Analysis of fusion fuel conditions and instability development in shocklessly compressed MagLIF implosion simulations

Magnetized liner inertial fusion (MagLIF)^1,2 implosions achieve thermonuclear conditions by quasi-adiabatic cylindrical compression of premagnetized, laser preheated³ deuterium fuel using pulsed magnetic fields generated by ∼20 MA peak current ∼100 ns rise time pulses supplied by the Z pulsed power accelerator.^4,5 The process of converting stored electrical energy from Z's capacitor banks to internal energy in the deuterium fuel suffers from many inefficiencies, many of which are unavoidable (e.g., losses associated with compression of the pulse from μs timescale to 100 ns necessary to drive sufficiently high implosion velocities). During the process of converting stored electrical energy in the capacitor banks to internal energy of the fusion fuel, one of the final steps occurs when the kinetic energy of the liner is converted to internal energy of the fuel and liner material.

To achieve higher thermonuclear yields in MagLIF implosions, one must increase the fuel ion temperature $T ion$ in conjunction with the pressure–confinement-time metric, $p stag τ$⁠, where $p stag$ is fuel stagnation pressure and τ is the confinement time of the fuel.⁶ While higher implosion velocities and higher fuel temperature are achievable with higher aspect ratio (AR =  $R out ( 0 ) / [ R out ( 0 ) − R in ( 0 ) ]$⁠) liners, magneto-Rayleigh–Taylor instabilities (MRTI)⁷ may become more problematic and reduce conversion of liner kinetic energy to fuel internal energy and degrade fuel confinement. Functionally, this places a limit on the liner $A R$ with typical MagLIF targets designed to be in the $A R = 4 – 9$ range. In general, implosion times are minimized as much as possible to reduce the amount of time over which MRTI can grow during implosion.⁸ This partially explains why MagLIF designs tend toward the shortest achievable current pulse that the Z accelerator can generate, namely, a ∼100 ns rise time “short pulse.”

Previous work exploring longer rise time current pulses in the context of MagLIF-like liner implosions⁹ suggests that the shape of the current pulse could potentially be modified to improve MagLIF performance due to several effects. Specifically, modifying the pulse to eliminate shocks^9,10 that are normally launched in the liner due to the high current rise rate associated with the short pulse configuration could result in benefits particular to the implosion process: (1) more efficient flux compression of pre-imposed axial magnetic field, and therefore, better thermal insulation of the fuel due to the higher average electrical conductivity of un-shocked, solid phase liner material compared to shocked, melted liner material; (2) higher hydrodynamic implosion efficiency due to elimination of shocks which increase the entropy within the liner material and reduce the compressibility of the overall fuel/liner system; (3) reduced mix of liner material into the fuel due to the solid phase of the inner liner surface (rather than the typical post-shock melted phase) when the blast wave from the laser energy deposition¹¹ reaches the liner–fuel interface. Additional benefits associated with the efficiency of delivering energy from the pulsed power generator to the target region through reduction of transmission line losses could also be realized, though we refrain from studying those topics here instead focusing on benefits to the magneto-inertial fusion (MIF) target implosion assuming similar driver-target coupling efficiency. Shockless compression of an initially solid cylindrical metallic tube¹⁰ is realized by reducing the current rise rate (i.e., lengthening the current pulse) sufficiently to prevent the transit speed of information from the outer liner surface to the inner liner surface at any time throughout the implosion from exceeding the material sound speed (in the reference frame of the liner material). In practice, this corresponds to introducing a current prepulse that raises the drive magnetic field at the outer liner surface gradually enough that “communication” with the inner liner surface occurs over a longer timescale than the sound speed transit time (preventing shocks from forming).

