Eigenvalues and Eigenfunctions of Landau Damping Oscillations in Very Weakly Collisional Plasma

TL;DR

This paper analytically and numerically investigates how rare collisions in weakly collisional plasmas transform Landau-damped oscillations from singular quasi-modes into discrete regular eigenmodes with complex eigenvalues, approaching collisionless Landau eigenvalues.

Contribution

It provides approximate analytical expressions for eigenvalue corrections and eigenfunction shapes due to rare collisions, validated by high-precision numerical calculations.

Findings

Eigenvalues approach Landau eigenvalues as collision frequency decreases.

Eigenfunctions exhibit increasing oscillations in the resonance region with decreasing collision frequency.

Analytical formulas accurately predict resonance width, peak, and oscillation period.

Abstract

Landau-damped oscillations in collisionless plasmas, described by van Kampen and Case, are quasi-modes, representing a continuous superposition of singular eigenfunctions, not true eigenmodes. Recent work by Ng et al. shows that even rare collisions replace these singular modes with discrete regular modes having complex eigenvalues for the phase velocity (or frequency), approaching Landau eigenvalues in the collisionless limit. We analytically derive approximate expressions for the eigenvalue correction due to rare collisions and for the shape of the eigenfunction describing DF perturbations in velocity space, demonstrating its increasing oscillations in the resonance region as the collision frequency tends to zero. We also obtain approximate expressions for the resonance region's width and peak value, and the oscillation period within it. We validate these analytical results with high-precision numerical calculations using a standard linear matrix eigenvalue problem approach.