A novel wire-array Z-pinch platform has been developed to study the effect of a pulsed magnetic mirror field on a collisional high energy density plasma. The mirror is driven in series with the Z-pinch target and, therefore, scales with driver current. Axial confinement is observed based on visible imaging and axial flow velocity measurements. The presence of axial compressing J  × B force is determined indirectly based on Thomson scattering and interferometry measurements and corroborated by three-dimensional extended-MHD simulations. Compared to non-magnetized wire array Z-pinch, a modified pulsed mirror configuration is observed to increase precursor plasma thermal energy density by about 30%. If optimized, such a configuration could potentially improve magnetized liner inertial fusion performance by reducing axial plasma end loss.

Inertial confinement fusion (ICF) experiments have recently demonstrated burning plasma,1 ignition,2 and scientific breakeven.3 However, more than an order of magnitude increase in fusion yield is required to achieve economical power generation. By suppressing electron thermal conduction and capturing α-particle energy, magnetization of ICF fuel boosts fusion neutron yield in both laser-driven experiments4–6 and current-driven magnetized liner inertial fusion (MagLIF).7 These magnetized configurations typically rely on external field coils to produce a spatially and temporally uniform axial magnetic field within the fuel prior to compression. The field constrains particle motion perpendicular to the field line while unconstrained axial motion manifests as thermal conduction loss and plasma end loss. A field configuration that can reduce axial loss is, therefore, desirable. Even for high-aspect ratio, fast implosion configurations like MagLIF, where the axial loss of α-particles is typically small compared to the radial losses,7 improved axial confinement of alpha-particles would relax the stringent requirement of short implosion time (<100 ns) and high laser preheat energy (∼6.5 kJ).8 

To improve axial confinement, we explore a novel application of a pulsed magnetic mirror field. Unlike standard steady-state mirror traps, the pulsed mirror theoretically provides an axially compressing J θ × B r force9 without the conservation of magnetic moment and can, therefore, be applied to even highly collisional plasmas (ωiτii < 1, where ω i is the ion gyrofrequency and τ i i is the ion–ion collision time). Due to the finite resistivity, the dynamic mirror field induces an azimuthal diamagnetic skin current ( j θ ) that interacts with the radial component (Br) of the mirror field to produce axial compression.9 Alternatively, the process can be viewed as a manifestation of Lenz's law in which an azimuthal image current is induced and repelled by the azimuthal current in the electrodes and the azimuthal current produced by the axially varying flux compression (Fig. 1). The Br, however, will also induce rotation in the Z-pinch, reducing its convergence ratio due to the centrifugal barrier effect.10 For an axial mirror field, the induced angular momentum is in opposite directions on the two sides of the array due to the opposite direction of Br. Ideally, there is no rotation at the array center due to the absence of Br and any rotation advected to the center will cancel out by mixing, which potentially provides an additional pathway for plasma heating.

FIG. 1.

Pulsed mirror axial compression scheme: The purple region and the blue lines represent the plasma and the magnetic field lines, respectively.

FIG. 1.

Pulsed mirror axial compression scheme: The purple region and the blue lines represent the plasma and the magnetic field lines, respectively.

Close modal

We emphasize that the induced axial J θ × B r force by the pulsed mirror is the focus of this study. Classical mirror traps based on the conservation of magnetic moment require magnetization of charged particles (i.e., ωiτii > 1), which does not apply in our highly collisional plasma. Throughout the rest of the article, the word mirror is used exclusively to refer to the field geometry of a traditional mirror trap, not its confinement mechanism.

To measure the effect of a pulsed mirror on collisional plasma, we developed a wire array Z-pinch platform that can simultaneously drive the mirror field and the plasma implosion. Early investigations of pulsed magnetic mirror have observed trapping of an electron ring,11 and qualitatively improved confinement of exploding single wire plasmas.12 Theoretical study has shown that the end loss of a linear theta pinch can be reduced by pulsed mirrors.9 To our knowledge, our experiments are the first direct measurements of collisional plasma dynamics inside a pulsed mirror on sub-microsecond timescales. A cylindrical eight-wire-array Z-pinch was chosen to be the plasma source because of its compressibility on a 1-MA university-scale machine, compatibility with auto-magnetizing electrodes, and well-studied implosion physics.13–19 Furthermore, while the density of the wire array Z-pinch precursor differs from the preheated fusion fuel in MagLIF (Table I), both plasmas exhibit high thermal to magnetic pressure ratios ( β 1), high magnetic Reynolds numbers (Rm 1) and low ion hall parameters (ωiτii 1). These similarities make wire array precursor plasma suitable for the study of dynamic magnetic field effects on MagLIF laser-preheated plasma.

