Simulation study of the influence of experimental variations on the structure and quality of plasma liners

The main goal of this work is to perform simulation studies of how experimental imperfections affect the structure and quality of plasma liners for plasma-jet-driven magneto-inertial fusion (PJMIF) and to validate simulations using experimental data. In the PJMIF concept, a spherically symmetric plasma liner, formed by the merger of a spherically distributed array of highly supersonic plasma jets, implodes on a magnetized target and compresses it to the fusion conditions.^1–3 Solving the so-called “stand-off problem” of solid liners, i.e., difficulties in assembling solid liners from a sufficient distance from the fusion hot spot, is among the main potential advantages of PJMIF compared to the solid liner-driven MIF. Simulation studies reported in this work have been performed in support of the Plasma Liner Experiment–ALPHA (PLX-α) project, the primary objective of which is to form and study a spherically imploding plasma liner with at least 36 and up to 60 merging plasma jets that are launched by an array of coaxial plasma guns from the periphery of a 2.74-m-diameter vacuum chamber. The first series of experiments operating with 6 jets have been recently completed.⁴ 

Plasma liners and the corresponding compression of plasma or gas targets have been studied by a number of analytic models and 1D hydrodynamic codes.^2,3,5–9,11 These works focused on the study of the averaged properties of idealized plasma liners, their ram pressures, Mach numbers, and hydrodynamic efficiencies, the dependence of these quantities on material parameters, atomic processes, and radiation, and the compression and fusion gains of plasma targets. As 3-dimensional processes that contribute to the deterioration of the quality of plasma liners are neglected in the spherically symmetric approximation, such works evaluate only an upper limit of the liner-target performance for each study case. First, 3D simulations of the detailed structure and the properties of argon plasma liners formed by the merger of 60 jets, reported in the study of Kim et al.,¹³ predicted a cascade of oblique shock waves during the jet merger process, seeding non-uniformities in plasma liners. This work continues detailed numerical simulations of plasma liners at conditions closely related to PLX, confirms previous predictions, presents comparisons with experiments, and studies the sensitivity of the structure and properties of liners to variations in experimental conditions. It must be emphasized that the present experiments are not intended to demonstrate fusion-worthy liner quality and uniformity, which will most likely require smaller merging angles between plasma jets and many more jets (e.g., hundreds of jets over a sphere of 3 to 4-m radius). The purpose of this work is to benchmark our models and codes such that they may be used to investigate and help design follow-on experiments with more jets at smaller merging angles.

The rest of this paper is organized as follows: Section II describes the numerical models and simulation setups. The analysis of numerical simulations that model two initial settings with different values of variations of the initial mass of gas in plasma guns and jet velocities and their role in the local and global properties of plasma liners is presented in Sec. III. The comparison of simulations with experiments is the subject of Sec. IV. Finally, we conclude this paper with a brief summary of our results and perspectives for the future work.

Simulation results reported in this paper have been performed using FronTier,¹² a code for multiphysics simulations of compressible hydrodynamic and low magnetic Reynolds number MHD flows that use the method of front tracking for the accurate resolution of material interfaces in multiphase flows. Plasma liner-specific physics models in FronTier include an equation of state for high-Z materials with atomic physics transformations, based on the Zeldovich average ionization model,⁸ and a radiation model in the thin optical limit approximation, in which the Planck emission opacity is calculated using data generated by the PROPACEOS code (see Appendix A in Ref. 14). Other details of numerical models and the use of FronTier for the simulation of PJMIF processes, including 3D simulations of the formation and implosion of plasma liners, are described in Refs. 13 and 15.

