Emergent kink stability of a magnetized plasma jet injected into a transverse background magnetic field

TL;DR

This study experimentally demonstrates that a sheared axial flow, induced by magnetic tension in a plasma jet within a transverse magnetic field, stabilizes the kink instability, highlighting a natural stabilization mechanism in magnetized plasma jets.

Contribution

It provides the first experimental evidence that emergent sheared axial flow can stabilize kink modes in plasma jets injected into transverse magnetic fields.

Findings

Sheared axial flow stabilizes the n=1 kink mode.

Kink instability occurs without background magnetic field.

Magnetic tension induces flow shear that stabilizes the jet.

Abstract

We report experimental results on the injection of a magnetized plasma jet into a transverse background magnetic field in the HelCat linear plasma device at the University of New Mexico [M. Gilmore et al., J. Plasma Phys. 81, 345810104 (2015)]. After the plasma jet leaves the plasma-gun muzzle, a tension force arising from an increasing curvature of the background magnetic field induces in the jet a sheared axial-flow gradient above the theoretical kink-stabilization threshold. We observe that this emergent sheared axial flow stabilizes the n=1 kink mode in the jet, whereas a kink instability is observed in the jet when there is no background magnetic field presented.

Equations

$$q=\frac{2\pi a{B}_z}{l{B}_{\phi }}<1,$$

$$F_{tension}=\frac{B_0^2}{{\mu }_0}\cdot \frac{\hat{R}}{R},$$

Full text

astrophysical jets, axial sheared flow, $n=1$ kink instabilities

The collimation and stability of a plasma jet injected into a transverse magnetic field are relevant to astrophysical jets,honda2002self  bipolar outflows associated with young stellar object,pelletier1992hydromagnetic  solar-wind evolution,schatten1969model ; jokipii1990polar  plasma beam focusing,rosenzweig1991acceleration  magnetotail physics,zmuda1970characteristics ; lyons2013quantitative  etc. In magnetic fusion, cross-magnetic-field injection to deliver fuel into the core of a tokamak is necessary for achieving more efficient utilization of deuterium–tritium fuel and optimize the energy confinement time for high fusion gain.brown1990spheromak ; brown1990current ; raman1997experimental ; olynyk2008development  The study of plasma motion across a transverse magnetic field has a long history.schmidt1960plasma ; baker1965experimental  Modern experimental investigations of plasma jets include those using pulsed-power-driven plasma guns,hsu2002laboratory ; hsu2003experimental ; hsu2005jets ; bellan2005simulating ; yun2007large  radial wire array Z pinches,lebedev2002laboratory ; lebedev2004jet ; lebedev2005magnetic ; ciardi2007evolution  and laser-produced plasma.mostovych1989laser ; ripin1990laboratory ; harilal2004confinement  However, nearly all of the experimental studies have been of plasma jets launched into vacuum. In addition, nonlinear ideal magnetohydrodynamic (MHD) simulations have modeled a plasma injected into a transverse magnetic field under varying conditions.liu2008ideal ; liu2009ideal ; liu2011ideal

We report here the experimental observation of emergent kink stabilization of a current-driven plasma jet injected into a transverse background magnetic field. Experiments have also been carried out in which a plasma-jet is injected into a background plasma (temperature $T_{e}\sim$ 1-5 eV, density $\sim 10^{17}$ m*-3*), desjardins2016dynamics ; kelly2016ari  but in such low-$\beta$ ($\sim 0.01$) plasmas, the magnetic field provides the dominant mechanism for kink stabilization. Both charge-coupled device (CCD) camera images and magnetic-field probe data show similar plasma-jet behavior for both magnetic-field-only and plasma cases. The data presented in this letter are taken from experiments with background magnetic field only.

The plasma jet is formed and launched by a compact magnetized coaxial gun,zhang2009design  and the background magnetic field is produced in the Helicon-Cathode (HelCat) linear plasma device at the University of New Mexico.lynn2009helcat ; gilmore2015helcat  The injection results in several observed phenomena: $\left(1\right)$ when launching the plasma jet into vacuum, a classical $n=1$ kink instability is observed, consistent with prior work;hsu2003experimental ; hsu2005jets  $\left(2\right)$ when injecting the jet into a transverse background magnetic field, a more-stable jet is observed; and $\left(3\right)$ analysis of magnetic-field measurements and high-speed-camera images during the injection indicate that the transverse background magnetic field exerts a magnetic tension force on the jet, allowing the formation of extended-length, kink-free jets. Although, the measured safety factor is below unity, implying kink instability, we show that the stability is a result of an emergent sheared axial flow due to the tension force of the background field acting on the jet.

