Increasing plasma parameters using sheared flow stabilization of a Z-pinch

Magnetic compression and confinement of plasma in a Z-pinch configuration date back to the earliest investigations of plasma physics. Z-pinch formation was described in a theoretical publication by Bennett¹ as the “self-focusing” effect of a current-carrying plasma column. A comprehensive review of the Z-pinch is provided by Haines.² 

It is well known that Z-pinches suffer from classical sausage and kink MHD instabilities.^3–5 High-voltage pulsed power technology has been used to quickly form the Z-pinch in an attempt to outpace instabilities. Z-pinches initiated from deuterium fibers were studied for their potential as a compact fusion reactor,⁶ but these plasmas were plagued by rapidly growing instabilities.⁷ Wire arrays,⁸ solid cylindrical liners,⁹ and gas puff valves¹⁰ have been used to mitigate instabilities by producing plasmas that can be quickly imploded and stagnated on axis before instabilities have grown significantly. As a result, the Z-pinch has experienced success as a radiation source^11,12 and is being investigated in application to compressing magnetized fusion fuel in MagLIF.^13–15 

Various means exist to mitigate and stabilize MHD modes, such as pressure profile control⁴ and applied magnetic fields.^16,17 In the context of fusion applications, sheared plasma flow is a promising mechanism for providing stability in that it does not require external magnetic field coils, and it does not result in heat loss through enhanced thermal conductivity. Shear flow stabilization, generated by differential E × B drift, has been used to improve tokamak confinement and to explain observed H-mode transitions.¹⁸ A general review of sheared flow stabilization and suppression of turbulence is given by Terry.¹⁹ 

Plasma columns have exhibited stable behavior when axial flows are present.^20–22 This article describes scaling the sheared flow stabilized Z-pinch to high plasma parameters. Section II presents the theoretical scaling relations that demonstrate that an increase in the plasma current in a stable Z-pinch decreases the plasma radius and increases the density, temperature, and magnetic field. Experimental and computational studies of a long duration and long length Z-pinch are presented in Sec. III. It is shown that the plasma's stability properties are consistent with the sheared flow stabilization theory. Spatially dependent viscosity²³ profiles are calculated from the experimentally measured plasma parameters and found to be consistent with the observed plasma flow profiles and plasma length. Section IV presents the experimental measurements of magnetic fluctuation levels, velocity profiles, and density structure indicating stable Z-pinch plasmas with decreased plasma radii and corresponding high densities and temperatures. A discussion of open questions and future directions is provided in Sec. V.

Attractive features of the Z-pinch configuration include its linear geometry, efficient utilization of the confining magnetic field, and simple one-dimensional equilibrium, which is described by a radial force balance. The plasma pressure is balanced by an axial current density and the resulting self-generated azimuthal magnetic field, such that in a charge-neutral two-fluid equilibrium

where n_i, n_e are the ion and electron number densities, T_i, T_e are the ion and electron temperatures, k_B is the Boltzmann constant, μ₀ is the permeability of free space, and $Bθ$ is the azimuthal magnetic field. The plasma pressure is composed of the sum of the ion and electron pressure

Note that the configuration has no axial magnetic field, i.e., it is not a screw pinch, and therefore needs no magnetic field coils. The average β of the Z-pinch is defined as

If the plasma pressure vanishes at a distant outer wall at radius r_w where the axial current returns, then $⟨β⟩=1$⁠, which represents the ideal magnetic confinement efficiency. See the  Appendix for the derivation.

Integrating Eq. (1) over the plasma volume yields the Bennett relation¹ 

where Z is the ionization state, I is the total axial plasma current, $⟨T⟩$ is the average temperature, assuming T_i = T_e, and N_i is the linear ion number density, defined as

The Bennett relation is valid for any equilibrium profile of density and magnetic field, and Eq. (5) leads directly to a temperature scaling relation

where subscripts 1 and 2 refer to two states of the system. For example, changing parameters (I₁, N₁) in a Z-pinch to (I₂, N₂) produces a change in the temperature from T₁ to T₂.

