Gas puff Z-pinches are intense sources of X-rays and neutrons but are highly susceptible to the magneto-Rayleigh-Taylor instability (MRTI). MRTI mitigation is critical for optimal and reproducible yields, motivating significant attention toward various potential mitigation mechanisms. One such approach is the external application of an axial magnetic field, which will be discussed here in the context of recent experiments on the Zebra generator (1 MA, 100 ns) at the University of Nevada, Reno. In these experiments, an annular Kr gas liner is imploded onto an on-axis deuterium target with a pre-embedded axial magnetic field Bz0 ranging from 0 to 0.3 T. The effect of Bz0 on the stability of the Kr liner is evaluated with measurements of plasma radius, overall instability amplitude, and dominant instability wavelength at different times obtained from time-gated extreme ultraviolet pinhole images. It was observed that the external axial magnetic field does not affect the implosion velocity significantly and that it reduces the overall instability amplitude and the presence of short-wavelength modes, indicating improved pinch stability and reproducibility. For the highest applied Bz0=0.3 T, the stagnation radius measured via visible streak images was found to increase. These findings are consistent with experiments reported in the literature, but here, the Bz0 required for stability, Bz0=0.13Ipk/R0 (where Ipk is the driver peak current and R0 is the initial radius), is lower. This could be attributed to the smaller load geometry, both radially and axially. Consistent with other experiments, the cause of decreased convergence cannot be explained by the additional axial magnetic pressure and remains an open question.

The dense Z-pinch, one of the longest-studied plasma configurations,1 is a compression scheme in which a large axial current is driven by a pulsed power generator on the surface of a cylindrical load; the current interacts with the self-generated azimuthal magnetic field to radially compress the column. Gas puff Z-pinches, where the initial load is a gas jet, are intense sources of X-rays and neutrons that have been studied since the 1970s.2,3

Z-pinch implosions are prone to the development of the magneto-Rayleigh-Taylor instability (MRTI) that can eventually disrupt the implosion symmetry, leading to reduced and/or nonuniform compression.4 Therefore, mitigation of the MRTI is essential to maximize the compression of a Z-pinch. Improving compression by mitigating MRTI is also important for thermonuclear Z-pinch neutron sources. One of these concepts is the gas-puff Staged Z-pinch (SZP),5 in which the load is made of two coaxial cylindrical plasma columns: an outer, hollow cylindrical shell made of high-Z plasma (the liner) and an inner deuterium (D) column on axis (the target). The underlying proposed idea was that a high-Z liner plasma would be highly radiative and remain relatively resistive during the implosion, allowing the magnetic field to diffuse through and be compressed between the liner and the target. The result would be a fast current transfer from the liner to the target near peak compression.6 For a gas puff Z-pinch in which the liner is thick and the liner-target interface is not a sharp boundary, this effect has yet to be definitively observed in experiments.

In this paper, we report on SZP experiments performed on the Zebra machine at the Nevada Terawatt Facility (University of Nevada, Reno), using Kr gas as the liner and D2 gas as the target.7 

Several techniques have been proposed and studied to reduce the impact of MRTI on Z-pinches, including snowplow stabilization8,9 using tailored or multishell density profiles and stabilization by magnetic field line tension using an externally applied axial magnetic field.4,10 Multishell gas-puff Z-pinch loads have drawn interest from both theoretical11,12 and experimental13–16 standpoints for their improved X-ray output. Previous works have shown that the interaction of the imploding outermost shell with the inner shell(s) can mitigate the MRTI, depending on the thicknesses and mass densities of the shells.9,17 If an axial magnetic field (Bz) is applied to the pinch, the field line tension acts against the MRTI growth as the implosion progresses. The Bz is typically applied with a set of coils before the implosion begins and is then compressed due to flux conservation. It has been shown that effective MRTI mitigation can be obtained with an initial axial magnetic field on the order of