TABLE I.

Summary of plasma parameters of wire array Z-pinch precursor plasma and preheated MagLIF plasma before implosion.

Al wire array precursor columna MagLIF fuel pre-implosionb
Bz (T)  1.0c  15.9 
ne (cm−3 1.0 × 1019  3.0 × 1020 
ni (cm−3 1.0 × 1018  3.0 × 1020 
Te (eV)  92  100–500 
Ti (eV)  190  100–500 
⟨Z⟩d  9.7  1.0 
λe  5.7  6.1 
λmfp (μm)f  1.3 × 10−2  8.0 × 10−1 
βg  370  95 
xeh  7.9 × 10−2  5.3 × 10−1 
xii  1.8 × 10−3  1.2 × 10−2 
Rm j  230  3700 
Al wire array precursor columna MagLIF fuel pre-implosionb
Bz (T)  1.0c  15.9 
ne (cm−3 1.0 × 1019  3.0 × 1020 
ni (cm−3 1.0 × 1018  3.0 × 1020 
Te (eV)  92  100–500 
Ti (eV)  190  100–500 
⟨Z⟩d  9.7  1.0 
λe  5.7  6.1 
λmfp (μm)f  1.3 × 10−2  8.0 × 10−1 
βg  370  95 
xeh  7.9 × 10−2  5.3 × 10−1 
xii  1.8 × 10−3  1.2 × 10−2 
Rm j  230  3700 
a

Shot 7046 with pulsed mirror field.

b

Bz and densities are based on Z facility shot z3289 reported in Ref. 7, and temperature ranges are from Ref. 8. Magnetic parameters are calculated assuming Te = Ti = 100 eV, a characteristic velocity of 100 km/s, and a characteristic length of 1 cm.

c

Average value along wire array axis obtained from Ansys simulation scaled to a load current of 0.62 MA.

d

Average ionization state based on the FLYCHK table.20 

e

Landau–Spitzer Coulomb log. Using PlasmaPy.21 

f

Mean free path. Using PlasmaPy.21 

g

Plasma beta. Using PlasmaPy.21 

h

Electron hall parameter. Using PlasmaPy.21 

i

Ion hall parameters. Using PlasmaPy.21 

j

Magnetic Reynolds number. Using PlasmaPy.21 

To produce the dynamic axial fields, twisted conduction paths are integrated directly into the Z-pinch electrodes and the wire arrays. Many such auto-magnetizing configurations have been studied, including auto-magnetizing MagLIF liners,22,23 Dynamic Screw Pinch,24 wire array screw pinch,19,25–27 magnetic cusps,28 dynamically magnetized single-wire Z-pinch,29 and magnetized hybrid X-pinch.30 By tailoring the design of the helical electrodes and twist in the wire array, a sub-microsecond pulsed mirror can be produced.

In Z-pinches, the interaction between an axial current passing through a cylindrical target (e.g., wire array, metal liner, gas puff) and the induced azimuthal magnetic field produces an inward radial pinching force. A standard cylindrical wire array Z-pinch can be divided into three phases: (1) ablation of wire surface toward the array axis, (2) implosion of the wire cores, and (3) stagnation of the imploding plasma on the axis. During the ablation phase, coronal plasma ablated off the wire surface converges on the array axis and forms a plasma precursor column. The precursor column is not confined by the J z × B θ force, but by the dynamic pressure of the continuous inward ablation stream.19 As a result, the column does not exhibit MRT instability and remains relatively stable throughout the ablation phase. The density of the precursor column is also significantly smaller than that of the stagnation column, enabling laser probing in the visible spectrum. These features make the plasma precursor a relatively diagnosable environment to study the effects of a pulsed mirror.

In Sec. II, we describe the experimental setup, including the COBRA pulsed-power generator, the auto-magnetizing wire array load, and the plasma diagnostics. In Sec. III, we discuss the experimental results from pulsed mirror configurations and compare them to axially magnetized non-mirror configurations. Section IV presents three-dimensional (3D) extended-MHD (XMHD) simulations of the experiments using the PERSEUS31 code. We conclude our findings in Sec. V.

The experiments were carried out on the COBRA pulsed power generator.32 It is a double Marx bank driver that stores approximately 100 kJ of electrical energy. In this study, the two generators were fired successively to deliver a 250-ns rise time 1-MA peak current pulse through the load. The average and maximum variation of machine current for the shots reported in this study are shown in Fig. 2(b), demonstrating good repeatability of current waveform up to and during the diagnostic time window (approximately 100–130 ns after current start). Due to a temporary load voltage reversal at approximately 145 ns into the current pulse, we focus our laser diagnostics to the period before this time when the magnetic mirror field has been monotonically rising.