In this manuscript, we describe simulations executed to compare implosions driven by a ∼100-ns rise time, ∼21.5 MA peak current short pulse (Fig. 1, red) and a ∼200-ns rise time, ∼21.5 MA peak current pulse designed to shocklessly implode a liner configuration first described by McBride et al.⁹ (Fig. 1, dashed black). In Sec. II, we describe results from one-dimensional (1D) radiation-magnetohydrodynamic (rad-MHD) HYDRA¹² simulations that utilized the short and shockless pulses to drive MagLIF targets with identical initial conditions. We compare the attained thermodynamic fuel conditions at the moment of peak neutron emission for the two cases in 1D HYDRA and discuss the benefits realized by using the shockless compression pulse. We also discuss differences in the conditions of the liner material and the implications for magnetic flux compression. In Sec. III, we show and discuss three dimensional (3D) MHD ALEGRA¹³ simulations which focused on studying the development of acceleration instabilities during the implosion of “empty” liners (i.e., without premagnetized, laser preheated deuterium). Finally, in Sec. IV, we provide concluding remarks and discuss potential next steps.

1D HYDRA¹² was used to simulate implosions of a MagLIF target made of initially solid beryllium (ρ = 1850  $kg / m 3$⁠) with initial inner liner radius $R in ( 0 )$ = 2.39 mm and initial outer liner radius $R out ( 0 )$ = 3.19 mm driven by the short and shockless current pulses shown in Fig. 1. The initial aspect ratio AR of the liner was 4. These liner dimensions match those that were fielded in the reference experiments from which the measured current pulses were taken.⁹ A 1 cm imploding height was assumed for 1D HYDRA simulations, reflecting typical MagLIF target heights.¹⁴ The simulated targets were filled with room temperature 1  $mg / cm 3$ deuterium gas, premagnetized with $B z ( 0 ) = 15$ T, and the fuel was preheated by depositing 2 kJ of energy over 3 ns in the central region of the deuterium gas (where r < 0.75 mm) 60 ns prior to peak neutron output time (Table I). All materials were initialized in pressure equilibrium to prevent generation of numerical noise from material discontinuities. These initial conditions and preheat energy insertion protocol were chosen to reflect those of contemporary MagLIF experiments.¹⁴ A generalized Ohm's law was used that includes effects such as Nernst advection, which can affect the magnetization of the fuel. The equation of state and the transport coefficients for the nonideal thermal and magnetic conduction of the deuterium fuel and the beryllium liner were taken from pregenerated LEOS and SESAME tables. The radiation field was modeled using implicit Monte Carlo photonics. HYDRA simulations were run with arbitrary Lagrangian–Eulerian mesh, which enabled dynamic mesh relaxation/optimization; however, the beryllium–deuterium material boundary was kept Lagrangian. We note that the reference Z experiments⁹ were not integrated MagLIF experiments; in other words, those experiments were not premagnetized and did not contain preheated fusion fuel.

The peak neutron output time or “bang time” $t bang$ was matched for the short pulse and shockless pulse simulations (Fig. 2) by shifting the time base of the current pulses in the manner shown in Fig. 1. Consequently, peak current of the shockless pulse on this time base is ∼5 ns later than in the short pulse. As shown in Figs. 2 and 3, the implosions proceed with similar inner and outer liner radius trajectories as well as in flight aspect ratio (IFAR =  $R out ( t ) / [ R out ( t ) − R in ( t ) ]$⁠) but display notable differences. Particularly, the inflection seen in the $R in ( t )$ trajectory for the short pulse driven simulation (Fig. 3, dotted–dashed red) is evidence of shock breakout, contrasting with the smoother inner liner surface trajectory curve from the shockless implosion (Fig. 3, dotted–dashed black). While IFAR in the shockless implosion is larger than the short pulse case in the early stages of the implosion (a result of the early current rise gently compressing the liner), the peak IFAR of the short pulse implosion is 18% higher than the shockless case (Fig. 3, Table I).

Here, the adiabatic index γ is assumed to be 5/3.