(1)

where Bst is measured in tesla, Ipk is the generator peak current in MA, and R0 is the initial plasma radius in cm.18 Noting that Ipk/R0 is an estimate of the strength of the azimuthal magnetic field, this condition simply requires the axial field to be of similar magnitude as the driving field. For example, an Ar gas puff Z-pinch on a 300 kA, 1μs generator was sufficiently stabilized with Bz0=0.4T=2.5Bst, but filamentation and twisting of the plasma column were observed.10 

A combination of snowplow and magnetic stabilization was used in experiments reported in the literature.19–24 A parameter scan of shell-on-target geometry and mass, and applied Bz, was performed with MHD simulations and experiments on the IMRI-5 generator (400 kA, 400 ns) in order to optimize K-shell radiation.19 It was found that a ratio of the outer shell radius to inner fill radius of around 5 and an outer to inner mass ratio of 0.5–1 were optimal for K-shell generation. Improved stability was observed using Ne gas and Bz0=0.066T=0.5Bst on time-integrated X-ray and visible streak images. On the larger GIT-12 generator (2.5 MA, 300 ns), Ar K-shell plasma radiation source experiments achieved MRTI mitigation with Bz0=0.14T=0.25Bst, at the cost of a reduced K-shell yield.20 In both these experiments, the X-ray emitting plasma radius was reduced with an applied Bz, suggesting that the conditions for K-shell production were hindered by the axial field.

In metallic plasma puff experiments on the IMRI-5 machine, a power-law density profile provided snowplow stability, but for Bz0=0.3T, the radiated energy increased due to a longer plasma dwell time at stagnation.21 However, when Bz00.3T was applied, the implosion time increased, the implosion velocity decreased, and the plasma outer boundary became less defined.22 Additionally, the initial plasma current radius also decreased monotonically, and the stagnation radius increased, for increasing Bz0.23 

A seed Bz0=0.1T=0.25Bst was also shown to qualitatively improve the stability of a triple gas puff on the 1 MA, 200 ns COBRA generator using Ar and Ne gases.24 

Recent experiments25,26 have studied the MRTI structure and growth and the use of mitigation techniques more in depth.

Data on MRTI mitigation on short-pulse (100 ns), 1 MA generators are less readily available. The present work focuses on systematically quantifying the overall plasma stability as a function of Bz during the implosion on a 1 MA, 100 ns driver. Due to the shorter rise time, a smaller initial radius can be used. A relatively short plasma length of 11 mm was used in this experiment as well, which is expected to prevent the growth of longer-wavelength instabilities.

The use of the deuterium target reduces the significance of the X-ray yield, given that the goal of these experiments was to produce neutrons; for the same reason, lower target-to-liner mass and diameter ratios were used, as described below, to favor target compression. The presence of the deuterium target is not expected to impact stability except near stagnation when it decelerates the imploding liner. The initial load mass and geometry were not changed throughout the experiment so that the effect of Bz could be isolated.

The present experiments studied the stability of the outer plasma shell for several values of pre-embedded axial magnetic in the range of Bz0=00.3T=00.4Bst. The results show that an external magnetic field above 0.1T=0.13Bst reduces the overall instability amplitude and the presence of short-wavelength modes.

This paper is organized as follows: in Sec. II, we describe the setup for the experiment, the diagnostics, and the data analysis procedures; in Sec. III, we discuss the stability analysis, and finally Sec. IV has concluding remarks.

Experiments were carried out on the Zebra machine, which stores about 200 kJ in a Marx capacitor bank. The Marx output is connected to an intermediate storage capacitor and a 1.9 Ω coaxial transmission line, delivering a peak current of 1 MA with a rise time of 100 ns to the load.27 The time convention throughout this paper is to reference all times to the beginning of the current pulse (t =0).

The Zebra machine applies a negative polarity voltage pulse to the load; the gas injection was connected on the anode (ground) electrode, and a cathode electrode with a honeycomb hole pattern was used to allow gas flow and expansion outside the load region. The anode-cathode gap was set to 11 mm. The design of the gas injector28 and its characterization29 were reported in previous publications. The liner and target gas initial conditions were selected on the basis of interferometric measurements. The load setup is shown in Fig. 1.

