High-Voltage Ionized Gas with Spherical Cathode Emission
Walter A. Strauss, Masahiro Suzuki
TL;DR
The paper provides a rigorous global bifurcation analysis for Townsend-type gas discharges in a spherical shell, proving the existence of a one-parameter family of radial steady states emanating from a sparking voltage when a gamma-emission mechanism is present. By linearizing around the trivial solution and studying the electron-ion coupling, the authors establish a simple eigenvalue crossing with a one-dimensional nullspace and a matching one-dimensional adjoint, enabling a transversality condition. They then apply an analytic global bifurcation theorem to obtain a continuous curve of steady states, investigate positivity and high-voltage asymptotics, and identify parameter regimes (notably Γ) where sparking occurs with dominant cathode emission. The results illuminate the structure of gas discharge states under geometry-induced variability and nonlocal boundary effects, with implications for understanding arcing thresholds and voltage-ionization dynamics in spherical configurations.
Abstract
We consider a plasma that is created by a high voltage difference, which is known as a Townsend gas discharge. The plasma is confined to the region between two concentric spheres, one of which is a cathode and the other an anode. Ion-electron pairs are created by collisions inside the plasma. Additional electrons enter the plasma by collisions of ions with the cathode. We prove under certain conditions that there are many steady states exhibiting gas discharge, beginning with a `sparking' voltage. In fact, there is an analytic one-parameter family of them that connects the non-ionized gas to a plasma with arbitrarily high ionization or arbitrarily high potential, or else the family ends at an `anti-sparking' voltage.