In this work, we study the merging process of 6 plasma jets and the formation and implosion of a section of the plasma liner by closely following the setup of the PLX-α experiment at Los Alamos National Laboratory. While the experimental chamber is capable of holding 60 plasma guns in a 4π configuration, the first series of experiments operated with 6 closest plasma guns, forming a small section of a spherical liner. The location of plasma guns in the experimental device is shown in Fig. 1. The simulation parameters are as follows. The radius of the chamber is 130 cm. We assume that the PLX chamber contains residual vacuum gas at a pressure of 1 mTorr and a density ρ₀ of ∼10^–9. The initial argon plasma jets have densities ρ = 2 × 10¹⁶ 1/cm³ = 1.327 × 10^–6 g/cm³, a temperature T = 2.5 eV, and a velocity of 34.7 km/s. The diameter and length of the jets are 8.5 and 10 cm, respectively. More details on experimental studies that quantify initial plasma parameters can be found in Ref. 4.

Ideally, identical plasma jets should be shot synchronously from the plasma guns at the periphery of the PLX chamber. In practice, however, there are variations in the injected mass and trigger times of the six guns. We are interested in the understanding of how such variations change the internal structure of plasma liners and their global properties. We study two different initial settings. In the first scenario, the initial masses of argon jets and their shot times experience random variations. In particular, we study the random variations of jet masses within ±10% and ±2%, together with random variations of shot time within 2 μs, 100, and 50 ns. Table I summarizes parameters for this series of simulations. For a better understanding of details of our numerical results, we list the exact parameters for each jet which were generated by random numbers.

In the second initial setting, the plasma guns are shot perfectly synchronously, but the amount of argon gas in the plasma guns varies by ±5%. We assume that plasma guns convert the same amount of energy into the kinetic energy of plasma jets and compute the corresponding variations in the initial jet velocity. Specific initial parameters are presented in Table II. Both simulation series are compared with the idealized simulation that implements identical initial states for each plasma gun.

Consider an array of 6 supersonic plasma jets, shot by synchronized plasma guns from the periphery of the PLX chamber. We start with a qualitative discussion of important processes occurring during the formation and implosion of liners that strongly influence their quality. As jets propagate towards the chamber center, they expand and cool down due to the adiabatic expansion and radiation losses. As the initial jet length is close to its diameter, the jets look spherical in shape in the middle of the chamber. PLX operates with somewhat longer plasma jets compared to those used in simulations. Since the main contribution of the liner to the target compression, to be explored in the future, is provided by the front part of the plasma jets, the reduction of the jet length in numerical simulations to 10 cm does not lead to essential differences in the estimation of main liner properties while significantly reducing the computational time. Justification for the above statement can be found in Ref. 7, which shows that using liners of increased thickness does not lead to higher target compression rates but reduces instead the hydrodynamic efficiency of liners, defined as the ratio of the internal energy of the compressed target to the initial energy of the liner.

Before discussing the details of the plasma liner evolution, we introduce definitions of the liner leading edge and the spherical slice. Let us take a complete 3D distribution of the plasma ram pressure ρv² at certain time during the simulation, where ρ is the plasma density and v is the velocity magnitude, and average it in angular coordinates within a spherical coordinate system that has its origin in the PLX chamber center. This results in a radial distribution of the averaged ram pressure with respect to the chamber radial coordinate. An example of such a radial distribution of ram pressure for 6 jets at time 36 μs is shown in Fig. 2. We define the liner leading edge R_L as the radial coordinate of the maximum value of the averaged ram pressure. For this particular example, the liner leading edge is at 7 cm.

We find it convenient to analyze typical states in the liner, its non-uniformity, and some global properties, by using distributions of hydrodynamic states on spherical slices drawn through the leading edge of the liner. An example of the density distribution in the leading edge of a plasma liner, demonstrating shock waves, is shown in Fig. 3. In the remainder of this paper, we will show states on spherical slices in the form of 2D plots (ignoring the curvature of spherical slices).

