Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence

TL;DR

This paper analyzes the stability and mode signatures of a Hamiltonian model for electron temperature gradient turbulence, revealing conditions for stability, the nature of energy modes, and the mechanism of ETG instability through bifurcation.

Contribution

It introduces a Hamiltonian framework for ETG turbulence, deriving invariants, stability criteria, and explicitly characterizing mode signatures and bifurcation mechanisms.

Findings

Stable equilibria can be energy stable or spectrally stable with negative energy modes.

ETG instability arises via a Krein bifurcation from merging energy modes.

Fast mode is always a positive energy mode, slow mode can be positive or negative.

Abstract

Stability properties and mode signature for equilibria of a model of electron temperature gradient (ETG) driven turbulence are investigated by Hamiltonian techniques. After deriving the infinite families of Casimir invariants, associated with the noncanonical Poisson bracket of the model, a sufficient condition for stability is obtained by means of the Energy-Casimir method. Mode signature is then investigated for linear motions about homogeneous equilibria. Depending on the sign of the equilibrium "translated" pressure gradient, stable equilibria can either be energy stable, i.e.\ possess definite linearized perturbation energy (Hamiltonian), or spectrally stable with the existence of negative energy modes (NEMs). The ETG instability is then shown to arise through a Kre\u{\i}n-type bifurcation, due to the merging of a positive and a negative energy mode, corresponding to two modified drift waves admitted by the system. The Hamiltonian of the linearized system is then explicitly transformed into normal form, which unambiguously defines mode signature. In particular, the fast mode turns out to always be a positive energy mode (PEM), whereas the energy of the slow mode can have either positive or negative sign.