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

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.

7 pages, 2 figures

References in corpus (4)

• Modes of a stellar system I: ergodic systems
• Damped perturbations in stellar systems: Genuine modes and Landau-damped waves
• Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero Collisions
• Dynamics of stellar systems with collisions: eigenvalues and eigenfunctions in nearly collisionless limit