FIG. 1.

Cross-sectional sketch of the SZP experiment load on the Zebra machine. The following elements are labeled: 1 = machine cathode, 2 = machine anode (gas injector), 3 = current return cage, 4 = liner gas valve plenum, 5 = target gas valve plenum, and 6 = axial magnetic field coils.

FIG. 1.

Cross-sectional sketch of the SZP experiment load on the Zebra machine. The following elements are labeled: 1 = machine cathode, 2 = machine anode (gas injector), 3 = current return cage, 4 = liner gas valve plenum, 5 = target gas valve plenum, and 6 = axial magnetic field coils.

Close modal

The liner was a hollow cylindrical krypton shell with a peak number density of nKr=2.2×1016 cm−3 centered at a radius of RKr=12.5 mm with a full-width at half-maximum of FWHMKr=5 mm, which produced a mass per unit length M/LKr=10μg/cm. As an estimate for the initial outer radius of the liner, we will consider R0=13 mm. The on-axis target jet was deuterium with a full-width at half-maximum of FWHMD2=8 mm and a peak number density of nD2=1.2×1018 cm−3, which results in a mass per unit length M/LD23μg/cm. Other density values were also used during the experiment, but the most complete dataset was obtained for this set of parameters.

Since the gas was injected from the grounded electrode toward the high-voltage power feed, the timing of the gas injection relative to the pinching pulse had to be kept short enough to avoid gas contamination of the vacuum-insulated high-voltage feed. The downside of a short gas flow time is that an axial density gradient can be present in the gas jet. For the experimental parameters discussed in this work, the axial gradient in both the liner and target mass can be up to 30%. This can lead to the well-known “zipper” effect.30 The images recorded during the experiments show acceptable axial uniformity, possibly due to a combination of gas expansion and flow stagnation at the cathode electrode.

The axial magnetic field was produced by a pair of coils placed outside the Zebra electrodes and driven by a long-pulse (10 ms) capacitor bank in order to allow field diffusion into the electrode region. From Eq. (1), we can calculate Bst = 0.77 T for our experiment; thus, the applied field was in the range of Bz0=00.3T=00.4Bst.

A typical Zebra current pulse is shown in Fig. 2(a) together with voltage traces from two filtered X-ray photodetectors. The particular shot displayed in this plot (shot 5306) had an initial applied axial magnetic field of 0.10 T.

FIG. 2.

(a) X-ray pulses and generator current for Zebra shot 5306 with Bz0=0.10 T. (b) X-ray transmission curves for the foil filters used.

FIG. 2.

(a) X-ray pulses and generator current for Zebra shot 5306 with Bz0=0.10 T. (b) X-ray transmission curves for the foil filters used.

Close modal

The first X-ray photodetector was a 1 mm2 diamond photoconducting detector (PCD), biased at −400 V and filtered with an 8 μm thick Be foil. The second detector was a 1 mm2 Si photodiode, biased at −50 V and filtered with a 100 μm thick Al foil. The transmission curves for the two filters are plotted in Fig. 2(b). The Be filter has a 1/e37% transmission cutoff at a photon energy hν=0.96 keV, therefore this detector provided a time signature of softer X-ray emission. The Al filter has a 1/e transmission cutoff at a photon energy hν=8.7 keV and transmits less than 0.01% radiation for hν<4.0 keV, so this detector provided a time signature of harder X-ray emission.

Radiation peaks of width 10 ns can be observed around peak current at t =100 ns, coincident with the plasma stagnating on axis. Subsequently, more intense peaks of harder X-rays are produced, likely by the pinch going unstable and disrupting. The plasma pinch produces a negligible inductive dip in the current waveform due to the high impedance (2Ω) of the Zebra generator. The hard X-ray emission occurring during the implosion phase could be due to several factors. The interaction with the Cu-W electrodes could introduce ablation impurities into the plasma and disturbances in the plasma flows due to electrode geometry. Additionally, runaway electrons can be accelerated either by the global voltage, d(L·I)/dt, applied at the electrodes, or by plasma instabilities. Note that the detectors were not absolutely calibrated for photon flux and that Si photodiodes typically have a much higher sensitivity than diamond PCDs, which makes a quantitative comparison between the two traces difficult.