1D HYDRA simulations indicate that the improved efficiency of flux compression is primarily due to the unshocked liner material retaining a higher average electrical conductivity throughout the implosion. In the beginning of the implosion, simulations indicate a sharp drop in electrical conductivity from the initial electrical conductivity of beryllium [∼3  $× 10 7$ (Ohm m)⁻¹ to below 10⁷ (Ohm m)⁻¹] at the inner liner surface can be observed in both short pulse and shockless pulse cases. This drop is associated with the melt phase transition. For the short pulse case, it occurs due to the breakout of the shock and precedes any significant movement of the inner liner surface. For the shockless case, melt of the inner liner surface occurs via densification/compression of the inner liner surface material layers (not due to shock breakout) and the associated drop in electrical conductivity does not occur until the liner has imploded to $C R i n ∼$ 1.2.

The average electrical conductivity in the liner material from $C R i n$ = 2 to $C R i n ∼$ 25 (when 1D HYDRA predicts $t bang$ occurs) is higher in the shockless pulse case according to 1D HYDRA (Fig. 8). The unshocked beryllium liner material has a $> 50 %$ higher average electrical conductivity from $C R i n$ = 2 to $C R i n ∼$ 25, helping to enable the 12% higher $Φ z / Φ z , 0$ (Figs. 6 and 7) evident in the shockless pulse case. Secondarily, interaction of the laser energy deposition blast wave with the unshocked solid phase inner surface in the shockless implosion simulation results in lower magnetic flux losses out of the fuel region compared to the short pulse case according to 1D HYDRA; the drop in $Φ z ( t ) / Φ z , 0$ in the fuel region resulting from the blast wave transmitting from the fuel region to the liner material is apparent in both simulations in Fig. 6 (solid color) near t = 3010 ns. Also notable is the slight increase in $Φ z ( t ) / Φ z , 0$ apparent in the short pulse case that occurs near t = 2990 ns due to the shock wave breaking out from the liner inner surface into the fuel region and injecting a small amount of additional B_z flux as a result. However, this additional B_z flux is quickly overwhelmed by flux loss from the fuel region out into the melted, lower average electrical conductivity liner material in the short pulse case.

More efficient magnetic flux compression in the shockless implosion simulation (12% higher $Φ z / Φ z , 0$⁠) results in better thermal insulation of the fusion fuel, corresponding to ∼13% higher $T ion$ averaged over the burn duration (Fig. 9). Furthermore, the 12% higher $Φ z / Φ z , 0$ in the hotspot for the shockless implosion compared to the short pulse implosion (Figs. 6 and 7; Table I) results in ∼9% reduced relative magnetic flux losses^17,18 from the hotspot due to Nernst thermoelectric effect, which is known to scale approximately as $∼ 1 / B z ( t )$⁠. Although simulations indicate an 8% decrease in confinement time τ for the shockless implosion, the 26% increase in burn-duration-averaged stagnation pressure $p stag$ (Fig. 10) results in a 16% increase in $p stag τ$ (Table I).

1D HYDRA simulations demonstrate that using a shockless current pulse instead of a short current pulse to drive a MagLIF implosion with identical target specifications and initial conditions can improve fusion fuel conditions and enhance thermonuclear performance. However, the shockless current pulse essentially adds ∼100 ns to the overall pulse length and, thus, increases the implosion time. Specifically, the outer liner radius begins imploding ∼20 ns prior to the short pulse case and the inner liner radius begins imploding ∼25 ns prior to the short pulse case (Fig. 2). One would expect this increase in implosion time and the correspondingly longer time spent in-flight subject to magnetic acceleration to result in longer time for MRTI to develop, although the lower peak acceleration of the shocklessly driven implosion counteracts the longer acceleration time. In the context of short pulse driven and shockless pulse driven MagLIF implosions explored in this study, the benefits to fuel conditions and yield evident from 1D simulations (which do not capture MRTI) could potentially be threatened by poorer implosion stability resulting from a longer implosion time in the shockless case. Since cumulative MRTI development depends on acceleration and time, and therefore, in the linear growth phase is directly dependent on distance traveled, one would also expect the shockless implosion which is more highly converged (i.e., has traveled a greater distance) by the time of peak neutron production to have developed more cumulative MRTI.

