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Recurrence Time

\[T_R = \frac{2\pi}{k\,\Delta v}\]

Variables

Symbol Meaning Units
\(T_R\) time at which a damped wave spuriously revives \(\omega_{pe}^{-1}\)
\(k\) wavenumber of the perturbation \(\lambda_D^{-1}\)
\(\Delta v\) velocity-grid spacing \(v_{th}\)

Assumptions

  • Collisionless Vlasov on a uniform velocity grid
  • No velocity-space filtering or artificial diffusion
  • Free-streaming phase mixing dominates (linear regime)

Derivation sketch

Free streaming shears a perturbation as \(f_1 \propto e^{ik(x - vt)}\), so the phase of the contribution from velocity \(v\) advances at rate \(kv\). Adjacent grid velocities, separated by \(\Delta v\), dephase at rate \(k\Delta v\). They return to a common phase when \(k\,\Delta v\,T_R = 2\pi\) — at which point the filaments realign, the velocity integral stops cancelling, and the field revives.

Notes

  • Depends only on the grid, not on the physics — which makes confirming it a genuine verification exercise.
  • \(T_R \propto N_v\): refinement postpones recurrence proportionally but never removes it.
  • Filtering or a collision operator suppresses the revival at the cost of reversibility, and sets a floor on the damping rates that can be honestly measured.
  • Distinguishing recurrence from nonlinear trapping: change \(N_v\) (moves recurrence, not trapping) or the amplitude (moves trapping, not recurrence).