Conical Wire Arrays in Pulsed-Power Plasma Research

• Conical wire arrays are pulsed-power setups where fine wires on a conical surface implode via azimuthal magnetic pressures to form high-energy-density plasmas.
• They generate plasma jets by converging ablative flows that collide to form shocks, with jet properties strongly dependent on cone geometry and opening angle.
• Nested conical arrays overcome single-array limitations by synchronizing implosion timing to achieve ultrafast jets with velocities up to 10^8 cm/s, offering applications in fusion and laboratory astrophysics.

Conical wire arrays are pulsed-power loads in which many thin wires are arranged on the surface of a cone and driven by a large, short current pulse. In z-pinch physics, the current produces an azimuthal magnetic field $B_\theta$, and the associated magnetic pressure $p_B = B^2/8\pi$ drives an inward MHD implosion at a characteristic velocity comparable to the Alfvén speed $v_A = B/\sqrt{4\pi \rho}$ (Winterberg, 2010). In laboratory high-energy-density plasma research, these arrays are also used as compact, repeatable sources of supersonic, magnetically driven plasma jets, including on the MAGPIE generator (Bocchi et al., 2011). The topic encompasses both standard single-cone jet sources and more specialized nested conical geometries proposed for ultrafast jet production, as well as recent diagnostic work showing how opening angle controls jet propagation velocity while leaving density and temperature profiles largely unchanged under fixed drive conditions (Izquierdo et al., 8 Jul 2025).

1. Definition and physical basis

In the standard pulsed-power and z-pinch context, a wire array is a set of many fine wires arranged around an axis and carrying a mega-ampere-scale pulsed current. A conical wire array is the corresponding non-cylindrical case: the wires lie on the surface of a cone with some opening angle $2\alpha$, so the magnetic drive is oblique rather than purely radial. The current still flows approximately along the wires, while the azimuthal magnetic field and magnetic pressure push the wire material inward toward the cone axis (Winterberg, 2010).

A basic scaling used in analytic treatments is that the average magnetic field varies as $B \propto 1/r$, while the average mass density of the wire array also varies as $\rho \propto 1/r$. This yields an Alfvén speed scaling $v_A \propto 1/\sqrt{r}$. In cylindrical arrays this produces a symmetric radial implosion, and such arrays have been used successfully to reach plasma velocities of order $10^7\ \text{cm/s}$ over centimeter scales. In conical arrays the same radial dependence is geometrically projected onto a convergent surface, making the axial convergence and timing strongly geometry dependent (Winterberg, 2010).

In experimental jet platforms, the implosion is not usually treated as a solid conical shell but as a system of ablated plasma streams launched from individual wires. On MAGPIE, tungsten conical wire arrays between two electrodes are driven by a pulsed current, the wires ablate, and the resulting streams are magnetically accelerated toward the axis, where they form a conical shock and an axial jet (Bocchi et al., 2011). This distinction between shell-like cone collapse and ablative-flow convergence is central to the modern literature.

2. Jet formation mechanisms and the single-array limitation

Two jet-formation pictures coexist in the conical-wire-array literature. The first is the classical shaped-charge picture, in which the sides of a cone collapse normal to the cone surface with speed $V_0$, giving jet and slug velocities

$$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$

and for $p_B = B^2/8\pi$0,

$p_B = B^2/8\pi$1

This predicts very large jet velocities for small opening angle, but only if the implosion velocity is approximately constant along the cone surface (Winterberg, 2010).

Winterberg’s analysis shows that a single electric-pulse-power-driven conical wire array does not satisfy that condition. Because $p_B = B^2/8\pi$2, the implosion velocity increases toward the apex, so the cone does not collapse self-similarly with uniform $p_B = B^2/8\pi$3. The consequence is that the ideal thin, high-velocity shaped-charge-like jet is not formed; instead, the converging material accumulates near the axis in a more compact “blob” that moves with the effective velocity of the cone apex (Winterberg, 2010). This result concerns the feasibility of obtaining an ultrafast, narrow jet by direct cone-shell collapse.

