Feasibility and Performance of the Staged Z-Pinch: A One-dimensional Study with FLASH and MACH2

Z-pinch platforms constitute a promising pathway to fusion energy research. Here, we present a one-dimensional numerical study of the staged Z-pinch (SZP) concept using the FLASH and MACH2 codes. We discuss the verification of the codes using two analytical benchmarks that include Z-pinch-relevant physics, building confidence on the codes' ability to model such experiments. Then, FLASH is used to simulate two different SZP configurations: a xenon gas-puff liner (SZP1*) and a silver solid liner (SZP2). The SZP2 results are compared against previously published MACH2 results, and a new code-to-code comparison on SZP1* is presented. Using an ideal equation of state and analytical transport coefficients, FLASH yields a fuel convergence ratio (CR) of approximately 39 and a mass-averaged fuel ion temperature slightly below 1 keV for the SZP2 scheme, significantly lower than the full-physics MACH2 prediction. For the new SZP1* configuration, full-physics FLASH simulations furnish large and inherently unstable CRs (>300), but achieve fuel ion temperatures of many keV. While MACH2 also predicts high temperatures, the fuel stagnates at a smaller CR. The integrated code-to-code comparison reveals how magnetic insulation, heat conduction, and radiation transport affect platform performance and the feasibility of the SZP concept.


I. INTRODUCTION
The Z-pinch concept is fundamentally a cylindrical plasma implosion onto the symmetry axis by a J × B force provided by a current pulse.There are many variations on the target plasma, such as foils, wire arrays, jets, gas-puffs, pre-filled cylinders, or combinations thereof. 1 Furthermore, additional materials can be used as liners to assist the implosion, with yet more variations on how the liner is created and which material is used.When such a system is driven by modern pulsed-power drivers, the current pinching the target can reach many MA, leading to plasmas that achieve keV temperatures at near-solid densities.
These plasmas are of interest to the fusion community and are useful scientific platforms for atomic physics, radiation transport, and laboratory astrophysics studies. 1,2e Z Machine at Sandia National Laboratories in Albuquerque (SNL) is the most powerful pulsed-power device in the world, providing up to 30 MA of peak current to a Z-pinch target. 3In recent years, the Magnetized Liner Inertial Fusion (MagLIF) concept has been a focus of research and development at SNL. MagLIF is a specific type of Z-pinch that utilizes an externally applied axial magnetic field to reduce thermal conduction losses and an on-axis laser to preheat the fuel (typically close to 100 eV), which reduces the implosion velocity required to reach ignition temperatures. 4,5The axial magnetic field is initially 10-20 T but is compressed to much larger values, which my help confine alpha particles when deuteriumtritium fuel is used.MagLIF typically uses an aluminum or beryllium liner to compress a deuterium target and require sufficient liner thickness to avoid significant degradation due to the Rayleigh-Taylor instability.
The staged Z-pinch (SZP) is an alternative fusion concept in which energy is transferred to the target plasma in stages.The SZP name was first used for a configuration with an on-axis cryogenic deuterium fiber (i.e., target) compressed by an argon or krypton liner. 6A current pre-pulse through the fiber would create the target plasma and pre-magnetize the liner, and a subsequent main Z-pinch current pulse would implode the liner.A theorized benefit of the SZP is the control and mitigation of the magneto-Rayleigh Taylor (MRT) instability at the fuel/liner interface 7 , however this point is beyond the scope of this work and will be addressed in future publications.Current SZP configurations typically employ a gas fill for the target load and high atomic number liners (gas-puff liners 8,9 or solid liners 10,11 ).The working hypothesis is that a high atomic number liner will radiate more efficiently, and the Feasibility and Performance of the SZP: 1-D FLASH and MACH2 resulting colder liner will allow more magnetic diffusion towards the fuel/liner interface.This would in turn result in a stronger magnetic pressure on the target plasma and potentially reduce thermal conduction losses.
It is well known that fusion output is severely inhibited when high atomic number impurities are mixed into the fuel plasma, because this increases radiative losses (i.e., reduces fuel temperatures).Therefore, the high atomic number liners used by the SZP concept will only perform well if the fuel/liner interface remains relatively stable during the implosion.
The magnetic, thermal, and radiation transport properties of the system become crucially important as they can all affect the time scales of the implosion and fuel heating and stability of the fuel/liner interface.We focus on the transport properties and their effects in the simulations presented in this work, but, as previously mentioned, we do not include a stability analysis as these simulations are one-dimensional.
In this paper, we have modeled two different SZP configurations: a new xenon gas-puff liner (SZP1*) with different initial conditions as compared to the original xenon gas-puff liner (SZP1 9 ), and the original silver solid liner setup (SZP2 10 ).Fig. 1 shows schematics of the SZP2 and SZP1* configurations with approximate dimensions.5][16] Most of the criticism has been aimed at the interpretation of key shock physics and the calculations behind the fusion energy output.The FLASH code can now contribute to this debate courtesy of our on-going collaboration between the Flash Center for Computational Science at the University of Rochester and MIFTI, made possible by funding from the ARPA-E BETHE program.For the present work, we focus on specific physics and code-to-code comparisons, and we exclude calculations and discussion of fusion yield and energy production, in part, because FLASH does not have this capability.This paper is written with several goals in mind: (1)  The structure of the paper is as follows: in Section II we describe the two codes used in this work, FLASH and MACH2.Then in Section III, we present results from two analytical test problems with SZP-relevant physics: a radiative shock problem and the Noh cylindrical implosion problem.We show FLASH results from an ideal equation of state (EOS) silver liner (SZP2) model in Section IV, and we briefly discuss how it compares to the originally published MACH2 SZP2 results.In Section V, we include more realistic EOS tables and physics to present both FLASH and MACH2 simulation results of a xenon gas-puff liner configuration (SZP1*).Lastly, we conclude our findings in Section VI.

