Characteristics and scaling of tungsten-wire-array $z$-pinch implosion dynamics at 20 MA

We present observations for 20-MA wire-array $z$ pinches of an extended wire ablation period of $57\%\pm 3\%$ of the stagnation time of the array and non-thin-shell implosion trajectories. These experiments were performed with 20-mm-diam wire arrays used for the double-$z$-pinch inertial confinement fusion experiments [M. E. Cuneo et al., Phys. Rev. Lett. 88, 215004 (2002)] on the $Z$ accelerator [R. B. Spielman et al., Phys. Plasmas 5, 2105 (1998)]. This array has the smallest wire-wire gaps typically used at 20 MA (209 $\mathrm{\mu m}$). The extended ablation period for this array indicates that two-dimensional $\left(r\text{−}z\right)$ thin-shell implosion models that implicitly assume wire ablation and wire-to-wire merger into a shell on a rapid time scale compared to wire acceleration are fundamentally incorrect or incomplete for high-wire-number, massive $(>2\mathrm{mg}∕\mathrm{cm})$, single, tungsten wire arrays. In contrast to earlier work where the wire array accelerated from its initial position at $\sim 80\%$ of the stagnation time, our results show that very late acceleration is not a universal aspect of wire array implosions. We also varied the ablation period between $46\%\pm 2\%$ and $71\%\pm 3\%$ of the stagnation time, for the first time, by scaling the array diameter between 40 mm (at a wire-wire gap of 524 $\mathrm{\mu m}$) and 12 mm (at a wire-wire gap of 209 $\mathrm{\mu m}$), at a constant stagnation time of $100\pm 6\mathrm{ns}$. The deviation of the wire-array trajectory from that of a thin shell scales inversely with the ablation rate per unit mass: $f_m\propto [d{m}_{\text{ablate}}∕dt]∕m_{\text{array}}$. The convergence ratio of the effective position of the current at peak x-ray power is $\sim 3.6\pm 0.6:1$, much less than the $⩾10:1$ typically inferred from x-ray pinhole camera measurements of the brightest emitting regions on axis, at peak x-ray power. The trailing mass at the array edge early in the implosion appears to produce wings on the pinch mass profile at stagnation that reduces the rate of compression of the pinch. The observation of precursor pinch formation, trailing mass, and trailing current indicates that all the mass and current do not assemble simultaneously on axis. Precursor and trailing implosions appear to impact the efficiency of the conversion of current (driver energy) to x rays. An instability with the character of an $m=0$ sausage grows rapidly on axis at stagnation, during the rise time of pinch power. Just after peak power, a mild $m=1$ kink instability of the pinch occurs which is correlated with the higher compression ratio of the pinch after peak power and the decrease of the power pulse. Understanding these three-dimensional, discrete-wire implosion characteristics is critical in order to efficiently scale wire arrays to higher currents and powers for fusion applications.

Figure 1

Schematic of the four-level, $Z$ magnetically insulated transmission line (MITL) section and plastic insulator stack (1.8 m radius). The inset shows the post-hole convolute which adds the current from the four transmission lines, with a 6-mm-wide final anode-cathode feed gap, and a typical 40-mm diam, 20-mm-long, wire array load.

Figure 2

Experimental wire array geometry and diagnostic lines of sight. (a) Wire array hardware in cross section shows the radial and axial diagnostic views with x-ray diodes (XRD), bolometers (BOLO), time-resolved x-ray pinhole cameras (XRPHC), transmission grating spectrometers (TGS), and time-integrated crystal spectrometers (TIXTL). The critical dimensions are given (in mm) for both 20-mm and 40-mm arrays; the 20-mm number is first. 12-mm arrays are fielded in the 20-mm hardware. Also noted is the anode-cathode (AK) power feed gap at the base of the pinch. (b) A cutaway view from the top showing the chordal lines of sight for the radial optical streak (ROS), the streaked CW laser, the TGS and filtered silicon diode array (FSDA) viewing the wires early in the pulse, and the position of the Be and Au markers for axial-self-backlighting.

Figure 3

Wire array load current and radiated power (average of four shots: 566, 567, 665, 667), from a typical 20-mm, 6-mg load, showing the definition of the stagnation time and the implosion phases: (I) is the wire initiation phase, (II) is the wire ablation phase, (III) the implosion or acceleration phase, and (IV) is the stagnation and thermalization phase.