Having defined the liner leading edge and the spherical slice, we discuss qualitatively the main phases of the plasma jet merger and liner formation process using idealized simulations with identical initial states of plasma jets. At time 22 μs, corresponding to R_L = 58.5 cm, plasma jets are positioned closely to each other but still beyond their interaction distance [Fig. 4(a)]. At this stage, the highest values of density and ram pressure are located in the center of each plasma jet. When R_L = 30 cm at time 30 μs, plasma jets collide with each other, resulting in the formation of oblique shock waves [Fig. 4(b)]. Locations of the highest values of density move into planes between the interacting plasma jets, i.e., into the primary shock waves. A 1D distribution of density along a circular line shown in Fig. 3, drawn through shock waves and the jet centers, is depicted in Fig. 4(c). We have verified that the density and pressure jumps are in agreement with the theory of oblique shock waves.¹⁶ Their large jumps are explained by the low values of the adiabatic index γ ∼ 1.2 caused by atomic processes (ionization) and radiation in the location of shock waves. In addition, the single-fluid approximation, used in all simulations, does not resolve small interpenetration of rarefied edges of plasma jets and possibly over-predicts the strength of shock waves.

When R_L = 7 cm at time 36 μs, the interaction of primary shock waves is observed [Fig. 4(d)]. The highest values of density are now located along the central axis of the group of 6 jets, corresponding to the center of the secondary shock wave. The density values in the secondary shock exceed those in the primary shock waves and in the jet centers. Such a difference in states between plasma jet central regions and oblique shocks is the main cause of non-uniformities in plasma liners. For idealized cases with identical initial jet states, cascades of oblique shock waves are observed in simulations with any number of jets.

We now discuss the influence of experimental variations on the shock-wave structure and global properties of plasma liners. In the first initial setting, described in Table I, the distribution of density states on spherical slices is shown in Figs. 5(b)–5(e)–8(b)–8(e). For comparison, states from idealized simulations with identical initial conditions are shown in Figs. 5(a)–8(a). Mostly, due to large variations in jet shot times in “variation 1” simulation, and with some contribution from large jet-mass variations, some plasma jets did not start interacting with other jets on R_L = 58.5 cm at time 22 μs. This caused a significantly different structure of shock waves at 30 μs. However, when errors in the gunshot time were reduced from ±2μs to ±100 ns and then to ±50 ns, simulation data corresponding to variations 2, 3, and 4 demonstrate that the structure of primary shock waves at 30 μs remains almost identical to the ideal case. We observe that results are more sensitive to shot-time variations compared to jet-mass variations.

The later stages of the liner implosion are more sensitive to the small values of the experimental variations of initial conditions. As expected, the structure of shock waves is completely different to the unperturbed case for variation 1 at 36 μs, when the leading edge of the liner implodes to the radius of 7 cm. However, there are also visible changes in shock waves in simulations corresponding to variations 2–4. This simulation prediction was confirmed by experimental observations described in Sec. IV. The distributions of other states on spherical slices, such as ram pressure, hydrostatic pressure, and temperature, show similar features.

We now investigate how experimental variations affect the global properties of plasma liners. The averaged Mach number and ram pressure of a liner are the most important global properties that characterize the ability of the liner to compress plasma targets. In particular, it is very important to maintain high Mach numbers of liners during their evolution in order to obtain high values of fusion energy gain.¹⁰ Of course, this can be achieved only for a spherically symmetric array of plasma jets. In the case of the self-implosion of six jets, jets continue their motion beyond the chamber center. Nevertheless, the dependence of such a global characteristic as the Mach number on experimental variations is important for a better understanding of the liner performance.

In Figs. 9 and 10, we plotted the evolution of the averaged Mach number and the averaged ram pressure in imploding liners for simulations with identical initial conditions and with experimental variations described in Table I. We observe that these variations have a very small effect on the averaged Mach number. For smaller values of variations (variation 2–variation 4), all plots are very close to each other, at least before the leading edges reach the chamber center at 38 μs, and exhibit a typical increase in the Mach number caused by the primary shock waves at 33–34 μs. For the largest experimental variation values (variation 1), the overall behavior is still similar, but the increase in the Mach number due to the formation of primary shocks is not observed: the shock structure is very different in this case as different pairs of shocks form at different times. Similar conclusions can also be obtained about the evolution of the averaged ram pressure on the leading edge of the liner: three lines corresponding to smaller variations are close to each other and to the line representing the simulation with identical initial conditions. The simulation with the largest variations (variation 1) reaches a much smaller maximum value of the averaged ram pressure because jets reach the chamber center at significantly different times.

