Assessing the performance of MagLIF with 3D MHD simulations

The Magnetized Liner Inertial Fusion (MagLIF) platform^1–5 requires three critical inputs to achieve robust fusion conditions, as illustrated in Fig. 1: (1) an applied axial magnetic field that provides thermal insulation, (2) laser preheat with the $2ω$ Z-beamlet laser (ZBL) that reduces convergence requirements to reach fusion conditions, and (3) a z-pinch implosion of a solid metal liner (a cylinder) that provides the compressive PdV work to heat the plasma, and confinement of the fuel through stagnation. In recent years, improving these capabilities has been a focus of the MagLIF efforts on the Z machine, enabling enhanced performance and scaling studies.^6,7 These improvements have been generally motivated by predictions from one- and two-dimensional (1D, 2D) radiation-magneto-hydrodynamics (rad-MHD) simulations.

As with the design of laser-driven ICF capsules,⁸ these relatively fast simulations play a critical role in the design of MagLIF experiments by enabling efficient searches through a large parameter space that specifies a given MagLIF configuration.^9–11 This includes (but is not limited to) the applied magnetic field strength (⁠ $Bz$⁠), the initial fuel density (⁠ $ρ0$⁠), the amount of laser preheat and how it is distributed, and the liner dimensions, including the height, radius, and thickness. More recent ideas also include multi-component liners with conductive^2,9 or non-conductive coatings,^6,12 as well as fuel ice layers to reduce the impact of liner mix and enable ice-burning configurations producing hundreds of MJ of fusion yield¹² on a future pulsed power machine. From these simulations, interesting places in parameter space can be identified and studied in higher fidelity.

In this paper, we present 3D HYDRA MHD^13–16 simulations of recent experimental configurations to address the 3D nature of the implosions and how these effects alter the dependency of target performance with respect to the experimental input parameters. The evolution of a “typical” implosion is shown in Fig. 1 from time sequences of a 3D HYDRA MHD simulation of the complete target with a $90°$ wedge removed to see into the fuel.

The laser enters the top of the target and primarily interacts with the fuel material. However, simulations show that stochastic filaments can divert into the liner or cushion, which could introduce mix and generate mass perturbations that seed subsequent hydrodynamic instabilities.¹⁷ Non-uniformity in the blast wave, caused by the energy deposition, can also contribute to instability seeding when it interacts with the liner inner wall. As illustrated by the streamlines in Fig. 1, the laser deposition also redistributes the initially uniform axial magnetic field via hydrodynamic and thermal gradients (e.g., the Nernst and cross-Nernst terms of extended MHD), leading to significant axial and azimuthal variation in fuel magnetization throughout the column. During compression, the field is continually moved by residual motion in the fuel introduced by the laser deposition, as described in Weis et al.¹⁷ 

While the laser introduces deviations from axisymmetry, this paper primarily focuses on the impact of liner perturbations caused by the development of acceleration phase magneto-Rayleigh–Taylor instabilities (MRT),^18–20 as these deviations dominate over the effects of laser asymmetries in the following simulations. The liner is essentially motionless up until the time of preheat (⁠ $∼60$ ns before stagnation), so the initial perturbation growth is dominated by the electrothermal instability.^12,21 As the liner accelerates, MRT grows the perturbations sufficiently on the outside surface of the liner to eventually “feedthrough”²² to the inner liner surface during compression, seeding deceleration phase instabilities. Feedthrough of MRT is a function of the liner thickness (⁠ $Δ=rout−rin$⁠) or alternatively, the aspect ratio $AR=rout/(rout−rin)$⁠, where thicker liners or lower AR liners are more robust to feedthrough.

It is not possible to image MRT, and associated feedthrough, via standard backlighting radiography²³ in fusion-producing MagLIF implosions (ZBL provides backlighting or preheat but cannot perform both functions during the experiment), but these processes have been studied extensively in solid liner implosions, without laser preheat.^24,25 Although 2D (r, z) simulations have been shown to reasonably model MRT in un-magnetized experiments, they do so with extensive tuning of the initial seed amplitude and mode spectrum²⁴ in order to mitigate the impact of the artificially enforced azimuthal correlation inherent in axisymmetric calculations. More problematically, introduction of the applied $Bz$ in MagLIF leads to MRT structures that are helical in nature. These structures are fully 3D and cannot be reproduced with an axisymmetric code. Finally, it is also expected that flute modes in the $(r,θ)$ plane, seeded by any of the above mechanisms, can grow aggressively during the deceleration phase.¹⁹ This requires 3D simulations to model the complete evolution accurately.

The experiments studied in this paper are restricted to recent configurations that field the most advanced load current and preheat delivery protocols, which both maximize performance on Z and are more predictable and controllable.⁶ A summary of the key parameters is shown in Table I. The main difference between the experiments is the increase in preheat energy for high energy preheat (HEP) relative to integration 18 (Int18), otherwise the other inputs are nominally the same.

The peak load current (⁠ $Imax$⁠) has been increased for a given target by reducing the inner MITL (magnetically insulated transmission line) inductance, which also reduces current losses in the final transmission line. The impact is such that the standard 85 kV charge voltage for Z can deliver $∼$20 MA to a standard 10 mm tall, AR6 liner, compared to $∼$16 MA on the previous, higher inductance, final transmission line.⁶ 

For the laser coupling, phase plate smoothing is used along with two different spot sizes: (Int18) A higher intensity $∼7×1013$ W/cm², 1.1 mm diameter spot that tends to propagate deeper into the target and (HEP) a cryogenically (cryo)-cooled higher-energy, lower intensity $∼4×1013$ W/cm², 1.5 mm diameter spot configuration. The propagation depth of the smaller spot limits the coupling to the fuel for the 10 mm tall, 1.05 mg/cc fuel cavity, which has been demonstrated in experiments^7,26 and simulations.¹⁷ The HEP platform substantially increases the coupled energy to the fuel by reducing loss to the window (the window is 3  $×$ thinner) and increasing the spot size to heat a larger fuel volume, stopping the laser within the target height. The laser preheat values quoted in Table I are determined from surrogate laser-only experiments with ZBL; there is no direct diagnostic of the laser coupling in fusion-producing experiments. HYDRA simulations of laser-only experiments reproduce the total coupled energy to 10%⁷ and show good agreement with the energy deposition as a function of height.

The remainder of this paper is organized as follows: Sec. II provides a brief overview of the 3D HYDRA simulation model; Sec. III discusses application of the 3D model to AR6 liners with varying preheat (0.9–2.7 kJ) and comparison to experiments; Sec. IV gives extrapolation of the 3D model to different applied magnetic fields which have yet to be fielded on Z; Sec. V covers modeling of the helical instability with the Hall term in HYDRA; and Sec. VI provides conclusions.

Here, we document key inputs and model selections employed by the simulations of the experiments described in Table I. To the simulations, the Int18 and HEP configurations are essentially identical with the exception of the preheat configuration. A laser ray-trace approach models the laser heating and incorporates the measured Z-Beamlet power history and intensity profile produced from distributed phase plate smoothing (approximately super-Gaussian) to capture the time-dependent propagation and spatial uniformity of the preheat.