Figure 4

(a)–(e). Axial x-ray pinhole framing camera images at various normalized times $(t∕{\tau }_{\mathrm{stag}})$ during the implosion of the 20-mm-diam, $6\text{−}\mathrm{mg}∕\mathrm{cm}$ wire array from shot 747. The position of the Au and Be markers are shown on part (a). The 18 Be spokes are apparent in (b)–(e). The precursor pinch is observed on axis in (a)–(d). Cold-ablated tungsten fills the center of the array in (a)–(c), obscuring the contrast between the Au and Be markers. The imploding shell implodes onto the cold-ablated tungsten at smaller radii in (b)–(d). The imploding shell has nearly merged with the precursor in (d). Trailing mass obscures the Au and Be markers in (c) and (d). The broad radial pinch profile at stagnation is shown in (e).

Figure 5

Comparisons of convolute voltages $V_{\mathrm{conv}}\left(t\right)$ (solid lines) with initial voltages $V_0\left(t\right)$ (dotted lines) indicating the times of array acceleration when $V_{\mathrm{conv}}>V_0$ for (a) 40-mm- (shot 846), (b) 20-mm- (shot 818), and (c) 12-mm-diam (shot 931) wire arrays. The pinch power pulseshapes (dashed lines) are shown for reference. The acceleration time is later in the pulse for heavier, smaller diameter arrays.

Figure 6

(Color) Data from a 20-mm-diam, $6\text{−}\mathrm{mg}∕\mathrm{cm}$ array showing (a) array trajectories versus normalized time $(t∕{\tau }_{\mathrm{stag}})$ from four methods. Trajectories are from a radial optical streak [ROS, blue circles, typical of four shots (665, 674, 684, 747), shot 674 shown], the edge of the x-ray pinhole framing camera radial profile (XRPHC edge, green circles, from shots 728, 747), and the peak of the moving shell from the radial profile (XRPHC peak, red circles, from shots 728, 747) and from the inductance unfolds (orange line, average of three shots: 724, 817, 818), compared with a thin-shell model (black line). Trajectory data show the delayed start of acceleration with respect to a thin shell (black line) but earlier arrival at the axis (red circles) than a thin shell and the current (orange line), indicating trailing mass and current. (b) Radial pinch power from two different measurements: x-ray diode’s ($P_{\mathrm{XRD}}$, the blue line is average of shots 566, 567, 665, 674, the black line is corrected for slot collimation), and silicon diode ($P_{\mathrm{SD}}$, light blue, average of shots 198, 199). The pinch power is on a log scale (left axis) and a linear scale (in red, right axis). (c) Precursor radius (black circles, from shots 728, 747) and XRPHC peak (red circles from shots 728, 747).

Figure 7

(Color) Data from a 40-mm-diam, $2.2\text{−}\mathrm{mg}∕\mathrm{cm}$ array showing (a) array trajectories versus normalized time $(t∕{\tau }_{\mathrm{stag}})$ from the same four techniques as in Fig. 6. The radial optical streak (ROS) is a composite obtained on shots 685 and 846. Inductance unfolds are from shot 846. As for the 20-mm array, trajectory data show delayed acceleration with respect to a thin shell (black line) but earlier arrival at the axis than a thin shell (red and green circles) and the current (orange line), indicating trailing mass and current. (b) Streak shadowgraphy (and fitted trajectory in light blue) from a 532-nm laser transmission through the edge of the array on shot 846, showing the individual wires early, transmission cutoff, and the gradual clearing of the lower density trailing plasma at the edge later. Where unmarked, curves are the same as in Fig. 6.

Figure 8

Radial profiles of axial x-ray pinhole framing camera images (from Fig. 4, for the 20-mm array from shot 747), plotted as power density $(\mathrm{TW}∕{\mathrm{cm}}^2)$ by normalizing the images to the measured axial power from an x-ray diode and bolometer. The numbers on the curves are normalized times $(t∕{\tau }_{\mathrm{stag}})$. (a) used 4-ns gates between $0.58{\tau }_{\mathrm{stag}}$ and $0.93{\tau }_{\mathrm{stag}}$; (b) used 2-ns gates between $0.90{\tau }_{\mathrm{stag}}$ and $1.05{\tau }_{\mathrm{stag}}$. The error is $\pm 20\%$ in power density. The precursor is observed on axis. A 1.9–3.1-mm FWHM shell-like object implodes onto the precursor. A plateau of hot trailing mass is observed outside the shell between $0.85{\tau }_{\mathrm{stag}}$ and $0.98{\tau }_{\mathrm{stag}}$. The radial extent of the pinch at peak power $\left(1.05{\tau }_{\mathrm{stag}}\right)$ is very broad.

