Global stability and local bifurcations in a two-fluid model for tokamak plasma

TL;DR

This paper analyzes a two-fluid model of tokamak plasma, proving stability conditions, identifying primary instabilities, and characterizing bifurcations leading to stable periodic solutions, supported by numerical simulations.

Contribution

It refines previous steady-state solutions, proves global stability outside instability regions, and characterizes bifurcations as supercritical Hopf bifurcations with stable periodic orbits.

Findings

Global stability outside certain parameter sets

Primary instabilities from lowest harmonics

Supercritical Hopf bifurcations lead to stable periodic orbits

Abstract

We study a two-fluid description of high and low temperature components of the electron velocity distribution of an idealized tokamak plasma. We refine previous results on the laminar steady-state solution. On the one hand, we prove global stability outside a parameter set of possible linear instability. On the other hand, for a large set of parameters, we prove the primary instabilities for varying temperature difference stem from the lowest spatial harmonics. We moreover show that any codimension-one bifurcation is a supercritical Andronov-Hopf bifurcation, which yields stable periodic solutions in the form of traveling waves. In the degenerate case, where the instability region in the temperature difference is a point, we prove that the bifurcating periodic orbits form an arc of stable periodic solutions. We provide numerical simulations to illustrate and corroborate our analysis. These also suggest that the stable periodic orbit, which bifurcated from the steady-state, undergoes additional bifurcations.