# Electron hole instability in linearly sub-critical plasmas

## TL;DR

This paper investigates how tiny seed fluctuations can destabilize linearly damped plasmas, leading to the formation of electron holes and complex nonlinear wave interactions beyond traditional Landau damping theory.

## Contribution

It introduces a mechanism for electron hole instability triggered by small seed fluctuations, involving nonlinear eigen-modes and trapped particle dynamics, expanding understanding of sub-critical plasma turbulence.

## Key findings

- Tiny seed fluctuations can trigger electron hole instability.
- Nonlinear eigen-modes explain the transition to unstable plasma states.
- Energy transfer from electrons to ion acoustic modes accelerates electron holes.

## Abstract

Electron holes (EH) are highly stable non-linear structures met omnipresently in driven collision-less hot plasmas. A mechanism destabilizing small perturbations into holes is essential for an often witnessed but less understood sub-critically driven intermittent plasma turbulence. In this paper we show how a tiny, eddy-like, non-topological seed fluctuation can trigger an unstable evolution deep in the linearly damped region, a process being controlled by the trapping non-linearity and hence being beyond the realm of the Landau scenario. After a (transient) transition phase modes of the privileged spectrum of cnoidal EH are excited which in the present case consist of a solitary electron hole (SEH), two counter-propagating "Langmuir" modes (plasma oscillation), and an ion acoustic mode. A quantitative explanation involves employing non-linear eigen-modes, yielding a non-linear dispersion relation with a forbidden regime and the negative energy character of the SEH, properties being inherent in Schamel's model of undamped Vlasov-Poisson structures identified here as lowest order trapped particle equilibria. An important role in the final adaption of nonlinear plasma eigenmodes is played by a deterministic response of trapped electrons which facilitates transfer of energy from electron thermal energy to an ion acoustic non-uniformity, accelerating the SEH and positioning it into the right place assigned by the theory.