Stability and energetics of Bursian diodes

TL;DR

This paper analyzes the stability, energy dynamics, and torque properties of a Bursian diode model using fluid equations, identifying stable and unstable solutions and their nonlinear evolution.

Contribution

It provides a detailed stability analysis, energy description, and a torque integral theorem for Bursian diodes within a one-dimensional Eulerian framework.

Findings

Stable and unstable equilibrium solutions identified.

Unstable solutions relax to stable states non-linearly.

A non-local torque integral theorem is derived.

Abstract

We present an analysis of the stability, energy and torque properties of a model Bursian diode in a one dimensional Eulerian framework using the cold Euler-Poisson fluid equations. In regions of parameter space where there are two sets of equilibrium solutions for the same boundary conditions, one solution is found to be stable and the other unstable to linear perturbations. Following the linearly unstable solutions into the non-linear regime, we find they relax to the stable equilibrium. A description of this process in terms of kinetic, potential and boundary-flux energies is given, and the relation to a Hamiltonian formulation is commented upon. A non-local torque integral theorem, relating the prescribed boundary data to the average current in the domain, is also provided. These results should prove useful for understanding Bursian diodes in general, as well as for control applications and benchmarking numerical codes.