HelCat is a 4-m-long and 50-cm-diameter cylindrical device with axial magnetic field, providing a background magnetic field. The plasma jet is formed by a compact, cylindrical magnetized coaxial plasma gun. As indicated in Fig. 1, the gun is mounted on a side port 1.8 m away from the helicon source, with an angle of $90^{\circ}$ with respect to the background magnetic field. The center electrode disk is 2.54 cm in diameter, and the cylindrical outer electrode is 4.75 cm in diameter. A fast gas valve puffs neutral argon gas into the annual electrode gap with a peak pressure of 0.25 Torr. An ignitron-switched 10-kV, 120-$\mu$F ($\sim$10-$\mu$s rise time) capacitor bank is employed to fully ionize the neutral gas via the discharge current. An initial radial current sheet is formed between the inner and outer electrodes. The resulting axial $\mathit{J\times B}$ force accelerates the current sheet along the gun axis, forming a plasma jet that propagates into the main vacuum chamber.

The gun is typically operated with a discharge current $I_{gun}$ = 60–100 kA and a 6.0–10 kV charge voltage on the capacitor bank. Experimental results to date have centered around time-resolved global imaging and probe measurements of the plasma-jet evolution. The images were taken using a Hadland Ultra UHSi 12/24 multiple-frame CCD camera, which takes up to twelve images per plasma discharge. The camera view is through a 39-cm $\times$ 18.5-cm observation window at the end of the chamber (labeled in Fig. 1) such that the gun port appears on the right-hand side of each image. The exposure time of each frame is 500 ns and the interframe time is set typically to 2.0 $\mu$s, which is of the order of an Alfvén transit time. False color is applied to the images for ease of viewing. Typically, the plasma is observed in unfiltered visible light. Although the exact relationship between emission intensity and plasma density is complex, we use the light emission (allowing for a generous uncertainty) as a proxy for inferring the radius and length of the bulk mass of the jet. We allow for a conservative $\pm 50$% error bar in the inferred radius and length to calculate the safety factor $q$. The physics conclusions remain valid over the entire range of the uncertainty.

A double Langmuir probe and a magnetic-probe array have been inserted into the chamber (as indicated in Fig. 1) to measure plasma parameters. Typical plasma-jet densities, electron temperatures and approximate peak magnetic field are $10^{20}$ m*-3*, 10 eV, and 0.1 T respectively. The jet propagates from the gun to the opposite wall of the chamber at near the local Alfvén velocity (about 3.5$\times 10^{4}$ m/s). For such a dynamic system, the jet characteristics exhibit very robust, reproducible measurements for a wide parameter regime. Each setup has explored hundreds of shots which show similar jet dynamics. The shot-to-shot standard deviation in the magnetic-field measurements over a large shot sample is about 15%.

First, the coaxial plasma gun is operated to launch a collimated magnetized plasma jet into the vacuum chamber without a background magnetic field, as shown in Fig. 2. As the length of the jet increases, the $n=1$ kink instability develops. The observed kinks in the plasma jet column have been analyzed and examined using Kruskal-Shafranov theory of current-driven instabilities, consistent with previous work.hsu2002laboratory ; hsu2003experimental ; hsu2005jets  The safety factor and condition for instability is given by

[TABLE]

where $\mathit{a}$ and $\mathit{l}$ are the jet radius and length, respectively.

Then, under the same operational settings of the coaxial gun, the plasma-jet is launched into a 500-G background magnetic field (transverse to the direction of jet propagation). The jet evolution for this case is shown in Fig. 3. The images indicate that $\left(1\right)$ the plasma jet penetrates into the background magnetic field and $\left(2\right)$ a more-stable jet is formed during the injection. As inferred from light emission in Fig. 3, the jet length is on the order of 50 cm, which is much longer than the approximately 10-cm length observed for the vacuum case in Fig. 2. The jet life-time is $\sim 3t_{Alfv\acute{\mathrm{e}}n}$ (where $t_{Alfv\acute{\mathrm{e}}n}\equiv l/V_{Alfv\acute{\mathrm{e}}n}$), which is prolonged compared to the vacuum case for which the plasma-jet lifetime is $\sim t_{Alfv\acute{\mathrm{e}}n}$. Comparing the experimental settings for these two cases, the only difference is the presence of the background magnetic field in the main vacuum chamber.

Theoretical analysis has shown that for a plasma jet to penetrate into a transverse magnetic field, the kinetic-energy density of the plasma jet must overcome the magnetic energy density of the background magnetized plasma.brown1990current ; zhang2016experimental  This requirement may be stated as $\rho_{jet}{V_{jet}^{2}}/2\geq{B_{0}^{2}}/2\mu_{0}$ , where $\rho_{jet}$ is the plasma jet mass density, $V_{jet}$ is the plasma jet velocity, and $B_{0}$ is the background magnetic field strength. In this experiment, substituting jet density $n=10^{20}$ m*-3*, $M_{argon}=6.62\times 10^{-26}$ kg, and $V_{jet}=3.5\times 10^{4}$ m/s into $\rho_{jet}{V_{jet}^{2}}/2$, one obtains $P_{jet}=4.1\times 10^{3}$ Pa. For ${B_{0}^{2}}/2\mu_{0}$, using $B_{0}=500$ G, one obtains $P_{B}=1.0\times 10^{3}$ Pa. Thus, $P_{jet}$ is greater than $P_{B}$, which means that the plasma jet can propagate into the transverse field of the background plasma.