By contrast, the MAGPIE experiments and simulations describe a different mechanism. As the current rises, Ohmic heating vaporizes and ionizes the wires, producing ablated plasma around each wire. The local $p_B = B^2/8\pi$4 force accelerates plasma roughly perpendicular to the wire, so the conical wire arrangement makes the flows converge toward the array axis. Their collision creates a converging conical shock, and because the opening angle of this shock is smaller than that of the original wire cone, the post-shock flow is redirected to be almost axial, producing a collimated, supersonic jet (Bocchi et al., 2011). A common misconception is therefore that all conical wire array jets are shaped-charge analogues; the literature instead separates ordinary ablative conical-shock jets from the more restrictive ultrafast-jet problem addressed by nested geometries.

3. Nested conical wire arrays and the ultrafast-jet proposal

To overcome the single-array limitation, Winterberg proposed a nested conical wire array with “opposite alternate opening angles” (Winterberg, 2010). The simplest configuration contains two nested arrays. The inner array is a divergent cone, starting on axis at point $p_B = B^2/8\pi$5 and expanding outward to point $p_B = B^2/8\pi$6. The outer array is a convergent truncated cone, starting at finite radius $p_B = B^2/8\pi$7 at point $p_B = B^2/8\pi$8 and ending at the same point $p_B = B^2/8\pi$9. The two arrays therefore share the same circular ring at $v_A = B/\sqrt{4\pi \rho}$0, but only the inner array reaches the axis at the left side.

When a high-voltage pulse is applied, the current initially flows primarily in the outer array because the inner array has higher impedance. The outer array implodes under magnetic pressure and eventually entrains the inner array material, compressing it inward. In the analytic model, the implosion velocities are written as

$v_A = B/\sqrt{4\pi \rho}$1

where the factor $v_A = B/\sqrt{4\pi \rho}$2 in $v_A = B/\sqrt{4\pi \rho}$3 arises from the density doubling during entrainment. Integrating $v_A = B/\sqrt{4\pi \rho}$4 gives

$v_A = B/\sqrt{4\pi \rho}$5

Requiring the outer cone to reach the axis only slightly before the entrained inner structure yields the geometric constraint

$v_A = B/\sqrt{4\pi \rho}$6

In the limiting case $v_A = B/\sqrt{4\pi \rho}$7, the implosion-generated cone would have zero opening angle; for $v_A = B/\sqrt{4\pi \rho}$8, the analysis gives

$v_A = B/\sqrt{4\pi \rho}$9

so the inner entrained cone has already collapsed to a very small radius when the outer array reaches the axis (Winterberg, 2010).

With a typical diode gap $2\alpha$0, this corresponds to a small cone angle of order $2\alpha$1 radians, and the shaped-charge scaling then gives

$2\alpha$2

If $2\alpha$3, the inferred jet velocity is of order $2\alpha$4. The paper therefore argues that a nested conical wire array can, in principle, generate ultrafast jets with velocities of order $2\alpha$5 (Winterberg, 2010).

The same analysis also gives the mass partitioning. For total cone mass $2\alpha$6,

$2\alpha$7

with

$2\alpha$8

For small $2\alpha$9,

$B \propto 1/r$0

The total kinetic energy satisfies

$B \propto 1/r$1

and the jet-efficiency expression

$B \propto 1/r$2

tends to unity as $B \propto 1/r$3. In this formulation, small opening angles simultaneously favor high jet velocity and high conversion efficiency, while the jet mass becomes small (Winterberg, 2010).

4. Experimental characterization and opening-angle dependence

Recent measurements have quantified how conical wire array geometry affects jet properties under fixed drive conditions. In experiments on the Llampudkén pulsed-power generator, conical arrays of 16 Al wires of $B \propto 1/r$4 diameter were driven at peak current $B \propto 1/r$5 with current rise rate $B \propto 1/r$6. The cathode diameter was fixed at $B \propto 1/r$7, the interelectrode gap at $B \propto 1/r$8, and the anode diameter was varied to produce opening angles $B \propto 1/r$9, $\rho \propto 1/r$0, and $\rho \propto 1/r$1, corresponding to anode diameters $\rho \propto 1/r$2, $\rho \propto 1/r$3, and $\rho \propto 1/r$4, respectively (Izquierdo et al., 8 Jul 2025).

A 3 mm-thick metallic lid over the anode with a 5 mm central aperture was used so that, for $\rho \propto 1/r$5 and $\rho \propto 1/r$6, flaring ablation flows from the wires did not directly pass through the aperture; only the axial jet emerging from the zippering process did. The plasma above the anode was then characterized using moiré schlieren deflectometry, visible self-emission spectroscopy, and optical Thomson scattering. These diagnostics provided spatially resolved density, flow velocity, ion temperature, and constraints on electron temperature and mean charge state (Izquierdo et al., 8 Jul 2025).

