Asymptotic state of nonlinear Landau damping in one-dimensional plasma

TL;DR

This study analyzes the long-time state of nonlinear Landau damping in a 1D collisionless plasma by combining a plateau-based quasi-linear dispersion framework with high-precision Vlasov–Poisson simulations using a second-order symplectic integrator. By extending the plateau dispersion relation into the complex plane, the authors identify scenarios where undamped and weakly damped modes persist, and reveal that the asymptotic state in realistic perturbations is a stable, multi-wave BGK structure. Spectral analysis shows discrete low-frequency peaks linked to phase-space vortices at low velocities, while a large Langmuir-velocity vortex exhibits an internal rotation whose period matches the envelope period of the electric-field energy. These findings emphasize the importance of strong nonlinear effects beyond weakly nonlinear theory and provide a framework for extending the analysis to higher dimensions and magnetized plasmas.

Abstract

In this work, the asymptotic state of nonlinear Landau damping in one-dimensional plasma has been examined using a quasi-linear model and a second-order symplectic integrator. The dispersion relation of the plateau distribution function for the steady-state solution of the quasi-linear mode is extended to the complex plane and compared with the nonlinear simulation. We determine that the asymptotic state of the collisionless plasma is a multi-wave BGK structure. This structure is characterized by multiple vortices in phase space, which correspond to distinct peaks in the frequency-wavenumber (ω, k) spectrum of the electric field

Figures (17)

• Figure 1: The plateau distribution function and its plateau area (zoomed area).
• Figure 2: Base-10 logarithmic contours of the dispersion function for a plateau distribution with parameters $k=0.4$,$v_0=3.21$ and $\Delta v_p=0.02,n_p=20$. (a) is the global dispersion function and (b) is the dispersion function near the plateau.
• Figure 3: Logarithmic contours of the dispersion function corresponding to the plateau distribution function with $k=0.4,v_0=3.60$ and $\Delta v_p=0.02$, $n_p=20$. (a) is the dispersion function near the plateau and (b) is near the weakest damping solution.
• Figure 4: The evolution of the electric field energy with small-amplitude initial perturbations, where the phase velocity of the velocity plateau (a) does and (b) does not coincide with that of the Vlasov dispersion relation, while (b) shows the case where it does not.
• Figure 5: Long-time evolution of the electric field energy with finite-amplitude initial perturbation with parameters $\beta=0.15 ,k_1=0.4$.