As shown in Fig. 1, the full target geometry is modeled, which includes extra volume to account for the laser propagation and end-losses, and the radial boundary of the simulations extends to $r=8$ mm to approximate the return can dimensions. The materials simulated (electrodes, laser entrance window, liner, gas) are initialized at ambient conditions (only the fuel is gaseous, initially solid materials begin at solid density) with the exception of the “vacuum” section, which begins at $ρ=10−5$ g/cc. The physics models (inverse Bremsstrahlung laser ray-trace absorption, multi-group implicit Monte Carlo photonics, anisotropic thermal conduction) and the meshing and mesh relaxation strategies are the same for all runs. HYDRA is run in arbitrary-Lagrangian–Eulerian (ALE) mode, allowing the mesh to move with the implosion. The initial resolution in the azimuthal and axial directions is $3°$ and $∼20 μ$m, respectively. The radial resolution varies but is initially $4 μ$m at the outer liner surface. The ALE nature of the code allows the radial (and azimuthal) mesh lines to generally follow the implosion.

HYDRA includes a full Braginskii extended Ohm's law formulation (see Refs. 27 and 28 for details on the physical equations) with fits for the transport coefficients based upon the original work of Epperlein and Haines,²⁹ updated by Sadler et al.³⁰ A magnetized plasma viscosity model is not available but may be important at stagnation as in un-magnetized spherical capsule implosions.³¹ This importantly includes advection of the magnetic field due to temperature gradients (Nernst term, $β∧$ coefficient) that can cause de-magnetization of the fuel. The cross-Nernst term (⁠ $β⊥$⁠) has also been shown to contribute to twisting of field lines in magnetized laser-heated gases.³² Simulations were run with all terms active, except Hall (this also zeroes the collisional Hall term, which is proportional to the $α∧$ in the resistivity tensor), which we discuss separately in Sec. V. Tabular equations of state and conductivities come from the LEOS and SESAME libraries.^33,34

The implosions are driven with a specified drive current derived from a combination of circuit-driven 2D simulations and velocimetry measurements from the experiments that reproduce the experimental stagnation times to within a few nanoseconds (that is, a few percent of the entire duration of a MagLIF implosion). Better agreement is difficult to achieve due to uncertainty in determining the post-peak current from velocimetry, and the level of instability development can alter the stagnation timing by $0.5−1.5$ ns. Generally, the instabilities lead to earlier stagnation, whether by truncating the pulse or compressing portions of the fuel earlier. Determination of the stagnation time comes from the peak of the neutron pulse in simulations and the peak of the x-ray pulse (measured from x-ray diodes) from experiments. Simulations suggest that the neutron and x-ray pulses are co-timed, although the width of the x-ray pulse depends upon the spectrum, filtration, and response of the diagnostic.

MagLIF implosions show helical MRTI perturbations. In these 3D simulations, we seed small-scale perturbations as was first proposed in Ref. 35 with the Eulerian GORGON code. In particular, the perturbation seeds are imposed by a combination of white “noise” (from a uniform distribution) and discrete (10) helices launched at a 10  $°$ angle. The angle is selected based on matching to the available experimental radiography data. The individual helices' launch from the base of the liner, with the initial azimuth randomly distributed. The seed specification is labeled as HXNY, where X corresponds to the helical groove amplitude and Y corresponds to the white noise amplitude (both in micrometers), e.g., H5N2.5 uses 5 μm helices and 2.5 μm (1.44  $μ$m RMS) white noise. For the majority of runs shown in this report, the “random” and helical pattern applied to the simulations is identical (e.g., identical random numbers) across simulations to allow consistent comparison across scans.

When following these prescriptions, we achieve similar agreement with radiography data as shown in Fig. 2, although we note that the simulations in this work are driven with a higher load current than the experimental images highlighted in the inset blue box in Fig. 2.

We observe reasonable agreement in the overall amplitude of the MRTI, although the helical modes appear more discrete in experiments. This seed could likely be further refined to better match these particular experiments, but the lack of data at other conditions (applied field, load current) makes this iterative approach to 3D simulations impractical to confidently extrapolate. We also acknowledge that the helical MRTI modes may change with different applied fields and load currents, but we lack experimental data to motivate any changes to the helical seed. To overcome this limitation, improved physical models suggest that low-density plasma undergoing a Hall interchange instability^36,37 or axial field flux compression³⁸ can lead to the physical development of the helices. Section V presents the first results from HYDRA with the Hall term that is able to generate helical modes without the need for specifying helices.

Notably, the RMS amplitudes used in these simulations to roughly match MRT growth are significantly larger than the typical $∼50−100$ nm surface finish. One reason could be under-resolving the electrothermal instability phase, which underestimates the impact of electric current enhancement²¹ and Joule heating on rippled surfaces or more isolated defects in the target. In particular, the simulations lack randomly distributed voids and high-density inclusions in the beryllium (often referred to as resistive inclusions²¹). Modern-day target characterization techniques identify these features as ranging from a few micrometers to >100  $μ$m, where the number of imperfections is inversely related to the size of the feature, although more statistical characterization is needed for assessing a typical liner. Nonetheless, recent scans show it is not uncommon to find $∼5−10$ inclusions on the order of 50–100  $μ$m. In some sense, the larger perturbations in these calculations account for these features in an ad hoc way. The random presence of larger scale mass inclusions could also contribute to the target variability and increase overall “effective” RMS surface uniformity. Future work intends to statistically include some of these features in 3D calculations to assess their impact relative to surface perturbations. Similar features are also known to be problematic in laser-driven capsules and have recently been examined as a source of yield variability at the mega-joule level.³⁹ 

Finally, it is important to note that while the 2D and 3D simulations presented include many important physical processes relevant to MagLIF, they do not include atomic mixing, a mix model or attempt to model turbulent mixing of the liner material into the fuel. In 3D simulations, the deceleration instabilities in these simulations degrade the yield primarily through conduction loss due to the larger surface area and direct disruption of the hotspot (e.g., bifurcation by a spike). These simulations otherwise maintain a continuous fuel liner interface with “mixed” zones confined near the interface and no mixed fuel/liner zones in the core of the fuel to radiate. Similarly, laser preheat interactions, such as the blast wave or direct interaction with the liner only, suggest possible sources of mix but subsequent mixing is not modeled in detail. One exception that has been observed in simulations⁴⁰ is the injection of laser entrance window mix, which behaves as a chunk mix. In this case, the window material absorbs energy from the hotspot, primarily through conduction, and re-radiates it, acting as a heat sink. However, in these simulations, the laser preheat has been tailored to avoid this scenario. Higher resolution simulations that capture secondary instabilities or different numerical approaches may lend more insight on the topic of mix in MagLIF. Importantly, there is evidence of the mix of liner material in MagLIF experiments, but the amount of mix and spatial distribution is not well-constrained.^41,42 With respect to the simulations presented, this suggests simulations that over-predict the fusion yield may not be unreasonable if we assume an additional degradation due to un-modeled mix.

Figure 3 plots the experimental and simulated (a) neutron yield, (b) fuel temperature, (c) fuel pressure, and (d) Lawson ignition parameter $χ$ as a function of deposited laser (preheat) energy in the fuel. Points below 1500 J are from Int18, and points above 2000 J are from HEP. The delivered laser energy in the integrated experiment is used to infer the deposited energy based on from surrogate laser only experiments.²⁶ The simulated values are directly computed in the simulation. In Eq. (1), the “burn width” $τbw$ is defined as the average full-width-half-maximum (FWHM) of the x-ray diode signals in experiments and FWHM of the neutron pulse in simulations (found to be virtually identical to synthetic x-ray pulses). The 3D simulations produce x-ray and neutron burn widths of $τ=1.29±0.043$ ns (the 1.1 and 2.7 kJ cases produce the lower and upper end of the widths, respectively), and the “clean” 2D simulations, that is simulations without interfacial MRT instabilities and no mix, produce $τ≈1.9$ ns. The two experimental configurations produce x-ray pulse widths of $∼1.75±0.3$ (Int18) and $2.15±0.06$ ns (HEP). The longer experimental x-ray pulse width is possibly related to mix or stagnation of the liner material rather than fusion production. To resolve this, nuclear burn or reaction histories are desired but will require the addition of a significant fraction of tritium to the fuel.