To assess the effects of the two different current pulse shapes shown in Fig. 1 on MRTI development, we executed 3D ALEGRA¹³ simulations of empty beryllium liners (i.e., without any deuterium gas) with $R in ( 0 )$ = 2.39 and $R out ( 0 )$ = 3.19 mm (Fig. 11). ALEGRA is an arbitrary Lagrangian–Eulerian resistive MHD code with thermal conduction and multi-material hydrodynamics that utilizes an unstructured mesh. Simulations employed an “Eulerian” mesh (a remap step moved the mesh back to the original location after each Lagrangian mesh deformation step) and utilized SESAME equation of state and coupled electrical/thermal conductivity material tables. The simulation geometry used for this study was a 1-mm tall simulation region with periodic top and bottom boundaries. Simulations employed a “radial trisection” mesh, with radial mesh cells (i.e., mesh cell edges oriented along the radial, azimuthal, and axial directions) that converted to Cartesian geometry near the axis. The mesh resolution (i.e., axially, radially, and azimuthally oriented mesh cell edge length) in the liner region was 20 μm. The axial mesh resolution is uniform and constant throughout the simulation. The mesh begins to convert from radial to Cartesian at $r ∼ 250$ μm. 3D ALEGRA was chosen for these 3D cylindrical liner implosion simulations due to its robust resistive MHD physics and its demonstrated utility in modeling similar magnetic implosion systems.^10,19

All initial conditions were made identical between the two series of simulations so that the only difference was the drive current pulse. MRTI was seeded identically in both simulation series by applying an initial surface roughness seed, a ±1.25 μm random radial node perturbation on the outer liner surface. Perturbations were not added to the inner liner surface. Focus was placed on studying acceleration-driven MRTI development on the outer liner surface rather than deceleration-induced perturbation growth on the inner surface, principally because 3D ALEGRA simulations did not contain preheated/premagnetized fusion fuel, and therefore, provided minimal hydrodynamic pressure that could induce liner deceleration prior to the liner material impacting the axis.

We quantitatively assess cumulative MRTI growth in 3D ALEGRA simulations by analyzing the standard deviation of the areal density $σ ( ρ r )$ (Figs. 12 and 13) for each simulation as a function of normalized distance moved of the inner liner surface, $Δ R$⁠. Here, $Δ R = [ 1 − R in ( t ) / R in ( 0 ) ] = [ 1 − 1 / CR in ]$⁠. A 2D map of the areal density ρr of the imploding liner was produced by radially integrating the mass density at each $( θ , z )$ coordinate in the simulation, where θ is the azimuthal coordinate and z is the axial coordinate.

Despite the ∼20–25 ns longer implosion time of the simulated shockless implosion, the cumulative MRTI is comparable (Figs. 12 and 13). The cumulative MRTI in the shockless case is lower when $Δ R ≳ 0.8$⁠; the short pulse case has ∼30% larger $σ ( ρ r )$ when the liners are near peak convergence captured in simulations, $Δ R ∼ 0.98$⁠. The uncertainty ranges for the $σ ( ρ r )$ analysis were assigned based on the range of $σ ( ρ r )$ observed at the point of highest convergence for an ensemble of three separate simulations with identical initial random surface roughness radial perturbation magnitude but different random number seed. The lower cumulative MRTI development evidenced by the $σ ( ρ r )$ analysis from 3D ALEGRA simulations indicates that the shockless pulse results in improved liner implosion stability. Therefore, not only does the simulated shockless pulse driven MagLIF implosion of this target design result in higher ion temperatures, stagnation pressures, and $p stag τ$ according to 1D HYDRA (Sec. II): the shockless pulse driven implosion of this liner design also appears to be more robust to MRTI according to 3D ALEGRA.