Figure 9

Radial profiles of the density (dashed lines) and corresponding synthetic axial x-ray pinhole camera (XRPHC) images (solid lines) from $2\mathrm{D}\left(r\text{−}z\right)$-RMHD simulations of a 20-mm, 6.0-mg array seeded by random density perturbations (RDP) for (a) a shell-like implosion with a 1% RDP and (b) a shell implosion onto a uniform, cold, prefill with 25% of the array mass, with a delayed trajectory and with a 4.5% RDP. ${\tau }_R$ is the Rosseland opacity of the material located radially between the axis and imploding shell, when viewed axially.

Figure 10

(Color) X-ray pinhole camera data with a radial view of the pinch (normalized exposure) from shot 665: pinhole images were filtered with 4.8 $\mu m$ Kimfol and 193 Å aluminum in parts (a)–(e) and with 4.8 $\mu m$ Kimfol and 1313 Å aluminum in parts (f)–(i). Times are given as time with respect to peak radiated power, rather than normalized to the stagnation time. Part (a) was obtained at 6.6 ns prior to peak power, parts (b) and (f) at $-4.6\mathrm{ns}$, parts (c) and (g) at $-1.1\mathrm{ns}$, parts (d) and (h) at 0.2 ns, and parts (e) and (i) at 2.4 ns. The images all use the same color table. Cold material surrounds a hot core region in (a)–(e). Only the hot core is observed with the thicker filter in (f)–(i). “Bubble” regions (cold material surrounded by hotter material) are labeled b1, b2, and b3.

Figure 11

(Color) (a) Axially averaged, normalized radial x-ray exposure profiles from Figs. 10a, 10b, 10c, 10d, 10e. Times given are with respect to peak radiated x-ray power in ns. (b) Normalized soft-filtered (solid black line) and hard-filtered (dotted black line) radial profiles at 0.2 ns [Fig. 10d]. The soft-filtered profile is fit by the sum of two Gaussians, showing a wing (blue), a core (green), and the total (red). Numbers on the plots are the FWHM. The pinch reaches maximum compression after peak power.

Figure 12

(Color) (a) Temperature distribution obtained from normalizing shot 665 radial x-ray pinhole camera image [0.2 ns after peak power, Fig. 10d] with the measured radial power and (b) 3D representation of the 2D image. The broad wings of the colder halo, the outside edge of the slot in the current return electrode, and the cold slot material expanding from the slot edges in a few places are clearly seen.

Figure 13

(Color) Spectral power from a wire array at peak power, viewed radially with a transmission grating spectrometer from shot 987 (circles with $\pm 20\%$ error bars). Unfolds assume the superposition of two Planckian distributions (dashed and dotted black lines) possibly representing a cold halo and hotter core. The red lines (dashed lines show $\pm 35\%$ error range) are the equivalent pinch spectrum obtained from normalized radial x-ray pinhole camera data from shot 719 (0.3 ns after peak power), using the same method as for Fig. 12a.

Figure 14

(Color) Wire array trajectory measurements compared with 2D-MHD models of wire array ablation for (a) Alegra simulations (hot core, cold core, and cold core plus $\sigma ∕10$ models) and 20-mm array data from Fig. 6a, (b) Gorgon simulations and 20-mm array data from Fig. 6a, and (c) Gorgon simulations and 40-mm array data from Fig. 7a.

Figure 15

(Color) (a) Comparison of wire array emission trajectory measurements corresponding to the brightest implosion front [peak of the radial profile from the axial x-ray pinhole camera (XRPHC peak) from the 20-mm array data from Fig. 6a] with a Lebedev rocket model fit. This trajectory involves $\sim 60\%\text{–}80\%$ of the initial array mass. (b) Comparison of wire array inductance trajectory measurements corresponding to the array edge [inductance, radial optical streak (ROS) and XRPHC edge from 20-mm data from Fig. 6a] with a rocket model fit. This trajectory involves 90%–100% of the initial array mass.

Figure 16

(Color) Comparison of measured convolute voltages for the 20-mm array [from Fig. 5b, shot 818] with models of convolute voltages for three cases: thin shell at a convergence of $10:1$ (blue line), emission at a convergence ratio of $10:1$ (purple line), and trajectory of inductance at a convergence of $4:1$ (red line). The load impedance is consistent with a smaller convergence of the current.

Figure 17

(Color) (a) Comparison of a composite pinch power history during the implosion phase [$P_{\text{composite}}$, black line, from Fig. 6b] with a 1D-resistive MHD model ($P_{\mathrm{G}\text{−}\mathrm{MHD}}$, green line), and a snowplow accretion model [$P_{\mathrm{SP}}$, purple lines, from trajectory of Fig. 15a]. Triangles correspond to an ablation velocity of $10\mathrm{cm}∕\mu \mathrm{s}$, circles to $14\mathrm{cm}∕\mu \mathrm{s}$, and squares to $20\mathrm{cm}∕\mu \mathrm{s}$. (b) Comparison of the measured increase in precursor diameter [black circles, from Fig. 6c] with a simple precursor energetics model (orange line). The peak of the radial profile of the axial x-ray pinhole camera radial profile peak is shown in red circles, rocket model trajectory shown with a purple line.