Target dynamics are challenging to diagnose from emission measurements due to the highly radiative liner. Some information about stagnation conditions can be inferred from our experiment by analyzing neutron data, both in terms of the total neutron yield and analyzing the shape of neutron time-of-flight detectors.31 However, this analysis will be presented in a separate publication, currently under preparation, which focuses specifically on neutron measurements. For the purposes of the present discussion on implosion stability, we note that the total neutron yields in the range of (1.32.5)×1010 were measured with no applied Bz, whereas yields in the range (0.71.2)×1010 were measured with Bz00.05 T. The larger variations in the yield without Bz are consistent with a larger contribution from beam-target neutrons.7 In the context of this work, this is a qualitative indicator of increased liner stability with Bz, which is consistent with the data presented from other measurements.

The available diagnostics provided adequate information about the liner dynamics during the implosion, but very little information about the inner target dynamics, particularly near stagnation. We therefore focus this analysis on the study of the outer liner boundary.

Two extreme ultraviolet and X-ray time-gated pinhole cameras, capable of recording 4–8 frames per shot, were used to measure the plasma stability. In the following, this diagnostic will be referred to as “XUV imaging.” The diagnostic is composed of two cameras, each consisting of a 40 mm round microchannel plate (MCP) intensifier with an Au photocathode divided into four independent quadrants, arranged along two perpendicular lines of sight. The exposure time for each frame was 3–5 ns, and the frames were taken at different times during the implosion, with interframe delays of 5–10 ns. The pinhole diameter was 100 μm, the image magnification was 0.5, and the spatial resolution was about 300 μm. The images were recorded on a photographic film (Kodak 400 T-Max black/white, ISO 400) placed directly on the MCP phosphor screen and later digitized.

To qualitatively and visually convey the stabilizing effect of the different external axial magnetic field values, we can compare individual images captured at approximately the same time during the implosion, as shown in Fig. 3. We can clearly distinguish the cases with Bz0<0.10 T, where more pronounced instabilities are observed, from the more stable cases for Bz00.10 T, with smaller differences at higher Bz0 values. For Bz0=0.200.30 T, the plasma column has a slightly larger radius than for Bz0=0.100.15 T, which may be an indication of a reduced convergence and an improved axial uniformity, especially near the electrodes.

FIG. 3.

Selection of images at t90 ns for increasing values of applied axial magnetic field Bz0, showing improved stability for Bz00.10 T. The anode is at the bottom of each figure, and gas injection occurs from bottom to top. The gas injection parameters were the same for all shots.

FIG. 3.

Selection of images at t90 ns for increasing values of applied axial magnetic field Bz0, showing improved stability for Bz00.10 T. The anode is at the bottom of each figure, and gas injection occurs from bottom to top. The gas injection parameters were the same for all shots.

Close modal

To illustrate the data analysis process more in detail, we will consider as an example shot 5306 with an applied magnetic field of 0.10 T.

A visible streak camera was used to follow the plasma implosion. The image of a horizontal slice of the plasma, 1 mm tall and centered vertically along the pinch (z=56 mm with a 0.5 mm position accuracy), was relayed to one axis of the streak image with a magnification of 0.6 and an optical resolution of 0.2 mm. The second dimension in the streak image recorded the time sweep with a duration of either 240 ns or 480 ns. A 250 MHz pulse comb generator was used to measure the time during the sweep. The streak time resolution was 1 ns for the 480 ns sweep and 0.5 ns for the 240 ns sweep.