If plasma compression occurs on a time scale such that no heat is transferred into the system, then the adiabatic approximation is valid, such that

where γ is the adiabatic index, and additional scaling relations can be derived.²⁴ 

Using the “sharp pinch” (see Ref. 3) as a convenient model, for an equilibrium with a uniform density, $ni=n, ne=Zn$⁠, and temperature, $Ti=Te=T$⁠, within the characteristic pinch radius a beyond which the density is zero, the average density scales as

and the characteristic plasma pinch radius scales as

Scaling relations given by Eqs. (7), (9), and (10) provide a theoretical foundation that shows how different regimes can be accessed. Of particular interest is the ability to reach high-energy density (HED)²⁵ parameters, and possibly thermonuclear fusion conditions, in a compact device. In particular, for a Z-pinch with a constant mass and therefore a constant linear density, increasing the plasma current from I₁ to I₂ increases the plasma temperature and density and decreases the plasma radius, leading to a smaller device. The scaling is illustrated in Fig. 1, which shows that by increasing the current, a Z-pinch plasma can theoretically reach high temperatures and small radii.

Based on the scaling properties, controlling the radial compression allows the Z-pinch equilibrium to reach the plasma parameters of interest. However, the scaling results are predicated on a stable plasma, in a configuration that is classically unstable to MHD modes.^3–5 To exploit the favorable scaling that the Z-pinch provides, a viable means of providing stability is needed.

The seminal work by Kadomtsev⁴ demonstrated that the diffuse static Z-pinch equilibrium can be unstable to internal MHD modes. The sausage mode has an azimuthal mode number m = 0 and is therefore axisymmetric. This mode can be stabilized by limiting the pressure gradient such that

where β is a local measure of the ratio of plasma pressure to magnetic pressure. With $γ=5/3$⁠, the equilibrium pressure satisfying the marginal stability condition of Eq. (11) is given in the parametric form by

where p₀ is the plasma pressure on axis. Note that there is an error in the parametric forms given in Ref. 4. Asymmetric modes, such as m = 1, can be stabilized by limiting the pressure gradient such that

Modes with $m≥2$ are stable if the m = 1 kink mode is stable. Therefore, the m = 0 sausage mode and m = 1 kink mode are the two primary MHD instabilities of concern for the Z-pinch. Unlike the m = 0 mode, no realizable pressure profiles exist that satisfy the m = 1 stability condition unless a hard core, i.e., a rigid conductor, is placed at the axis.²⁶ 

The Z-pinch can be stabilized against the m = 1 mode by embedding an axial magnetic field²⁷ with a sufficient strength, given by the Kruskal-Shafranov limit.^16,17 The stability condition limits the axial plasma current relative to the embedded axial magnetic field, which significantly reduces the average β. If electrodes at the end of the plasma column are used to supply the Z-pinch current, the applied axial field can decrease the energy confinement time through parallel thermal conduction to the electrodes. Without an axial magnetic field, thermal conductivity in the Z-pinch is everywhere perpendicular except at the magnetic field null at the axis. Parallel thermal conductivity can be orders of magnitude larger than the perpendicular thermal conductivity,²³ $κ∥/κ⊥∝(ωcτ)2$⁠, making sustained confinement of a high-temperature plasma challenging. Note that the stabilizing effect of an embedded axial magnetic field is distinct from the successful magnetic insulation effect used in magnetized target fusion concepts,²⁸ including MagLIF.^13,14,29

Stabilization of the m = 1 kink mode in the Z-pinch can also be provided by surrounding the plasma with a conducting wall.^30,31 The stabilizing effect is only realized when the wall is within 20% of the pinch radius, $rw/a<1.2$⁠, which does not provide an adequate separation to sustain a high-temperature plasma.

Thus while scaling relations based on the Bennett relation and the adiabatic approximation suggest ways in which Z-pinch plasmas can be scaled to smaller radii and high energy density conditions, instabilities need to be addressed. Traditional means of stabilization are inadequate for long-lived HED plasmas. This calls for an alternative means of stabilization, like sheared flow.

Sheared flows have the potential to stabilize the MHD modes in a Z-pinch without the deleterious effects of the approaches described in Sec. II. The configuration is called the sheared flow stabilized Z-pinch. Since the axial flows do not alter the equilibrium radial force balance given by Eq. (1), sheared axial flows $vz(r)$ can be introduced without reducing $⟨β⟩$⁠. Theoretical analysis suggests that the m = 1 kink mode in a Z-pinch can be stabilized when the flow shear exceeds a threshold value,³² $dvz/dr>0.1kVA$⁠, where k is the axial wave number, and V_A is the Alfvén speed. The analysis assumed a uniform flow shear, which is not realizable in practice at the axis due to viscosity. Nevertheless, numerical simulations and experimental investigations of the ZaP Flow Z-Pinch^31,33–37 have demonstrated that sheared flow stabilization of a Z-pinch behaves as theoretically predicted.

