Plasma exhaust in a sheared-flow-stabilized Z pinch

Plasmas relevant to nuclear fusion and space propulsion can be described by the ideal magnetohydrodynamics (MHD) equations which predict that plasma is frozen into the magnetic flux.¹ In space propulsion, thrust is only achieved if plasma escapes the source. However, magnetic fields tied to external electrodes can prevent plasma exhaust as the plasma follows the divergence-free field.

In order for a plasma to escape a magnetic field, non-ideal MHD processes must exist in the system. First, plasma resistivity can be sufficiently high to allow diffusion of the magnetic field to take place on a fast enough timescale.² Second, plasma can demagnetize if the Larmor radius becomes sufficiently large compared to the scale length of the system.^3,4 These two phenomena are studied extensively in the context of magnetic nozzle research. Third, flux can escape if the magnetic field reconnects outside of the flux-conserving region.⁵ However, for all the aforementioned effects to take place, a plasma trapped within a flux-conserving region has to first break free of the confining forces.

This paper focuses on how plasma exhaust can be achieved by changing the ratio of plasma ram and thermal pressures to magnetic pressure at the partially open boundary of a flux conserver. Specifically, plasma properties from multiple pulses are analyzed from the Fusion Z-pinch Experiment (FuZE) at the University of Washington to assess how the flux conserver boundary geometry affects the plasma exhaust.

FuZE produces a 0.3 cm radius by a 50 cm long Z pinch between the end of the inner electrode of a coaxial plasma gun and an end wall 50 cm away. The plasma column is stabilized for thousands of instability growth times by an embedded radially sheared axial plasma flow, a method proposed in Ref. 6 and demonstrated experimentally in previous sheared flow-stabilized Z-pinch devices, ZaP⁷ and ZaP-HD.⁸ FuZE uses a similar geometry and gas injection as the ZaP experiment, while improving the power supply and gas injection system. Z pinches are formed and sustained by plasma from a coaxial accelerator coupled to an assembly region. The assembly region is an extension of the plasma accelerator outer electrode. The accelerated plasma is compressed on axis, which forms the Z pinch. The accelerator continues to supply an axially flowing plasma to the Z pinch, maintaining the sheared flow necessary for stabilization.

The coaxial accelerator is derived from the Mashall gun,⁹ the same geometry used for spheromak formation.¹⁰ Unlike the application for spheromaks, the sheared-flow-stabilized Z pinch does not have any applied magnetic field, leading to an ideal magnetic confinement efficiency.¹¹ 

Ion temperatures of 1–2 keV and peak electron densities of $1×1023 m−3$ are obtained in a 200 kA Z-pinch sustained for 5–8 $μs$⁠.¹² Based on force balance, the electron temperature is approximately equal to the ion temperature.⁸ Recently, FuZE demonstrated sustained generation of neutrons over thousands of instability growth times,¹² suggesting the sheared flow Z-pinch concept's potential as a thermonuclear fusion core for energy production by operating at higher current.

The flux conserver boundary is provided by the outer electrode end wall located 50 cm downstream of the end of the coaxial acceleration region, illustrated in Fig. 1. The end wall provides a fixed return current path from the outer electrode, trapping the magnetic flux in the pinch assembly region. For plasma to escape the flux conserver region, it has to first overcome the magnetic forces at the end wall, and subsequently detach from the magnetic field, or the magnetic field has to escape.

This paper analyzes the Z-pinch current and plasma density measured from two different end wall geometries: a “central hole” end wall and a “spoked” end wall. The end wall is shown to affect the plasma current when the device is operated at high bank energy and high gas injection. The plasma magnetic Reynolds number is high enough such that ideal MHD should properly describe the plasma dynamic. However, a model shows that increasing the plasma ram and thermal pressures relative to the magnetic pressure leads to exhaust as plasma overcomes the magnetic tension at the end wall spokes. The conditions for which increased plasma exhaust is predicted are satisfied experimentally in the high bank energy conditions.

