# Hierarchy of instabilities for two counter-streaming magnetized pair beams: influence of field obliquity

## TL;DR

This paper extends the analysis of unstable modes in counter-streaming pair plasmas to include arbitrary magnetic field angles, revealing how obliquity influences the hierarchy of instabilities in relativistic and non-relativistic regimes.

## Contribution

It provides a comprehensive numerical study of how magnetic field obliquity affects the hierarchy of instabilities in counter-streaming pair plasmas across different flow regimes.

## Key findings

- Weibel instability dominates at high obliquity and relativistic speeds.
- Hierarchy simplifies and becomes field-independent at large angles and non-relativistic speeds.
- The two-stream instability prevails regardless of obliquity in the non-relativistic regime.

## Abstract

The hierarchy of unstable modes when two counter-streaming pair plasmas interact over a flow-aligned magnetic field has been recently investigated [PoP \textbf{23}, 062122 (2016)]. The analysis is here extended to the case of an arbitrarily tilted magnetic field. The two plasma shells are initially cold and identical. For any angle $\theta \in [0,\pi/2]$ between the field and the initial flow, the hierarchy of unstable modes is numerically determined in terms of the initial Lorentz factor of the shells $\gamma_0$, and the field strength as measured by a parameter denoted $\sigma$. For $\theta=0$, four different kinds of mode are likely to lead the linear phase. The hierarchy simplifies for larger $\theta$'s, partly because the Weibel instability can no longer be cancelled in this regime. For $\theta>0.78$ (44$^\circ$) and in the relativistic regime, the Weibel instability always govern the interaction. In the non-relativistic regime, the hierarchy becomes $\theta$-independent because the interaction turns to be field-independent. As a result, the two-stream instability becomes the dominant one, regardless of the field obliquity.