For fixed preheat, the yield increase from high amplitude to low amplitude seed is $∼35$% for Int18 (1.4 kJ) and $∼78$% for HEP. Initial repeat simulations of HEP with different numerical random number seeds for the noise background (same RMS radial perturbation) show $∼15$% yield variation with $∼5$% variation in pressure and temperature, suggesting that a change in effective RMS amplitude is needed to significantly impact the yield and associated plasma conditions (for details see  Appendix A). Ideally, this would be checked across different inputs (preheat, $Bz$⁠, current) but this is outside the scope of this effort.

The yield dependence on preheat depends on the seed amplitude. For the high amplitude seed (H5N5), there is essentially no effect on yield past the 1.4 kJ simulation, but there are hints of a cliff as the preheat energy drops below 1.0 kJ. This is likely forcing the implosion to higher convergence and exacerbating instability growth to compensate for the poor preheat. The lower amplitude seed (H5N2.5) shows the expected 2D yield increase from 1.4 to 2.1 kJ (⁠ $∼30$%); however, there is no noticeable increase from 2.1 to 2.7 kJ (expected to be $∼25$% from clean 2D simulations). Additional simulations could elucidate whether the yield is more sensitive to the seed amplitude as the preheat is increased. The remaining fuel parameters from 3D simulations also qualitatively follow the expected 2D trends with preheat; e.g., $Tion$ increases and fuel density decreases with preheat energy, although again, their absolute values are significantly degraded relative to the ideal 2D values. Consequently, the “3D” values of $χ$ are significantly smaller.

Overall, the 3D yields are factors of $5−10×$ below clean 2D simulations, which is primarily a result of the liner instabilities in these calculations. We have verified that 3D simulations run without outer surface perturbations approach the unseeded 2D conditions but, importantly, require cylindrically symmetric preheat deposition. 3D simulations run without outer surface perturbations but including laser asymmetry from filamentation and off-axis propagation show yield degradation of $∼25$%, $∼0.4$ ns shorter burn width, and $∼10$% lower stagnation plasma parameters compared to clean 2D simulations (second row of Table II). In this case, the laser deposition leads to inner surface perturbations in the liner (that grow rapidly during deceleration) and azimuthal flow in the fuel (amplified by conservation of angular momentum) that lead to a disrupted, and incompletely stagnated, hotspot. Repeating this case, but utilizing an idealized source deposition mostly recovers the clean 2D performance (third row of Table II). Finally, comparing the laser and idealized source deposition method when including the H5N2.5 MRT perturbation, the idealized source produced $∼15$% yield increase and $10$% increase in stagnation plasma conditions, although the burn width was unaffected. This suggests deceleration instabilities, whether from MRT feedthrough or preheat seeding, are primarily responsible for limiting the burn width in these calculations. We note that for H5N2.5 HEP (2.1 kJ at 1.5 mm) with cylindrically symmetric deposition, no change in yield is found, and plasma conditions show a reduction (increase) in $Tion$ (P) by 10%. The simulation shows that this is attributable to more $Bz$ flux loss in the uniform preheat case, leading to more cooling of the fuel.

In addition to some of the reasons outlined above, it may also be possible for the simulations to produce a higher burn-averaged temperature and pressure, while maintaining a similar fusion yield, if the implosion is more unstable such that it is dominated by smaller hot spots, rather than a larger volume, cooler stagnation. This scenario suggests an issue with the seeding method or MRT development in the simulations. If the Nernst effect is overestimated in the simulations (e.g., more $Bz$ is retained in the fuel), this could also raise the fuel temperature; however, without Nernst advection, simulations actually find the fuel pressure to be lower due to the substantial increase in magnetic pressure (and contribution to the total pressure) stagnating the liner.

Although it is clear MRT plays a significant role in degrading the fusion yield, explaining the discrepancies in stagnation conditions between simulations and experiments will require additional efforts. Two experimental priorities in this effort are (1) the development of tritium capabilities at the Z-facility, which will enable burn and reaction history measurements to refine the burn width, and (2) spectroscopy and imaging of krypton dopants in the fuel to assess the fuel volume (from orthogonal imaging), temperature (to cross-check with $Tion$⁠),^43,44 and potentially, pressure.⁴¹ We are also exploring performing the Bayesian inference on the HYDRA simulation itself to understand systematic issues as well as alternative analysis methods, particularly with respect to the fuel volume, which we discuss next.

In this section, we examine the fuel convergence and volume from experiments and 3D simulations using x-ray emission images as a rough surrogate for neutron emission. Understanding these parameters is important for determining the convergence of the target, compression of the magnetic field, and trapping of fusion products. The primary diagnostic is a time-integrated crystal imager that captures primarily $∼6$ and $9$ keV emission and produces $∼20 μ$m resolution 2D images⁴⁵ of the full height of the stagnation. Figure 4 displays the experimental images from Int18 and HEP along with synthetic x-ray images from the simulations (generated with the Spect3D code⁴⁶).

The experiments show obvious differences in morphology, which we attribute to variable MRT development rather than the details of the preheat. In the experiments, although the intended change is the difference in preheat energy, the MRT seed is not controlled, and the resulting experimental images are most consistent with a change in MRT development. This is emphasized by the simulation images, which compare emission from fixed MRT seed amplitude and varied preheat, and fixed preheat and varied MRT seed amplitude. The resulting images show highly reproducible morphology for the change in preheat and obviously different structure for the change in seed amplitude. Although there is ongoing work to quantify “stability” from these images, the remainder of this section focuses on estimating the fuel compression.

To determine the fuel radius from experiments, two methods are employed: (1) the “Bayesian” method employs a hotspot model for the emission width using a down-sampled version of the emission image (details in Knapp et al.⁴²) and (2) a “contour” method follows emission contours using the complete image. The simulated fuel radius is determined from the contour tracing method applied to synthetic x-ray emission using the same 10% contour (this is above the minimal threshold set by the noise in the experimental image). The fuel volume can be estimated from the total emission height and effective radius, which can then be compared to the “true” simulated fuel volume.

The results of the analysis are shown in Fig. 5. For each simulation, additional analysis of an orthogonal view is plotted and shows small differences in the average radius. Although not available at the time of these experiments, orthogonal views have been fielded in subsequent experiments and used to better constrain the fuel volume.⁴⁷ There are two major observations with respect to the calculated fuel radius from the x-ray emission. First, the Bayesian versions produce significantly smaller radii with $CR∼50−60$⁠, while the contour method produces radii with $CR∼30$⁠, which is in good agreement with the simulations analyzed with this method. Second, given the large error bars (a measure of the axial variation), there is no obvious change in convergence as the preheat energy increases, although the average radius increases with preheat. Analysis of the simulations with the contour method also shows a similarly weak trend in the average fuel radius as a function of preheat energy. We also find that the higher amplitude seed, for fixed preheat, produces slightly larger average fuel radii, although the variation is similar, so the metric does not strongly quantify stability.