This paper first presents the measured plasma current and plasma density in the assembly region and their evolution with the end wall geometry change. Then, the magnetic Reynolds number is calculated. An equilibrium model at the end wall is presented and the results are used to show that the increase in plasma ram and thermal pressures to the magnetic pressure corresponds to an increase in plasma exhaust.

This section compares the plasma current and density between end wall designs. The current measurement is the closest plasma diagnostic to the end wall and therefore shows the largest sensitivity to the geometry change. The density is used to compute the plasma ram and thermal pressures, while the current is used to compute the magnetic field pressure which are the parameters that determine plasma exhaust, as shown in Sec. III. The location of the different diagnostics are shown in Fig. 1.

The central hole end wall has a 2 cm diameter exit located on the axis that opens a total of 1.1% of the end wall area. This design was used on the ZaP and ZaP-HD experiments as well as the initial FuZE experiment. The spoked end wall has eight openings creating a ratio of end wall open area of 68.1%. The spoked geometry was chosen to maximize the open area while maintaining the structural integrity of the end wall necessary to sustain the electromagnetic forces.

The line-integrated density is measured at 35 cm, 1 cm above the pinch axis. This axial location also has an azimuthal array of magnetic field probes measuring the pinch current. Finally, axial plasma velocity and ion temperature are obtained from ion Doppler spectroscopy at 15 cm.

Two operating conditions are tested. The first operating condition uses a low neutral gas mass through a single gas valve and a low bank energy of 80 kJ, obtained from a 4 kV charge voltage. This will be referred to as “case 1.” Current and density data for case 1 are shown in Figs. 2(a) and 3(a), respectively. The second operating condition is chosen to be similar to the neutron producing conditions used in Ref. 12. These conditions use higher neutral gas mass injected through five valves and a higher bank energy of 125 kJ, obtained from a charge voltage of 5 kV. This set of pulses will be referred to as “case 2.” Current and density data for case 2 are shown in Figs. 2(b) and 3(b) respectively. The gas used is a mixture of 95% $H2$ and 5% $CH4$ by pressure. The $CH4$ is added as a diagnostic impurity to produce the carbon lines and CV lines used for spectroscopic analysis. Between 15 and 20 pulses were taken for each condition to collect enough data for ion Doppler spectroscopy and to limit effects from pulse to pulse variation. The set of plasma pulses is repeated after changing the end wall. A set of pulses for case 1 and a set for case 2 is performed for each end wall configuration, for a total of four sets.

Pinch current is measured at $z=45 cm$ through azimuthal arrays of eight magnetic field probes. Each plot in Fig. 2 shows the average current at 45 cm as well as a region encompassing one standard deviation above and below the average value, computed from the plasma pulses produced with the same experimental settings. For case 2, the current at all axial locations between 0 and 35 cm agree to within one standard deviation for the two end wall configurations. However, the current at 45 cm exhibits a greater change between end wall designs compared to all other axial locations. For case 1, the current stays within one standard deviation for all axial locations, including at 45 cm. This indicates a change in plasma behavior that results from the end wall geometry in case 2.

Density data are obtained from a He–Ne interferometry system.¹³ Figure 3 compares the central hole end wall density signal from all the pulses from case 1 and all the pulses from case 2. The location of the line-integrated density measurement is indicated in Fig. 1. The average assembly region density is obtained by dividing the line integrated density by the path length of the interferometer chord inside the assembly region. The path length is 19.2 cm. The density is measured at $z=35 cm$⁠, which is coincident with the eight-probe magnetic field measurement array closest to the end wall with an optically accessible path. The initial density for case 1 is about $0.5×1021 m−3$⁠, while the initial density for case 2 is $5.0×1021 m−3$⁠. The temporal evolution of density is not affected by the end wall change for any of the run conditions.