The streak images were used to measure the plasma implosion velocity. The majority of the visible radiation comes from the liner boundary, as can be seen in Fig. 4(a). A MATLAB code was used to determine the radial locations where the emission intensity is 50% of the maximum, on the right and left boundaries of the plasma. The plasma radius and implosion velocity were then calculated, as shown in Fig. 4(c). A moving average window with a 2 ns width was applied to the radius vs time data. The uncertainty in the radius measurement is determined by compounding the 0.2 mm spatial resolution, a 5% error in the radius measurement, and a ±1 ns time uncertainty. The uncertainty in the velocity is obtained by propagating the uncertainties in radius and time and can be as high as ±25% right before stagnation. The implosion takes place over approximately 40 ns, the average implosion velocity is vr=360±50 km/s, and the peak velocity can exceed 500 km/s. From these data, we can also calculate the plasma acceleration, in order to estimate the MRTI growth rate of a particular mode; the average acceleration is 1013 m/s2. The typical plasma outer radius at stagnation measured from streak images is 1.3 ± 0.3 mm, i.e., a liner convergence ratio CR=Rinitial/Rmin=10±3.

FIG. 4.

Example of imaging data from shot 5306. All images were adjusted in brightness and contrast for visualization. (a) Streak image of imploding plasma; the dashed curves represent the plasma radius. (b) Sequence of time-gated XUV pinhole images. The solid lines represent the plasma edges xL/R(z), and the yellow dashed lines indicate the linear fit SL/R(z). (c) Plasma radius from the streak image (red solid line). The black dashed line is the implosion velocity, calculated as the time derivative of the radius. The shaded areas represent the uncertainty intervals. The blue dots represent the average radius R from the XUV gated images.

FIG. 4.

Example of imaging data from shot 5306. All images were adjusted in brightness and contrast for visualization. (a) Streak image of imploding plasma; the dashed curves represent the plasma radius. (b) Sequence of time-gated XUV pinhole images. The solid lines represent the plasma edges xL/R(z), and the yellow dashed lines indicate the linear fit SL/R(z). (c) Plasma radius from the streak image (red solid line). The black dashed line is the implosion velocity, calculated as the time derivative of the radius. The shaded areas represent the uncertainty intervals. The blue dots represent the average radius R from the XUV gated images.

Close modal

To analyze the XUV pinhole images, a computer code was used to determine the radial locations where the emission intensity is 50% of the image maximum, at each axial position along the image. This results in a left and a right plasma boundary as a function of axial position, xL(z) and xR(z), respectively. Images that did not produce a well-defined boundary, due to an image blur or a low contrast, were discarded from the analysis. An example of four images with the calculated plasma boundaries is shown in Fig. 4(b), where xL(z) is plotted in red and xR(z) is plotted in blue.

For each image, we fitted the plasma edges xL(z) and xR(z) with a straight line SL/R(z), in order to account for flaring or zippering of the plasma column. The dashed yellow lines in Fig. 4(b) indicate the linear fit SL/R(z) of the left/right plasma edges. We can then calculate the axially averaged mean plasma radius R, the flaring angle α, defined as the aperture angle of the two linear edges SL/R(z), and the overall instability amplitude ΔR, expressed as the standard deviation of x(z)S(z), at the time of each frame,

(2)
(3)
(4)

where L is the length of the pinch.

The average radius R calculated from these images is also plotted as dots in Fig. 4(c). The error bars in these data points were calculated as the quadratic sum of the spatial resolution (0.2 mm) and the standard deviation ΔR. The uncertainty in the time of each image is considered to be half the exposure time, i.e., 1.5–2.5 ns. The discrepancy in the measured radius from the XUV images and streak images could be attributed to a small systematic error in the timing signatures, to the different lines of sight of the instruments, or to the different spectrum recorded on the two images: the XUV pinhole images represent a higher energy portion of the plasma emission than the visible streak images, typically resulting in a smaller measured radius. For the purpose of this work, we only require sufficient agreement between the two diagnostics to estimate the implosion velocity, acceleration, and final radius.