Finally, Fig. 6 compares the simulated x-ray and neutron volumes. The x-ray volume uses the contour method above and is plotted for two different intensity thresholds using the average radius and $∼8$ mm height: 10% contour (purple “X”) used above and a tighter 25% (yellow “X”) that best matched the bulk neutron volume. The neutron volume is directly computed from the simulation data, where the squares, diamonds, and circles correspond to the simulated fuel volumes producing $90%,95%,100%$ of the total neutron yield.

First, we find that a large fraction of fuel does not produce significant neutron yield, which is also generally true in 2D simulations. This is fuel on the order of the radiation temperature (⁠ $∼200$ eV), near the liner wall, that neither produces neutrons nor x-rays that can escape the surrounding liner $ρR$⁠. On average, the results show that 95% (diamonds) of the fusion yield comes from roughly 60%–70% of the total fuel volume. We also find the fuel volume increases with preheat and is generally larger for larger amplitude seeds. The neutron fuel volume increases for two reasons: (1) the volume is physically larger due to lower convergence, and (2) a larger fraction of the fuel volume is neutron-producing. The fraction of neutron producing fuel tends to increase with both preheat energy and stability; however, even the most extreme cases (e.g., high preheat, low amplitude seed vs low preheat, high amplitude seed) produce fractional volumes within $∼5%−10%$⁠. The 100% contour demonstrates most clearly that the CR is decreasing with preheat.

Despite the approximations in the x-ray image analysis, the contour volumes generally produce sensible values and both 10% and 25% thresholds show trends consistent with the neutron volumes as the preheat energy increases. We expect the emitting volume to be less than the total fuel volume (due to “cold” fuel); however, the 10% threshold (purple) clearly produces an upper bound as it closely matches the total simulated fuel volume. Since the fuel is not generally cylindrically symmetric, it is not surprising that the volume could be overestimated due to the singular view. The fuel volume could also be computed from individual axial slices along the emission height. Various models of the hotspot shape could also be constructed to calculate the fuel pressure, similar to the Bayesian process.

The analysis in this paper differs in some important ways. First, the experiments considered in this work are quite different than the past experiments, given the higher peak current (20 vs 16 MA), higher initial field (15 vs 10 T), higher fuel density (1.05 vs 0.68 mg/cm³), as well as the laser preheat configuration, which alters the preheated volume and thermal gradients in the fuel. Second, this study is conducted with more complex simulations than have been used in analysis before that include 3D effects and laser deposition physics that more accurately capture the preheat phase. Figures 5 and 6 show that fuel convergence changes with preheat affecting the strength of the compressed $Bz$⁠. Furthermore, thermal filamentation of the beam, which is stronger for the larger spot size and higher density, can also disrupt the Nernst mechanism¹⁷ compared to completely uniform deposition of energy. According to the simulations this is due to more randomly oriented and smaller scale thermal gradients in hot spots that form within the laser heated plasma, which disrupts direct radial transport of flux due to the Nernst term.

In experiments, the compressed magnetic field within the fuel can be determined from the magnetic field-radius product (BR), which in turn is determined from the secondary DT yield and the nTOF spectra.^48–50 Unfortunately, the nTOF data were poor for these experiments, but we can query the burn-averaged fuel magnetization (⁠ $xe$⁠) from the simulations to see how insulation properties vary with preheat. The result is plotted in Fig. 7 for the Int18 and HEP simulations.

With the higher intensity beam in Int18, there is a weak decrease in magnetization as a function of preheat, from 1 to 1.5 kJ due to a reduction in field strength within the fuel. This is due to a combination of Nernst advection and reduction in CR (⁠ $Bz∼CR2$⁠). Here the laser beam is intense enough to avoid some of the filamentation effects so Nernst maybe more important. As the liner stability improves (from H5N5 to H5N2.5), the liner is able to converge more, which increases $Bz$ and $xe$⁠. $xe$ also increases from the 1.4 kJ 1.1 mm (Int18) configuration to the 2.1 kJ 1.5 mm (HEP) configuration. Despite the increase in preheat energy and small reduction in convergence (according to Fig. 5), both configurations reach similar magnetic field strengths of 6 to 7 kT. Because the fuel temperature is also significantly higher for HEP, the fuel magnetization is also higher. To highlight the impact of the laser, when uniform deposition of energy is used for H5N2.5 HEP, the magnetic field drops to 6 kT and $xe∼43$⁠, similar Int18. Notably, the stagnation temperature is also lower $2.4$ keV, despite no change in yield, which is further from the experimental result. Finally, no change in $xe$ is observed between 2.1 and 2.7 kJ simulations due to a reduction in $Bz$⁠, despite the increased $Tion$ and decreased density. The reduction in magnetic field strength for the 2.7 kJ simulation is most likely due to a combination of reduced convergence and compression of the magnetic field and loss from the Nernst effect.

Although the simulations predict significant changes in thermal insulation, $Bzrfuel$ is expected to stay approximately constant. Using the radii from Fig. 5 and assuming similar 6–7 kT $Bz$⁠, as found in the simulations, BR is estimated to be $<0.55$ MG $·$ cm (⁠ $rf/rα<2$⁠), which would be the largest values attained in MagLIF. However, even if the smaller Bayesian radius values are used, $rf/rα>1$ when assuming the same $Bz$ field strength. Overall, these results suggest the current preheat configurations are robust to de-magnetization of the fuel from higher preheat.

In this section, we examine the performance of seeded 3D simulations as a function of applied $Bz$ strength up to 30 T. Such a scan has not been performed on Z as coils are not currently able to reach 30 T so the results here are purely computational. The motivation for increased fields is the enhanced thermal insulation provided at stagnation, higher fuel temperature, and, more fundamentally, to provide insight into magnetized transport models and magnetic flux loss from the fuel via BR inference.^48,50 In future ignition and “high yield” designs, fuel magnetization will also strongly influence how $α$ particles are stopped in the fuel. We use the Int18 simulation model (H5N2.5) and perform a magnetic field scan to supplement the preheat study from Sec. III. The smaller spot was selected to minimize the impact of the laser deposition scheme, but we plan to investigate $Bz$ scaling for the HEP configuration in the future. We also include a few results from clean 2D and 3D simulations. As a point of clarification, a clean calculation is one without seeded outer surface MRT instabilities or mix. In the clean 3D simulations, deceleration phase instabilities—primarily flute modes in the $(r,θ)$ plane—may still develop due to preheating as discussed in Sec. III. This is an important distinction, as these modes are not present in a 2D axisymmetric code.

Figure 8 plots the neutron yield, ion temperature, fuel (thermal) pressure, and fuel total pressure (thermal + magnetic) for the scan. Compared to the 2D clean simulations, the seeded 3D simulations produce significantly lower yields (⁠ $∼5−10×$⁠), temperatures (⁠ $∼1.5×$⁠), and pressures (⁠ $∼2×$⁠), although the degradation is least severe at lower initial fields where thermal insulation is more marginal. The 2D simulations also show continued yield increase up to 20 T, while the seeded 3D simulations show small decreases in yield past 10 T.

