Effects of parallel magnetic fields on sheaths near biased electrodes in a highly collisional Z-pinch plasma

TL;DR

The paper addresses how a parallel magnetic field alters sheath formation and current flows near biased electrodes in a Z-pinch plasma. Using a 1X-2V continuum Boltzmann-Poisson model with artificially enhanced collisions, the authors find a classical sheath formed by electrons within $\rho_e$ while ions are partially de-magnetized, leading to a non-monotonic potential profile and limited axial current ($\sim 10^{6}$ A m$^{-2}$) compared with unmagnetized theory ($\sim e n_0 c_s/2$). Radial currents, driven by $\mathbf{E}\times\mathbf{B}$ and diamagnetic drifts, remain large ($\sim 10^{9}$ A m$^{-2}$), and parallel flows establish significant radial shear despite small axial transport. The results demonstrate that parallel magnetic fields act as high resistivity for cross-field current and emphasize the dominance of ion motion in axial transport under these magnetized, collisional conditions. These insights inform design considerations for magnetized Z-pinch interfaces and guide future work toward more realistic collisionality regimes.

Abstract

Sheath formation near biased electrodes in magnetic fields parallel to the wall is an understudied topic, especially within the context of Z-pinch fusion experiments. We perform 1X-2V Boltzmann-Poisson simulations of an axial cut at the pinch radius of a Z-pinch plasma between two biased electrodes with a magnetic field parallel to the wall. The collision frequencies are artificially increased to enhance thermalization of the plasma in the smaller simulation domain versus the actual experiment size; this increases the perpendicular mobility and partially de-magnetizes the ions resulting in non-monotonic sheath profiles with the potential increasing away from the wall to a peak before decaying. A classical sheath forms within an electron gyroradius from the wall not due to the natural thermal motion of the electrons, but due to the magnetized electrons gyrating into the wall; therefore, the sheath structure does not significantly change with bias potential or between electrodes. With increasing bias potential, a current is induced perpendicular to the wall due to changes in ion flow, differing from unmagnetized cases where current is induced by changes in electron flow. The magnetic field acts as a high resistivity with the perpendicular current density being three orders of magnitude lower than unmagnetized theoretical predictions. There is, however, significant flow parallel to the wall from the force balance between the pressure tensor and Lorentz force. These parallel flows induce a parallel current density three orders of magnitude larger than the perpendicular current density.

Figures (6)

• Figure 1: Panel (a) shows the radial normalized Bennett pinch profiles of the density (pink) and magnetic field (orange) based on Eqs. \ref{['eq:n_Bennett']} and \ref{['eq:B_Bennett']}, respectively. Panel (b) shows how important length scales, normalized by the Debye length, vary with radius: ion gyroradius (solid red line), electron gyroradius (solid blue line), ion mean free path (dashed red line), electron mean free path (dashed blue line), and artificial simulation mean free path (dash-dot green line). The ion and electron mean free paths are based on the total collision frequencies in Eqs. \ref{['eq:nu_e']} and \ref{['eq:nu_i']}, respectively. The artificial simulation mean free path is the same for ion and electron self-collisions at $50\sqrt{2}\lambda_D$. Panel (c) shows the perpendicular mobility of the ions (red) and electrons (blue) based on realistic Z-pinch parameterszhangSustainedNeutronProduction2019 (solid line) or with artificially increased collisions (dashed line). The vertical black dashed line corresponds to the pinch radius, which is the radius at which the simulations in this paper are run.
• Figure 2: Plots of the fractional charge density (a-b), normalized axial electric field (c-d), and normalized axial ion drift velocity (e-f) at the anode (a,c,e) and cathode (b,d,f) for bias potentials of 0 (solid black line), 5 (dashed blue line) and 10kV (dotted yellow line). Minimal changes in sheath behavior are observed between these cases and between the electrodes.
• Figure 3: Plot of the potential profile (b) for bias potentials of 0 (solid black line), 5 (dashed blue line), and 10kV (dotted yellow line). The lightly shaded regions in Panel (b) are presented in expanded scales in Panels (a) and (c). The vertical dashed lines correspond to important length scales, annotated above the plots.
• Figure 4: Plots of the axial ion (solid red line) and electron (dashed blue line) particle fluxes for bias potentials of 0 (a), 5 (b), and 10 (b) kV. Both particle fluxes slightly increase in magnitude with increasing bias potential. The ion particle flux becomes more positive with increasing bias potential resulting in a net axial current toward the cathode.
• Figure 5: Plots of the radial ion (a) and electron (b) velocities for the 10kV case. The radial velocities (solid black line) occur due to a force balance between the pressure tensor and Lorentz force (dashed blue line), as described by Eq. \ref{['eq:uy_force_balance']}. Also plotted are the individual drifts: inertial drift (solid green line), diamagnetic drift (solid purple line), and $\mathbf{E}\times\mathbf{B}$ drift (solid gray line).