A scan of the externally applied axial magnetic field was performed, with values of the initial magnetic field Bz0=0,0.05,0.10,0.15,0.20, and 0.30 T. Repeated shots were collected for each value of the magnetic field in order to obtain an average of 30 images per Bz0 value. Statistical trends were calculated for the mean radius and overall instability amplitude, and the results are presented in Sec. III.

Average trends over multiple shots repeated under the same conditions can provide a more quantitative analysis of the plasma stability. Since the majority of the images were collected around fixed time points during the implosion, we can average images taken at sufficiently close times (±3 ns). The plots in Fig. 5 report the average plasma radius R, the average instability amplitude ΔR, and the average zipper angle α, where the brackets represent the average over multiple shots. The shaded areas represent the error bars, calculated as the quadratic sum of the shot-to-shot standard deviation. Uncertainties calculated this way are ±0.5 to ±1 mm for R and ±2 to ±3 ns in time.

FIG. 5.

(a) Plasma radius R averaged over multiple shots at different Bz0 values. (b) Overall instability amplitude, expressed as the radius standard deviation ΔR. (c) Average flaring angle α. The shaded areas represent the error bars. The data for Bz0=0.15 T were omitted to improve readability.

FIG. 5.

(a) Plasma radius R averaged over multiple shots at different Bz0 values. (b) Overall instability amplitude, expressed as the radius standard deviation ΔR. (c) Average flaring angle α. The shaded areas represent the error bars. The data for Bz0=0.15 T were omitted to improve readability.

Close modal

The average plasma radius R as a function of time, measured from XUV images and averaged over repeated shots, is shown in Fig. 5(a). Though we note that pinches with no pre-embedded axial magnetic field (Bz0=0) tend to disrupt around t =100 ns, it is apparent from the data that there is neither a significant correlation between Bz0 and implosion time nor between Bz0 and implosion velocity. From first principles, this observation is not surprising: in the context of gas-puff Z-pinches with pre-embedded Bz0, the experiment discussed here is in a “weak-field” regime, i.e., the strength of the compressive J×B force (Bθ2) is significantly larger than the outward pressure from the embedded Bz, even at higher convergence ratios. For example, compressing a load with R0=1.3 cm by a factor of 10 with Ipk = 1 MA and Bz0=0.3 T, the highest value used here, gives Bθ150 T and Bz = 30 T at stagnation, assuming no axial magnetic flux loss. The corresponding pressure ratio, Bz2/Bθ2, is 4%.

Recent experiments22,26 in a similar “weak-field” regime found a correlation between implosion velocity and Bz0: as Bz0 was increased, the implosion slowed down. Since the Bz pressure is not enough to reduce the implosion velocity, another explanation was needed. In Ref. 26, it was determined that in the presence of Bz0, only a fraction of the current flows in the imploding pinch: the remainder of the current flows in what the authors called a low-density plasma (LDP) in the outer region. This explained the lower-than-expected Bθ measurements on the pinch surface, i.e., Bθ<(μ0I)/(2πR). We note that these experiments were performed with a single-shell, hollow Ar gas puff load (which is not expected to alter the implosion dynamics substantially) and with comparatively higher values of Bz0=0.62.5Bst than our experiment. Although, based on our available measurements, we have no evidence of a delayed implosion due to the current flow in the LDP, spatially resolved Bθ measurements in future experiments would help establish whether this effect is present.

As the pressure of the external Bθ relative to the internal Bz remains large through stagnation, Bz back-pressure alone cannot explain a possible increase in the stagnation radius (decrease in the overall compression). This observation is well-documented in the literature,20 where the authors noted a decrease in the K-shell yield consistent with reduced compression and heating, but greater than would be expected from Bz pressure alone. In the present experiment, we have measured the shot-averaged stagnation radius at the time of peak X-ray production. The results are shown in Fig. 6: for Bz0 up to 0.2 T, the stagnation radius is 1.2–1.6 mm, while for Bz0=0.3 T, it increased to 2.1 mm. Recall that with Bz0=0.3 T, the expected magnetic pressure ratio, Bz2/Bθ2, at stagnation is on the order of a few percent, which cannot explain the decreased compression. Measurements of plasma density, temperature, as well as Bθ and Bz profiles near stagnation could provide extremely useful insight.