The magnetic Reynolds number, which represents the ratio between magnetic advection and magnetic diffusion, is $R_{M}=\mu_{0}lV/\eta\sim~{}100$, where the characteristic length $l=0.5$ m, the characteristic velocity $V=3.5\times 10^{4}$ m/s, and the magnetic diffusivity $\eta=21.4$ m2/s (the plasma-jet conductivity $\sigma=1.7\times 10^{7}$ S/m, using $T_{e}=10$ eV). The value suggests that magnetic diffusion is relatively unimportant on the length scale l. The background magnetic field lines are then advected with the plasma jet propagation, as indicated in Fig. 4. The curved magnetic field induces a magnetic tension force to act on the plasma jet as:

[TABLE]

where $\hat{\textit{R}}$ is the unit vector pointing from the magnetic field line to the center of curvature, and R is the radius of curvature of the field line labeled in Fig. 4.

It is clear from Fig. 4 that the background magnetic tension force is against the axial $\mathit{J\times B}$ force which is from the main capacitor-bank discharge current, and this configuration induces a sheared axial flow of the plasma jet. The Kruskal-Shafranov criterion has been employed for the jet-instability analysis in this case. Figure 5 shows the calculated safety factor, q, from Eq. 1, using the measured magnetic-field data and plasma-jet dimensions from the CCD camera images. As shown in Fig. 5, with $\pm$ 50% error in inferring jet length and radius from light emission, the calculated ‘localized’ safety factors, where the B-dot probe is placed, are below unity. Based on the Kruskal-Shafranov criterion, the plasma jet should experience the kink instability, which will deform and break the jet column. However, the image data shown in Fig. 3 indicates a globally more-stable plasma jet column. Thus, the Kruskal-Shafranov criterion is not sufficient in this case to assess the plasma kink stability. The axial sheared flow, which is caused by the background magnetic-tension force, contributes to the global jet-stabilization process. The effect of sheared flow on current-driven MHD instabilities has previously been investigated theoretically and demonstrated experimentally.shu1995sheared ; shumlak2001evidence ; shumlak2003sheared ; golingo2005formation ; shumlak2006plasma ; shumlak2012sheared ; shumlak2017increasing  Linear MHD calculations show that a sheared axial flow has a stabilizing effect on the kink mode, while a uniform axial flow has no effect on the instability growth. The main prior conclusion is that an axial plasma flow with a linear shear of $dV_{z}/dr>0.1kV_{A}$ is required for jet stabilization, where k is the axial wave number and $\textit{V}_{\textit{A}}$ is the Alfvén velocity.shumlak2003sheared

The radial profile of the plasma axial flow velocity in this experiment is calculated based on the time delay between the initial rise of the magnetic field measured by the magnetic probe array, which is placed at various positions along the jet propagation axis in the main chamber at 10-cm intervals. After the radial velocity profile obtained at different positions in the vacuum chamber, the axial sheared flow $dV_{z}/dr$ is determined and the result is plotted in Fig. 6 (top). As shown in this plot, for the main region of the chamber ($z=15$–35 cm), the magnitude of the axial shear is above the threshold value $0.1kV_{A}$, consistent with the theoretical analysis of the axial sheared flow stabilization. As a comparison, the axial flow shear for plasma-jet injection into vacuum is also calculated and the result is shown in Fig. 6 (bottom). There is no flow shear for plasma-jet propagation into vacuum, and the uniform axial flow has no stabilizing effect on the kink instability.

In summary, we have demonstrated the phenomenon of emergent kink stability of a magnetized plasma jet by injecting the jet into a transverse background magnetic field. After the plasma jet leaves the plasma-gun muzzle, a background magnetic-tension force arises because the jet bends the transverse background magnetic field, and a global more-stable jet column is observed from the image data. Evaluation of the Kruskal-Shafranov criterion based on the measured safety factor indicates that the jet should be kink unstable. However, the evolution of the axial velocity profile shows an axial velocity shear that is above the theoretical threshold kink-stabilization threshold $0.1kV_{A}$, resulting in the kink stabilization of the plasma jet.

The authors thank Dr. Glen Wurden for loaning the CCD camera and Dr. Kevin Yates for assisting with its setup. This material is based upon work supported by the National Science Foundation under Grant no. AST-0613577 and the Army Research Office under award no. W911NF1510480.