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Vlasov–Poisson: Landau Damping, Recurrence, Trapping

Learning goal

Watch a wave damp in a system with no dissipation whatsoever. Nothing in the Vlasov equation is irreversible; the solver conserves particle number to one part in \(10^8\) and total energy to a few hundredths of a percent. And yet the field energy falls exponentially at precisely the rate linear theory predicts. Where the energy goes — and how the grid eventually gives it back — is the entire lesson.

What is being solved

\[\frac{\partial f}{\partial t} + v\frac{\partial f}{\partial x} - E\frac{\partial f}{\partial v} = 0, \qquad \frac{\partial E}{\partial x} = 1 - \int f\,dv\]

normalised to \(\omega_{pe} = 1\), \(\lambda_D = 1\), electrons on a fixed ion background. The scheme is the one from Chapter 2: Strang splitting into two exact shifts,

\[\underbrace{\tfrac12\Delta t \text{ in } x}_{f(x,v)\,\to\,f(x - v\Delta t/2,\,v)} \;\to\;\underbrace{\text{field solve}}_{\text{spectral}} \;\to\;\underbrace{\Delta t \text{ in } v}_{f(x,v)\,\to\,f(x,\,v + E\Delta t)} \;\to\;\underbrace{\tfrac12\Delta t \text{ in } x}\]

Each half-step is a pure translation, which is why the scheme is stable at any \(\Delta t\) and why it conserves particles so well. Interpolation is cubic Lagrange; the field solve is a direct DFT.

Things to try

  1. Start at the default (\(k\lambda_D = 0.5\), \(A = 0.01\)). The measured \(\gamma\) settles near \(-0.1538\) against the exact root \(-0.15336\) — a fraction of a percent. If your own solver does not reproduce this number, Chapter 2 is blunt about the consequence: stop and debug, because nothing built on top of it can be trusted.

  2. Watch the phase-space panel, not the energy plot. The perturbation shears into finer and finer diagonal stripes. That is phase mixing: the free energy is not destroyed, it is carried into velocity-space structure of ever-shorter scale. The field forgets, because the field only sees \(\int f\,dv\), and finely striped structure integrates to nothing.

  3. Change \(k\lambda_D\) and watch the damping explode. From 0.3 to 0.6 the rate goes \(-0.0126 \to -0.264\): a factor of twenty for a factor of two in \(k\). Long waves barely damp; short waves die before they can propagate. This extreme sensitivity is why Landau damping is simultaneously the most famous result in plasma theory and the hardest to measure in a laboratory.

  4. Find the recurrence. Run past the amber \(T_R\) line and the "damped" wave comes back from the dead. This is not a bug and not physics — it is the grid. Discrete velocities at spacing \(\Delta v\) get back into phase after $\(T_R = \frac{2\pi}{k\,\Delta v}\)$ and the filaments reassemble. The prediction depends only on the grid, and the widget hits it within a few percent. Then raise \(N_v\): the recurrence moves later, in exact proportion, and never goes away. You cannot solve this by refining — you can only push it beyond the time you care about, or add a little velocity diffusion and admit you have broken reversibility.

  5. Break linear theory. Slide the amplitude to 0.1, then 0.5. Early on the damping still matches theory, but at late times the decay stops, and at \(A = 0.5\) the field energy starts growing again. Particles have become trapped in the wave trough and are sloshing at the bounce frequency \(\omega_B = \sqrt{kE}\); a trapped particle no longer exchanges net energy with the wave, so the damping mechanism switches itself off. Compare the readout's bounce time with the damping time — trapping wins when \(\omega_B \gtrsim |\gamma|\).

  6. Check the conservation readout as you push \(\Delta t\) and amplitude. Particle number holds to \(10^{-8}\) because each split step is an exact translation. Energy holds only to the order of the splitting — it is not an invariant of the scheme. Chapter 2's classification of invariants into "holds to round-off" and "holds to the order of the method" is exactly this distinction, and it matters when you use conservation as a correctness check.

Why this is the hardest thing in the course to get right

The scheme has three places to go wrong that all produce plausible-looking output: the departure-point indexing in the interpolation, the sign of the velocity shift, and the DFT convention in the field solve. Each error yields a picture that damps something at some rate. The only defence is the benchmark: at \(k\lambda_D = 0.5\), \(\gamma\) must be \(-0.15336\), and it must stay right when you change \(k\).

Vlasov–Poisson system · Semi-Lagrangian solver · Landau damping numerically · Recurrence · Nonlinear trapping · Landau damping theory (PC368)