# Quasi-linear model for the beam-plasma instability: analysis of the self-consistent evolution

## TL;DR

This paper re-examines the quasi-linear model for beam-plasma instability, comparing theoretical predictions with simulations, identifying discrepancies, and proposing corrections to improve the model's accuracy during early evolution stages.

## Contribution

It introduces a correction to the quasi-linear growth rate by accounting for the distribution function's time dependence, enhancing the model's predictive capability.

## Key findings

- Diffusive asymptotic dynamics are well predicted.
- Early mesoscale transport is less accurately modeled.
- A revised growth rate improves spectral evolution predictions.

## Abstract

We re-analyze the quasi-linear self consistent dynamics for the beam-plasma instability, by comparing the theory predictions to numerical simulations of the corresponding Hamiltonian system. While the diffusive features of the asymptotic dynamics are reliably predicted, the early temporal mesoscale transport appears less efficient in reproducing the convective feature of the self-consistent scenario. As a result, we identify the origin of the observed discrepancy in the underlying quasi-linear model assumption that the distribution function is quasi-stationary. Furthermore, we provide a correction to the instantaneous quasi-linear growth rate based on a linear expansion of the distribution function time dependence, and we successfully test this revised formulation for the spectral evolution during the temporal mesoscale.