Flow Z-pinches are generated in the ZaP experiment by coupling a coaxial accelerator with a pinch assembly region where the flow Z-pinch forms and is sustained. The ZaP machine drawing is shown in Fig. 2. Figure 3 schematically illustrates the experimental operation and formation of a sustained flow Z-pinch in the ZaP experiment. Neutral gas, typically hydrogen, is injected at the neutral gas injection plane at t₁. After the gas expands into the annulus of the coaxial acceleration region, a capacitor bank with 144 kJ of stored energy at 10 kV is discharged across the inner and outer electrodes at t₃ leading to gas breakdown and initial plasma acceleration. The Lorentz force accelerates the plasma axially until it reaches the end of the inner electrode at t₄. The plasma transitions from the inner electrode to the axis where the plasma assembles into a Z-pinch configuration at t₅. Inertia and gun currents maintain the axial plasma flow through a deflagration process^35,38 and resupply plasma that exits the Z-pinch through the hole in the electrode end wall. The flow Z-pinch configuration is sustained until the injected gas in the accelerator is depleted, the capacitor bank current diminishes, or the plasma becomes unstable.

Magnetic probes are embedded in the outer electrode to measure the time-dependent magnetic structure along the length of the outer electrode and azimuthally in the assembly region. Magnetic fluctuations for the m = 1, m = 2, and m = 3 azimuthal modes are computed from a Fourier analysis of data from eight magnetic probes that are located at the same axial location and different azimuthal angles. The fluctuation level is computed by normalizing the asymmetric component of the magnetic field by the symmetric component, $Bm/B0$⁠, measured at the same axial location. Analyzed data are shown in Fig. 4. Large magnetic fluctuations exist during the Z-pinch formation. The fluctuation levels for all resolved azimuthal mode numbers subside to a low level during an extended quiescent period, which lasts 27 μs for the data shown. The theoretical linear growth time for the m = 1 kink mode in a static plasma with the measured parameters is 20 ns assuming an axial wavelength equal to the plasma diameter. Therefore, a stabilizing effect must be present.

The evolution of the plasma flow profile is determined by recording the radiation spectra through a view port that is oblique to the axis³⁹ and measuring the Doppler shift of impurity line emission.³³ Spectra are recorded using an intensified charge-coupled device (ICCD) detector with a 200 ns gate. Comparisons between different pulses are facilitated by defining a time normalized by the quiescent period such that the quiescent period begins at τ = 0 and ends at τ = 1. As described in Refs. 35 and 36, the experimentally observed quiescent periods are correlated with the presence of sheared flows that exceed the theoretical threshold given in Ref. 32. The flow profiles do not have a uniform shear. Nonlinear simulations³⁶ indicate that a uniform shear is not necessary and that a shear located at the edge of the plasma is sufficient to provide stabilization.

Flow profiles with a shear localized at the plasma edge are expected due to spatially dependent viscous damping. Magnetic fields introduce anisotropy into the ion viscosity, such that viscosity perpendicular to the field can be much lower than that parallel to the field. The flow Z-pinch has a radially sheared axial flow in the presence of an azimuthal magnetic field, which is zero at the axis and peaks at the pinch radius. The ion viscosity appropriate for the flow Z-pinch geometry is given as²³ 

which peaks at the axis where the ion cyclotron frequency, ω_ci, is zero, and the product of the ion pressure and the ion collision time, τ_i is maximum. The high viscosity at the magnetic field null leads to a flat velocity profile close to the axis. By contrast, the high magnetic field near the characteristic pinch radius produces a low viscosity and a concentration of the velocity shear. A viscous damping distance, i.e., the distance at which viscous effects are expected to eliminate the flow shear, is computed as $Lμ=vznimiL2/μ$⁠, where the shear scale length is assumed to be the plasma radius, L = a. For ZaP plasma parameters, the viscous damping distance is over 200 cm at the plasma radius, which indicates that the sheared flow profile can be sustained throughout the length of the assembly region.

Data from the ZaP Flow Z-Pinch experiment indicate typical plasma parameters of $ne=2×1016$ cm^–3, B_a = 1 T, and T_e = 75 eV, with a plasma radius of a = 1 cm and a length of 100 cm. Evidence of stability along the full length of the Z-pinch plasma is provided by computing the magnetic field variations along the length using the magnetic field probes that are located at 5 cm intervals in the pinch assembly region. A stable plasma should have a uniform magnetic field value. Figure 5 shows the magnetic field variation measurements from an experimental configuration with a 126 cm-long pinch assembly region, $z=$ −17 cm to 109 cm. See Fig. 2 for reference. Before the quiescent period begins, $τ<0$⁠, the variations are in excess of 30%, but during the quiescent period, $0<τ<1$⁠, the variations are significantly reduced and mostly less than 10% along the 126 cm pinch length.