II. NUMERICAL METHODS
FLASH 17 is a publicly-available, parallel, multi-physics, adaptive mesh refinement (AMR), finite-volume Eulerian hydrodynamics and MHD code, developed at the University of Rochester by the Flash Center for Computational Science (for more information on the FLASH code, visit: https://flash.rochester.edu).FLASH scales well to over a 100,000 processors, and uses a variety of parallelization techniques like domain decomposition, mesh replication, and threading, to optimally utilize hardware resources.The FLASH code has a world-wide user base of more than 4,350 scientists, and more than 1,300 papers have been published using the code to model problems in a wide range of disciplines, including plasma astrophysics, combustion, fluid dynamics, high energy density physics (HEDP), and fusion energy.
Over the past decade and under the auspices of the U.S. DOE NNSA, the Flash Center has added in FLASH extensive HEDP and extended-MHD capabilities 18 that make it an ideal tool for the multi-physics modeling of the SZP platform.These include multiple state-of-the art hydrodynamic and MHD shock-capturing solvers, 19 , three-temperature extensions 18 with anisotropic thermal conduction that utilizes high-fidelity magnetized heat transport coefficients, 20 heat exchange, multi-group radiation diffusion, tabulated multi-material EOS and opacities, laser energy deposition, circuit models, 21  The FLASH code and its capabilities have been validated through benchmarks and codeto-code comparisons, [25][26][27] as well as through direct application to numerous plasma physics experiments, [28][29][30][31][32][33][34] leading to innovative science and publications in high-impact journals.For pulsed-power experiments, FLASH has been able to reproduce past analytical models 35 , is being applied in the modeling of capillary discharge plasmas 36 , and is being validated against gas-puff experiments at CESZAR 37 .The Flash Center is also collaborating with Los Alamos National Laboratory (LANL) in the modeling of laser-driven experiments of cylinder implosions 38 at the Omega Laser Facility at the University of Rochester and the National Ignition Facility at Lawrence Livermore National Laboratory, in a successful integrated inertial confinement fusion (ICF) verification and validation (V&V) effort with xRAGE. 39,40e Multi-block Arbitrary Coordinate Hydromagnetic (MACH2 ) code 41 is a single-fluid, multi-material, three-temperature resistive MHD code, developed by the Center for Plasma Theory and Computation at the Air Force Research Laboratory (AFRL), Phillips Research Site.It solves the usual set of MHD equations: mass conservation, momentum conservation, electron, ion, and radiation energy, and Faraday's law of induction for the magnetic field.
One fundamental difference between FLASH and MACH2 lies in the formulation of the total energy equation.Although MACH2 advances the total energy in a non-conservative manner, this has been proven to not impact the code's ability to capture MHD shocks, provided that adequate grid resolution is used. 42Radiation is calculated using a single energy group (Gray radiation), with a flux-limited, non-equilibrium model.The EOS and transport coefficients (opacities, thermal conductivities, magnetic resistivity) can be obtained from the LANL SESAME tables.The code also contains options to use a gamma-law EOS and certain analytical transport coefficients (e.g., Spitzer thermal conductivity).This code has an adaptive mesh algorithm, which can alter the computational grid every time step according to user-specified criteria.Its Arbitrary Lagrangian-Eulerian (ALE) framework allows simulations to be run in pure Lagrangian, pure Eulerian, or a combination of the two methods.In the pure Eulerian mode, the code is still taking a Lagrangian step, but maps the result back to the fixed Eulerian mesh.The grid spacing is potentially adjusted by the adaptive algorithm, depending on magnetic or fluid pressure gradients (or both), which can provide increased accuracy in regions of interest while saving computational time.The Eulerian method, where the computational grid is fixed in space for the entire duration of Feasibility and Performance of the SZP: 1-D FLASH and MACH2 a simulation, is perhaps the easiest to conceptualize and analyze.However, it may require increased resolution in certain regions to properly model important phenomena driving the system dynamics.New MACH2 results in this work use the pure Eulerian method for comparison with FLASH.
MACH2 contains a self-consistent circuit model, which is intended to represent the refurbished Z pulsed power machine at Sandia National Laboratory (SNL). 21The input opencircuit voltage profile and other circuit parameters are described in a previous paper. 10This same circuit model is also now implemented in FLASH.
MACH2 has been successfully used for a variety of studies, which supports its use as a code that has gone through an extensive amount of V&V.These studies include, but are not limited to, explosive magnetic generators, plasma opening switches, 43,44 compact toroid schemes, [45][46][47] ICF and alternative fusion concepts, 48 and Z-pinches with solid liners. 49,50Some have questioned whether previous SZP simulations used MACH2 correctly with appropriate boundary conditions and sufficient spatial resolution.The code-to-code comparisons reported in this paper are intended to build confidence that these codes can accurately model Z-pinches and help guide experiments.
MACH2 is also actively used and developed at the Naval Research Laboratory (NRL), and newer versions of the code may have significant differences from the version used in this study.One of the purposes of this work is to assess the SZP platform with FLASH within the context of MIFTI's previous and current research using the version of MACH2 in their possession.Therefore, it is not our intention to fix any potential errors that may be discovered in MACH2.Any code-to-code discrepancies described in this work pertain specifically to this version of MACH2 and should not have any bearing on newer versions of the code being used by other research groups.