The clean 3D simulations perform close to, but still below, the clean 2D simulations. At the time of this writing the clean 3D 30 T simulation result not available, but we do not believe it alters the general conclusions of the work. As stated earlier, without MRT, the remaining degradation involves perturbations introduced by the laser preheat phase, which are generated by the feedback between thermal insulation (provided by $Bz$⁠) and thermal filamentation¹⁷ and off-axis propagation, so it is not surprising disagreement between clean 2D and clean 3D generally is slightly worse for higher field strengths. The resulting perturbations impart azimuthal motion to the fuel and produce an asymmetric blast wave leading to seeding of flute mode inner surface instabilities, which grow aggressively during deceleration. This also includes Kelvin-Helmholtz instabilities due to the combination of azimuthal flow and shear near the fuel/liner interface. In simulations, these issues are addressed by utilizing a uniform energy source deposition (not affected by $Bz$⁠) or a smoother, higher frequency laser. Simulations may also overestimate these effects due to the statistical ray-based approach and the lack of viscosity that could dissipate the azimuthal kinetic energy. Nonetheless, in reality, the resultant effects must also compete with the effects of MRT feedthrough, so constraining their impact is a significant challenge.

With respect to the stagnation conditions, the seeded simulations show that the ion temperature is lower and increases more slowly as a function of $Bz$⁠. The fuel pressure is lower overall compared to the clean calculations and generally decreases with $Bz$⁠. However, both quantities qualitatively follow the trends from the clean simulations. The increasing $Tion$ and decreasing p with $Bz$ imply the fuel density decreases with $Bz$⁠, with burn-averaged values ranging from $0.75$ g/cm³ at 5 T to $0.34$ g/cm³ at 30 T. The reduction in fuel density and pressure with $Bz$ is consistent with the additional finding that the total fuel volume increases as a function of $Bz$ (not shown). This suggests that as the applied $Bz$ increases, the overall compression of the fuel (and associated PdV work) is reduced due to the additional work required to compress the applied field (up to $∼10$ kT for the initial 30 T seed field). This is further illustrated in Fig. 8(d), which plots the burn-averaged thermal pressure and total pressure (⁠ $Pth+Pmag$⁠) as a function of initial $Bz$ for the 3D simulations. The total pressure in the fuel remains approximately constant with $Bz$⁠, but the total pressure is increasingly supported by the magnetic pressure as the applied field is increased. For $Bz>25$ T, the plasma beta $β$ approaches unity.

As noted in Sec. III A, removing the Nernst term tends to increase the ion temperature at the expense of the fuel pressure due to the presence of more axial field in the fuel (and higher $Pmag$⁠). As the applied field increases, Nernst becomes increasingly less important due to high $xe=ωceτei$ [see Fig. 9(a) below], keeping more field in the fuel, which may help to explain the divergence between the pressure contributions as the initial field increases. Finally, the simulations show that as the applied field is increased, the contribution of magnetic pressure to the total fuel pressure increases. Thus, from the standpoint of the Bayesian inference the total pressure may not be an accurate representation of the fuel thermal pressure for higher applied field strength.

As can be surmised from the conditions presented in Fig. 8, the increase in applied field produces increasingly magnetized fuel conditions with electron Hall parameters ranging from $xe∼$ 4 at 5 T to $xe∼120$ at 30 T, as shown in Fig. 9(a). For $xe=4$⁠, which is similar to the preheat phase, the reduction in perpendicular thermal conductivity is still significant (⁠ $κ⊥/κ∥≈0.05$⁠), but weaker magnetization is more susceptible to the Nernst effect, whose coefficient, $β∧$⁠, will be close to maximum (see Epperlein-Haines²⁹ and Sadler et al.³⁰ for plots). For the high field cases, where electron conduction is negligible, ion conduction will also be important as the ion Hall parameter $xi∼2$⁠.

Interestingly, we also find the clean 3D simulations produce the highest $ωτ$⁠, which is also a by-product of the laser coupling and the Nernst effect. In an ultra-smooth beam, the temperature gradient (and subsequent Nernst advection) is purely radial, while for a filamented beam, the thermal gradients become isotropized in (⁠ $r,θ$⁠) and can trap flux in between hotspots. This keeps more field in the initial spot area, and then azimuthal motion keeps the field stirred throughout the volume during compression due to frozen-in advection. Despite the significant magnetization of the fuel, the seeded 3D simulations only show a weak increase in the burn width with $Bz$⁠, and the clean 3D simulations also do not reproduce the 2D burn width as plotted in Fig. 9(b).

Clean 2D simulations show that thermal insulation will increase the burn width as the liner stagnates, whereas in the perturbed case, the neutron pulse ends due to disruption before thermal insulation is lost. In particular, the clean 3D simulations show that deceleration instabilities will also limit the burn time, which is an important consideration for high yield ice-burning targets that must be confined long enough for the burn to reach the intact ice layer.

According to these 3D HYDRA simulations, fielding higher $Bz$ in experiments at 20 MA appears unlikely to improve overall fusion performance as MRTI limits the benefits of increased thermal insulation. However, the plasma conditions are still significantly modified, such as the achieved fuel pressure, density, as well as the magnetic transport properties. Similarly, with the level of instabilities studied, the optimal $Bz$ is in the 10–15 T range for 3D compared to $∼20$ T for 2D. The yield is limited at large $Bz$ by the significant contribution of the magnetic pressure to the total stagnation pressure. Targets with higher preheat and initial fuel densities should make better use of larger initial fields. For example, 2D simulations of HEP show that the optimal field could be as high as 25 T, although 3D instabilities may still moderate the improvement. To leverage any future improvements to capabilities in the applied $Bz$⁠, this numerical study suggests that a priority of the MagLIF program is to improve the stability of the platform.

Experiments by Ampleford et al.,^6,51 utilizing a dielectric-coated liner (although lower peak current of 15 MA), show that there are accessible experimental regimes where the applied field does impact performance with more stable implosions. The experiments showed an 80% increase in primary neutron yield and a 30% increase in ion temperature by increasing the initial $Bz$ from 10 to 15 T. Unfortunately, these experiments are among the more challenging to model as they utilized the higher inductance power feed and the unconditioned laser spot. Future work should re-examine the 3D scaling shown here for other preheat configurations and more stable platforms. Finally, secondary D-T reactions, induced from the trapping of tritons by the compressed $Bz$⁠, are not computed in these simulations; however, it is clear that the increased field could increase the D-T yield as a function of $Bz$⁠.

Sections III and IV have leveraged a simple helical pattern to understand 3D MagLIF implosions but asserted the seed is constant. Although we discussed the issues with using a constant “random” noise seed, the helices could also change depending on the circumstances generating them. To summarize, there are a number of issues with the current seeding methodology: (1) obvious differences in the morphology of the x-ray emission, (2) uncertainty in how the helical modes change with applied $Bz$ and drive current, and (3) the overall number of free parameters to specify the seeding (number of helices, width, and depth). To attempt to address some of these issues, in this final section, we show the first 3D HYDRA calculations with the Hall term included, which has been shown in other codes to naturally generate helical instabilities from noise in the presence of low-density plasmas (LDP).^36–38 Running the Hall term in HYDRA then directly enables assessment of the helical instability (generated from white noise) on integrated performance, although with some limitations we discuss next.

HYDRA currently uses an explicit formulation with a time step based on the whistler wave speed that introduces some practical limitations for Hall runs with respect to cell sizes and the minimum density (or so-called density floor) due to the decrease in time step size. More specifically, the primary effects are (1) the time step decreases quadratically with cell size, rather than linearly, and (2) the Hall velocity may be significantly faster than the Alfvén velocity (⁠ $Va∼1/ρ$⁠). Nonetheless, the prospect of self-consistently modeling the helical instability in MagLIF within a fully developed design code is enticing, so for this reason, the following runs have attempted to stay as close to typical resistive MHD settings as possible, allowing for the fact they may not be optimal for modeling Hall effects.

