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Nonlinear Trapping

Source: PHY653B Ch. 2

Intuition

Linear Landau theory assumes particles stream past the wave, exchanging a little energy on the way. If the wave is strong enough, some of them stop streaming: they get caught in the wave's potential trough and slosh back and forth. A trapped particle spends equal time gaining and losing energy, so it stops giving the wave a net drag — and the damping switches itself off.

The bounce frequency

A particle near the bottom of a trough of a wave with field amplitude \(E\) oscillates at

\[\omega_B = \sqrt{kE}\]

(in the normalised units of the course; dimensionally \(\omega_B = \sqrt{qkE/m}\)). This is the only new timescale in the problem, and the whole of nonlinear Landau physics is a competition between it and the damping rate:

  • \(\omega_B \ll |\gamma|\) — the wave damps away before a particle can complete a bounce. Linear theory holds.
  • \(\omega_B \gtrsim |\gamma|\) — particles complete bounces during the damping time. Trapping arrests the damping.

Setting \(\omega_B = |\gamma|\) gives the amplitude at which linear theory dies: \(E_{\rm crit} \sim \gamma^2/k\).

What you actually see

In the widget, at \(k\lambda_D = 0.5\):

Amplitude Early \(\gamma\) Late \(\gamma\) Behaviour
0.01 −0.154 −0.153 textbook linear damping throughout
0.1 −0.17 −0.047 damping visibly arrests
0.5 −0.20 +0.086 decay stops and reverses — the field grows

At large amplitude the field energy oscillates rather than decaying: energy sloshes back and forth between the wave and the trapped population at roughly \(\omega_B\). This is the classic O'Neil result, and the phase-space signature is unmistakable — a cat's-eye vortex forms at the resonant velocity, a closed island of trapped orbits rotating in \((x,v)\).

Watch the phase-space panel at \(A = 0.5\) and the cat's eye is plainly visible. It is one of the most satisfying things a kinetic code produces, and no fluid model can show it.

Why the early rate is steeper than linear theory

A detail worth noticing in the table: at large amplitude the early damping is faster than \(-0.153\), not slower. Harmonics generated by the nonlinearity carry energy out of the fundamental in addition to Landau damping. Only later, once trapping establishes itself, does the decay arrest. Nonlinearity is not simply "linear theory plus a correction" — it changes the behaviour in both directions at different times.

Common mistakes

  • Testing a code's Landau damping at large amplitude. \(A = 0.1\) already contaminates the measurement. Use \(A \sim 0.01\) for the benchmark, and then raise it deliberately to study trapping.
  • Confusing trapping arrest with recurrence. Trapping scales with amplitude and is insensitive to \(N_v\); recurrence is the reverse.
  • Reporting a single fitted \(\gamma\) for a nonlinear run. There is no single rate — the whole point is that it changes. Report the window, or report the curve.

Knowledge graph position

Prerequisites: Landau damping, Vlasov–Poisson. Leads to: phase-space holes, BGK modes, nonlinear saturation of beam instabilities.

Quiz

Q1 (computational). A wave has \(k = 0.5\) and field amplitude \(E = 0.05\) in normalised units. Is linear theory valid?

Answer

\(\omega_B = \sqrt{kE} = \sqrt{0.025} = 0.158\). The linear damping rate at \(k\lambda_D = 0.5\) is \(|\gamma| = 0.153\). Since \(\omega_B \approx |\gamma|\), this sits right on the boundary — particles complete about one bounce per damping time, so trapping will be visible and linear theory is marginal at best.

Q2 (conceptual). Why does trapping stop the damping rather than reversing it permanently?

Answer

A trapped particle is no longer streaming past the wave; over a full bounce it gains as much energy as it loses, so its net drag averages to zero. The resonant population that was responsible for the damping is removed from the exchange. Energy then sloshes back and forth between wave and trapped particles at \(\omega_B\), producing oscillation rather than monotonic decay.

Q3 (MCQ). The cat's-eye structure in phase space is:

  • (a) a numerical artefact of the velocity grid
  • (b) a closed island of trapped orbits at the resonant velocity
  • (c) the signature of recurrence
  • (d) evidence of a bug in the field solver
Answer

(b). It is the separatrix-bounded region of trapped particles centred on \(v = \omega/k\) — real, physical, and reproduced by any correct kinetic solver.