FIG. 2.

Experimental setup: (a) laser diagnostics setup around a wire array. The pink cylinder on the axis and the blue spheres indicate the approximate locations of the plasma precursor structure and Thomson scattering (TS) collection volumes, respectively. (b) Average and maximum variation of the current pulses from experiments and the model current function used in the extended-MHD (XMHD) simulations.

FIG. 2.

Experimental setup: (a) laser diagnostics setup around a wire array. The pink cylinder on the axis and the blue spheres indicate the approximate locations of the plasma precursor structure and Thomson scattering (TS) collection volumes, respectively. (b) Average and maximum variation of the current pulses from experiments and the model current function used in the extended-MHD (XMHD) simulations.

Close modal

The cylindrical wire-arrays used in the experiments consist of eight 17-μm diameter Al wires threaded through CuW (25% Cu:75% W) 1.6-mm diameter tube electrodes. The wire array length is set by the vertical displacement between the electrodes to be 9.4 ± 0.3 mm. The array diameter is 12.9 ± 0.1 mm and the inter-wire spacing is 4.9 ± 0.1 mm based on the wire–electrode contact points. The uncertainty reported here is one standard deviation of measurements. The helical conduction paths that generate the axial magnetic field are formed by effectively twisting the tube and wire section independently [Fig. 3(a)]. By adjusting the twist angles of wires (θw) and of tubes (θt), we create a variety of axial field profiles. The θw used in this study is no more than 20°, which results in a maximum diameter reduction of ∼2% at the waist of the wire arrays. In addition to the azimuthal twist, the tubes have an approximately 4° inward radial tilt to ensure consistent wire contact. The wires ablate at constant velocity15 and do not implode during the first 150 ns of the COBRA current pulse, providing a relatively stable environment for measurements.

FIG. 3.

Auto-magnetizing wire array: (a) a photo of a θ t = 40 ° , θ w = 20 ° configuration wire array. θt and θw denote the twist angles in degree of the tubes and wires, respectively. The markers in (b) are d B d t probe array measurements of the axial field inside over-massed W wire arrays with three different combinations of twist angles. The dashed lines in (b) are Ansys Maxwell 3D simulations, which do not include plasma. All Bz measurements and simulations are scaled to 0.62 MA assuming linear scaling. (c)–(e) Simulated magnetic field streamlines for the three twist combinations. The 3D images are clipped at the x-plane.

FIG. 3.

Auto-magnetizing wire array: (a) a photo of a θ t = 40 ° , θ w = 20 ° configuration wire array. θt and θw denote the twist angles in degree of the tubes and wires, respectively. The markers in (b) are d B d t probe array measurements of the axial field inside over-massed W wire arrays with three different combinations of twist angles. The dashed lines in (b) are Ansys Maxwell 3D simulations, which do not include plasma. All Bz measurements and simulations are scaled to 0.62 MA assuming linear scaling. (c)–(e) Simulated magnetic field streamlines for the three twist combinations. The 3D images are clipped at the x-plane.

Close modal

The axially varying magnetic fields formed by the wire arrays can be characterized by the mirror ratio, rmirror, which is defined in this article as the ratio of Bz at the two ends of the array to the Bz at the center of the array axis. A rmirror> 1 corresponds to a pulsed mirror field, while a rmirror ≤ 1 corresponds to a non-mirror field. To characterize the magnetic field produced by the arrays and the tubes, the initial vacuum magnetic fields at multiple axial positions are measured by an axial d B d t probe array for three twist (θw and θt) combinations using over-massed W wires that minimize plasma formation. 3D Ansys Maxwell33 simulations of wire arrays with the same twist combinations are performed using the Eddy Current Solver with 2.5 MHz AC as the source of excitation. Figures 3(c)–3(e) show the simulated magnetic field streamlines for the three twist combinations. In Fig. 3(b), the on-axis Bz from the experiments and that from Ansys simulations are both scaled to 0.62 MA assuming linear scaling and plotted against the axial position as markers and dashed lines, respectively. We see general agreement between the simulations and the experimental data, confirming the formation of a vacuum magnetic mirror field. A higher field measured in θ t = 40 ° , θ w = 0 ° configuration is consistent with the advection of off-axis magnetic field flux by the converging plasma flow. This effect is less pronounced in the other configurations due to the reverse flux produced by the wire twist. Disagreement at z = 12.2 mm could be due to the exclusion of flux by plasma formed between the anode opening and the probe in the experiments. Throughout the rest of the article, the vacuum magnetic fields reported for different experimental array configurations are based on the Ansys simulations, which do not include the plasma effect.