The typical floor settings for 2D and 3D MagLIF simulations (as used in Secs. III and IV) use a minimum density (dfloor) of 1.0 × 10⁻⁵ g/cc with a 4× multiplier such that 4 × 10⁻⁵ g/cc is essentially the minimum density that is affected by MHD forces. Additionally, the vacuum is set to the minimum density. The combination of these settings leads to a generally benign vacuum region, which also maximizes the MHD time step. As the multiplier is reduced to 1.0, more of the vacuum may participate in the dynamics of the simulation, and the time step generally decreases as lower density material contributes to the evolution of the simulations. In particular, for a 1× multiplier, the full vacuum is subject to MHD forces and the resistivity is no longer forced to the maximum value. If the material model does not return a fairly high resistivity at ambient temperature for the vacuum, then it will participate in the implosion almost instantly and divert current from the liner. Additionally, the 1× treatment will continually replenish mass in the “vacuum” as it advects out of the cell. For this study, simulations used the Integration 18 setup (20 MA drive, $∼1−1.4 kJ$ preheat), with only a white noise background (2.5  $μ$m).

The applied magnetic field was tested at values of 0, 10, and 20 T to demonstrate the impact of the Hall term on the MRT. All simulations used a density floor $=10−5$ g/cc, which is equal to the initial vacuum density, a floor multiplier of 1.1 (i.e., the initial vacuum density material is below the MHD floor), and the vacuum material as beryllium, which is resistive as a vacuum material in the table. Synthetic 6.151 keV radiographs of the implosion at $CR∼10$ are shown in Fig. 10.

The 0 T case shows no sign of helical modes rather displaying the expected azimuthal correlation. Both 10 and 20 T runs produce the expected development of helical modes from the white noise, whereas the 20 T run without Hall [Fig. 10(d)] shows no helical modes, less azimuthal correlation, and shorter wavelengths than the 0 T Hall simulation. From 10 to 20 T, the pitch angle of the helices does not look obviously different and the modes are not as clear as observed in the original radiography data shown in Fig. 2 which we aim to explore in the future. We can also see the implosion stability varies between the magnetized and unmagnetized cases, with the 0 T case showing the most disrupted inner surface. This is consistent with reduced feedthrough from helical modes or a stabilizing presence of the compressed $Bz$⁠.⁵² Overall, these results demonstrate the applied magnetic field is necessary to produce the helical modes which was the key observation made in 2013.⁵³ Additional comparisons with and without the Hall term are shown in  Appendix B.

The helical modes, in these configurations, are generated by the slow accumulation of LDP around the liner that first forms near the liner outer surface as the liner melts and ablates, rather than being sourced from the significantly larger return can radius (or even in the upstream transmission line), which precludes significant compression of the applied field. The material density (log scale) and $Bz$ contours at 7, 15, and 20 MA for a portion of a slice through a 3D simulation (⁠ $Bz,0=20 T$⁠) are shown in Figs. 11(a)–11(c) to illustrate this evolution.

At early times [Fig. 11(a)], LDP is localized to the liner surface, and the axial field (and azimuthal field—not shown) shows large perturbations along the height of the liner that produce localized fields roughly twice the initial strength (⁠ $20 T$⁠), which induce current perturbations consistent with observations of the initial phase of the Hall interchange instability.³⁷ As the liner begins to implode [Fig. 11(b)], more tenuous LDP has coalesced around the liner, and the perturbations in the axial field have combined into larger structures, peaking at $∼$ 60 T. At this point, there may be some flux compression component in amplifying the field as a result of LDP motion, but the peak fields are significantly lower than many hundreds of Teslas quoted in other work that sources plasma from large radius.³⁸ At peak current [Fig. 11(c)], the MRT is fully developed in the liner with a surrounding LDP halo. Synthetic 6.15 keV radiography of the full height of the liner [Fig. 11(d)] shows that the MRT is primarily helical, and the low-density halo is not visible. Finally, Figs. 11(e) and 11(f) plot a full 3D rendering of the beryllium mass density at peak current panel (e) and near stagnation panel (f). The LDP is removed from the foreground portion of the image (⁠ $ρ<0.02$ g/cc) to clearly show the helical MRT in the dense metal, which is visible in the radiography diagnostic. The helical bubbles and spikes act as x-ray filters modulating emission from the fuel, which is observable to some extent in the x-ray images shown below.

Despite the invisibility of the halo to radiography, the presence of this plasma and subsequent coupling with the Hall term is critical in HYDRA, as Hall runs with a typical floor multiplier of 4× disable the LDP evolution too effectively and do not produce helical modes with similar results to “no-Hall” simulations. Additionally, simulations without Hall do not show such significant amounts of plasma at large radius. This could be a result of a force-free configuration or simply a stronger interaction with the low-density material. The plasma density in these simulations drops as a function of radius and ranges from $∼0.5×1019$ to 2 × 10¹⁹ /cc, which is in good agreement with PIC calculations for expected plasma densities near the load.^54,55 The plasma is also hot (⁠ $∼1−2$ keV), conductive, and well magnetized, leading to some drive current carried in the plasma past peak current. In fusion-producing calculations, this delays the bang time by a few ns as shown below but does not clearly impact the yield. With respect to the current loss, these calculations do not include any form of anomalous resistivity or collision frequency modifications (such as Bohm) proposed in other work,^55–58 which could alter the current distribution. Finally, we note the LDP is very dynamic with significant axial and azimuthal velocities, on the order of hundreds of km/s, which is another 3D feature. Unfortunately, this LDP simulation model is not well constrained by experimental data outside of radiography, but optical imaging with shadowgraphy, Thomson-scattering, as well as soft x-ray emission offer potential diagnostic methods to address this.

To conclude this work, we report on the fusion performance from the first fully integrated 3D HYDRA simulations with the Hall term and self-generated helical modes. The simulations utilize the Integration 18 configuration (1.4 kJ, 20 MA) run with 20 T, and two different white noise amplitudes: 1.25 and 2.5  $μ$m. Although we are working on an exact A/B comparison, it is reasonable to compare these simulations as (1) without Hall, stagnation conditions are similar between 15 and 20 T (see Fig. 8) and (2) without Hall, the applied field has little effect on MRTI growth and stagnation morphology.

The burn-averaged stagnation conditions for the “Hall” and comparable no-Hall runs are shown in Table III. Generally, the majority of stagnation parameters are similar between the Hall and no-Hall runs, particularly when the fusion yield is comparable. As with the no-Hall simulations from Sec. III, the Hall simulations show significant yield differences for the factor of two change in seed amplitude, and reduction in $Tion$ and P with the larger amplitude seed. More importantly, both Hall simulations show a clear increase in burn width compared to the no-Hall simulations, which is in better agreement with the experimental x-ray widths. However, this also means that the no-Hall simulations tended to produce roughly 50% higher neutron powers. One possible explanation for the longer burn duration is variability in timing of stagnation as a function of height due to LDP shorting. Qualitatively, the process is similar to “zippering” in gas puff z-pinch implosions.⁵⁹ For otherwise similar conditions, the increased burn width leads to an increase in $χ$ for the Hall simulations, although the experiments still produce larger values of $χ$ while producing lower yields, so the remaining plasma parameter discrepancies remain important to explore.

