# Electron inertia and quasi-neutrality in the Weibel instability

## TL;DR

This paper compares various reduced kinetic models of the Weibel instability, highlighting the importance of electron inertia and quasi-neutrality assumptions in accurately capturing the instability's behavior at sub-electron skin depth scales.

## Contribution

It analyzes the limitations of existing models neglecting electron inertia and proposes fully kinetic theories assuming quasi-neutrality to address these issues.

## Key findings

- Current models have drawbacks in neglecting electron inertia.
- Quasi-neutrality models can overcome lengthscale restrictions.
- Full kinetic theories provide a more complete description.

## Abstract

While electron kinetic effects are well known to be of fundamental importance in several situations, the electron mean-flow inertia is often neglected when lengthscales below the electron skin depth become irrelevant. This has led to the formulation of different reduced models, where electron inertia terms are discarded while retaining some or all kinetic effects. Upon considering general full-orbit particle trajectories, this paper compares the dispersion relations emerging from such models in the case of the Weibel instability. As a result, the question of how lengthscales below the electron skin depth can be neglected in a kinetic treatment emerges as an unsolved problem, since all current theories suffer from drawbacks of different nature. Alternatively, we discuss fully kinetic theories that remove all these drawbacks by restricting to frequencies well below the plasma frequency of both ions and electrons. By giving up on the lengthscale restrictions appearing in previous works, these models are obtained by assuming quasi-neutrality in the full Maxwell-Vlasov system.