A multi-chord interferometer is installed in the PLX chamber to probe the liner plasma. The interferometer measures the laser light phase shift that is proportional to the integrated electron number density along the interferometer chords. Figure 11 gives a schematic of the interferometer chord placements. For a reference, a density distribution on a liner spherical slice is shown on the right at the correct angle to the interferometer schematic. This implies, for example, that chords 1 and 5 should measure signals from primary shock waves, chords 3 and 7 should probe plasma along jet centers, and the chords closer to the center should detect the secondary shock waves or combinations of primary and secondary shocks. At present, chords 7–12 are not installed in the PLX chamber, but we present simulation data along them as well.

Figure 12 presents an example of computed signals along the interferometer chords using simulations with initial states described in Table I. The purpose of this example is not to give comprehensive information on all interferometer readings at all time but rather demonstrate their sensitivity to the variations of initial jet parameters. Our results show that interferometer signals are quite sensitive to experimental variations, especially at earlier times, due to the fact that shock waves form initially thin structures in plasma liners: if the location of a primary shock is shifted, the corresponding interferometer signal changes significantly (see interferometer signals for the variation 1 case). However, if the structure of shock waves is more stable, interferometer signals on chords 1 and 3 may change by 10%-30%. Signals on the chords corresponding to the jet centers depend on jet lengths and thus are stronger for longer jets used in experiments.

In this section, we briefly summarize simulation results for the second initial setting, in which the initial mass of argon in plasma guns changes by ±5%, but the kinetic energy of jets remains the same and all jets are shot perfectly synchronously (see Table II). Figure 13 depicts density [images (a) and (b)] and ram pressure [images (c) and (d)] on spherical slices of the plasma liner at 30 μs (left column) and 36 μs (right column). As before, we observe that the structure of primary shock waves is unchanged under small variations, but the later stages of the evolution are more sensitive to variations (see images at 36 μs).

Figure 14 shows the averaged Mach number evolution in imploding liners with identical initial conditions (blue line) and with variations of initial conditions described in Table II (green line). This result and the evolution of the averaged ram pressure (Fig. 15) in the liner leading edge are also consistent with the previous simulations and show that the global properties of plasma liners are not sensitive to small experimental variations. Despite the relative stability of primary shock waves, interferometry signals corresponding to primary shocks (chords 1 and 5 in Fig. 16) are sensitive to small experimental variations at early time because of small displacements of narrow shock wave structures. The discrepancy between interferometry signals related to shocks reduces with time as the oblique shock regions widen [see Fig. 16(b)].

In this section, we confirm our simulation predictions by comparing numerical results with experimental data. The high-quality CCD images of imploding liners formed by six plasma jets have been obtained in recent experiments. An accurate comparison of CCD images with simulations is a rather difficult task as experiments and simulations operate with very different sets of quantities. In this paper, we adopt a simplified approach. Using simulation data, we approximate the experimental CCD images by assuming that each pixel is proportional to a line integral of the radiation power across the entire simulation domain in the normal direction to the CCD camera element. This approach ignores the spectral sensitivity of the CCD camera element and absorption along the light path. With such an interpretation of the CCD images, the comparison of simulations with experiments is presented in Fig. 17. The left image for every time represents the integrated radiation power distribution obtained from the simulation initialized as in Table II, and the middle image represents the same quantity obtained from the simulation initialized as in the variation 1 entry of Table I. Experimental results are shown in the right images.