Comparing the 20 T simulations with only random noise, we observe that the no-Hall run with 2.5  $μ$m seed outperforms the same simulation run with Hall (although the burn width is shorter). There are a few potential reasons for this: (1) Hall enhances the MRT growth rate and correlation of the “noise” modes such that the feedthrough is larger, (2) overall MRT growth is similar, but the helical modes are more damaging, and (3) the superposition of the helical modes and noise modes produces a larger effective perturbation. From these results, it is difficult to conclusively say, whether the helical modes are beneficial (better stability) or damaging (asymmetric ram pressure). Similarly, the improved performance observed for the no-Hall simulation without the prescribed helices could be a result of less overall MRT rather than an indictment of the helical modes. As discussed above, the Hall runs also produce slightly delayed implosions due to some redistribution of current to the LDP; otherwise, the remaining stagnation metrics are fairly similar to the no-Hall runs, suggesting the impact of the instabilities and overall kinematics are quite similar between the runs.

Finally, we compare the impact of Hall and associated helical modes on stagnation morphology. Figures 12(a) and 12(b) plot experimental time-integrated x-ray self-emission (HRCXI) images for Int18; (c) synthetic image for the no-Hall H0N2.5 simulation discussed in Table III, (d) no Hall 15 T Int18 simulation (H5N2.5) and panels (e) and (f) the 20 T, 1.25, and $2.5 μ$m Hall runs. Metrics from the contour tracing analysis show that the average radius for the Hall runs is $≈7−15 μ$m larger than the no-Hall runs; however, the standard deviation is also larger (⁠ $∼20$ vs $13 μ$m) due to the more complex variation in emission. The total height is $∼1$ mm shorter with Hall, but still clearly longer than the experiments. Arguably, the total emitting height, particularly for the 2.5  $μ$m Hall run, is also significantly shorter than the total height of the column from more severe dim regions.

Figures 12(c) and 12(d) mainly compare the effect of removing the prescribed helices, from which it is clear that the column becomes significantly straighter due to lacking the long wavelength kink mode present in Fig. 12(b). The helical modes produced by Hall present more localized kink structures in the stagnation images, resembling those in the experiments panels (a) and (b), rather than the large-scale kink mode produced by the prescribed helices. Future work will seek to address, which simulations are more representative of the experimental images through more detailed quantification of the images.^51,60

In these images, the brightest regions correlate with the neutron emission, so the Hall runs appear to show more concentrated neutron production and larger regions of dudded fuel, while both no-Hall images show similar axial emission length and display high-frequency axial modulation from MRT. The lower yield Hall simulation is dominated by $∼5$ bright spots separated by $∼1$-2 mm spacing. This is similar to the no-Hall runs, although the Hall run shows much more severe dips in emission. The higher yield Hall simulation shows a much longer $∼3$ mm tall region at the top of the target, with a smaller hotspot further down. Despite the obvious differences in structure, images (c) and (e) are both high yield, producing $>1.3×1013$⁠, while $>1.3×1013$ (b), (d), and (f) produce roughly half the yield, highlighting the many different realizations of the target morphology that can perform similarly.

The dim sections of these Hall simulations correspond to disruption of the column by instability feedthrough. This is better illustrated in Fig. 13, which plots slices in the (x,z) plane through both Hall simulations at bang time with an inset plot of the time-integrated self-emission (stretched radially for clarity). The slices plot the fuel temperature, mass density of the liner material, and contours of $Bz/Bθ$⁠. The fuel temperature at this time clearly reasonably represents the emission structure and can be directly correlated with the stability of the liner. Interestingly, the 1.25  $μ$m simulation shows an obvious difference in convergence, and perhaps stability, as a function of height, where the most continuous and brightest portion of the column near the top of the target corresponds to lower convergence and later bang time. The earlier stagnation of the more unstable portion, plus the later stagnation of the more stable portion, thus can lead to an apparent increase in burn width. The simulations produce this effect by virtue of the LDP evolution and associated redistribution of magnetic field.

It is reasonable to assume a similar effect could be at play in the experiments, but it is also possible that dim regions in the experiments could be due to mix of contaminants and cooling. Similarly, the experimental images could include emission contributions from mixed contaminants, rather than emission from neutron producing fuel, which has obvious implications for understanding the achieved fuel conditions. We are pursuing 1D (see Ricketts et al.⁶¹ and Ampleford et al.⁵¹) and eventually 2D neutron imaging, along with the inclusion of krypton dopants⁶² to help to address this issue. Additionally, time-resolved imaging diagnostics could play a critical role in understanding the uniformity of the burn history of the implosion.

We first studied the effect of preheat, and later $Bz$⁠, variations including the impact of MRTI, on fusion performance for standard AR6 liners driven with 20 MA. We found that the seeded 3D simulations, with a reasonable MRTI seed (⁠ $∼2.5$– $5 μ$m), are able to reproduce experimental fusion yields within a factor of two, and the simulated yield is generally most sensitive to the seed amplitude used, rather than the preheat energy. Comparison with the associated plasma conditions shows it is difficult to consistently match both the neutron yield along with the inferred stagnation pressure and temperature; e.g., the simulated pressure and temperature are typically lower for comparable yield; or vice versa, matching the ion temperature produces too much yield compared to experiments. In all cases, the simulated burn widths are also shorter than x-ray diode measurements. Finally, the Bayesian estimate of the fuel convergence (radius) in experiments was significantly higher (smaller) than simulations, which may partially explain the higher inferred fuel pressure compared to the simulated values. We found good agreement between the simulated and experimental fuel volume, outside of the Bayesian framework, using a different analysis technique. These results present two different “hot spots,” (1) the Bayesian results suggest smaller volume, higher pressure and (2) the simulations and more detailed analysis of the experimental images show a larger volume, lower pressure, hotspot. Using the thermonuclear yield equation, the different results are explained by a simple inverse relationship between the pressure and radius and suggest the Bayesian inference overestimates the fuel convergence. More conclusively resolving the differences involves a few potential approaches, including (1) alterations to the Bayesian inference, (2) direct Bayesian inference from the HYDRA simulations, and (3) more generally, constraining the experimental fuel volume.

MRTI is not only the dominant fusion yield degradation mechanism in these calculations but also marginalizes the overall importance of the preheat energy and applied magnetic field strength for enhancing fusion yield in MagLIF. This complicates many of the expected trends from clean 2D simulations. For example, 3D simulations extrapolated to higher preheat show only modest improvements to stagnation conditions with current MRTI growth. Similarly, despite significantly increasing the fuel magnetization and thermal insulation, increasing the applied $Bz$ does not substantially increase the MagLIF burn width (or fusion yield) as expected in 2D simulations. Although the field is necessary to initially reach fusion conditions, MRT disrupts the fuel column before the full benefits of thermal insulation can be realized. Conversely, as the stability of the outer surface and associated feedthrough improves, deceleration instabilities seeded by other means, such as a non-uniform blast wave, may also become more important to the burn history.

Although the MRTI seed used in most 3D simulations seems to generally reproduce experimental conditions, we have begun investigating using the Hall term in HYDRA to better reproduce the helical instability in a more self-consistent and potentially more predictive way. This becomes more important when extrapolating to new configurations such as higher load currents or substantially different applied axial fields. Initial results show that the helical instability can be produced in HYDRA but is sensitive to the floor settings and the amount of low-density plasma produced. Integrated Hall simulations, with self-generated helical instabilities, do not dramatically alter the stagnation performance relative to no Hall simulations, but there appears to be some evidence that the stagnation morphology is sensitive to how the helical modes form and evolve, as well as some minor current redistribution to LDP. Unfortunately, this small subset of calculations still does not resolve whether self-consistently generating these helical modes is actually faithful to the true physics or critical to understanding MagLIF performance and scaling to high yield, but nonetheless lays the foundational model to begin to address these issues. Time-resolved imaging diagnostics, including soft x-rays from the liner and harder x-rays from the fuel, provide one method to address this.

