Landau damping from the root of the dispersion relation

Adam Zettel · September 25, 2026 · written with an agent

1 + \frac{1 + ζ Z ( ζ )}{k ^{2} λ _{D}^{2}} = 0, ζ = \frac{ω}{2 k v _{t h}}

A Langmuir wave in a collisionless plasma still decays: electrons moving near its phase velocity take energy from it. Newton's method on the kinetic dispersion relation, with the plasma dispersion function from lib/zfunc, puts the least-damped root at $k λ_{D} = 0.5$ at $ω = 1.415662 - 0.153359 i$ (in units of $ω_{p e}$ ), matching a 30-digit mpmath root to $1 0^{- 12}$ . The usual small- $k$ damping formula is 1.3% off there and 59% off at $k λ_{D} = 0.3$ .

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A Langmuir wave is an oscillation of the electrons in a plasma against the ions, at about the plasma frequency $ω_{p e}$ . In a plasma with no collisions at all it still decays. Electrons moving a little slower than the wave are pushed forward by it and take energy; electrons moving a little faster are held back and give energy. In a Maxwellian distribution the slower ones outnumber the faster ones at any speed above the mean, so the wave loses. This is Landau damping (Landau, 1946).

The rate comes out of the kinetic dispersion relation. For a Maxwellian with thermal speed $v_{t h}$ and Debye length $λ_{D}$ , a wave $e^{i (k x - ω t)}$ exists when

ε (ω) = 1 + \frac{1 + ζ Z ( ζ )}{k ^{2} λ _{D}^{2}} = 0, ζ = \frac{ω}{2 k v _{t h}},

where $Z$ is the plasma dispersion function. Its roots are complex, $ω = ω_{r} - iγ$ , and $γ$ is the damping rate. $Z$ comes from lib/zfunc, which computes it by Weideman's rational expansion and is tested there against scipy.special.wofz. Units: $ω_{p e} = λ_{D} = v_{t h} = 1$ , so $k$ means $k λ_{D}$ . posts/refs/landau_root.py finds the same roots with scipy and with mpmath at 30 digits.

The root at $k λ_{D} = 0.5$

A good starting guess is the Bohm-Gross frequency $ω = 1 + 3 k^{2}$ , the fluid-theory answer, on the real axis. From there Newton's method moves the guess down into the lower half plane. The cell runs 30 steps; the step size is at the level of rounding after six.

Printed

1.41566

-0.153359

v_phase: 2.83132

Tests 3 of 3 pass

at k lambda_D = 0.5 the root is 1.4156618886045364 - 0.15335946690960483 i (mpmath, 30 digits) (passes)
scipy.optimize.newton with scipy.special.wofz finds the same root to 1e-12 (passes)
the wave's phase velocity is 2.83 thermal speeds, out in the tail of the Maxwellian (passes)

Where the roots are

The dispersion relation has more than one root. The heatmap shows $lo g_{10} ∣ ε (ω) ∣$ over part of the complex $ω$ plane at $k λ_{D} = 0.5$ ; roots are the dark points. The one found above is the highest, the least damped, and it is the one a wave settles into. The others lie deeper below the axis and die out faster: Newton's method started at the next dark point, near $1.8 - 1.1 i$ , converges to $ω = 1.789571 - 1.144143 i$ , which decays 7.5 times faster than the first. Re $ω$ is across and Im $ω$ up.

Printed

1.78957

-1.14414

48 × 64 grid

Tests 1 of 1 pass

the next root, 1.7895705949216291 - 1.1441430104277659 i (mpmath), is damped 7.5 times faster (passes)

Against the small- $k$ formula

For long waves, $k λ_{D} ≪ 1$ , the phase velocity is far out in the tail and the damping can be expanded. The usual result is

γ \approx \frac{π}{8} \frac{ω _{p e}}{( k λ _{D} ) ^{3}} exp (- \frac{1}{2 k ^{2} λ _{D}^{2}} - \frac{3}{2}) .

The cell follows the root from $k λ_{D} = 0.2$ to $0.6$ , each Newton solve starting from the previous root. At $0.5$ the formula gives $0.1514$ against the exact $0.1534$ , 1.3% low. That agreement is an accident of where the error changes sign: at $0.3$ the formula gives $0.0200$ against $0.0126$ , 59% too high, and at $0.2$ it is still 18% high. The real part is above Bohm-Gross by 0.5% at $0.2$ and by 7% at $0.5$ .

Printed

5.51074e-05

0.0126204

1.58637

0.98714

weak_err_05: -0.0128602

weak_err_03: 0.586373

weak_err_02: 0.181972

Damping rate of the least-damped Langmuir wave

exact rootsmall-k formula

Frequency of the least-damped Langmuir wave

exact root

Tests 4 of 4 pass

along the curve the roots at k = 0.2, 0.3, 0.5 are the mpmath roots to 1e-11 (passes)
the small-k formula is 1.3% low at k = 0.5, 59% high at 0.3 and 18% high at 0.2 (passes)
the small-k formula gives 0.1514 at k = 0.5 and 0.0200 at k = 0.3 (passes)
Re omega exceeds Bohm-Gross by 0.54% at k = 0.2 and 7.0% at k = 0.5 (passes)

The wave itself

The root says how fast a wave decays; the last cell watches one decay. It solves the linearized Vlasov equation for one Fourier mode, $f = f_{0} (v) + f_{1} (v, t) e^{ik x}$ at $k λ_{D} = 0.5$ , on 512 velocities from $- 8$ to $8$ :

\partial_{t} f_{1} = - ik v f_{1} + E_{1} f_{0}^{'} (v), E_{1} = \frac{i}{k} \int f_{1} d v,

from a density bump $f_{1} (v, 0) = f_{0} (v)$ , by fourth-order Runge-Kutta with $Δ t = 0.05$ . Nothing in it knows about $Z$ or the root. The field oscillates and its peaks fall on a straight line in $lo g ∣ E_{1} ∣$ ; the slope of that line is the measured damping rate, and the spacing of the peaks, half a period, gives the frequency. Both are compared with the root above.

Printed

n_peaks: 14

gamma_measured: 0.153359

gamma_root: 0.153359

omega_measured: 1.41566

gamma_rel_gap: 2.77736e-06

omega_rel_gap: 1.94462e-07

The field of a Langmuir wave at k lambda_D = 0.5, from the linearized Vlasov equation

Re E1Im E1decay at the root's rate

Tests 1 of 1 pass

the measured damping rate and frequency are the root's within 1e-4, relative (passes)

The root at kλD​=0.5

Where the roots are

Against the small-k formula

The wave itself

The root at $k λ_{D} = 0.5$

Against the small- $k$ formula