the very least, given this added perturbation seed from the material microstructure, not to mention the possibility of other as yet unidentified small-scale perturbations, the short-wavelength mix results shown in Fig. 1 should be taken as a lower bound on the potential for high-mode interface mixing.

As shown in Fig. 5, the fill tube and 3-D x-ray flux asymmetries are the two largest degradations to performance in current implosions, and uncertainties in the magnitudes of these effects are hence a strong lever on the simulated performance. Work is ongoing to further constrain fill tube modeling against dedicated radiography, self-emission imaging, and spectroscopy experiments and reduce uncertainties. Likewise, parallel efforts are underway to better understand the origins and magnitudes of the 3-D modulations seen in fNAD sky maps. While the simulations in Sec. III, as well as a growing body of supporting evidence, suggest that NIF laser power imbalance is the source of these modulations, other origins cannot be ruled out, and better quantifying the magnitudes of these modulations will further constrain modeling. Similar modeling validation work is also underway for other perturbations such as the support tent and the as yet unexplained bright “meteors” that occasionally appear in x-ray self-emission images from NIF hot spots. Similar to the fill tube perturbation, the brightness of these meteors indicate that small, localized quantities of the doped ablator material are being injected deep into the hot spot and heated to temperatures of several keV. Recent work has focused on flakes of high-Z debris adhering to the ablator surface or other fine-scale defects (micron-scale pits on the ablator surface or similarly sized voids in the ablator bulk) as the likely sources. Experiments and simulations are ongoing to resolve whether these, or possibly other perturbations, are responsible for the appearance of the meteors.

There are also a number of physical data uncertainties that contribute to uncertainties in the modeling results. These include EOS, opacity, thermal conductivity, stopping power, electron-ion coupling, and plasma viscosity uncertainties. Modeling results testing different EOS models for DT^79–81 have shown remarkably little sensitivity in simulated performance, but a similar exercise using different HDC EOS models^82–84 does show some sensitivity. The possible role of magnetic fields in NIF implosion hot spots has also been debated for some time. Recent simulation results⁸⁵ suggest that the impact when all magnetohydrodynamic (MHD) effects (Biermann battery, Nernst, and Righi-Leduc)⁸⁶ are included is not a significant change in yield or hot spot temperature; however, effects such as MHD that appear small at the current scale could be more significant at the threshold of ignition.

Finally, the numerics and physical models in the simulations can always be improved. As described in Sec. II, the current model resolves features at scales characteristic of Legendre mode numbers up to ℓ ∼ 100 and must rely on surrogate perturbations to account for the short wavelength features of the tent, fill tube, and high-mode fuel-ablator interface mix. Directly resolving these features would require capturing mode numbers as high as ℓ ∼ 1000 in 3-D over the full capsule extent. This would require a resolution increase of an order of magnitude, or a ∼10⁴ increase in total compute requirements. Such resolutions are well beyond today's capabilities, but as exa-scale computing becomes available, this ultrahigh resolution in 3-D will eventually become feasible and would enhance the predictive capability of the current model. AMR is another possibility for improving capabilities that is ideally adapted to resolving fine scale features embedded in a larger smoothly varying flow, such as the tent and fill tube in the larger capsule implosion. This capability does not currently exist in HYDRA but is available in other codes such as xRAGE and ARES.⁸⁷ Of course, care must be taken to ensure that any AMR does not artificially seed perturbations due to numerical symmetry breaking.

Hand in hand with higher resolution simulations, there is also the need for better information to initialize those simulations, namely, better metrology of implosion initial conditions. This includes better characterization of surface roughness or isolated defects on all capsule interfaces, of debris on the capsule surface, and of the details of features such as the fill tube counterbore and glue fillet or the tent liftoff and curvature. Finally, current simulations rely on diffusive radiation transport,^88–90 both for reasons of computational speed and low memory overhead, and also due to its low numerical noise characteristics. 2-D simulations testing the diffusion approximation against implicit Monte Carlo⁹¹ or discrete ordinate⁹² radiation transport show the diffusive approximation is generally well validated with the notable exception of modeling discrete features like the fill tube. Nevertheless, relaxing the constraint of the diffusive approximation and moving to either discrete ordinate or implicit Monte Carlo transport would further reduce modeling uncertainties.

