Hierarchy of instabilities for two counter-streaming magnetized pair beams

TL;DR

This paper analyzes the hierarchy of instabilities in counter-streaming magnetized pair beams, deriving a phase space map of dominant modes and identifying a triple point where three modes grow equally.

Contribution

It provides a comprehensive instability hierarchy map considering magnetic fields and beam energies, including analytical expressions for growth rates and phase boundaries.

Findings

Identified four regions of dominant unstable modes in the parameter space.

Derived analytical expressions for growth rates and phase boundaries.

Discovered a triple point where three modes have equal growth rates.

Abstract

The instabilities triggered when two counter-streaming pair beams collide are analyzed. A guiding magnetic field is accounting for, while both beams are considered identical and cold. The instability analysis is conducted over the full \textbf{k}-spectrum, allowing to derive the hierarchy map of the dominant unstable modes, in terms of the initial beams energy $\gamma_0$ and a magnetic field strength parameter $\Omega_B$. Four different regions of the $(\Omega_B,\gamma_0)$ phase space are identified, each one governed by a different kind of mode. The analysis also unravels the existence of a "triple point", where 3 different modes grow exactly the same rate. A number of analytical expressions can be derived, either for the modes growth-rates, or for the frontiers between the 4 regions.