FIG. 6.

Shot-averaged stagnation radius as a function of applied Bz0.

FIG. 6.

Shot-averaged stagnation radius as a function of applied Bz0.

Close modal

We now turn the discussion to liner stability. Figure 5(b) compares the overall plasma instability amplitude, expressed as the standard deviation of the radius ΔR, averaged over the shot ensemble. The error bars are the sum of the radial resolution limits and standard deviation over the different images, ranging from ±0.2 to ±0.4 mm. We observe a reduction in the maximum absolute instability amplitude for Bz00.10T=0.13Bst, as well as a later onset of the significant relative instability amplitude (ΔR/R>0.5), compared to Bz0=0. This agrees with the qualitatively significant MRTI growth mitigation in Fig. 3 for Bz00.10 T. An additional qualitative indicator of increased stability, albeit for Bz00.15 T, is a pronounced decreased in the shot-to-shot variation in the standard deviation of time of the X-ray pulse: t=99±8 ns for Bz0=0 T vs t=102±4 ns for Bz0=0.15 T, with comparably small variations for higher Bz0 values.

The zipper (flaring) angle α vs axial field strength is reported in Fig. 5(c). This angle exceeds 15° only in the case Bz0=0.05 T, while for all other cases, it remains around 10° and seems fairly insensitive to the applied magnetic field.

For each individual XUV time-gated image, a fast Fourier transform (FFT) routine was applied to the plasma boundary (xR(z)xL(z))/2 in order to find the dominant wavelengths of the MRTI. Analyzing the instability growth over a single shot, thereby removing shot-to-shot variations, would be very useful if one or a few individual modes can be isolated. The single-image spectra in this dataset, however, included a multitude of wavelengths; in this case, a more general picture can be extracted by analyzing the average data. The collection of the spectral data, averaged over the shot ensemble with the above-mentioned criteria, is shown in Fig. 7. The higher wavenumbers k4 rad/mm (λ1.5 mm) have reduced amplitude, while the longer wavelengths are only moderately affected by the applied magnetic field.

FIG. 7.

Fourier analysis of the plasma boundary with normalized amplitude. The color scale is logarithmic. The dashed lines correspond to the wave-number k =4.0 rad/mm and the wavelength λ=0.8 mm (k =7.9 rad/mm), as discussed in the text.

FIG. 7.

Fourier analysis of the plasma boundary with normalized amplitude. The color scale is logarithmic. The dashed lines correspond to the wave-number k =4.0 rad/mm and the wavelength λ=0.8 mm (k =7.9 rad/mm), as discussed in the text.

Close modal

We can compare this result with the simple planar MRTI theory if the curvature effects can be neglected.32–34 For example, if we consider the stability of the (arbitrarily chosen) wavelength λ=0.8 mm for t80 ns, then the plasma outer radius R4 mm (CR4) and kR301, which justifies the planar MRTI approximation. The growth rate can be written as γ2=kg(kBz)2ρμ0, where g is the acceleration, k=2π/λ is the wave-number, μ0 is the magnetic permeability, and ρ is the mass density of the fluid. From the streak images, a conservative estimate for the acceleration is g5×1012m/s2. Mass and magnetic flux conservation, within the snowplowed liner plasma, can be used to write ρ and Bz as functions of the outer radius R and liner width W, respectively: ρ(t)=ρ(0)[R(0)W(0)R(t)W(t)],Bz(t)=Bz0[R(0)W(0)R(t)W(t)]. The growth rate can hence be expressed as a function of known quantities and the unknown W(t),

(5)

The assumption of frozen flux into the liner is a conservative approach for our argument, because any lost flux would further reduce the expected stabilizing effect of Bz. Other experiments performed at a higher applied axial field21 indeed suggest that some flux is lost during the compression. From experimental observations, we can estimate conservatively γ(30ns)1, since over the 60 ns implosion time, we did not measure any noticeable presence of the λ=0.8 mm wavelength for Bz00.1 T.