The plasma properties of the wire array precursor are probed with a combination of three-axis Thomson scattering (TS) and laser interferometry. The Thomson scattering system uses a 10-J Nd:YLF laser at 526.5 nm with a pulse FWHM of 2.3 ns34 that passes horizontally through the array axis. The laser radiation scattered orthogonally to the propagation direction is captured by three separate fiber bundles as shown in Fig. 2(a). Each fiber bundle consists of a linear array of 100-μm diameter fibers that are 125 μm apart center-to-center. The two side-on fiber bundles have a total field of view of 2.2–2.5 mm along the laser path centered on the array axis, while the axial fiber bundle consists of twice the number of fibers and a total field of view of 6–9 mm along the laser path. The intersection of the laser path and the fiber view defines the scattering volumes. The fiber bundles are coupled to two 0.75-m Czerny–Turner spectrometers and the spectra are captured by gated intensified CCD cameras. The laser interferometer is set up in the Mach–Zehnder configuration using a 150-ps 532-nm or 355-nm laser pulse. The interference pattern obtained during the experiment is compared with a reference taken immediately before the shot to extract phase information. By interpolating the phase values between interference fringes, a 2D map of line-integrated electron density, n ¯ e, over the cross section of the laser beam is obtained. Both the TS and the interferometry lasers are aligned with respect to the same metal pin before the experiments to ensure spatial alignment. Due to random jitter, the TS and interferometry lasers have an average time offset of 1.8 ns and a maximum offset of 8.6 ns for all shots reported in this article.

Due to the limited coverage of the side-on n ¯ e map and deviation of the discrete wire array geometry from cylindrical symmetry, local ne cannot be satisfactorily obtained by standard inverse Abel transform. Instead, local ne’s were determined by fitting a density profile to the total scattering intensity profile obtained from adjacent scattering volumes along the TS laser path. Due to the eightfold rotational symmetry of the load array, we can assume the n ¯ e along both the TS and the interferometry laser path to be the same for a given radial and z position. This equivalence allows us to normalize the density profile obtained from TS by the n ¯ e from interferometry, resulting in spatially resolved ne’s along the TS laser path.

Since the total scattering intensity is proportional to the product of ne and total scattering cross section, S(k), the dependence of S(k) on ne and plasma temperatures is accounted for by a two-pass fitting procedure and using Salpeter approximation for S(K).36,37 In the first pass of fitting, the scattering intensity is approximated to be proportional to local ne only. Using Te obtained from the first pass and assuming Ti = Te, we construct an intensity profile function that depends explicitly on local ne’s of all the scattering volumes along the laser path. This profile function is then fitted to the measured scattering intensity in the second pass to obtain the final values of plasma ne’s. These electron densities are then included in the fitting of TS spectra to determine plasma temperatures and velocity. Uncertainty on the density is determined by the difference between the values obtained by different fiber bundles. The difference in the observed scattering intensity is partially attributed to the refraction of the scattered light.

From the Doppler shift of the Thomson scattering spectra collected from three orthogonal directions, velocity along three linearly independent vectors can be obtained. Two opposing fiber bundles in the r θ plane allow for the measurement of velocity parallel and perpendicular to the laser wave vector in this plane. The axial fiber bundle captures downward scattering that contains Doppler shift due to plasma movement parallel and perpendicular to the laser wave vector in the r z plane. Noting that the velocity component parallel to the laser wavevector is the same in both measurement planes, all three orthogonal components of the flow can be resolved.

Te and Ti are obtained by fitting the scattered spectra with a collisionless Thomson scattering form factor convolved with the instrumental function and a Gaussian profile.34 The Gaussian profile is used as a first-order approximation to account for additional broadening due to collisions, velocity gradient, and turbulence.38 An additional laser profile is also included in the fit to account for the specular reflection of the laser off solid wires or aluminum vapor. There are eight fit parameters: Te, Ti, fluid velocity, Gaussian profile width, electron–ion relative drift velocity, relative intensity of the reflected laser signal, a highly constrained laser wavelength, and an intensity scaling factor. The error bars on the fit parameters are estimated using a Monte Carlo technique described in Banasek et al.34 We note that laser can heat plasma through inverse bremsstrahlung absorption and affect the time-gated TS temperature measurements.34,35

In addition to laser probing, we employ a suite of imaging and electrical diagnostics to monitor the plasma throughout the implosion. A 12-frame high-speed visible camera (400–850 nm) is used to monitor the dynamics of the wire-array plasma throughout the current pulse. Extreme-UV cameras with 5-ns gate time and sensitivity to emission between 10 and 100 eV are positioned to capture the precursor column on the axis with a direct line-of-sight both side-on and end-on. A Rogowski coil and a voltage monitor located below the array monitor load current and voltage respectively. While x-ray measurements are not presented here, multiple filtered diamond photoconducting detectors (PCD) are used to monitor x-ray output.