Velocity profiles are recorded at 15 cm through ion Doppler spectroscopy^14,15 using a telescope angled at 45° with respect to the z axis and measuring CIII impurities emission (229.7 nm). The spectrometer records spectra through 20 chords spanning a total of 3.88 cm perpendicular to the pinch axis. One 0.5 μs exposure is recorded per pulse for the ion Doppler spectroscopy analysis. Velocities and temperatures are averaged across multiple pulses. No significant difference is found between the two end walls for the CIII velocities and temperature. Average velocities of 110 km/s and 140 km/s are calculated using the Doppler shift of the CIII line for case 1 and case 2, respectively. An average assembly region temperature of 80 eV is calculated from the Doppler broadening of the CIII lines in both cases.

In both cases, no change is apparent from the measured properties upstream of 40 cm, including the density signals shown in Fig. 3. For case 1, the current traces stay within a standard deviation across all axial locations between end wall configurations, showing that the geometry change has little effect on the upstream plasma. However, for case 2, the current traces close to the end wall exhibit a change as illustrated by Fig. 2(b).

This section shows that the plasma is linked to the magnetic flux such that it should not escape the assembly region flux conserver under ideal MHD. However, we also show that the equilibrium at the end wall will let plasma escape when the plasma thermal and ram pressures increase relative to the magnetic pressure. Finally, the ratio of ram and thermal pressures to magnetic pressure, β_ram, is calculated and we show that it is much greater for case 2, where an upstream current change is detected compared to case 1 where the currents remain unchanged.

The magnetic Reynolds number, Rm, quantifies the relative importance between advective effects and resistive diffusion effects. For a plasma where advection dominates, resistivity can be neglected and ideal MHD applies. In a flux conserver, an ideal MHD plasma is frozen in flux and no particles can escape the region. The magnetic Reynolds number is given by

where η is the plasma resistivity, μ₀ is the vacuum permeability, a is the scale length of the system, and v is the characteristic speed. The end wall thickness of 7.6 cm is the distance that the plasma has to travel in order to be outside the volume of the flux conserver and is chosen to be the scale length a. v is chosen to be the plasma velocity, which is approximately $110 km/s$ for case 1 and $140 km/s$ for case 2.

In SI units, the perpendicular Spitzer resistivity¹⁶ for hydrogen is

where T_e is the electron temperature in eV and $ln Λ$ is the Coulomb logarithm. The magnitude only depends on the plasma temperature as $ln Λ$ is nearly constant for a large range of densities and temperatures. A value of $ln Λ=17$ is used for the Coulomb logarithm. The average background plasma temperature in the assembly region will be lower than the peak Z-pinch temperature, which is approximately 1 keV for case 1.¹² The lower bound for the plasma temperature is obtained from Doppler broadening of CIII impurities. The CIII impurity wavelength is associated with colder ions,¹⁷ characteristic of the background, off-axis, plasma. The resulting range of magnetic Reynolds numbers is shown in Fig. 4.

At the lowest estimated temperature, the magnetic Reynolds number is still well above unity, supporting the frozen-in flux condition. The resistivity of the plasma at 20 eV is comparable to the resistivity of the spoked end wall, made of graphite. The resistive diffusion timescale, $τ=μ0L2/η$⁠, is on the order of $500 μs$⁠, much longer than the $40 μs$ time span of interest shown in the current and density data of Figs. 2 and 3. Therefore, we can assume that the magnetic field is frozen to the plasma and that the end wall acts as a perfectly conducting boundary.

The end wall closes the flux conserver, providing the return current path. Consequently, as shown by the previous calculation, no plasma is allowed to escape the assembly region as long as an end wall providing electrical continuity is present, independent of its open area. The frozen in flux condition explains why no large difference is found along most of the Z-pinch length between the two end wall configurations. Both end walls provide equivalent boundary conditions for the magnetic field. However, data taken just upstream of the end wall for case 2, shown in Fig. 2(b), suggest an effect from the geometry change. This effect could be due to plasma ram and thermal pressures exceeding the magnetic pressure, leading to plasma exhaust and expansion beyond the end wall where non-ideal MHD effects can take place.