While the ZaP experiment successfully investigated the sheared flow stabilization of a Z-pinch, the achievable plasma parameters were limited by the power supply and neutral gas injection. As the plasma current was increased, the plasma in the accelerator would deplete more rapidly and thereby shorten the quiescent period.

Plasma current in the ZaP experiment is divided between the Z-pinch plasma and accelerator plasma. The ZaP-HD Flow Z-Pinch High Density experiment uses a modification of the ZaP design to introduce a third coaxial electrode to improve control of the plasma current. Machine drawings of ZaP and ZaP-HD are shown in Fig. 6. The triaxial electrode design uses two independent power supplies that separate the control for plasma acceleration and plasma compression. Plasma acceleration is controlled by a capacitor bank connected to the inner and middle electrodes. Plasma compression is controlled by a capacitor bank connected to the inner and outer electrodes. The ZaP-HD assembly region length is 50 cm.

Plasma acceleration and Z-pinch plasma compression in the ZaP-HD experiment have been simulated with the Mach2⁴⁰ nonlinear, time-dependent, resistive MHD code. The geometry corresponds to the ZaP-HD experiment shown in Fig. 6 with the triaxial electrode design. Mach2 uses a block-structured mesh that can model complex geometries. The simulation includes the currents from the two separate capacitor banks using experimentally measured current waveforms and applied between the corresponding electrodes. The plasma pressure evolution is shown in Fig. 7. A Z-pinch configuration is established at 30 μs along the entire 50 cm length of the assembly region. The plasma maintains a high pressure with a structure that exhibits a high degree of axial uniformity. The sustained Z-pinch plasma has a radius of approximately 0.5 cm.

Stable behavior is experimentally observed in ZaP-HD with a quiescent period similar to previous ZaP results. The ZaP-HD experiment has more magnetic field probes than ZaP, which allows for simultaneous monitoring of the azimuthal fluctuation levels at nine axial locations (every 5 cm) in the assembly region. Figure 8 shows the magnetic fluctuation level for the m = 1 mode for a typical ZaP-HD plasma pulse. The magnetic fluctuation at the nine axial locations is represented as a mean and standard deviation for each recorded time. The data are presented as a solid line for the mean with error bars for the standard deviation, i.e., $x¯±σ$⁠. Thereby, the plotted magnetic fluctuation values indicate stability along the entire length of the Z-pinch plasma. A quiescent period is evident from 24 μs to 72 μs where the mean value of the magnetic fluctuations is reduced to a low level. The diminished standard deviation during the quiescent period suggests the stability of a 50 cm long Z-pinch plasma with high axial uniformity.

Axial velocity profiles measured in ZaP-HD are shown in Fig. 9. The velocity is determined from the Doppler-shifted centroids of chord-integrated C-III line (229.7 nm) emission recorded with the ICCD spectrometer³³ using a 200 ns gate. The data are presented in a chord-integrated form to demonstrate the reproducible centering of the plasma. The data still indicate the evolution of the velocity profile. The plasma velocity is high and uniform before the quiescent period begins. A large flow shear develops coincident with the beginning of the quiescent period. The flow shear remains large throughout the quiescent period, except for a transient reversal that has been previously observed.³⁶ The flow shear reversal is conjectured to be caused by a stagnation of the outer-radii flow on the electrode end wall. The evolution of the plasma flow, the plasma stability behavior, and their correlation are consistent with and similar to observations from the ZaP experiment.³⁶ 

With improved control, a larger current can be supplied to the ZaP-HD Z-pinch plasma. Smaller plasma pinch radii are observed as compared to ZaP measurements. Density structure is measured using a digital holographic interferometer.⁴¹ The technique uses a pulsed Nd:YAG laser and a consumer-grade digital SLR (single-lens reflex) camera arranged in a Mach-Zehnder configuration. The recorded holograms are numerically reconstructed using a Fresnel transform that is able to accurately resolve the two-dimensional structure of the complex-valued electric field strength. Chord-integrated electron number density is related to the phase shift computed from the complex-valued wave field

where e is the electron charge, c is the speed of light, m_e is the electron mass, ϵ₀ is the permittivity of free space, λ is the laser wavelength, and $Ñe=∫nedl$ is the chord-integrated electron number density.