We experimented with a variety of electrode and wire array twist combinations ( θ t , θ w ) to determine the effect of the pulsed mirror. This article focuses on a set of non-mirror configurations with non-twisted θ t = 0 ° tube arrays as electrodes and a set of mirror configurations with twisted θ t = 40 ° tube arrays as electrodes. By twisting the wire array θ w , we modify the axial field profile to achieve either a higher mirror ratio or non-mirror (rmirror < 1) axial field. Our array hardware and loading procedure allow fine control of θ t and θ w with a precision of ± 1°, reducing shot-to-shot variation.

Figure 4 shows visible images of an implosion of the wire array with and without a pulsed mirror field. Here we can see the axial confinement present when the mirror field is applied. The pulsed mirror field modifies the structure and dynamics of the wire array precursor plasma, resulting in an axially compressed density distribution. In Fig. 4(b), the precursor plasma inside the pulsed mirror ( θ t = 40 ° , θ w = 20 ° , rmirror ≈ 19) is initially concentrated about the center of the array and gradually forms a hollow hourglass structure. The hourglass structure increases in radius over time but is axially compressed when compared to the precursor column of the standard Z-pinch ( θ t = 0 ° , θ w = 0 ° ). Furthermore, we observed no axial end loss in the pulsed mirror as indicated by the plasma jet that extends beyond the initial wire array length in standard Z-pinch. These qualitative observations suggest a trade-off between radial compression and axial compression in mirror configurations. To determine the mechanism for the formation of the hourglass structure and axial concentration of plasma at the array center in the pulsed mirror configuration, we directly measure the local velocity field.

FIG. 4.

Effect of an axial pulsed mirror: (a) computer aided wire array drawing and (b) visible emission images of the standard non-magnetized wire array Z-pinch (top row) and the pulsed mirror configuration (bottom row), respectively.

FIG. 4.

Effect of an axial pulsed mirror: (a) computer aided wire array drawing and (b) visible emission images of the standard non-magnetized wire array Z-pinch (top row) and the pulsed mirror configuration (bottom row), respectively.

Close modal

Figure 5 shows Thomson scattering (TS) and interferometer measurements taken during the ablation phase. They describe the density distributions and flow velocities just before voltage reversal. Figure 5(a) shows line-integrated electron density contours with r-z plane velocity projection overlayed for multiple shots with the same twist combination θ t = 40 ° , θ w = 20 °. The TS laser passes horizontally through the array axis at 1.6–0.3 mm below the axial position of the peak plasma density. The vertical position of the TS laser relative to the peak density, z*, has an uncertainty of ±0.2 mm. The reported times correspond to the starts of the TS laser pulses. The density contours show shot-to-shot variation but demonstrate reproducibility of the hourglass precursor structure. The measured velocities are averaged across the scattering volumes for each shot and plotted against z* to show trends in the plasma flow field.

FIG. 5.

Plasma flow and axial confining pressure inside the precursor structure under pulsed mirror field: (a) contour maps of line-integrated electron density about the neck of precursor from multiple shot experiments with the same wire array configuration θ t = 40 ° , θ w = 20 ° with velocity projection in the r-z plane from TS at the reported times overlayed. The absence of color indicates the absence of analyzable interference patterns in those regions. The dashed black line and the green arrow represent the array axis and the TS laser, respectively. Red dots indicate the locations of peak density. Green error bars represent uncertainty in the vertical position of the TS laser. (b) Averaged velocity components for each shot plotted against TS position relative to the peak density. (c) Averaged axial J × B force and its components calculated assuming dynamic MHD equilibrium and azimuthal symmetry. The error bar on the ⟨ ( J × B ) · z ̂⟩ is defined as the contribution from the term containing v z z.

FIG. 5.

