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Landau Damping as a Benchmark

Source: PHY653B Ch. 2

Intuition

Landau damping is the benchmark of kinetic plasma simulation, for three reasons: it is collisionless (so it tests the physics, not a dissipation model), it has an exact answer to five digits, and it is unforgiving — the rate depends so steeply on \(k\lambda_D\) that a code which is subtly wrong cannot fake it across a range of \(k\).

The numbers your code must reproduce

The least-damped root of \(\epsilon(k,\omega) = 0\) for a Maxwellian:

\(k\lambda_D\) \(\omega_r\) \(\gamma\) Recurrence-limited run length
0.30 1.15985 −0.01262 ~210
0.40 1.28506 −0.06613 ~1100
0.50 1.41566 −0.15336 ~2560
0.60 1.54571 −0.26411 ~4400

\(k\lambda_D = 0.5\) is the standard test. The wave damps by a factor of ~20 over \(20\,\omega_{pe}^{-1}\) — enough decades to fit a clean exponential, few enough that a modest run captures it before recurrence arrives.

Chapter 2 is blunt: if your solver does not return \(\gamma = -0.1534\) to within a few per cent at this \(k\), stop and debug. Nothing built on top of it can be trusted.

Note the sensitivity: tripling \(k\) from 0.2 to 0.6 changes \(\gamma\) by more than three orders of magnitude. Long waves barely damp; short waves die before they propagate. A code with a small systematic error will match at one \(k\) by luck and fail at the others — which is why the test is a scan, not a single run.

The factor of two

The most common error in the whole exercise. Linear theory gives the damping of the field amplitude: \(E \sim e^{\gamma t}\). But the natural diagnostic is field energy, \(\mathcal{E} = \int E^2/2\,dx\), which therefore decays as \(e^{2\gamma t}\).

\[\text{slope of } \ln\mathcal{E} \text{ vs } t = 2\gamma\]

Forget the 2 and you report exactly half the damping rate — and because \(-0.0767\) is a perfectly plausible-looking number, nothing else in the run will tell you.

How to measure it properly

The field energy oscillates at \(\omega_r\) and decays at \(2\gamma\), so a naive least-squares fit to \(\ln\mathcal{E}(t)\) is contaminated by the oscillation. Fit the envelope: locate the local maxima and regress \(\ln\mathcal{E}\) against \(t\) through those peaks only. The widget does exactly this and lands within a few tenths of a percent.

Choose the fit window deliberately:

  • Too early and you catch the initial transient. The initial condition is not the least-damped eigenmode; other roots contribute and decay away first.
  • Too late and you catch recurrence, or nonlinear trapping.

A defensible measurement states its window and shows the result is insensitive to reasonable changes in it — see growth-rate fitting.

Where the damping comes from

There is no dissipation. The energy goes to particles near the resonant velocity \(v \approx \omega/k\), which surf the wave. Because a Maxwellian has more particles just below the phase velocity than just above (\(\partial f_0/\partial v < 0\) there), slightly more particles gain energy than lose it, and the wave pays. Reverse the slope — put a bump on the tail — and the sign flips: the wave grows, which is the bump-on-tail instability.

This is why the phase-space panel matters more than the energy plot: you can watch the free energy move into fine velocity-space filaments rather than disappearing.

Common mistakes

  • The factor of two. See above.
  • Truncating velocity space too tightly. \(v_{\max} = 3v_{th}\) cuts out the resonant particles at \(\omega/k \approx 2.8v_{th}\) for \(k\lambda_D = 0.5\). Use \(5\)–\(6\,v_{th}\).
  • Using an amplitude that is not small. \(A = 0.1\) already shows trapping at late times. Linear theory needs a genuinely linear perturbation; \(A \sim 0.01\) is the safe choice.
  • Reporting agreement at one \(k\). Scan.

Knowledge graph position

Prerequisites: Landau damping theory, Vlasov–Poisson. Leads to: validated kinetic solvers; the same measurement in PIC, where noise makes it far harder.

Quiz

Q1 (computational). A student fits \(\ln\mathcal{E}\) vs \(t\) and reports \(\gamma = -0.0767\) at \(k\lambda_D = 0.5\). What went wrong?

Answer

They forgot the factor of two. Field energy decays at \(2\gamma\), so their slope of \(-0.1534\) is \(2\gamma\) and the correct \(\gamma\) is \(-0.0767 \times 2 = -0.1534\) — i.e. the code is right and the analysis is wrong. Exactly half the true rate is the signature.

Q2 (conceptual). Why does Landau damping test a code more severely than, say, a plasma oscillation frequency?

Answer

The frequency is set by the bulk of the distribution and is robust; the damping rate is set by \(\partial f_0/\partial v\) at the resonant velocity — a small population in the tail. Any numerical diffusion, velocity truncation or interpolation error corrupts it, and its extreme sensitivity to \(k\lambda_D\) means a scan cannot be faked by tuning.

Q3 (MCQ). In a collisionless simulation the wave energy lost to damping goes:

  • (a) into numerical dissipation
  • (b) to particles near the resonant velocity \(v \approx \omega/k\)
  • (c) into heat via collisions
  • (d) nowhere — it is destroyed
Answer

(b). It is a transfer, not a loss. Slightly more particles just below the phase velocity gain energy than just above lose it, because \(\partial f_0/\partial v < 0\) there.