FR31 · Which electrons take the energy?

Landau Damping and Trapping

When a plasma wave fades, which electrons take its energy? Electrons moving at almost the wave speed take it. The wave pushes slightly slower ones forward and holds slightly faster ones back. In a Maxwellian the slower ones outnumber the faster ones, so these electrons gain energy and the wave loses it. A strong wave traps them instead, and they bounce inside its troughs.

Launch a weak or strong wave and see where the kinetic energy goes. A weak wave hands it to electrons near the wave speed; a strong wave traps them in rotating eddies. A fast bump or two beams make the wave grow instead.

Model assumptions

Electrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. The box holds one wavelength. A standing wave there is two traveling waves, each with half the field amplitude; the trapping band and bounce time use that half amplitude at the largest recorded field and are estimates. The growth or damping rate printed for comparison is the leading root of the dielectric function, computed separately.

Measured early rate

\gamma_{\mathrm{fit}}

-0.302\,\omega_p

Dielectric-root rate

\gamma_{\mathrm{root}}

-0.153\,\omega_p

Bounce time at peak field

\tau_b

16.2\,\omega_p^{-1}

Trapping half-width

\Delta v_{\mathrm{trap}}

1.55\,v_{\mathrm{th}}

Field strength over time

simulated fielddielectric-root rate (reference)

Where the kinetic energy went

energy gainedenergy lostwave speed

Equations, approximations and sources

Electron distribution with fixed ions, in units of the plasma frequency, thermal speed and Debye length:

\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}-E\frac{\partial f}{\partial v}=\nu\frac{\partial}{\partial v}\Big[(v-u)f+T\frac{\partial f}{\partial v}\Big],\qquad \frac{\partial E}{\partial x}=1-\int f\,dv

The program splits each step into streaming in position, a kick from the field and, when switched on, a weak-collision step. The comparison rate is the root of the Maxwellian dielectric function:

1-\frac{1}{2k^2}Z'\!\left(\frac{\omega}{\sqrt2\,k}\right)=0

Trapping estimates use the field of one traveling half of the standing wave at the largest recorded amplitude:

\tau_b=\frac{2\pi}{\sqrt{kE_w}},\qquad \Delta v_{\mathrm{trap}}=2\sqrt{E_w/k}

This is a numerical model with one space and one velocity dimension. It omits magnetic fields, ion motion, and the experiment geometry of the original echo measurements. Exported records include every parameter, the model version and the full field history.

Read the kinetic plasma course: from phase-space stripes to wave echoes

Start with the essentials

Focus question: When a plasma wave fades, which electrons take its energy, and when does the wave catch them?
One-sentence intuition: The sign of the slope of the velocity distribution at the wave speed decides whether resonant electrons absorb energy from the wave or feed it.

Core mathematical model

Weak-wave rate near the Bohm–Gross frequency

\gamma\approx\frac{\pi}{2}\,\frac{\omega_p^{3}}{k^{2}}\,\left.\frac{\partial f_0}{\partial v}\right|_{v=\omega/k}

A negative slope at the wave speed gives damping; a positive slope, as on the rising side of a bump, gives growth. The Lab uses the full dielectric root, not this approximation.

Bounce time of a trapped electron

\tau_b=\frac{2\pi}{\sqrt{kE_{\text{wave}}}},\qquad \Delta v_{\text{trap}}=2\sqrt{\frac{E_{\text{wave}}}{k}}

An electron within the trapping half-width of the wave speed oscillates in a wave trough. Linear damping holds while the bounce time is much longer than the damping time.

Kinetic energy gained at each velocity

\Delta K(v)=\frac{v^{2}}{2}\,L\,\Bigl[\langle f\rangle(v,t)-\langle f\rangle(v,0)\Bigr]\,\Delta v

The angle brackets mean an average over the box. Summed over velocity, this equals the field energy the wave has lost.

Common difficulties

Every electron is resonant

Typical misconceptionAll electrons trade energy with the wave equally.

Better mental modelMost electrons slosh back and forth with the wave and return what they take. Only those near the wave speed keep a net gain or loss.

Model scale

Typical misconceptionThe trapping band is measured from the run.

Better mental modelElectrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. The box holds one wavelength. A standing wave there is two traveling waves, each with half the field amplitude; the trapping band and bounce time use that half amplitude at the largest recorded field and are estimates. The growth or damping rate printed for comparison is the leading root of the dielectric function, computed separately.

Run the experiment

01
Find where the energy goes
Choose A weak wave just damps and watch the energy chart.
What to observe: Amber bars rise near the gold wave-speed lines: the resonant electrons gain the energy the field loses.
02
Trap the electrons
Choose Electrons get trapped, then drag a test electron into the shaded band.
What to observe: The distribution curls into eddies at the wave speed and the field stops decaying, then pulses with roughly the bounce time.
03
Turn damping into growth
Choose A fast bump feeds the wave or Two beams roll up.
What to observe: The field grows at close to the dielectric-root rate until trapping saturates it.

Landau Damping and Trapping

Field strength over time

Where the kinetic energy went