We have shown that MRT is the strongest lever on fusion performance, but we have also neglected some important aspects of the physics of the implosion. In particular, the mix of contaminants into the fuel can also degrade the fusion yield sufficiently to account for some discrepancies in yield (and may increase the x-ray burn width). Previous preheat simulations¹⁷ show that there is the potential for increased mix due to higher preheat from the blast wave and direct laser interaction with the metal. In these scenarios, the higher intensity co-injection pulse could steer off-axis into the cushion or liner, reducing the coupled energy to the fuel (it goes to the metal) and producing a plume of mix. The lower intensity HEP pulse can lead to more diffuse deposition into the liner or cushion from low energy filaments. Although these HYDRA simulations capture these effects, they do not capture the detailed mixing of the metal into the fuel. Differentiating the relative degradation mechanisms still remains a challenge, both experimentally and computationally, that should continue to be explored with anti-mix layers (D2 ice or wetted foam) or buried layers of tracer material. In other words, although the simulations presented here show MRT is a pressing issue, mix should not be discounted or ignored.

The 3D HYDRA simulations presented show that the stability of the MagLIF platform is significantly impacting current MagLIF performance and must be improved in order to make significant progress toward high yield configurations on a reasonably sized driver. 3D simulations play a critical role in assessing this due to the presence of flute modes at stagnation, not present in 2D simulations, that could quench ice burning configurations. To that end, we are pursuing a number of options to improve stability. These include revisiting dielectric coatings compatible with cryogenic, high preheat configurations, which have already shown favorable stability properties at lower currents and preheat;^51,62 graded density liners (a discrete coating is a simple version of this concept), which is a common approach in other areas of ICF; as well as more unique concepts such as the dynamic screw pinch.^65,66 Although we are investigating them separately, the technologies are not mutually exclusive and could be combined. We are also investigating improved fabrication techniques to reduce the size and number of impurities (voids, inclusions) in our liners to also reduce the seed for the electrothermal and magneto-Rayleigh–Taylor instability. Such improvements in target quality have proved an essential part of the achievement of ignition on the NIF.^39,63

Simulation renderings prepared with VisIt.⁶⁴ M. R. W. would like to acknowledge helpful discussions with E. P. Yu, J. R. Fein, O. M. Mannion, M. Schaeuble, Sandia High Perfromance Computing (HPC) for computing support, and the anonymous referees for their time and suggestions.

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology \& Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration (DOE/NNSA) under contract DE-NA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title and interest in and to the written work and is responsible for its contents. This paper describes objective technical results and analysis.

Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This manuscript has been authored in part by Lawrence Livermore National Security, LLC under Contract No. DE-AC52-07NA27344 with the US. Department of Energy. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan.

The authors have no conflicts to disclose.

Matthew Robert Weis: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). David Ampleford: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal). K. Beckwith: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Joseph Koning: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Daniel Edward Ruiz: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Matthew R. Gomez: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Adam James Harvey-Thompson: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Christopher A. Jennings: Conceptualization (equal); Investigation (equal); Methodology (equal). David Yager-Elorriaga: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). William E. Lewis: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Stephen A. Slutz: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Luke Shulenburger: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We review the results of three simulations of HEP utilizing different random number seeds to generate different surfaces patterns with the same effective RMS amplitude, while keeping the helices constant, in order to provide a sense of run to run variability. The different realizations used the more unstable $5 μm$ noise seed. Table IV shows a summary of the achieved stagnation conditions along with the previous simulations of HEP shown in the main text. The different realizations show fairly consistent performance compared to the change in amplitude (H5N2.5). The yield varies by $∼15$%, the plasma conditions variation is $∼5$%, and the burn widths are within 100 ps.

However, there are significant changes in the morphology of the stagnation as the seed is changed. Figure 14(b) show slices through the 3D simulations of HEP with H5N2.5 (1.1 × 10¹³), H5N5 (6.57 × 10¹²), and H5N5B (7.28 × 10¹²) seeds at peak neutron power. The fuel variables plotted are $Tion$ (keV), $Pth$ (Gbar), $Pmag$ (Gbar), and the normalized yield rate. Note that the color scales for $Tion$ and $Pth$ are adjusted for the differing peak values between the seed amplitudes, the other scales are identical. The liner mass density is displayed on a consistent log scale from $10−5−50$ g/cm³.

Referring to Fig. 3(b) and 3(c), we can see that the peak fuel temperature and pressure are $∼50%−60%$ higher than the burn averaged quantities and the overall variation throughout the fuel is obviously substantial. Despite the axial variation due to MRT, we still observe end losses impact the conditions near the top and bottom of the target leaving the peak temperatures located toward the mid-point of the target. With respect to the isobaric approximation, the simulations show thermal pressure variations of $≈5%−10%$ across the radial dimension of the hotspot where the deficit from approximately uniform pressure is due to the magnetic pressure contribution.

This section reviews additional supporting 3D HYDRA simulations and synthetic data run with the Hall term to supplement the main text. Figures 15 and 16 show synthetic radiographs at $Bz=20$ and 0 T, respectively. Panels (a) and (b) of each figure show inner convergence ratios, $CR∼2.3$⁠, and panels (c) and (d) show $CR∼10$⁠.

There are three important observations: (1) the Hall run clearly produces helical modes while the run without Hall does not; (2) the runs without Hall show higher frequency MRTI compared to the run with Hall; (3) the implosions with Hall are slightly delayed compared to the no Hall run. Additionally, for the no Hall simulations, there is very little difference in the MRT between $Bz=0$ and 20 T, although detailed analysis may reveal slightly longer wavelength structures in the 0 T simulation. In ideal MHD, this is explained by short wavelength stabilization by $Bz$ that effectively slows the mode cascade to longer wavelengths.

In the 0 T case, it is easy to see that the hall term leads to longer axial wavelengths (⁠ $∼250 μ$m) and more obvious $∼m=0$ azimuthal bands, which then become pitched with the application of the applied field.

We note that the need to include some amount of azimuthal correlation in resistive MHD simulations of unmagnetized implosions has been realized since McBride 2012²⁴ and the Hall term provides a physical mechanism to produce this bias.

Finally, we considered a change in the seed amplitude, which was reduced from $2.5$ to $1.25 μ$m as shown in Fig. 17. This was done in the spirit of comparing the fusion performance (shown in the main text) similar to the earlier H5N2.5/H5N5 comparisons; however, the change in MRT development was also interesting.

In particular, the long wavelength helical modes appear to persist even with the lower amplitude seed. The main difference is in the reduction of the higher frequency modes. This suggests that the helical modes can be partially decoupled but yet still communicate with the surface evolution of the solid liner.

Beyond these relatively small changes, additional variations may provide insight on the development of the helical modes, such as examining the effect of increasing the azimuthal resolution, changing the density floor and introducing some amount of flux compression. The latter two could benefit from additional experimental measurement of LDP. However, these different approaches will tend to significantly increase the runtime, which is already long (many weeks). However, some combination of these approaches may help to more faithfully reproduce the instability structure and pitch angle as a function of $Bz$⁠, but nonetheless provides a starting point for studying the effect of helical modes on MagLIF. In future work, we hope to benchmark the model to a collection of experimental helical instability scaling data.