Plasma flow and axial confining pressure inside the precursor structure under pulsed mirror field: (a) contour maps of line-integrated electron density about the neck of precursor from multiple shot experiments with the same wire array configuration θ t = 40 ° , θ w = 20 ° with velocity projection in the r-z plane from TS at the reported times overlayed. The absence of color indicates the absence of analyzable interference patterns in those regions. The dashed black line and the green arrow represent the array axis and the TS laser, respectively. Red dots indicate the locations of peak density. Green error bars represent uncertainty in the vertical position of the TS laser. (b) Averaged velocity components for each shot plotted against TS position relative to the peak density. (c) Averaged axial J × B force and its components calculated assuming dynamic MHD equilibrium and azimuthal symmetry. The error bar on the ⟨ ( J × B ) · z ̂⟩ is defined as the contribution from the term containing v z z.

Close modal

The radial velocity, ⟨ v r⟩, near the neck of the precursor structure is observed to be predominantly compressional (<0) and increases in magnitude as the measurement position moves closer to the peak density [Fig. 5(b)]. The gradual reversal of ⟨ v θ⟩ from clockwise to counterclockwise looking top down as z* increases indicates a reversal of the J z × B r torque. Since J z is always pointing downward, this torque reversal can be explained by the reversal of B r, which is consistent with the axial mirror field profile. The observed rotational velocity, ⟨ v θ⟩, can support a centrifugal barrier force, which explains the formation of the hourglass structure, since the internal axial field pressure alone cannot balance the compressing force at the observed convergence ratio.

Axial compression (⟨ v z⟩ greater than 0 at z* < 0) is directly observed in shot 7046 with TS measurements at z* = −0.4 ± 0.2 mm [Fig. 5(a), fifth plot from the left]. Measurements at nearby locations in other shots, however, do not demonstrate compression flow, which could be due to shot-to-shot variation. Nevertheless, the absence of compression flow does not exclude the presence of an axially compressing J × B force. To calculate the average J × B force in the z direction, ( J × B ) · z ̂ , we solve the MHD equation of motion along the z-axis by assuming dynamic equilibrium ( v z t = 0), cylindrical symmetry ( y = 0 on the x-axis, which is defined to be the TS laser vector), isotropic thermal pressure, and negligible plasma viscosity. Upon elimination of all the terms with a zero value, the equation of motion is reduced to a force balance equation:
J × B · z ̂ = ρ v x x + v z z v z + P z ,
(1)
where the angle brackets indicate averages over the radial positions. ρ and v are known from interferometry and TS. To obtain ρ, we first calculate the ni by assuming quasi-neutrality and using the FLYCHK table20 to determine the average ionization state. v z x can be obtained from adjacent TS scattering volumes along the laser vector. P z can be approximated by taking the line-integrated electron density values from interferometry on the array axis and assuming thermal equilibrium (i.e., T z = 0, and T i = T e, which is also obtained from TS). This procedure leaves v z z as the only unknown, which can be approximated as < v z > z. Using the v z in Fig. 5(b) and assuming linear dependence on z, we estimate the spatial derivative to be 5.4 × 10 7 1 s. We plot ρ v x x v z , P z , and J × B · z ̂ in Fig. 5(c) with the absolute value of the approximated ρ v z v z z as the error bars on J × B · z ̂ . Shot 6933 [The second set of markers from right in Figs. 5(b) and 5(c)] lacks sufficient interferometry data, and its P z is inferred based on data from shot 6856. We see that multiple shot experiments exhibit a compressing axial J × B force [i.e., J × B · z ̂ greater than 0] and an overall average compression force of 8 × 10 10 P a m. The two experiments with expanding force (shots 6927 and 7045) show the lowest density, consistent with the hypothesis that the axial J × B force is responsible for plasma compression. We note that the uncertainty associated with the ρ v z v z z can be significant compared to other terms and could affect the determination of compression vs expansion for some shots.

Figure 6 shows the Thomson scattering and interferometer measurements of precursor plasma density and temperature and their dependence on mirror ratio and axial magnetic field strength. The introduction of an axial magnetic field can reduce the radial compression ratio and thus thermal energy density. This trend is observed by keeping θ t = 0 ° and gradually increasing θ w from 0 ° to 20 ° [Fig. 6(a), three plots on the right]. To isolate the effect of the axial mirror field profile from the axial field in general, a series of mirror and non-mirror configurations were explored. The vacuum Bz at the array center for these configurations was simulated with Ansys Maxwell 3D33 and plotted in Fig. 6(b) with their rmirror along the y-axis. The set of angle combinations allows us to explore a wide parameter space in the axial magnetic field profile, both mirror (rmirror > 1) and non-mirror (rmirror < 1 or non-magnetized).

FIG. 6.