In this paper, we presented simulations of initially solid cylindrical liner implosions driven by two different measured current pulses from historical Z experiments:⁹ a ∼100 ns rise time ∼21.5 MA short pulse and a ∼200 ns rise time ∼21.5 MA peak current pulse designed to shocklessly implode the liner. Historically, MagLIF experiments on Z have almost exclusively been driven with ∼100 ns rise time current pulses, and the exploration of longer rise times has been dismissed due to concerns with longer implosion times and their associated (presumably negative) effect on implosion stability. 1D HYDRA simulations were executed to compare stagnated fusion fuel conditions and thermonuclear yield for the two cases. These simulations demonstrate that ion temperature, fuel pressure, $p stag τ$⁠, BR, and thermonuclear yield are all significantly higher for the simulated implosions driven by the shockless current pulse. We postulate that improved performance is primarily due to the higher average electrical conductivity of the un-shocked liner material throughout the implosion, which corresponds to more efficient flux compression and better thermal insulation of the fuel. We also postulate that the lower adiabat of the liner material and resultant higher compressibility of the fuel/liner system enables the evident improvements in fuel conditions. More qualitatively, the shockless compression provides a more efficient method for implosion, enabling more energy to be coupled to the fuel for the same peak current.

3D MHD ALEGRA simulations of initially solid metallic liner implosions driven by these current pulses indicate that although the shocklessly driven implosion has an overall longer implosion time that conventional wisdom says should correspond to longer time for implosion instabilities to grow, the cumulative magneto-Rayleigh–Taylor instability (MRTI) growth seen in simulations is comparable for both cases. Indeed, 3D ALEGRA simulations indicate that the shocklessly driven implosion results in lower cumulative MRTI and an overall more stable implosion. Furthermore, 3D MHD modeling and experiments will be necessary to more precisely assess the difference in MRTI between these two cases, particularly the feedthrough of MRTI and the resultant effects of MRTI feedthrough on fuel compression and confinement. In addition, simulations discussed in this study seed MRTI growth from a nonphysical surface roughness and did not attempt to resolve the sub-micrometer spatial scales necessary to capture known surface-roughness-seeding processes, such as the electrothermal instability (ETI).^20,21 ETI is expected to become worse as the time during which Joule heating of the liner surface increases, suggesting potential concerns with longer current pulses. If future simulations or experiments indicate that shockless implosions develop more cumulative MRTI, mitigation strategies such as dielectric coatings²² or dynamic screw pinches^19,23 could be leveraged to improve implosion stability and realize the benefits associated with shockless implosion demonstrated in this study.

Given the benefits of shocklessly compressed MagLIF implosions evident from simulations, further evaluation alongside other design choices and scaling pathways would be prudent. Simulations presented here indicate that performance benefits could be realized in MagLIF experiments on Z for a particular target, but combining shockless implosion with other design modifications could be pursued either to ease constraints (e.g., more efficient flux compression could reduce required initial axial magnetic field from external field coils) or to compound benefits (e.g., combine higher preheat energies with more efficient flux compression to reduce Nernst losses). Future efforts will focus on 3D rad-MHD modeling to evaluate use of pulse shaping in implosions utilizing contemporary MagLIF target dimensions^14,16 and alternative design pathways intended to realize the broader benefits of pulse shaping in combination with other design modifications. Once the design physics basis has been established regarding potential benefits and pathways to explore, experiments on the Z facility will be necessary to assess the fidelity of modeling predictions and guide further study.

The authors gratefully acknowledge a detailed review of this article by W. Lewis as well as helpful discussions with C. A. Jennings, P. F. Schmit, J. Niederhaus, A. C. Robinson, E. P. Yu, M. R. Martin, M. R. Weis, D. C. Lamppa, B. T. Hutsel, O. M. Mannion, L. J. Stanek, and R. D. McBride. The authors also gratefully acknowledge programmatic support and encouragement from L. Shulenburger, D. Ampleford, G. Rochau, and D. Sinars.

One of the authors (G. A. Shipley) was supported in part by an appointment to the Sandia National Laboratories Truman Fellowship in National Security Science and Engineering, which is part of the Laboratory Directed Research and Development (LDRD) Program (Project No. 226067). Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. 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. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. 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.

Gabriel Alan Shipley: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). Daniel Edward Ruiz: Formal analysis (supporting); Methodology (supporting); Writing – review & editing (supporting). Andrew Porwitzky: Conceptualization (equal); Methodology (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.