III. ANALYTICAL TESTS
In published MACH2 simulations of the SZP, shock waves were identified as crucial in preheating the target plasma and piling up liner mass at the liner/target interface. 14e interpretation of these shock waves has come under some criticism in recent years. 12,13vertheless, these shock waves are present in the MACH2 simulations and are complex phenomena as they develop in a magnetized medium with important radiative effects.For these reasons, we decided to test both FLASH and MACH2 with simpler analytical problems in which Z-pinch-relevant shock physics is important.In the subsections that follow, we present test results from a radiative shock problem and from the cylindrical Noh problem.
One purpose of these tests is to help build confidence in each code's ability to accurately model constituents that make up the fluid-and thermo-dynamics of the complicated SZP.

A. Radiative Shock Problem
Radiative shocks (and radiation in general) are essential elements of SZP simulations.
As reported by Ruskov et al. 16 (cf.Fig. 9 therein), MACH2 simulations indicate that a more radiative (higher Z ) liner is compressed more for the same liner mass and driver; consequently coupling its kinetic energy into target internal energy more efficiently; ultimately resulting in higher yield.
The radiative shock problem presented here follows the prescription described in Lowrie and Edwards 51 , and its simulation setup is also described in the FLASH user's guide. 52It is an analytical solution to a 1D, steady, radiative shock in which electron and ion temperatures are in equilibrium, but the radiation temperature differs.Constant opacities are used for a single radiation energy group (gray).The purpose of the problem is to test a code's radiation transfer and shock-capturing capabilities, both of which are important for the modeling of the SZP concept.
The Planck opacities (absorption and emission) are set to approximately 423 cm -1 , and the Rosseland opacity (transport) is approximately 788 cm -1 .An ideal EOS is used with adiabatic index γ = 5/3, atomic number Z = 1 (also the constant ionization state), and atomic mass A = 2 amu.Electron, ion, and radiation temperatures are initially in equilibrium, and their upstream (pre-shock) value is 100 eV.The electron and ion temperatures remain in equilibrium throughout the domain for the duration of the simulation due to heat exchange with an enforced reduction to the equilibration time, but radiation temperature can change.The upstream density is set to 1.0 g cm -3 , and the remaining upstream and all downstream conditions are set appropriately to maintain a steady shock with Mach number Fig. 2 shows the analytical solution for electron (same as ion) and radiation temperatures as well as simulation results from both codes.The grid resolutions used for these simulations were approximately 0.146 µm and 0.293 µm for FLASH and MACH2, respectively.A resolution convergence study was conducted with MACH2, and the results did not change with higher resolution.Note that Fig. 2 does not contain every data point from either code to avoid over-crowding the plot.At a relatively early time of 2 ns, it appears that both FLASH and MACH2 recover the analytical solution, providing confidence that the radiation transfer algorithms give accurate results.However, at later times, we see that Feasibility and Performance of the SZP: 1-D FLASH and MACH2 only FLASH captures the exact position of the shock, whereas MACH2 shows an increasing positional offset in the electron/ion temperature jump.This result shows that MACH2 does not capture electron-radiation coupling as accurately as FLASH, especially in the presence of shocks.This kind of discrepancy will play a role in the SZP1* simulations presented later in this work (Section V).
In this version of MACH2, the radiation transport calculation occurs before the hydrodynamic advection calculation, which is a typical operator-split approach.However, the MACH2 result shows a small positional error on the order of a fraction of a cell width, which accumulates over time, hence the position of the shock continuously drifts farther away from the analytical solution at later times.FLASH utilizes a similar operator-split approach, but does not exhibit the same error.Additional tests were conducted to match the order of operations, but the results did not change.It is also important to note that MACH2 took on the order of 100,000 computational time steps for the radiative shock test problem, but the SZP1* simulation discussed later, required over 2.6 million steps, thus the aforementioned error accumulation could be significant in the SZP1* simulation.