Effect of vacuum field strength on precursor structure plasma: (a) line-integrated electron density maps of the precursor structures for different combinations of tube twists and wire twists. The absence of color indicates the absence of analyzable interference patterns in those regions. (b) Simulated vacuum field strength at the array center and the mirror ratio for these configurations at I = 0.62 MA. (c) and (d) Electron densities and thermal energy densities averaged from TS measurements within a radius of 1 mm for each shot experiment. All measurements are taken within a span of 15 ns during the current rise.

FIG. 6.

Effect of vacuum field strength on precursor structure plasma: (a) line-integrated electron density maps of the precursor structures for different combinations of tube twists and wire twists. The absence of color indicates the absence of analyzable interference patterns in those regions. (b) Simulated vacuum field strength at the array center and the mirror ratio for these configurations at I = 0.62 MA. (c) and (d) Electron densities and thermal energy densities averaged from TS measurements within a radius of 1 mm for each shot experiment. All measurements are taken within a span of 15 ns during the current rise.

Close modal

The spatial averages of electron density and thermal energy density from these experiments are plotted against their simulated vacuum center Bz in Figs. 6(c) and 6(d), respectively. All plasma properties are averaged across a span of 2 mm about axis, and the error bars represent the difference between two fiber bundles for ⟨ne⟩ and the average of standard deviations of local values for thermal energy density within this span. Measurements are at 1.8 mm above midplane for all θ t = 40 ° configurations and between 0 and 1.8 mm for all θ t = 0 ° configurations. All measurements are taken within a span of 15 ns during the current rise. Among all experiments, shot 7056  θ t = 40 ° , θ w = 15 ° produced the highest electron density. The thermal energy density is 30% higher than the non-magnetized shot 7055. The increase in thermal energy density is partially attributed to the increase in plasma density. We note that not all shots with mirror profile achieved higher thermal energy density when compared to non-mirror configurations with comparable axial field strength. One factor that contributes to the relatively low measured density is the deviation of the TS laser from the neck of the hourglass. Furthermore, flux compression could significantly alter the axial field profile, especially for mirror configurations with reverse wire twist.

3D extended-MHD (XMHD) simulations using the PERSEUS code31 were conducted to determine the plasma current and the dynamic magnetic field. While classical mirror trapping is a kinetic behavior that cannot be simulated by PERSEUS, the axial J × B force can be captured by the XMHD code. The 3D geometry of the wire array was modeled after the experiments, and the current pulse waveform used in the simulations was obtained by fitting a double-Gaussian function to the experimental current trace [as shown in Fig. 2(b)]. The code uses a composite resistivity model consisted of a reduced Spitzer resistivity and a density dependent term that fits molecular dynamics calculations by Desjarlais et al.39 

The XMHD code lacks a separate equation for Te, electron thermal conduction, and radiation transport. These limitations can contribute to disagreement between the wire array Z-pinch simulations and experiments. As the wire array ablation streams converge on axis, ions gain significantly more thermal energy and becomes hotter compared to electrons due to their larger mass.13 Over time, Te and Ti equilibrate and the thermal energy are radiated away by the electrons.13 The absence of electron thermal conduction and radiative cooling would, therefore, result in a hotter precursor plasma structure with higher thermal pressure and a reduced resistivity. These discrepancies can affect the simulated axial J × B force indirectly by modifying the magnetic field dynamics inside the wire array.

The simulation results for the θ t = 40 ° , θ w = 20 ° configuration array at 120 ns are shown in Fig. 7, demonstrating qualitative agreement with experimental plasma dynamics. The simulation reproduces the hourglass precursor structure as indicated by the log electron density contour [Fig. 7(b)]. Away from the neck we see counter-rotating plasma flows [Fig. 7(d)], which is consistent with the experimental measurements. From the qualitative agreement in plasma dynamics, we infer similar agreement in magnetic field dynamics between the simulations and the experiments.

FIG. 7.

PERSEUS extended-MHD simulation of θ t = 40 ° , θ w = 20 ° configuration at t = 120 ns: (a) 3D plasma structure, r-z plane slices of (b) log ne, (c) Bz, (d) vy, and (e) Jy. The surface streamlines in (c) represent magnetic field line. (f) z component of J × B force density on r-z plane (f-i), z = 1.6 mm plane (f-ii), and z = −1.6 mm plane (f-iii). (g) Lineouts along green and purple line segments in (f). The simulation reproduces the hour-glass structure, rotation reversal observed in the experiments, and predicts axial compressing J × B force.

FIG. 7.