Here, we can use the above-mentioned numbers from our conservative estimations, as well as ρ(0)=3μg/cm3,R(0)=15 mm, and W(0)=5 mm. Solving for W(t) gives that, in order to obtain sufficient stabilization, the liner would have to be compressed via supersonic snowplow to W(t)0.1 mm. Recent simulations suggest that the shock-compressed layer at the liner-target interface could have a comparable thickness.7 However, given the instability amplitude 1 mm measured in this work, such a thin structure would likely disrupt during the implosion.

If we assume a liner thickness comparable to the instability amplitude, W(t)1 mm, then the growth rate predicted by planar MRTI theory is γ(5ns)1, which would be clearly observable over the implosion time scale. It is worth noting that these simple estimates assume ideal MHD, but it is known that resistive diffusion can increase or decrease instability growth: Bθ diffusion can possibly reduce MRTI growth below ideal MHD estimates, particularly for shorter wavelengths; on the other hand, Bz diffusion out of the liner would reduce the field line tension and reduce the stabilizing effect. Estimates of resistivity via temperature and density measurements would help clarify the importance of field diffusion.

Additional work is necessary to characterize the plasma parameters during the implosion, as well as the stability properties of different liner materials. Recent experiments25 have shown different MRTI growth rates between Ar and Kr gas puff Z-pinches, with the latter gas being more stable. We plan to perform more direct measurements of the density and magnetic field profiles via interferometry, Faraday rotation,35,36 and Zeeman spectroscopy,26,37–39 when similar experiments are conducted on a linear transformer driver (LTD) machine currently under testing at UC San Diego.40,41 This LTD machine will produce an 0.8 MA peak current in an 150 ns rise time, fairly similar to Zebra; the plasma parameter space accessed in future experiments should be close to the one presented here.

Liner-on-target gas puff Z-pinch (SZP) experiments with a Kr liner and a D target were performed on the 1 MA, 100 ns Zebra generator. XUV time-gated images were used to measure the stability of the outer plasma boundary when an externally applied axial magnetic field in the range of Bz0=00.3 T was applied.

Averaging over a number of repeated shots, it was observed that the external axial magnetic field does not produce a noticeable slow-down of the implosion, but it reduces the overall instability amplitude and the presence of short-wavelength MRTI modes.

For Bz0=0.3T, the stagnation radius measured via visible streak images was found to increase to 2.1 ± 0.2 mm, up from 1.2 to 1.6 mm measured at lower Bz0. As with other experiments, the cause of decreased convergence cannot be explained by the additional axial magnetic pressure and remains an open question.

The instability mitigation appears to be higher than what a simple planar MRTI model suggests with conservative estimates for the relevant parameters, unless the liner is compressed via snowplow to a very thin layer with a thickness of W0.1 mm, which seems unrealistic.

Our findings are generally consistent with experiments reported in the literature, but here, we have found a lower field strength to reduce instabilities at Bz0=0.1T=0.13Bst. This lower value could be attributed to the smaller initial radius, allowed by the faster generator rise time, and a shorter plasma length. The maximum MRTI amplitude is a function of the initial radius because the average acceleration can be approximated simply in terms of implosion time and initial radius. On the other hand, a longer plasma length allows longer-wavelength instability modes to develop, which require higher Bz to stabilize.

The present and other experimental results suggest that there are other effects at play on MRTI stabilization and stagnation dynamics that cannot solely be explained by simple ideal MHD estimates and the presence of Bz. More direct measurements of the density and magnetic field profiles, and plasma parameters near stagnation, are needed to offer a more complete answer to the physical mechanisms that reduce convergence when the added pressure from the axial magnetic field is negligible.

This work was funded by ARPA-E, Grant No. DE-AR0000569, and partially supported by the Department of Energy, National Nuclear Security Administration under Award No. DE-NA0003842.

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