FR31 · Where does a plasma wave go?

Phase-Space Filaments

Why does a ripple in a hot plasma fade with no collisions at all? Electrons with different speeds drift apart, so the bunch of charge that made the field is smeared into finer and finer stripes in velocity. The field fades even though nothing is lost from the electrons. How fast it fades depends on the few electrons moving close to the wave speed. This collisionless fading is Landau damping.

Start a small density ripple and watch its electric field die away while the electrons fold into ever finer stripes in velocity. A coarse velocity grid brings the ripple back at the wrong time.

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 of the ripple. The comparison rate is the complex root of the dielectric function of a Maxwellian, computed separately from the simulation. Collisions are off unless the collision slider in the advanced settings is raised.

Measured damping rate

\gamma_{\mathrm{fit}}

-0.154\,\omega_p

Dielectric-root rate

\gamma_{\mathrm{root}}

-0.153\,\omega_p

Velocity cells per stripe at the end

2\pi/(kt\,\Delta v)

3.83

Grid recurrence time

T_R

268\,\omega_p^{-1}

How the field fades

simulated fielddielectric-root rate (reference)grid recurrence

Stripes in velocity get finer

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

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: Why does a ripple in a hot plasma fade with no collisions at all?
One-sentence intuition: Each electron keeps its speed, but electrons with different speeds slide apart, so the charge bunch spreads into velocity stripes and its field averages away.

Core mathematical model

Electron Vlasov–Poisson equations in these units

\frac{\partial f}{\partial t}+v\,\frac{\partial f}{\partial x}-E\,\frac{\partial f}{\partial v}=0,\qquad \frac{\partial E}{\partial x}=1-\int f\,dv

Electrons carry charge minus one, so the field pushes them with acceleration minus E. Ions are a fixed background of density one.

Free streaming winds the stripes

f_1(x,v,t)=\alpha\,M(v)\cos\bigl(k(x-vt)\bigr),\qquad \Delta v_{\text{stripe}}=\frac{2\pi}{kt}

Without a field, a ripple keeps its height but its velocity stripes get narrower in proportion to one over time.

Damping rate from the dielectric root

1+\frac{1}{k^{2}}\Bigl[1+\zeta Z(\zeta)\Bigr]=0,\qquad \zeta=\frac{\omega}{\sqrt{2}\,k}

Z is the plasma dispersion function. For $k\lambda_D=0.5$ the root is $\omega=1.4157-0.1534\,i$ in units of the plasma frequency.

When a velocity grid fakes a return

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

A velocity grid with spacing $\Delta v$ cannot tell a stripe that narrow from no stripe at all, so the density rebuilds itself at this recurrence time.

Common difficulties

Damping without friction

Typical misconceptionThe wave fades because electrons lose energy in collisions.

Better mental modelThe run has no collisions. Electrons near the wave speed gain the energy the field loses; the total energy stays fixed.

Model scale

Typical misconceptionThe grid reproduces a laboratory plasma.

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 of the ripple. The comparison rate is the complex root of the dielectric function of a Maxwellian, computed separately from the simulation. Collisions are off unless the collision slider in the advanced settings is raised.

Run the experiment

01
Watch the field fade
Choose A ripple fades and keep the self-consistent field on.
What to observe: The cyan field shrinks along the gold reference line while the phase-space picture fills with tilted stripes.
02
Remove the field
Choose Streaming without a field.
What to observe: The density still fades, but faster and with a curved logarithmic history: phase mixing alone gives a Gaussian decay, not one fixed rate.
03
Starve the velocity grid
Choose A coarse grid brings it back, then compare the stripe chart with the gold line.
What to observe: The measured stripe wave number hits the top of the chart and wraps around; the field returns at the coral recurrence line, a numerical artifact.

Phase-Space Filaments

How the field fades

Stripes in velocity get finer