The central density result is that the on-axis electron density peaks at approximately $\rho \propto 1/r$7 in the lower central region of the jet and decreases with height exponentially with characteristic length

$\rho \propto 1/r$8

For $\rho \propto 1/r$9, the fitted profile is

$v_A \propto 1/\sqrt{r}$0

with $v_A \propto 1/\sqrt{r}$1 in mm. The paper explicitly states that this behavior is reproducible and independent of the conical array geometry within experimental uncertainties (Izquierdo et al., 8 Jul 2025).

The strongest geometry dependence appears in the axial velocity. The measured flow velocity increases approximately linearly with axial position over the range $v_A \propto 1/\sqrt{r}$2–$v_A \propto 1/\sqrt{r}$3, and the slope increases by $v_A \propto 1/\sqrt{r}$4 for each $v_A \propto 1/\sqrt{r}$5 increase in opening angle. At $v_A \propto 1/\sqrt{r}$6, the reported values are

$v_A \propto 1/\sqrt{r}$7

At $v_A \propto 1/\sqrt{r}$8, the average velocities are $v_A \propto 1/\sqrt{r}$9, $10^7\ \text{cm/s}$0, and $10^7\ \text{cm/s}$1 for $10^7\ \text{cm/s}$2, $10^7\ \text{cm/s}$3, and $10^7\ \text{cm/s}$4, respectively (Izquierdo et al., 8 Jul 2025).

The internal thermodynamic structure is comparatively insensitive to opening angle. The ion temperature decreases from approximately $10^7\ \text{cm/s}$5 near the base of the jet to about $10^7\ \text{cm/s}$6 at higher axial positions, while the electron temperature lies in the range $10^7\ \text{cm/s}$7–$10^7\ \text{cm/s}$8 and increases with height. Typical mean charge states are $10^7\ \text{cm/s}$9–4. At $V_0$0, the sonic Mach numbers are $V_0$1, $V_0$2, and $V_0$3 for $V_0$4, $V_0$5, and $V_0$6, respectively, again reflecting the geometry-driven velocity change rather than a major change in thermodynamic state (Izquierdo et al., 8 Jul 2025).

5. Numerical modeling, instability, and hardware modifications

Three-dimensional resistive-MHD simulation has become the standard tool for interpreting conical wire array jet experiments. Bocchi and collaborators modeled MAGPIE conical wire arrays with the code GORGON, a fully 3D, single-fluid resistive-MHD code with separate ion and electron energy equations, Ohmic heating, optically thin radiation losses, and a vacuum treatment for regions below $V_0$7 (Bocchi et al., 2011). The simulated MAGPIE geometry used electrode separation $V_0$8, cathode diameter $V_0$9, anode diameter $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$0, and half-opening angle $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$1, with typical loads of 16 or 32 tungsten wires. The imposed current waveform was

$$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$2

with $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$3 and $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$4 (Bocchi et al., 2011).

A principal result of that work is the resolution requirement for instability studies. The jet diameter measured experimentally is $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$5, and simulations at $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$6 cell size produced a smooth, almost axisymmetric, artificially stable jet. At $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$7 cell size, the jet became corrugated and non-axisymmetric in a way consistent with experimental shadowgrams. The conclusion is that a minimum resolution of approximately $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$8, corresponding to roughly $$v_j = \frac{V_0}{\sin\alpha}(1+\cos\alpha), \qquad v_s = \frac{V_0}{\sin\alpha}(1-\cos\alpha),$$9 cells across the jet, is required to retrieve the unstable character of the jet (Bocchi et al., 2011).

The number of wires is a second major control parameter. Arrays with fewer wires produce more unstable jets, and the study attributes this effect to magnetic origin. With 16 wires, the larger inter-wire gaps permit stronger and more inhomogeneous local magnetic fields to penetrate toward the jet, and the region with $p_B = B^2/8\pi$00 extends up to $p_B = B^2/8\pi$01. With 32 wires, $p_B = B^2/8\pi$02 extends only to $p_B = B^2/8\pi$03, and the jet is more stable and more symmetric. The main changes are in density distribution and stability rather than bulk axial acceleration (Bocchi et al., 2011).

