Spatial-temporal evolution of the current filamentation instability

TL;DR

This paper analyzes the spatial-temporal evolution of the current filamentation instability, deriving analytical solutions and validating them with simulations, revealing how the instability transitions from spatial to temporal growth depending on beam parameters.

Contribution

It provides a novel analytical framework for understanding the spatial-temporal growth of the filamentation instability and its dependence on beam velocity and relativistic effects.

Findings

The instability grows spatially from the beam front to a critical length.

The critical length increases linearly with time and depends on beam velocity.

In the ultra-relativistic regime, the instability is purely temporal across the beam.

Abstract

The spatial-temporal evolution of the purely transverse current filamentation instability is analyzed by deriving a single partial differential equation for the instability and obtaining the analytical solutions for the spatially and temporally growing current filament mode. When the beam front always encounters fresh plasma, our analysis shows that the instability grows spatially from the beam front to the back up to a certain critical beam length; then the instability acquires a purely temporal growth. This critical beam length increases linearly with time and in the non-relativistic regime it is proportional to the beam velocity. In the relativistic regime the critical length is inversely proportional to the cube of the beam Lorentz factor $\gamma_{0b}$. Thus, in the ultra-relativistic regime the instability immediately acquires a purely temporal growth all over the beam. The analytical results are in good agreement with multidimensional particle-in-cell simulations performed with OSIRIS. Relevance of current study to recent and future experiments on fireball beams is also addressed.