PERSEUS extended-MHD simulation of θ t = 40 ° , θ w = 20 ° configuration at t = 120 ns: (a) 3D plasma structure, r-z plane slices of (b) log ne, (c) Bz, (d) vy, and (e) Jy. The surface streamlines in (c) represent magnetic field line. (f) z component of J × B force density on r-z plane (f-i), z = 1.6 mm plane (f-ii), and z = −1.6 mm plane (f-iii). (g) Lineouts along green and purple line segments in (f). The simulation reproduces the hour-glass structure, rotation reversal observed in the experiments, and predicts axial compressing J × B force.

Close modal

The simulation suggests more complicated field dynamics than a monotonically rising mirror field. A reversed axial field is observed in the simulation [Fig. 7(c)], which is likely due to the advection of downward magnetic flux by the ablation stream. J θ and axial J × B force density are plotted on the r-z plane in Figs. 7(e) and 7(f-i), respectively. Additionally, axial J × B force on z = ±1.6 mm plane (z = 0 corresponds to array midplane) are plotted in Figs. 7(f-ii) and 7(f-iii), respectively, to show azimuthal dependence. A strong clockwise azimuthal current surrounds the neck of the precursor structure, leading to an axially compressing J × B force both above and below the array midplane. By taking lineouts in these two planes [Fig. 7(g)], we see a compressing force density up to 9 × 1010 Pa/m, which is in reasonable agreement with the experimental average.

The simulated plasma and field dynamics is observed to strongly depend on the resistivity model. As we increase the resistivity, we observe an increase in upward-pointing magnetic flux about the array axis. This change is consistent with increased diffusion of the axial flux from the electrodes. The strong effect of varying the resistivity suggests that field diffusion is non-negligible despite the high magnetic Reynolds number calculated from the experimental plasma parameters. Nevertheless, we can reproduce an axial-compressing J × B force at up to 4.5 times the resistivity used in the simulation that produced Fig. 7.

Axial compression of the wire array Z-pinch is observed in this pulsed mirror configuration through side-on interferometry measurement of electron density distribution and TS measurement of compressional flow. Axial confining J × B force is indirectly measured in experiments, supporting the end-plugging effect of the pulsed mirror. Plasma thermal energy density at the center of the precursor structure is observed to be up to 30% higher in mirror configurations when compared to the non-mirror configurations. The 3D XMHD simulation of the pulsed mirror experiments demonstrates the presence of axial confining J × B force when the initial mirror field profile has been distorted by the converging plasma streams.

Based on the observed axial confinement, the pulsed mirror could potentially reduce the end loss in MagLIF following Laser preheat and convert the cylindrical implosion into a quasi-spherical implosion if the mirror field strength is properly scaled up. Furthermore, we expect the classical mirror effect to be significant in higher energy density regimes such as the implosion and stagnation phases of MagLIF on Z machine when hall parameters can be orders of magnitude higher. The classical pulsed mirror effect can potentially further improve axial confinement and adiabatically heat11 the plasma.

More generally, with careful design of the field geometry, the pulsed mirror configuration demonstrates a potential pathway to reduce the asymmetry introduced by the axial field in magnetized implosion. The XMHD simulation suggests that significant azimuthal current can be induced in the plasma due to flux compression alone. This may relax the requirement for a fast rise time of the externally applied axial magnetic field for short timescale (<10 ns) laser-driven ICF experiments.

Research is supported by NNSA Stewardship Sciences Academic Programs under DOE cooperative Agreements Nos. DE-NA0003764 and DE-NA0004148. We thank Todd Blanchard for fabricating the wire array hardware and tube electrodes, Harry Wilhelm, and Dan Hawkes for COBRA operation and maintenance. We thank Dr. Jacob Banasek for the Thomson scattering analysis code (thomsonpy) used for fitting the experimental spectra. We thank Dr. Hannah Hasson for the code used to align the high-speed camera images. We also thank Professor David Hammer and Dr. John Greenly for the helpful discussions. We acknowledge the use of Swanson Simulation Lab at Cornell University for Ansys Maxwell simulation and the use of the Ansys software for CAD drawing.

The authors have no conflicts to disclose.

C. Chen: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Project administration (lead); Software (supporting); Writing – original draft (lead); Writing – review & editing (lead). E. S. Lavine: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (supporting); Writing – original draft (equal); Writing – review & editing (equal). W. M. Potter: Investigation (equal); Methodology (equal); Writing – review & editing (equal). C. E. Seyler: Software (lead); Writing – review & editing (equal). B. R. Kusse: Conceptualization (equal); Funding acquisition (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

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

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