B. Magnetized Noh Problem
The classical Noh problem 53 provides a test to benchmark the accuracy of hydrodynamic codes to capture shock dynamics in a convergent geometry.Initially, an infinite mass is set with a homogeneous inward velocity in cylindrical coordinates.Due to the singularity at the origin, an accretion shock wave is generated that propagates outwards, decelerating the incoming fluid mass.The magnetized extension of the problem, derived by Velikovich et al. 42 , is well-reproduced by FLASH. 54We have repeated the simulation here since we are using a newer version of FLASH, and we compare with results from the relevant version of MACH2 used in the published SZP simulations [9][10][11]14 . A smulation setup for this test is also included in the release version of FLASH and described in the user's guide.52 It was observed that, for both codes, the numerical results converge towards the analytical solution with increased resolution.Fig. 3 shows that both codes approximately reproduce the analytical solution for mass density.The key features that are modeled accurately are the peak density and the location of the shock.Note that this test only involves hydromagnetic advection, and therefore, the MACH2 result does not suffer from the same error

IV. SILVER LINER MODELS
We first present simulations of the silver liner on a DT target configuration proposed in Wessel et al. 10 , referred to as SZP2.This configuration was chosen because it was used in Ruskov et al. 14 in response to SZP criticism, 12 and is therefore employed here in the FLASH calculations.
Our setup consists of a flat density profile for both fuel (ρ fuel = 9.8 × 10 −3 g cm −3 ) and liner (ρ liner = 0.6 g cm −3 ).Initially, the fuel region extends from r = 0 to r = 0.2 cm, and the thickness of the liner is 0.1 cm.The computational domain consists of a uniform grid of 1,024 points that extends to r = 0.4 cm.This ensures a resolution of 10 cells to describe the fuel region at stagnation.At t = 0, the system is in thermal equilibrium at an initial temperature of 2 eV.In this first comparison, we have taken an ideal-EOS approach, in which a gamma-law EOS with γ = 5/3, constant ionization of Z = 1 for the fuel and Z = 10 for the liner, and analytical formulas for radiation opacities corresponding to freefree electron transitions (Bremsstrahlung radiation) 55,56 are assumed.More precisely, we are taking the Planck mean opacity K f f P for the emission and absorption opacities and the Rosseland mean opacity K f f R for the transport opacity, where K f f P and K f f R are (in c.g.s.): Here, A refers to the mass number.
The motivation behind the ideal-EOS approach is to verify that FLASH can accurately solve for the fundamental physics that model and govern a pinch implosion.Particularly, appropriate treatment of the vacuum region is essential in an Eulerian code like FLASH.The vacuum region is modeled as a low-density fluid whose task is to transfer the magnetic field from the outer boundary, placed at r = 0.4 cm, to the outer surface of the liner while adhering to a current-free profile, B ∼ 1/r.To ensure this, an artificially high value of magnetic diffusivity in the vacuum region was used, namely, η vac ∼ 10 11 cm 2 s −1 .Additionally, a temperature ceiling was imposed in the vacuum to avoid potential build-up of thermal pressure that could affect pinch dynamics.The temperature ceiling has the side benefit of keeping thermal conduction low in the vacuum, hence reducing liner-vacuum heat losses.A signature of correct behavior of the vacuum is the robustness of the implosion dynamics to changes in the parameters that model the vacuum region.This is shown in Fig. 4, where the temporal evolution of mass-averaged ion temperature is minimally affected by the value of the initial vacuum density (provided that it is sufficiently low), vacuum diffusivity (provided that it is sufficiently high), and temperature ceiling.
The dynamics of the implosion are sketched in Fig. 5, where the evolution of the fuel radius and the implosion velocity are shown for runs with radiation physics on (black) and off (red).Initially, the fuel is slowly compressed as a result of a pressure imbalance, present in the initial conditions.This is a result of the three regions (fuel, liner, and vacuum) being Feasibility and Performance of the SZP: 1-D FLASH and MACH2 initialized with a homogeneous temperature of 2 eV.At t = 80 ns, the trajectories of the two runs depart.In the run with radiation physics turned off, a jump-off velocity of the liner of 7 cm/µs is observed when the leading shock breaks out into the fuel at t = 109 ns.This is consistent, albeit slightly above, the ∼ 6 cm/µs implosion velocity reported both in Lindemuth et al. 12 , cf.Fig. 4, and in Ruskov et al. 14 , cf.Fig. 3(b).Thermal and magnetic pressure profiles at the time of shock breakout are depicted in Fig. 6(a) for this run.The run where radiation physics is turned on shows more complex dynamics.Radiation keeps the liner colder, which allows for more magnetic field to diffuse into the fuel.At its interface with the liner, significant magnetic pressure is built up, and the magnetic piston thereby formed drives the initial stages of fuel compression.This stage occurs between After shock breakout, the shock travels in the fuel, heating it non-adiabatically (viz., shock preheating).Eventually, the shock reaches the symmetry axis, rebounds off, and propagates outward in the form of a weaker shock or sound wave.Subsequent shock preheating can take place due to shock rebound at the inner surface of the liner, or due to additional shocks launched by the driver.Shock preheating ends when the fuel temperature is raised to a value where further compression becomes subsonic.
Feasibility and Performance of the SZP: 1-D FLASH and MACH2 The evolution of the ion temperature as a function of the convergence ratio provides insights into the implosion dynamics.Throughout this work, the convergence ratio (CR) is defined as the ratio of the initial outer fuel radius to the compressed outer fuel radius.This is plotted in Fig. 7 and compared to its counterpart full-physics MACH2 run from Ruskov et al. 14 .Highlighted in light gray is the stage of the implosion driven by magnetic pressure, whereas the darker gray region denoted the period during which the main shock propagates in the fuel.It can be seen that, despite the different early dynamics previously described, both radiation-off and radiation-on simulations depart from approximately the same fuel temperature after shock preheating, CR = 5.At the later stage of subsonic compression, the the radiation-off run closely follows an adiabatic trajectory, while the radiation-on run deviates significantly.This indicates that radiation losses dominate over thermal conduction losses in the SZP2 configuration, as was anticipated in Lindemuth et al. 12 , cf.Table III.Feasibility and Performance of the SZP: 1-D FLASH and MACH2 Comparing the MACH2 and FLASH simulations, it can be observed that the fuel adiabat after shock preheating is significantly lower in FLASH, resulting in lower fuel temperatures and a lower convergence ratio at stagnation.In this aspect, the FLASH run is more consistent with the results reported in Lindemuth et al. 12 , cf.Fig. 7: stagnation temperatures slightly below 1 keV at CR ∼ 50.However, it should be highlighted that adopting an ideal-EOS framework resulted in different early dynamics.In the MACH2 run, the occurrence of secondary shock preheating, absent in the simulations in Lindemuth et al. 12 , allows the fuel adiabat to rise and attain fusion conditions.In the FLASH run, only primary shock preheating is observed, preceded by compression due to magnetic pressure.