The number density profiles, $ne(r)$⁠, at each axial location of the hologram are calculated through an Abel inversion⁴² by assuming axial symmetry and iterating to locate the axis. A plot of the number density along a plane through the plasma axis, i.e., $ne(x=0,y,z)$⁠, obtained for a plasma pulse is shown in Fig. 10. The presentation of the data retains the measured displacement of the pinch relative to the geometric axis of the experiment. The displacement is seen to vary less than ±0.05 cm. The electron number density peaks at approximately $2×1017$ cm⁻³ and has a pinch radius of approximately 0.3 cm along the axial extent of the image.

Radial profiles of the number density are extracted from the two dimensional plot at three axial positions and are presented in Fig. 11 including error bars associated with the experimental measurement and data analysis. The density profiles can be further analyzed by integrating additional diagnostic measurements to yield profiles of additional equilibrium plasma parameters, such as magnetic field and temperature.

Ampère's law and magnetic field probe data (located at the same axial location as the recorded holograph in Fig. 10), are used to compute the Z-pinch plasma current. Assuming a uniform drift speed between the ions and electrons, $vd=vi−ve$⁠, and matching the experimentally measured magnetic field at the outer electrode allows the calculation of the magnetic field profile from the electron number density

When the electron number density profiles of Fig. 11 are analyzed, the magnetic profiles of Fig. 12 are generated. The magnetic field peaks at 8.5 T at the plasma pinch radius, and all three profiles are similar indicating the axial uniformity.

Integrating the radial force balance equation, Eq. (1), provides a calculation for the electron temperature profile

assuming T_e = T_i, which is consistent with the expected collisionality. Figure 13 shows the temperature profiles calculated using this method. The electron temperature peaks at 1 keV at the axis and has a steep gradient where the magnetic field becomes high. The electron number density profiles and the analysis of that data indicate a small-radius Z-pinch plasma that is axially uniform and has high magnetic fields and high temperatures. The results suggest that the sheared flow stabilized Z-pinch could reach sustained HED conditions.

The experimental and computational results are encouraging, but there are open questions that require further investigation. The scaling relations derived in Sec. II assume that compression of the Z-pinch plasma is an adiabatic process. While the compression is slower than the fast Z-pinch implosion technique, shocks may occur during the initial formation of the flow Z-pinch. Determining the polytropic index is important to predict the plasma parameters that result from increasing the current.

Sheared flow stabilization appears to be robust over a range of plasma parameters as shown in Secs. III and IV. However, scaling to even higher parameters may introduce phenomena that cannot be stabilized or mitigated by sheared flows. For example, drift instabilities may appear as the plasma is compressed with higher currents and drift speeds. If sheared flows can replace magnetic shear to stabilize the Z-pinch, there may be opportunities to generalize the technique. Flow shear may be able to replace or augment magnetic shear in other plasma confinement configurations. Computational simulations using high-fidelity plasma models are essential to shed light on the sheared flow phenomena in the Z-pinch and in other configurations.

Scaling relations that build on the derivation presented in Sec. II show that the sheared flow stabilized Z-pinch can scale to fusion conditions in a compact device with modest power supplies. Scientific breakeven may be possible with currents of 650 kA²⁴ provided the plasma remains stable.

The sheared flow stabilized Z-pinch creates plasmas with high density and high temperature on a scale that facilitates diagnostic measurements. Further successful scaling could result in a compact and inexpensive fusion reactor.

The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency – Energy (ARPA-E), U.S. Department of Energy, under Award No. DE-AR-0000571, the U.S. National Nuclear Security Administration under Grant No. DE-NA0001860, and the U.S. Department of Energy under Grant No. DE-FG02-04ER54756.

Average beta as defined in Eq. (4) provides a measure of magnetic plasma confinement efficiency, since volume-averaged plasma pressure indicates confinement, and magnetic pressure at the outer wall indicates the cost of producing the magnetic field. The volume-averaged plasma pressure is given as

where p is the sum of ion and electron pressure, and the integral is performed over the confinement volume. Integrating the right-hand side by parts and then substituting the pressure gradient from Eq. (1) yields

If the plasma pressure is zero at the outer wall, then the first term on the right-hand side vanishes. The remaining integral simplifies to

which is equivalent to $⟨β⟩=1$ in Eq. (4).