Experimental Visualization

Figure 3 depicts a set of schlieren images with a vertical optical cutoff corresponding to select times during a pulse driven by 2.3 kJ of stored energy. The Supplementary Information includes the remaining frames spanning the entire discharge duration. Following the time delay required to accelerate the plasma down the gun volume (∼2 μs), a broad axial pinch attaches to the central cathode at the exit of the accelerator. At early times after pinch formation, a number of coherent, long-wavelength instabilities appear, resembling the ‘kink’ MHD modes depicted in Fig. 1. These instability modes are observed to shift the entire plasma column vertically in a quasi-periodic manner. To capture their spatial characteristics, the quasi-periodic ‘features’ that result in radial perturbations of the plasma column are identified and tracked over time, an example of which is shown in Fig. 3. The tracked modes are prevalent early in the jet evolution, specifically 3.5–4.5 μs with between 3–8 features tracked per frame, and evolve to have peak axial wavelengths ranging from λ z ≈ 1.2–2.2 cm. As these dominant length scales are more than twice the axial spatial resolution, measurements indicate that the shorter wavelengths are being damped within the flow. At later times during the discharge, the surface perturbations of the plasma column disappear, indicating a stable plasma jet has formed.

Figure 3 Schlieren Image Sequence: Schlieren images of the plasma jet with an external energy supply of 2.3 kJ at select times with 50 ns exposures. The flow is observed to be initially unstable to long-wavelength MHD instabilities that appear as perturbations on the envelope of the plasma column. At later times and with higher current flowing through the pinch, the flow appears to be stabilized over the observed window. Full size image

To illustrate the formation process of the plasma jet, vertical slices of the jet at fixed axial locations are organized into streak images depicted in Fig. 4. At both 0.3 cm and 1.5 cm axially downstream of the electrodes, the prevalence and amplitude of both streaks and inhomogeneities in the plasma column decrease over the lifetime of the jet. Integration over the top half of the streak images, shown in Fig. 4, also indicate that the high amplitude oscillations in the schlieren signal during the initial formation of the jet (2–5 μs) are damped in time. A stable jet is maintained until a sudden expansion around 8.5 μs when the drive capacitors no longer supply any current to sustain the quasi-steady flow. Although the drive current as measured at the capacitor bank is smaller once a stable jet forms, in situ measurements detailed in ref.26 indicate that the current flow within the pinch peaks 2–4 μs later and is thus greater than at earlier times when instabilities are more prevalent. The phase lag between the current passing through the pinch and the drive current is also demonstrated by the fact that the effective radius of the pinch decreases over the lifetime of the jet. Thus, the radial Lorentz force, and therefore axial current increases over time, while the observed perturbations decrease over time. Since the pinch appears to become more stable as the current supplied to it is increased, these results indicate the possibility of increasing both the duration and current density of the discharge to produce relatively long-lived stable plasma jets.

Figure 4 Jet Formation Process: Streak images of the plasma jet at fixed axial locations (a,b) formed by taking vertical slices of the schlieren image stack. Integrated schlieren signals of the top half of each of the streak images (c,d) illustrate that the inhomogeneities in the plasma column that are initially prevalent within the jet decrease in frequency over time. Full size image

Stability Analysis

We carry out a linear analysis of the ideal MHD equations to identify the instabilities observed experimentally in Fig. 3 from 3.5–4.5 μs and quantify the role that flow properties have on the resulting stability spectrum. The ideal limit of these equations is taken as the fastest timescales of the system are of specific interest in our analysis. To justify this assumption, we note that for conditions of interest in the core of the plasma, the ratio of the resistive and Alfvénic timescales approaches ∼100. The quantitative basis for this assertion including specific plasma properties is included in the Methods section. Without loss of generality, the Eulerian primitive form of the perturbed variables (ρ 1 , v 1 , p 1 , and B 1 ) are recast in terms of the plasma displacement vector ξ to reduce system dimensionality27. After normal mode analysis has been performed in time, ξ(r,t) = ξ(r)exp(−iωt), the system of equations can be written as,