V. XENON GAS-PUFF LINER MODELS
We have developed a new configuration of a xenon gas-puff liner to enact a direct comparison between FLASH and MACH2 simulations of the SZP platform.Previously published SZP models of Xe gas-puff liners, sometimes referred to as SZP1 9 , use different initial conditions.To avoid confusion, we refer to this new configuration as SZP1*.The main difference is a lower liner density, which was chosen to generate a faster implosion and thus stronger shocks in the hopes of achieving higher temperatures.
Another motivating factor for SZP1* is that the dominant thermal loss mechanism is thermal conduction rather than radiation as in SZP1.We can estimate the ratio of thermal conduction losses to radiation losses for DT with the following formula: where T e is electron temperature in keV, CR is fuel convergence ratio, K is a coefficient that accounts for the effect of the electron Hall parameter, χ e , on the electron thermal conductivity, and n i is the ion number density in 10 24 cm -3 .This equation is derived from the work of Lindemuth and Siemon 57 and represents an estimate of the ratio of the rate of electron thermal conduction to the rate of Bremsstrahlung radiation, specifically for DT.
For hydrogen, K(χ e ) = (4.664χ 2 e + 11.92)/(χ 4 e + 14.79χ 2 e + 3.77) was used.The input parameters (from the FLASH simulations) and resulting ratios for SZP1 and SZP1* are summarized in Table I.For the original SZP1 scheme, we calculated this ratio to be < 1 near stagnation, making radiation the primary heat loss mechanism.Conversely, for SZP1*, Feasibility and Performance of the SZP: 1-D FLASH and MACH2 we estimate this ratio to be > 57, thus thermal conduction losses dominate near stagnation.This configuration is potentially advantageous because thermal conduction losses can be reduced if sufficient magnetic field is diffused into the fuel.The main contributing factors for this difference, according to Eq. ( 3) and Table I, are the CR and n i attained.Since SZP1* reaches higher CR than SZP1 because of its lower density, the different thermal loss regimes are ultimately a result of the different densities.
The tail of the liner Gaussian from r = 1.722 to r = 2.016 cm is modeled as a "vacuum" region, with a floor density of ρ min = 1 × 10 −7 g/cm −3 enforced for the duration of the simulation.Fig. 8 shows the initial mass density profile for the SZP1* configuration.The mass density and magnetic field profiles at CR = 87 are shown in Fig. 10.This is the maximum CR attained in the MACH2 run, whereas the FLASH simulation continues to compress.While a cursory inspection of Fig. 10 may conclude that the simulations match fairly well, there are a two key differences to note.The liner in the MACH2 simulation has compressed to larger densities than in the FLASH simulation.Further, in the MACH2 simulation we see a significant built-up of magnetic field just inside the fuel abutting the fuel/liner interface, which lowers thermal conductivity, insulates the fuel, and reduces thermal losses.This disparity in magnetic field accumulation in the fuel is identified as the main cause for the observed difference in the maximum convergence ratios, at stagnation, between simulations.The relatively larger magnetic field in MACH2 leaks into the fuel at approximately the same time or shortly after the main shock in the liner breaks out into the fuel.This occurs relatively early (∼ 127 ns) at CR ∼ 1.11, and the simulations begin to diverge after this point.After shock breakout, the MACH2 simulation predicts a thin, cold region in the fuel next to the liner.The temperature drop leads to an increase in magnetic resistivity which, in turn, allows more magnetic field to diffuse inwards, further inhibiting thermal conduction.
In the limit of large magnetization, the perpendicular thermal conductivity is proportional to T 2.5 e /χ 2 e , where χ e is the electron Hall parameter.This thin fuel region next to the liner is more magnetized in the MACH2 simulation than in the FLASH run as shown in Fig. 11, with peak values of χ e ≈ 3138 and χ e ≈ 58.55, respectively.Taking also into account the different temperatures, we estimate that the thermal conductivity in this part of the fuel is more than 120 times greater in the FLASH simulation than in the MACH2 run.This observation explains why thermal conduction losses are higher in the FLASH simulation and is consistent with the continued compression of the fuel to higher CR.Also note that the Feasibility and Performance of the SZP: 1-D FLASH and MACH2 magnetic field spike inside the fuel in the MACH2 result (see Fig. 10) would require a return current at this location, and we do not generally expect to see return currents inside the fuel in Z-pinches.Nevertheless, the MACH2 result shows how increased fuel magnetization can benefit the SZP1* configuration by reducing thermal losses, in turn leading to higher temperatures and larger, more stable CR values.The electron, ion, and radiation temperature profiles at CR = 87 are shown in Fig. 12.
Here we see a much clearer discrepancy between the two simulations.The fuel in the MACH2 run has a much higher electron temperature, which helps explain why the implosion stagnates earlier than in the FLASH simulation.It is also noteworthy that, in the FLASH result, we have a fuel whose T e < T i .Conversely, in the MACH2 simulation, at stagnation, T e > T i .
