Finite gyro-radius multidimensional electron hole equilibria

TL;DR

This paper analytically and numerically investigates how finite electron gyro-radius affects the structure and stability of electron holes in magnetic fields, revealing limits on their equilibrium widths and the role of gyro-bounce-resonance.

Contribution

It provides the first self-consistent analysis of gyro-bounce-resonance effects on electron hole equilibria, combining analytical calculations with orbit integration.

Findings

Gyro-averaging reduces trapped electron deficits.

Equilibrium widths smaller than gyro-radius are prevented.

Gyro-bounce resonance causes orbit loss over multiple bounce periods.

Abstract

Finite electron gyro-radius influences on the trapping and charge density distribution of electron holes of limited transverse extent are calculated analytically and explored by numerical orbit integration in low to moderate magnetic fields. Parallel trapping is shown to depend upon the gyro-averaged potential energy and to give rise to gyro-averaged charge deficit. Both types of average are expressible as convolutions with perpendicular Gaussians of width equal to the thermal gyro-radius. Orbit-following confirms these phenomena but also confirms for the first time in self-consistent potential profiles the importance of gyro-bounce-resonance detrapping and consequent velocity diffusion on stochastic orbits. The averaging strongly reduces the trapped electron deficit that can be sustained by any potential profile whose transverse width is comparable to the gyro-radius $r_g$. It effectively prevents equilibrium widths smaller than $\sim r_g$ for times longer than a quarter parallel-bounce-period. Avoiding gyro-bounce resonance detrapping is even more restrictive, except for very small potential amplitudes, but it takes multiple bounce-periods to act. Quantitative criteria are given for both types of orbit loss.