Topology of the Warm plasma dispersion relation at the second Harmonic Electron Cyclotron Resonance Layer

TL;DR

This paper introduces a novel control method to stabilize and analyze the topology of the warm plasma dispersion relation at the second harmonic electron cyclotron resonance, especially for oblique waves and higher densities.

Contribution

It proposes a feed-forward controller to improve convergence of the iterative solution for the dispersion relation, enabling detailed topology analysis at the second harmonic resonance.

Findings

Stabilizes solutions at the second harmonic resonance layer.

Enhances robustness of the iterative method for wave coupling analysis.

Allows evaluation of wave coupling at oblique angles.

Abstract

The Warm Plasma Dispersion Relation, for waves in the electron cyclotron resonance range of frequencies, can be cast into the form of a bi-quadratic equation for $N_\perp$, where the coefficients are a function of $N_\perp^2$ and an iterative procedure is required to obtain a solution. However, this iterative procedure is not well understood and fails to converge towards a solution at the second harmonic resonance layer. In particular at higher densities where the wave can couple to an electron Bernstein wave. This paper focuses on a solution to the poor convergence of the iterative method, enabling determination of the topology of the dispersion relation around the second harmonic using a fully relativistic code for oblique waves. A feed-forward controller is proposed with the ability to adjust the rotation of a step of $N_\perp^2$ within the complex plane, while also limiting the step-size. It is shown that implementation of the controller stabilizes unstable solutions, while improving overall robustness of the iteration. This allows the evaluation of the coupling between the fast extraordinary mode and electron Bernstein waves at the second harmonic electron cyclotron resonance layer, for non-perpendicularly propagating waves.