Experimental data confirm the main features of the detailed liner structure predicted by simulations. At 26 μs, we observe the formation of primary shock waves. The brighter spots in the centers of six jets in experimental images can be explained by longer jets used in experiments compared to 10 cm long simulated jets. The primary shocks are observed in both simulation images at 26 μs, but the image corresponding to larger variations (the middle image) is a closer match to the experimental image. As the liner continues to implode, the structure of primary shock waves remains stable. At later times (38 and 41 μs), secondary shock waves are clearly visible in the center of both experimental and simulation images. While the shape of secondary shocks in the simulation with smaller variations of initial conditions remains symmetric star-like, only the strongest shocks are clearly visible in simulations with large initial variations and in experiments. In addition, most dominant shocks may change the direction in time (secondary shocks created by other pairs of primary shocks may start dominating at later time), as we see in simulations with large initial variations (top right images) at 38 and 41 μs. Such a dynamics of secondary shocks depends on a specific realization of random initial settings and does not always occur in simulations.

Experimental images confirm our prediction that the primary shock wave structure is much less sensitive to the experimental variations compared to the structure of secondary shocks. As the maximum values of the variation of jet parameters in real experiments are unknown, comparison to simulations may provide useful information on their amplitude.

New and improved interferometer data with significantly better jet-to-jet mass balance compared to that reported in the study by Hsu et al.⁴ will be reported in a separate paper. Both simulation and experimental interferometry results overlap due to their large error bars. The large error bars of the experimental data from the multi-chord interferometer can be explained by our simulation observation that interferometer signals are sensitive to experimental errors. However, because the current CCD images and interferometry data are available only for different experimental runs, their direct comparison at this point is not justified.

We have performed simulation studies of the merger of 6 plasma jets and the formation and implosion of plasma liners at conditions that model the PLX-α experiment at Los Alamos National Laboratory. The sensitivity studies of the detailed structure of plasma liners and their global properties to experimental variations in plasma guns that change the initial conditions of plasma jets and the comparison of simulations with experiments were the primary goals of this paper. Simulations predict that the primary oblique shock wave structure is preserved at small experimental variations. If initial conditions are too different in all jets (variation 1 entry in Table I), the topology of primary shocks is also changed. At later phases of the liner implosion, primary shocks and, especially, secondary shocks are more sensitive to experimental variations. The liner structure is more sensitive to the variations of jet shot times compared to the variations of jet masses. Small displacements of shock wave structures, which are confined to relatively narrow layers of high density and pressure plasma, cause significant changes in the interferometer readings, computed using the simulation data, especially at early time. As shock wave regions widen at later time, the interferometry readings become less sensitive for the small values of variations. Our studies also showed that the global properties of plasma liners (averaged Mach number and averaged ram pressure along the leading edges of plasma liners) are less sensitive to experimental variations.

Experimentally observable synthetic quantities have been computed using simulation data and compared with the available experimental data. Simulations have achieved a good agreement with experimental CCD camera images. The comparison of experimental CCD images with the corresponding images of the line-integrated radiation power obtained from simulations confirms our predictions of shock structures in plasma liners and their sensitivity to experimental variations. The shifting of shock wave locations due to variations in the initial jet states may also be responsible for large error bars reported in the previous studies of experimental interferometry data.⁴ The detailed studies of synthetic interferometry data will be continued, and the comparison of simulation with the relevant experimental data will be presented in a future work.

It must be emphasized that the present experiments are not intended to demonstrate fusion-worthy liner quality and uniformity, which will most likely require smaller merging angles between plasma jets and many more jets (e.g., hundreds of jets over a sphere of 3 to 4-m radius). The purpose of this work is to benchmark our models and codes such that they may be used to investigate and help design follow-on experiments with more jets at smaller merging angles. Future simulations will focus on the properties of plasma liners formed by the 4π-symmetric array of plasma jets, varying from 36 to 200 jets, and the compression of gaseous and plasma targets. A recently developed Lagrangian particle method, a consistent and convergent particle-based numerical method for hyperbolic hydrodynamic equations¹⁷ and elliptic problems,¹⁸ will also be used in future liner simulations due to its property of continuously adaptive resolution.

This research was supported by the Accelerating Low-cost Plasma Heating and Assembly (ALPHA) Program of the U.S. Department of Energy's Advanced Research Projects Agency-Energy (ARPA-E).