The bounce period of an electron trapped in a plasma wave

Landau damping is the linear theory of a wave losing energy to the electrons that move at nearly its phase velocity. It assumes the wave is small enough that those electrons go straight through it. O'Neil (Phys. Fluids 8, 2255,

1. asked what happens when the wave is large: the resonant electrons are

caught in its potential troughs and bounce back and forth there, and the damping stops after about one bounce. The time scale of that nonlinear stage is the bounce period, and this post computes it.

In the frame moving with the wave, the potential is static, $\varphi (x)={\varphi }_0\cos kx$, and an electron obeys $m\ddot{x}=-ek{\varphi }_0\sin kx$. That is the pendulum equation. In units with $k=m=e=1$ it reads $\ddot{x}=-{\varphi }_0\sin x$, and small oscillations about the bottom of a trough have the bounce frequency ${\omega }_B=\sqrt{{\varphi }_0}$.

The program integrates the equation with velocity Verlet and measures the period. The references are a closed form, the complete elliptic integral $K$, which the program computes with the arithmetic-geometric mean, and posts/refs/trapped-electron-bounce-period.py, which gets $K$ from scipy's ellipk and the period from scipy's solve_ivp without either of the program's methods.

The period at ${\theta }_0=0.1$

A pendulum released from rest at angle ${\theta }_0$ has the period

where $K(m)={\int }_0^{\pi /2}d\phi /\sqrt{1-m{\sin }^2\phi }$. For a small release angle the series $K(m)=\frac{\pi }{2}(1+m/4+9m^2/64+\dots )$ gives $T=\frac{2\pi }{{\omega }_B}\left(1+{\theta }_0^2/16+11{\theta }_0^4/3072+\dots \right)$: the period grows with amplitude because the restoring force $\sin x$ is weaker than $x$.

At ${\theta }_0=0.1$ and ${\varphi }_0=1$ the elliptic integral gives $T=6.2871145$, and $2\pi (1+{\theta }_0^2/16)=6.2871123$ is short by $2.25\times 10^{-6}$, the size of the next term. The Verlet run lands within $10^{-6}$ of the elliptic value: its phase error is $({\omega }_B\Delta t{)}^2/24=1.7\times 10^{-7}$ of a period, and the test allows twice that.

A deeper trough bounces faster

The only frequency in the problem is ${\omega }_B=\sqrt{{\varphi }_0}$, so quadrupling the wave amplitude halves the bounce period at the same release angle. The program runs ${\varphi }_0=4$ with the time step halved, which keeps ${\omega }_B\Delta t$ the same, and the period comes out half of the ${\varphi }_0=1$ period to rounding.

At large amplitude the period diverges at the separatrix

An electron released at rest near the crest, ${\theta }_0\to \pi$, takes longer and longer to start moving, and $K(m)$ grows like $\frac{1}{2}\ln \frac{16}{1-m}$ as $m\to 1$. The first figure puts the Verlet periods beside $4K$ at six release angles; the program checks each against the elliptic integral, and the reference script checks $4K$ against solve_ivp at the same angles.

Trapped and passing electrons

The energy $E=v^2/2-{\varphi }_0\cos x$ is conserved. An electron at the bottom of a trough ($x=0$) with speed $v_0$ reaches the crest ($x=\pm \pi$, where the potential energy is $+{\varphi }_0$) only if $v_0^2/2-{\varphi }_0>{\varphi }_0$, that is $v_0>2\sqrt{{\varphi }_0}=2{\omega }_B$. The curve $E={\varphi }_0$, $v=\pm 2{\omega }_B\cos (x/2)$, is the separatrix: inside it electrons are trapped and bounce, outside they stream past the crests. The trapped band is $4{\omega }_B/k$ wide in velocity, and it is the electrons in this band that O'Neil's theory follows.

The program finds the boundary without using the energy: it launches electrons from the trough, marks an electron as passing when it gets past $x=\pi$, and bisects on $v_0$. The boundary comes out at $2{\omega }_B$ to the bisection's tolerance.