Generally, in Z-pinch experiments, one may expect the ion temperature to be higher than the electron temperature, since electrons lose energy via radiation, thermal conduction, and heat exchange with the ions, whereas ions are also subject to shock heating.Nevertheless, the temperature inversion observed in the MACH2 result is not necessarily nonphysical, given the large fuel magnetization.In such regimes, ions can be more thermally conductive than electrons, so it is possible for ions to lose more thermal energy and remain colder than electrons.Also, we again observe a discrepancy at the fuel/liner interface where the temperatures in the MACH2 run sharply decrease to liner values before the interface is reached, while in the FLASH profile the temperatures decrease after the interface an inside the liner.This helps further explain the aforementioned presence of a larger magnetic field values in the fuel in the MACH2 simulation: The lower-temperature region just inside the fuel/liner interface results in higher magnetic resistivity, which in turn allows for more magnetic field to diffuse into the fuel.The fuel/liner interface is marked by a short-dash vertical line.In the FLASH result, the fuel has T e < T i , whereas in MACH2, at stagnation, T e > T i .Also, the temperatures in the MACH2 run decrease to liner values before the interface is reached, while in the FLASH profile the temperatures decrease after the interface inside the liner.As a result, in the MACH2 run, the magnetic resistivity and the magnetization of the fuel adjacent to the interface are larger than in the the FLASH simulation, insulating the fuel from heat conduction losses.
The FLASH model continues to compress and reaches a peak T ion of about 18 keV, on-Feasibility and Performance of the SZP: 1-D FLASH and MACH2 axis, at CR = 100, which occurs at 144.75 ns.After this peak, the FLASH model compresses further, for ∼ 265 ps, and reaches CR of approximately 388.This latter compression is accompanied by thermal losses that result in lower-than-peak temperatures.Fig. 13 shows a comparison of the mass density and azimuthal magnetic field from the FLASH model at CR = 100 and CR = 388 (stagnation).The fuel density has increased by an order of magnitude, which is consistent with the decrease in volume from a radius of 50 µm to 13 µm.The magnetic field in the fuel has also increased, but the plasma beta is still much larger than unity due to the high thermal pressure.Despite having the same initial conditions, circuit model, transport coefficients, and EOS and opacity tables, we were not able to reproduce the MACH2 result with FLASH simulations.We see that the fuel stagnates at lower CR in the MACH2 run because the latter reaches much higher temperatures and thus has enough thermal pressure to halt the implosion.The ability of the fuel to retain its thermal energy (i.e., high temperatures) depends on its thermal losses via radiation and thermal conduction.The fact that discrepancies start becoming apparent after the shock breakout may call into question the codes' shockcapturing capabilities.We have shown that the version of MACH2 used in this work does not reproduce the analytical solution of the radiative shock test problem as accurately as FLASH (see Section III A).The integrated SZP1* simulations are more complicated than the simple benchmark problem, and the ability to accurately model radiative shocks at material interfaces in SZP1* is crucial for accurately predicting thermal conduction losses in the fuel.The MACH2 SZP1* simulation should have a similar error accumulation as observed Feasibility and Performance of the SZP: 1-D FLASH and MACH2 in Fig. 2, once the shock breaks out into the fuel.However the error is potentially larger due to the greater number of computational time steps (∼ 2.65 M).
Another interesting observation from the MACH2 result is that the fuel electron temperature remains higher than than the ion temperature (see Fig. 12).We see the opposite relation in the FLASH model, because the electrons are losing more energy without the thin, highly magnetized layer to insulate them.This layer, observed only in the MACH2 result, develops after shock breakout, and is therefore susceptible to the errors associated with radiative shock modeling discussed in Section III A. One would expect the electrons to be radiating, losing energy via thermal conduction, and transferring energy to the ions, while the ions are also subject to compressional heating.Due to aforementioned thermal loss mechanisms, the FLASH model is allowed to reach higher CR values, where thermal conduction losses become even more important.
The MACH2 code has been successfully used for and validated against several plasma, inertial confinement fusion, and high energy density physics experiments.However, the modeling of the SZP1* platform, with specific settings to compare with FLASH, is a challenging problem for the particular version of MACH2 used in this work, due to its issues modeling radiative shocks.This deficiency, in this version of MACH2, leads us to conclude that FLASH gives more physically sensible results for SZP1*, even though the FLASH -predicted CR values are too large to be experimentally stable.