From a more global perspective, how good does the simulation model need to be? Is there a point of diminishing returns in modeling development when the model becomes “good enough?” In answering this, it bears remembering that many performance projections were made prior to the start of NIF experiments. These included statistical assessments of the likelihood of ignition using hundreds to thousands of then state-of-the-art simulations capturing the perturbation sources known at that time.⁹³ These predictions suggested a high probability of ignition using the point design target.⁹⁴ Regrettably, these projections proved highly inaccurate, in large part, because they overlooked the importance of the capsule support tent that proved to be the dominating perturbation for those early NIF implosions.²¹ Looking forward, it is the presence of similar unknown, or at least underappreciated, perturbation sources that represent the greatest uncertainty. Until NIF or other implosions reach the threshold of ignition, it will be difficult to rule out the possibility of currently unappreciated effects or unknown perturbations that could be controlling performance at the threshold of ignition.

In this sense, the modeling of NIF implosions will only be proven adequate once it is validated against an igniting experiment. Obviously, this is a circularity since the model is only fully validated once the very result it is trying to predict (ignition) is achieved. Nonetheless, until ignition is achieved, there is still considerable utility in an even partially validated model to provide guidance and prioritization for future experiments. Indeed, guidance provided by models similar to those described above in testing smaller fill tubes⁴⁵ and alternate tent configurations⁹⁵ has so far largely been borne out. These models have already marked a path to higher performance on the NIF, even if their exact predictions were inaccurate. In the oft-quoted aphorism of George Box,⁹⁶ “All models are wrong, but some are useful.” Whether the longer-range projections discussed here are correct can only be verified in some future experiment, but in the interim, it is hoped they will still provide useful guidance.

This prospectus article has summarized the current state of 3-D modeling of recent high-performing NIF implosions and preliminary results on hydrodynamically scaling those implosions to the threshold of ignition. Projections where the implosion quality is improved by reducing the largest perturbation sources in the implosions are also described.

Current 3-D simulations of NIF implosions capture many of the observables from the experiment fairly well, even if not strictly within the experimental error bars. For shot N170601, this includes the measured burn-averaged hot spot ion temperature, neutron down scatter ratio, and neutron image size. The primary neutron yield and burn width are within 10 and 20% of the measured values, respectively. There is also clear qualitative correspondence between the simulated and experimental x-ray and neutron imaging, in particularly, with respect to the presence of a jet penetrating the hot spot due to the fill tube, and between simulated and experimental fNAD sky maps. Additionally, simulations of experiments where the fill tube size was increased and separately where the delivered laser energy was increased capture the observed trends in the neutron yield.

Based on these results, hydrodynamic scaling projections and projections of implosion performance with reduced perturbation seeds have been made. These projections begin an assessment of the requirements for reaching ignition based on experience gained from NIF. An important result of this study is the significant error made in extrapolations based on simplified 2-D simulations compared to 3-D. More realistic 3-D simulations show a significantly higher threshold for ignition than the equivalent 2-D simulations. This is a consequence of the added degree of freedom for hot spot deconfinement that is allowed in 3-D, and shows the importance of realistic 3-D modeling for reliable extrapolations, as well as the importance of 3-D effects in current experiments. At present, though, these projections still have large uncertainties. Further refinements of the models will continue in an effort to both reduce these uncertainties and quantify how uncertain current projections are.

P. L. Volegov was gratefully acknowledged for providing the neutron imaging data for N170601. K. L. Baker, R. L. Berger, L. F. Berzak Hopkins, D. A. Callahan, E. L. Dewald, T. R. Dittrich, L. Divol, T. Döppner, W. A. Farmer, J. E. Field, S. W. Haan, B. M. Haines, D. E. Hinkel, H. Huang, O. A. Hurricane, C. Kong, O. L. Landen, S. Le Pape, J. D. Lindl, A. G. MacPhee, N. B. Meezan, A. Nikroo, L. A. Pickworth, N. G. Rice, H. A. Scott, V. A. Smalyuk, M. Stadermann, D. J. Strozzi, C. A. Thomas, R. E. Tipton, and R. P. J. Town were acknowledged for the valuable assistance and discussions. The essential contributions of the whole of the NIF diagnostic and operations teams were also greatly appreciated.

This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.