As shown in Sec. III A, the plasma and the magnetic flux are linked. The end wall needs to act as a barrier to the plasma flow as well as the return current path, conserving both the magnetic flux and the plasma. To do so, the plasma must be in force balance at the end wall. The ratio of ram pressure and thermal pressure over the magnetic pressure at the end wall is defined as

where $ρ0,vz,0,p0$⁠, and B₀ are the plasma parameters at the upstream face of the end wall.

Sections III B 1 and III B 2 show that more plasma can pass through the spoked end wall as β_ram increases. Case 2 has a higher β_ram compared to case 1, leading to an increased transparency which explains the difference in plasma current observed in Fig. 2 for case 2.

We can represent the spoked end wall as a configuration of purely radial current and a pressure varying along the z axis. The plasma is confined by the end wall spokes, preventing the azimuthal magnetic field from freely expanding. The interaction of the plasma with the end wall spokes is similar to magnetic draping of the solar wind as it interacts with planets.¹⁸ In particular, the orientation of the magnetic field relative to the obstacle is important.¹⁹ The radial end wall spokes are perpendicular to the azimuthal magnetic field, as shown in Fig. 5. Using the steady state MHD momentum equation, Ampère's law, and assuming that $v→=vzẑ$⁠, we get the following force balance equation:

In Eq. (4), ρ is the plasma density and v_z is the plasma axial velocity. At the midpoint between two adjacent spokes, the plasma pressure as well as the magnetic pressure gradients are directed to expand the plasma out of the flux conserver, while only the magnetic tension holds it in equilibrium. The directions of the forces at the inter-spoke midpoint are indicated in Fig. 5. At the midpoint, the terms of the LHS all contribute to the plasma exhaust, while only the term on the RHS provides confinement. In this configuration, the confinement is provided by the magnetic curvature, while the outward forces come from the ram pressure, the thermal pressure, and the magnetic field pressure.

The forces at the spoke mid point are expressed as the following equation:

where R is the radius of curvature and every parameter varies as a function of the axial distance z. Rearranging Eq. (5), we find an expression for the radius of curvature necessary to obtain force balance at the end wall

In a vacuum, the radius of curvature R can expand infinitely, providing a progressively weaker confinement force. A plasma with a small radius of curvature will keep expanding due to the sum of ram, thermal, and magnetic pressures. At the spoked end wall, the conducting spokes act as a barrier through which the magnetic field cannot expand. As such, the radius of curvature between the spokes throughout the end wall depth, EW_depth, is limited by the spoke spacing, D_spoke. Any small radius of curvature will therefore expand until it reaches the conducting end wall spokes.

From Sec. III A, the magnetic flux is frozen in the plasma. The magnetic field must decrease with the plasma density and as a consequence, the plasma pressure must reach zero at the same location as the magnetic field pressure. Therefore, the total plasma pressure and the magnetic field pressure are assumed to follow the same profile, that is $B22μ0∝f(z)$ and $ρvz2+p∝f(z)$⁠. Equation (6) now becomes

One particular profile that satisfies Eq. (7) is

where L is the scale length of the profile, determining the e-folding distance. The profile from Eq. (8) is chosen because it removes the dependence on axial distance z from the radius R. Equation (7) now becomes

As β_ram increases, the required radius for equilibrium decreases. We can rearrange Eq. (9) using the condition that $R≥Dspoke/2$⁠. This radius represents the largest semicircle that can fit between the end wall spokes, as illustrated in Fig. 6 

Equation (10) dictates the smallest possible e-folding distance as a function of β_ram. From the spoked end wall geometry, we know that the maximum spoke gap is $Dspoke=4.7 cm$⁠. As the e-folding distance L decreases, more plasma is confined within the end wall thickness. Figure 6 shows two plasma profiles with different e-folding distances as well as the corresponding end wall and magnetic field geometries associated with the model.