B. High-fidelity FLASH simulations of the SZP1*
We ran two additional SZP1* models with FLASH to determine effects of using higherfidelity physics implemented in FLASH.These include newer, higher-fidelity transport coefficients 20,24 , and multi-group radiation diffusion, neither of which are available in MACH2.These newer transport coefficients are more complicated functions of atomic number and the Hall parameter, and they more accurate than Spitzer coefficients.The multi-group radiation diffusion model also used the newer transport coefficients, as well as 40 radiation energy groups, spanning the same energy range as the single-group (gray) models.We denote the FLASH runs in this subsection as follows: SP is the single-group run with Spitzer transport coefficients (the same run discussed in the previous subsection), DW 1G is the run with the newer transport coefficients and one radiation group, and DW Feasibility and Performance of the SZP: 1-D FLASH and MACH2 40G is the run with the newer transport coefficients and 40 radiation groups.
Table II gives a summary of key results in terms of CR, stagnation time, and massaveraged fuel ion temperature at stagnation.Note that for all FLASH simulations, these stagnation temperatures are lower than the peak ion temperatures.We observe that with the newer coefficients, SZP1* converges slightly faster and to a smaller radius, but the ion temperature is slightly lower.The multi-group model converges the fastest and to the highest CR values encountered in this work, CR ∼ 560.At stagnation, the multi-group radiation diffusion run is hotter than both single-group FLASH runs, but its peak temperature, which occurs prior to stagnation, is lower.Fig. 15 shows the mass-averaged fuel ion temperature as a function of CR for all SZP1* models.There are several important features to note in this figure : (1) all models show fuel preheating early (CR < 2), (2) all FLASH models continue to compress to higher CR values after peak T ion , whereas the MACH2 model does not, (3) the FLASH models with newer transport coefficients reach higher CR values, and (4) the multi-group model is on a lower adiabat and has a lower peak T ion than all single-group models.The significance of shock preheating was discussed in Section IV in reference to ideal-EOS SZP2 models, and similar points apply to SZP1* as well.However, in SZP1* there is also a radiation wave that provides significant additional fuel preheating.This was seen when analyzing the early-time behavior of the simulations, and by executing a separate test run with radiation transport switched off, in which the wave was absent.This radiation wave and the initial shock break-out into the fuel effectively set the adiabat of the compression.
Point ( 2) is essential for understanding the differences between the FLASH and MACH2 Feasibility and Performance of the SZP: 1-D FLASH and MACH2 models.Thermal losses, which are more significant in the FLASH simulations, cool the fuel and allow for higher CR values.The reasons for the discrepant thermal losses were discussed in the previous subsection.
Points (3) and ( 4) are specific to the FLASH models.Use of the newer transport coefficients leads to more thermal conduction losses, which results in higher CR values and lower T ion .The multi-group model converges slightly more than its single-group counterpart, while its stagnation temperature is higher.This result indicates that the liner is radiating more efficiently; a colder liner is easier to compress and will subsequently act as a more effective piston for compressing the fuel.Also, some of the increased liner radiation goes into the fuel, keeping it hot for a longer period.This speaks to the benefit of using a high atomic-number liner and broadly supports the viability of the SZP concept.At these observed high temperatures, alpha particle heating could be significant, but this physics capability is not available in FLASH so we did not explore it with MACH2 either.
Any additional heat source or insulation, or increasing the initial fuel density, would help stagnate the fuel at a lower CR value, thus improving stability.It should be emphasized that experiments of other SZP configurations, at smaller-than-Z pulsed-power facilities, have proven to be stable, and SZP1* is a theoretical platform in a different regime that may be more difficult to stabilize.
We eventually want to use FLASH to simulate the entire spatial and temporal evolution of the SZP with a reactor-level drive current in three dimensions, taking full advantage of the extended-MHD and transport capabilities of the code.The next immediate step is to conduct two-dimensional simulations of the models discussed in this work and in previous publications. 14,16Future work will assess the stability of the pinch (liner and target) to MHD instabilities in the presence/absence of axial magnetic fields, and explore how FLASH 's extended-MHD terms can affect implosion dynamics and plasma conditions at stagnation.This will shed light on the importance of previously unexplored physical processes at play in the SZP concept and contribute to the evaluation of the feasibility of the concept to achieve fusion.

