FR31 · A field that comes back

Plasma Wave Echo

Why does a plasma echo appear after two pulses have both faded? The first pulse leaves stripes in the electron velocities that wind tighter as time passes. The second pulse adds stripes at a different wave number, which unwind part of that winding at a different rate. At one later moment the two windings cancel, the electrons bunch again and a field reappears with no third pulse.

Kick the electrons with two short potential pulses of different wavelengths. Each response fades; later a field reappears at the difference wave number, when the velocity stripes untwist. Change the delay and the pulse pair to predict when.

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. Both pulses are instantaneous standing-wave potentials in a box four pi Debye lengths long. The predicted echo time is the free-streaming result of Gould, O'Neil and Malmberg (1967); the self-consistent field shifts the measured peak by about one percent. Collisions use a Dougherty operator that keeps particle number, momentum and energy. On a coarse velocity grid each response returns at the recurrence time, which can look like an echo.

Predicted echo time

\tau'

40\,\omega_p^{-1}

Measured echo peak

t_{\mathrm{peak}}

40.5\,\omega_p^{-1}

Echo over first response

E_3/E_1

0.117

Earliest grid recurrence

T_R

154\,\omega_p^{-1}

The echo-mode field

simulated fieldpulsespredicted echogrid recurrence

Stripes wind, then unwind

measured stripe powerfree-streaming prediction

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

Each pulse is an instantaneous potential kick. Free streaming between the pulses predicts the echo time and mode:

\tau'=\frac{k_2\,\tau}{k_2-k_1},\qquad k_3=k_2-k_1

A velocity grid cannot hold stripes finer than one cell, so the free-streaming field comes back at the recurrence time, where the last factor is the cell width:

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

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: How can two pulses that both fade away make a third signal appear later?
One-sentence intuition: Stripes from the two pulses wind at rates set by their wave numbers; the echo appears in the difference wave number when the combined winding returns to zero.

Core mathematical model

Echo time and wave number (Gould, O’Neil and Malmberg 1967)

k_3=k_2-k_1,\qquad \tau'=\frac{k_2\,\tau}{k_2-k_1}

The first pulse at time zero has wave number $k_1$ ; the second, at $\tau$ , has $k_2$ . When $k_2=2k_1$ , the echo comes at twice the delay.

Winding of the second-order stripes

\kappa(t)=k_2\,(t-\tau)-k_1\,t

The stripe wave number in velocity crosses zero exactly at the echo time; the stripe chart draws this line in gold.

Weak collisions erase the echo

C[f]=\nu\,\frac{\partial}{\partial v}\Bigl[(v-u)f+T\,\frac{\partial f}{\partial v}\Bigr]

This Dougherty operator keeps particle number, momentum and energy. A stripe of velocity wave number $\kappa$ decays at the rate $\nu T\kappa^{2}$ , so fine stripes die first.

Common difficulties

An echo is a reflection

Typical misconceptionThe echo is the first wave bouncing back from a wall.

Better mental modelThe box is periodic and has no walls. The field reappears because the velocity stripes realign, and it appears at a new wave number.

Model scale

Typical misconceptionThe pulse strengths match an experiment.

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. Both pulses are instantaneous standing-wave potentials in a box four pi Debye lengths long. The predicted echo time is the free-streaming result of Gould, O'Neil and Malmberg (1967); the self-consistent field shifts the measured peak by about one percent. Collisions use a Dougherty operator that keeps particle number, momentum and energy. On a coarse velocity grid each response returns at the recurrence time, which can look like an echo.

Run the experiment

01
Predict, then watch
Choose An echo at twice the delay, predict the echo time, then compare with the readouts.
What to observe: The echo-mode field rises out of the noise near the gold marker, and the stripe chart shows the gold line crossing zero there.
02
Change the pulse pair
Choose An echo at three delays, then Swap the pulses: no echo.
What to observe: With the smaller wave number second, the winding never returns to zero and no echo forms.
03
Tell physics from numerics
Choose Weak collisions erase it, then A coarse grid fakes a return.
What to observe: Collisions shrink the echo at the predicted time. The coarse grid gives a large peak near the coral recurrence line instead, which disappears on a finer grid.

Plasma Wave Echo

The echo-mode field

Stripes wind, then unwind

Plasma Wave Echo

Core mathematical model

Echo time and wave number (Gould, O’Neil and Malmberg 1967)

Winding of the second-order stripes

Weak collisions erase the echo

Common difficulties

An echo is a reflection

Model scale

Run the experiment

Predict, then watch

Change the pulse pair

Tell physics from numerics

Related experiments and articles

Kinetic plasma from phase-space stripes to wave echoes