Figure 18

Plot of $t∕{\tau }_{\mathrm{stag}}$ vs $g_Z$ [Eq. (24)] for $Z$ tungsten arrays. Array mass and radii were changed to keep the measured implosion time roughly constant ($\Pi$ roughly constant).

Figure 19

(Color) (a) Mass ablation fraction as a function of time for $Z$ (solid lines) and Magpie arrays (dotted lines). The value of $V_a$ was adjusted to produce agreement with the measured time of acceleration at the appropriate value of the ablation fraction. Ablation fractions are noted with horizontal lines (41% for the 40-mm array, 47% for the 20-mm and 12-mm arrays, and 40% for Magpie; see Table ). Magpie curves and acceleration times are shifted earlier by 40 ns so that the $t=0$ point is defined in a similar fashion for both $Z$ and Magpie. (b) Mass ablation fraction as a function of normalized time $(t∕{\tau }_{\mathrm{stag}})$ for $Z$ (solid lines) and Magpie arrays (dotted lines). The Magpie acceleration and stagnation times from Table were decreased by 40 ns in calculating the $t_a∕{\tau }_{\mathrm{stag}}$ ratio. Magpie arrays ablate and burnthrough later in the pulse than $Z$ arrays.

Figure 20

(Color) Comparison of the effective velocity of current compression (dotted lines) with the normalized power pulses (solid lines) for the 40-mm (blue, shot 846), 20-mm (green, shot 818), and 12-mm-diam (red, shot 931) arrays. We observe a later acceleration time for smaller radius arrays. Faster initial acceleration of the current and a longer time between peak velocity of the current and peak power for the 12-mm array may indicate a higher trailing mass compared to the other arrays. Faster deceleration of the current near peak velocity for the 40-mm array may indicate less trailing mass compared to the other arrays.

Figure 21

(a) Comparison of the trajectory of the effective position of the current from the current scaling experiment of Ref. 20. Wire array experiments at 13 MA (dashed line, average of shots 725, 819) and 19 MA (average of shots 724, 817, 818). Higher convergence is noted for the lower-current case. (b) Effective velocity of current compression for the trajectories from part (a). A higher peak velocity and a more rapid deceleration of the velocity near peak power is observed for the low-current data.

Figure 22

(a) Comparison of the pinch FWHM obtained from the soft-filtered radial x-ray pinhole camera data for low-current (dashed lines with diamonds, shots 647 and 648) and high-current (solid line with circles, shots 646 and 665) configurations. (b) Pinch power pulse at low current (648) and high current (646), both with otherwise identical hardware configurations and AK gaps (1 mm). The low-current arrays implode faster, correlated with a narrower, faster rising pinch power pulse.

Figure 23

(a) Comparison of normalized radiation power pulse shapes for the current scaling experiments of Ref. 20. Normalized power pulses from 3- and 4-mm shots are averaged. High-current shots are 594, 683, 723, 724, 817, 818. Low-current shots are 647, 725, 819. (b) Comparison of normalized radiation power pulse shapes for three different initial array diameters: 12 mm (dashed line, shot 931, 3-mm AK gap), 20 mm (dotted line, average of shots 719, 726, 728, 747, 749, 3-mm AK gap), and 40 mm (solid line, average of shots 160, 161, 165, 168, 169, 234, 235, 269, 281, 326, 394, 5-mm AK gap). The power pulses are aligned with the 50% point on the leading edge at 100 ns. Peak radiated power for the 12-mm array is $\sim 50\mathrm{TW}$, for the 20-mm array $\sim 125\mathrm{TW}$, and for the 40-mm array $\sim 165\mathrm{TW}$. Error bars are reduced by $\sqrt{N-1}$ for a sample of $N$ independent measurements. Larger relative tail amplitude is observed for arrays with greater trailing mass.

Figure 24

(a) Equivalent circuit for the four-level $Z$ transmission lines from the water-vacuum insulator stack convoluted into a single feed to the load. VA, VB, VC, and VD are the stack voltages for each level. The horizontal inductors are the transmission line inductances from the insulator stack to the convolute. The vertical inductors are the coupling inductances between the transmission lines in the convolute. (b) Equivalent circuit for the $Z$ accelerator with a single constant impedance voltage source $V_{\mathrm{OC}}$, a variable impedance element to model current loss in the convolute $Z_{\mathrm{conv}}\left(t\right)$, and a variable inductance array $L_a\left(t\right)$.