Critical velocity-space mode scalings in linear and nonlinear Landau damping for the Vlasov--Poisson system

TL;DR

This paper derives analytical scalings for velocity-space mode requirements in simulating Landau damping in the Vlasov--Poisson system, validated by extensive numerical simulations.

Contribution

It provides the first unified analytical scalings for critical velocity-space modes in linear and nonlinear Landau damping, validated against numerous simulations.

Findings

Derived scalings depend on bounce frequency, wavenumber, and collisional frequency.

Strong agreement between predictions and 800 simulations.

Provides guidance on velocity-space resolution for kinetic plasma simulations.

Abstract

The velocity-space resolution required to accurately simulate kinetic phenomena in the 1D-1V Vlasov--Poisson system is generally not known a priori. In this work, we determine the upper bound on the resolution requirement for linear and nonlinear Landau damping mediated by collisional diffusion, deriving analytical scalings for the critical Fourier and Hermite velocity-space mode numbers using a unified cascade-balance argument. The resulting scalings depend on the bounce frequency $\omega_b$, wavenumber $k\lambda_D$, and electron-electron collisional frequency $\nu$. We validate these predictions against an ensemble of 800 Vlasov--Fokker--Planck simulations, finding strong agreement with the predicted $\omega_b$ and $\nu$ dependencies.