$$\omega {\rho }_{0}{\boldsymbol{\xi }}={\rho }_{0}{\boldsymbol{u}}-i{\rho }_{0}{{\boldsymbol{v}}}_{0}\cdot

abla {\boldsymbol{\xi }}$$

$$\omega {\rho }_{0}{\boldsymbol{u}}=-\,i\omega {\rho }_{0}{{\boldsymbol{v}}}_{0}\cdot

abla {\boldsymbol{\xi }}-G({\boldsymbol{\xi }})$$

where \(G({\boldsymbol{\xi }})={{\boldsymbol{J}}}_{1}\times {{\boldsymbol{B}}}_{0}+{{\boldsymbol{J}}}_{0}\times {{\boldsymbol{B}}}_{1}-

abla {p}_{1}-{\rho }_{0}{({{\boldsymbol{v}}}_{0}\cdot

abla )}^{2}{\boldsymbol{\xi }}+

abla \cdot [{\boldsymbol{\xi }}({\rho }_{0}{{\boldsymbol{v}}}_{0}\cdot

abla {{\boldsymbol{v}}}_{0})]\) is the generalized force operator and subscripts indicate the equilibrium (0) and perturbed quantities (1). A variable substitution has been introduced to reduce the order of the resultant stability equation and to recast the system in terms of the characteristic eigenvalue equation ωA⋅x = B⋅x. This facilitates the use of linear eigenvalue solvers to calculate the resultant spectrum28,29. We solve the equations in a cylindrical geometry, and assume mode decomposition in θ and z according to \({\boldsymbol{\xi }}({\boldsymbol{r}})=({\xi }_{r}(r)\,\hat{{\boldsymbol{r}}}+{\xi }_{\theta }(r)\hat{{\boldsymbol{\theta }}}+{\xi }_{z}(r)\hat{{\boldsymbol{z}}})\exp (im\theta +ikz)\). The radial variation of each vector component is discretized using mixed finite element basis functions according to the Appert projection30 to minimize spectral pollution. This analysis approximates the jet as an axisymmetric plasma column with uniform axial properties. Further details about the numerical implementation are included in the Supplementary Information.

The equilibrium profiles used in the analysis, shown in Fig. 5a–c and detailed further in the Methods section, are determined from an assumed analytic Bennett profile given plasma density and velocity fields specified from a combination of experiments and numerical simulation. At the edge of the domain, a vacuum boundary condition closes the system of equations. Nominal plasma density and temperature are obtained from experimental Stark broadening measurements21 and simulations22 respectively. The shape of the density profile is inferred from Abel inverted density profiles in ref.25. The axial velocity profile dips from a value of V z,∞ in the surrounding flow to a value of 0.2V z,∞ on axis just downstream of the central cathode, consistent with detailed resistive MHD simulations of the accelerator geometry22 that show higher velocity plasma is not entrained in the central pinch. The peak velocity of the simulations are also consistent with time-of-flight measurements made by feature tracking of the leading edge of the plasma plume, as highlighted in ref.25. The functional form and amplitude of the magnetic field is detailed in the Methods section and is inferred from equilibrium arguments of the flow configuration. Each calculation was performed using 1000 elements for a variety of peak axial velocities, azimuthal mode numbers, and axial wavenumbers. For each condition, the largest growth rates were recorded by tabulating the largest imaginary eigenvalues, \({\rm{\Gamma }}\equiv \Im (\omega )\), in the spectrum.

Figure 5 Stability Analysis: Results of the linear stability analysis of current-driven plasma jets. (a–c) Details of the equilibrium Bennett profiles used to model the axial pinch formed in the jet. (d,e) Show the m = 0 and m = 1 instability mode calculations as a function of k, the axial wavenumber, while (f) shows how the dominant spectral modes vary over k. Within the plots, n 0 , R 0 , and V z,∞ are the are the peak number density, radius, and free stream velocity respectively in the simulated profile and \({V}_{ti}=\sqrt{2{k}_{b}T/{m}_{i}}\) is the ion thermal velocity where a temperature of k b T = 25 eV was assumed22. Full size image