The same paper also examined a conical shield above the array, oriented perpendicular to the wires and provided with an 8 mm aperture. The shield reduces the presence of unwanted low-density plasma flows above the array, yielding a cleaner jet region, but the resulting jet is shorter and less dense. The axial velocity profile is barely altered, while areal electron density is significantly reduced along most of the axis. This creates a tradeoff between suppressing background flow and preserving jet momentum for jet-target interaction experiments (Bocchi et al., 2011).

6. Applications, preconditioning, and limitations

The most ambitious applications of conical wire arrays arise in Winterberg’s nested-array proposal. The first is supersonic shear flow stabilization of a dense z-pinch or magnetized target. In the modified concept discussed there, a plasma jet from a nested conical wire array is shot into liquid DT in an externally applied axial magnetic field. The paper argues that a jet with $p_B = B^2/8\pi$04, density $p_B = B^2/8\pi$05, and length $p_B = B^2/8\pi$06–$p_B = B^2/8\pi$07 could satisfy the quoted Lawson criterion $p_B = B^2/8\pi$08 and the magnetized-target condition $p_B = B^2/8\pi$09 (Winterberg, 2010). The second application is fast ignition of a pre-compressed DT target, including the possibility that a very fast jet might condense into small solid or dust-like particles and thereby implement an “impact ignition” variant using pulsed power (Winterberg, 2010).

In laboratory astrophysics, conical wire arrays are already established as analog platforms for supersonic, magnetized, radiatively cooling jets. The MAGPIE work was explicitly designed to strengthen the link between laboratory and astrophysical jets, particularly jets from young stellar objects, by combining experiments with validated 3D simulations and by examining control parameters such as wire number and shielding (Bocchi et al., 2011). The more recent opening-angle study further suggests a practical experimental virtue of conical geometry: it provides effective control over jet propagation velocity while leaving the internal density and temperature structure largely unchanged within the explored parameter space (Izquierdo et al., 8 Jul 2025).

A separate line of work concerns wire preconditioning. Experiments on the Qin-1 double-pulse current generator used two parallel Al wires rather than conical arrays, but the accompanying analysis states that the physics is directly transferrable to conical arrays. A prepulse of $p_B = B^2/8\pi$10 with $p_B = B^2/8\pi$11 rise time, followed by a $p_B = B^2/8\pi$12, $p_B = B^2/8\pi$13 main current, was shown to suppress core-corona structures, ablation, and trailing mass by gasifying the wires before the main implosion. The seeds for the MRT instability formed from the inhomogeneous ablation were suppressed, but the magneto Rayleigh-Taylor instability during the implosion was still significant (Wu et al., 2017). This suggests a possible route for cleaner conical-array implosions, but not a complete solution to instability.

The limitations in the literature are explicit. Winterberg’s nested-array treatment is largely analytic and assumes ideal MHD or hydrodynamics, cylindrical symmetry and radial dependence only, and simple current-path assumptions, while neglecting detailed treatment of Rayleigh-Taylor, sausage, and kink instabilities, resistive diffusion, anomalous transport, and radiation losses in the main derivation (Winterberg, 2010). The MAGPIE simulations demonstrate that insufficient numerical resolution can qualitatively change conclusions about jet stability (Bocchi et al., 2011). The Thomson-scattering study is restricted to one current waveform, one wire material, one wire diameter, and opening angles $p_B = B^2/8\pi$14, $p_B = B^2/8\pi$15, and $p_B = B^2/8\pi$16, so extrapolation to very different regimes must be made with care (Izquierdo et al., 8 Jul 2025).

Taken together, the literature presents conical wire arrays as a family of pulsed-power loads whose physics depends strongly on the distinction between ordinary ablative jet formation, idealized cone-collapse jetting, and nested-geometry synchronization. Standard single conical arrays are established sources of collimated, supersonic plasma jets for laboratory astrophysics, with geometry-dependent velocity and well-characterized instability behavior (Bocchi et al., 2011, Izquierdo et al., 8 Jul 2025). Nested conical arrays were proposed to recover the small-angle shaped-charge mechanism and thereby access jet velocities of order $p_B = B^2/8\pi$17 for fusion-related applications (Winterberg, 2010). The open problems are not conceptual definition but controlled realization: current sharing, implosion timing, three-dimensional symmetry, and the suppression or management of MHD instabilities remain the central technical issues across the field (Winterberg, 2010, Wu et al., 2017).