Feasibility
FIG. 1. Schematics showing the SZP2 (a) and SZP1* (b) configurations.The SZP2 configuration uses a DT fuel and a silver solid liner, whereas SZP1* uses DT fuel and a xenon gass-puff liner.
to introduce FLASH 's new capability of modeling Z-pinches, (2) to further verify both FLASH and MACH2 against analytical test problems and with direct code-to-code comparisons of SZP simulations, (3) to provide a new SZP configuration (SZP1*) for additional verification and future experimental validation, and (4) to shed more light on some of the previously published work by presenting SZP2 Feasibility and Performance of the SZP: 1-D FLASH and MACH2 results from FLASH.
and numerous synthetic diagnostics 22 .FLASH 's newest algorithmic developments include a complete generalized Ohm's law that incorporates all extended MHD terms of the Braginskii formulation 23 .The new extended MHD capabilities are integrated with state-of-the-art transport coefficients, 24 developed with Feasibility and Performance of the SZP: 1-D FLASH and MACH2 support from the BETHE program.

FIG. 2 .
FIG. 2. Analytical solution of electron (same as ion, black) and radiation temperatures (red) for the radiative shock test problem as compared to FLASH and MACH2 simulation results.The analytical solution is shown as a solid line whereas the FLASH and MACH2 results are circle-dot and cross symbols, respectively.The top panel (2 ns) appears to show good agreement, but for later times (4.5 ns and 7 ns) we see an increasingly discrepant position in the MACH2 result.

Feasibility
FIG. 3. Analytical solution of mass density to the magnetized Noh problem as compared to FLASH and MACH2 simulation results.The analytical solution is shown as a solid line whereas the FLASH and MACH2 results are circle-dot and cross symbols, respectively.Both codes recover the expected profile.

FIG. 5 .
FIG. 5. Fuel radius (a) and implosion velocity V i (b) as a function of time in the ideal-EOS SZP2 run.The black line denotes a simulation in which the radiation transport operates normally, whereas the red line denotes the simulation in which the radiation transport is artificially switched off.

FeasibilityFIG. 6 .
FIG. 6. Profiles of thermal pressure (solid) and magnetic pressure (dashed) at the time of shock breakout in the ideal-EOS SZP2 runs, for (a) radiation physics turned off (t = 109 ns), and (b) radiation physics turned on (t = 107 ns).

FIG. 9 .
FIG. 9. Comparison of shell trajectories (i.e., fuel outer radius) from SZP1* simulations with MACH2 (dashed black) and FLASH (solid cyan).The load current resulting from the circuit model in FLASH (red) is also shown.

FIG. 10 .
FIG. 10.Comparison of mass density (solid) and magnetic field (dashed) at CR = 87 from SZP1* simulations from MACH2 (red) and FLASH (black).The fuel/liner interface is marked by a shortdash vertical line.

FIG. 11 .
FIG. 11.Comparison of the electron Hall parameter at CR = 87 from SZP1* simulations with MACH2 (red) and FLASH (black).The fuel/liner interface is marked by a short-dash vertical line.The fuel adjacent to the fuel/liner interface in the MACH2 simulation is significantly more magnetized than in the FLASH result.

Fig. 14
Fig.14shows a comparison of the electron, ion, and radiation temperatures from the FLASH model at CR = 100 and CR = 388 (stagnation).From this comparison, we observe that thermal losses have begun to dominate beyond CR = 100.These are primarily due thermal conduction, as we estimated in the previous analysis at the beginning of this section (see Eq. (3) and TableI).Meanwhile, density increases due to compression, eventually causing the fuel to stagnate when the pressure is sufficiently high.Note that the radiation

Feasibility
and Performance of the SZP: 1-D FLASH and MACH2 ble sources of discrepancy are differences in the codes' algorithms.In general, the FLASH SZP1* simulations reach higher (potentially unstable) CR values than MACH2 simulations, and MACH2 simulations reach higher temperatures than all FLASH simulations.The discrepant results highlight the sensitivity of the SZP1* configuration to heat transport processes (i.e., thermal conduction and radiation).The high CR values are the result of significant fuel thermal conduction losses.As previously discussed, the SZP1* concept would benefit from decreasing thermal conductivity via fuel magnetization, as was shown (perhaps erroneously) in the MACH2 model.Such fuel magnetization could be achieved experimentally by applying an axial magnetic field to the configuration.Despite the different results, all SZP1* simulations with both FLASH and MACH2 generally agree on reaching peak fuel ion temperatures above 15 keV.The highest-fidelity run, the FLASH multi-group diffusion model, reaches the lowest peak ion temperature (see Fig.15), which in turn shows the importance of accurate radiation transport modeling for SZP1*.