With Eqs. (8) and (10) using $z=EWdepth=1.5Dspoke$⁠, an end wall transparency is calculated and illustrated in Fig. 7 as a function of β_ram. The end wall transparency quantifies the ratio of plasma density at the downstream edge over the plasma density at the upstream edge of the end wall. It is the fraction of plasma that escaped the flux conserving region to expand beyond the end wall

Figure 7 shows that as β_ram increases, the fraction of the profile that will be confined reduces. That is, more plasma can exhaust through the end wall. Increasing β_ram increases the e-folding distance L necessary for equilibrium, as shown in Eq. (10). As the end wall has a finite thickness, a longer e-folding distance L means that a smaller fraction of the profile will be confined through the end wall depth.

Even in the case of a very low value of β_ram, plasma is still expected to escape the assembly region as the configuration described by Eq. (6) is susceptible to interchange instabilities due to bad curvature at the spoke midpoint.

As shown in Sec. III B 1, a larger value of β_ram leads to an increase in plasma exhaust. Figure 8 compares the value of β_ram for the spoked end wall. From Sec. II, a total temperature of 80 eV is used in both cases and velocities of 110 km/s and 140 km/s are used for case 1 and case 2, respectively. The plasma density ρ in Eq. (3) is obtained from the density measurements at 35 cm, averaged over the assembly region area, shown in Figs. 3(a) and 3(b).

The value of β_ram is much larger for case 2 than for case 1 before approximately 27 μs. For example, according to Eq. (11) and Fig. 7, an increase in β_ram from approximately $βram=3$ at 20 μs for case 1 to $βram=15$ at 20 μs for case 2 leads to a threefold increase in end wall transparency.

In Fig. 2, the current for case 2 is different after the end wall change, while the current for case 1 remains unchanged. The difference between case 1 and case 2 is the value of β_ram. While the end wall geometry changed in both cases, the increase in pinch current is only observed for case 2. Both cases share the same end wall geometric parameters, and only the plasma parameters have been changed. In particular, the values of plasma density, axial velocity, and magnetic field are different for case 2 such that its value of β_ram is initially much higher.

Equation (11) shows that an increase in β_ram should be increasing the plasma exhaust through the spoked end wall. The larger β_ram values for case 2 lead to a larger plasma exhaust as a consequence of Eq. (11). Therefore, the observed change in axial current for case 2 is linked to the increased plasma exhaust.

Greater compression of the plasma produces higher measured current due to the magnetic field being frozen-in. Plasma compression increases with ram and thermal pressures, therefore increasing the measured current for the operating conditions with higher β_ram.

The end wall acts as the downstream boundary of the flux conserver on FuZE. The open area increased 60-fold after changing the end wall to a spoked design. No apparent increase in plasma exhaust is observed for case 1, but an increase is shown to occur for case 2. We show that the plasma is frozen to the magnetic flux and therefore the geometry change should have had minimal effect. However, we also show how the confinement at the end wall passes the plasma when β_ram increases. The plasma that passes the end wall can then expand beyond the flux conserver where diffusion, detachment, or magnetic reconnection can take place.

It is found that the charge voltage condition that exhibits enhanced plasma exhaust also has a much higher initial value of beta compared to the condition without enhanced exhaust. We conclude that plasma exhaust can be achieved by increasing the plasma beta inside the flux conserver. In FuZE, this could be achieved by increasing the plasma density or bulk velocity. Increased plasma exhaust can also be achieved by maximizing the space between the fixed conducting path of the flux conserver, allowing for a larger magnetic radius of curvature R. On the end wall, this would translate into using larger spoke spacing and narrower spokes.

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 (DOE) under Award No. DE-AR-0000571 and the National Nuclear Security Administration under Grant No. DE-NA0001860.

The data that support the findings of this study are available within the article.