Landau Damping Simulator

Introduction: Landau damping in collisionless Langmuir waves

Landau damping describes how an electrostatic wave in a collisionless plasma loses amplitude when resonant particles exchange energy with the wave. Rather than relying on viscosity, resistance, or particle collisions, the process transfers wave energy to electrons moving near the wave phase velocity $v_p=\frac{\omega }{k}$. This kinetic mechanism matters in fusion plasmas, space plasmas, and accelerator beams, where weak electric fields can influence the particle distribution. Direct laboratory observation can be difficult, so this Landau damping simulator connects the kinetic-rate expression to a visible decaying Langmuir wave. It animates a wave whose amplitude follows the calculated damping rate, allowing the effect of density, electron temperature, and wavenumber to be examined in time. The canvas is not a full plasma simulation; it is a compact visualization of the linear damping model and its exponentially changing wave envelope. Its initial normalized amplitude is $A_0=1$.

Landau damping variables and model assumptions

This Landau damping calculation models a homogeneous, unmagnetized Maxwellian electron plasma with immobile ions. Its inputs are electron density $n_e$, electron temperature $T_e$, and wavenumber $k$. From these it calculates the plasma frequency ${\text{ω}}_p$, thermal velocity $v_{\mathrm{th}}$, Debye length ${\text{λ}}_D$, and Landau damping rate $\gamma$. The dimensionless quantity controlling the rate is $k{\text{λ}}_D$. The wave is electrostatic and longitudinal, so magnetic forces are not represented. Amplitude begins at unity, while the displayed normalized energy is proportional to amplitude squared. All entries use SI units: density in m⁻³, temperature in kelvin, wavenumber in m⁻¹, and time step in seconds. The form rejects non-positive or non-finite values, so the entered conditions must satisfy $n_e>0$ and $T_e>0$. Since this is a linear Landau-damping model, it does not model distribution changes caused by a large-amplitude wave. Wave-packet dispersion, nonlinear trapping, and multidimensional structure are also outside the visualization.

Formula: Landau damping rate and Langmuir-wave equations

For the plasma described by this calculator, the electron plasma frequency is $${\text{ω}}_p=\sqrt{\frac{n_e{e}^2}{{\text{ε}}_0{m}_e}}$$. Here $e$ is the elementary charge, ${\text{ε}}_0$ is vacuum permittivity, and $m_e$ is electron mass. The thermal velocity is $v_{\mathrm{th}}=\sqrt{\frac{k_B{T}_e}{m_e}}$, giving the Debye length ${\text{λ}}_D=\sqrt{\frac{{\text{ε}}_0{k}_B{T}_e}{n_e{e}^2}}$. The calculator uses the following weak-damping expression for the Langmuir-wave damping rate:

Formula: γ = - sqrt(π) / 2 ω_p kλ_D^-3 e^-1/2kλ_D^2

$$\gamma =-\frac{\sqrt{\pi }}{2}{\text{ω}}_p{k{\text{λ}}_D}^{-3}{e}^{-\frac{1}{2}{k{\text{λ}}_D}^2}$$

The Langmuir-wave frequency used for the animated phase is $\omega =\sqrt{{{\text{ω}}_p}^2+3k^2{v_{\mathrm{th}}}^2}$. The envelope amplitude $A$ obeys $\frac{\mathrm{dA}}{\mathrm{dt}}=\gamma A$, whose solution is exponential decay $A\left(t\right)=A_0{e}^{\gamma t}$. Energy density scales as $E\propto A^2$. The simulator advances the amplitude equation and uses an illustrative sinusoidal trace $\phi (x,t)=A\left(t\right)\sin (kx-\omega t)$; its canvas coordinate is a display coordinate rather than a calibrated physical distance. The striped bar reports the normalized energy ${\text{E/E}}_0$.

Landau damping numerical stepping

This Landau damping animation advances amplitude with explicit Euler stepping: $A_{\text{n+1}}=A_n+\gamma A_n\mathrm{\Delta t}$. For the Euler approximation to closely follow the Landau-damped envelope, the time step should satisfy $\left|\gamma \right|\mathrm{\Delta t}\ll 1$; the form clamps it to ${10}^{-7}\le \mathrm{\Delta t}\le {10}^{-3}$ s. The wave phase advances as ${\varphi }_{n+1}={\varphi }_n+\omega \mathrm{\Delta t}$. Although the amplitude equation has an analytical exponential solution, the numerical update makes the impact of the selected step visible. A step that is too large can distort the expected decay and, in extreme cases, change the sign of the numerical amplitude. Check the reported damping rate and choose a time step small compared with the damping timescale before interpreting the animation.

Worked example: interpreting a calculated Landau damping run

A useful Landau damping run starts by choosing density, temperature, and wavenumber appropriate to the plasma being represented, then inspecting the displayed plasma frequency and negative damping rate after reset. The magnitude $\left|\gamma \right|$ sets how rapidly the normalized amplitude changes in the Euler update, while its negative sign represents damping in this model. The wavenumber and Debye length occur together as $k{\lambda }_D$, so entering a wavenumber without considering the plasma conditions can give a regime where the weak-damping approximation is not informative. Use the canvas to verify that the orange envelope contracts, and use the caption and CSV button when a time-by-time record of amplitude and squared-amplitude energy is needed. The downloaded record contains $t,A,A^2$ for each stored step. If a parameter change produces almost no visible decay, that may reflect an extremely small calculated rate at the display precision rather than a failure of the animation.

Landau damping parameter comparison

Rather than combining unlike input quantities into a single score, compare Landau damping scenarios by holding a clear set of plasma conditions fixed and changing one physical input at a time. Electron density changes the plasma frequency and Debye length; electron temperature changes thermal velocity and Debye length; and wavenumber enters the dimensionless product $k{\lambda }_D$ that appears strongly in the rate expression. Reset the simulation after each change and compare the reported $\gamma$, the decay of amplitude in the caption, and the energy-bar width. In this display the bar represents $\frac{E}{E_0}=A^2$ because the initial amplitude is one. This approach keeps the comparison tied to the kinetic damping formula instead of treating density, kelvin, and inverse metres as quantities that can be added together.

Reading the Landau-damped wave animation

The Landau damping canvas draws an illustrative Langmuir-wave potential as an orange curve oscillating around its midline. As each simulation step advances phase, the curve continues to oscillate while its envelope changes according to the calculated damping rate. The striped bar below the canvas is the remaining normalized wave energy, equal to the squared amplitude relative to the initial value. When the wave canvas has focus, press the Space key to start or pause this Landau-damping animation. The caption reports current time, amplitude, and energy fraction, and supplies the same changing information to screen-reader users. The run stops when or when the simulated time exceeds 0.01 s.

Landau damping model limitations

This Landau damping simulator assumes a Maxwellian electron distribution and omits magnetic fields, nonlinear wave-particle trapping, relativistic effects, and spatial inhomogeneity. Its exponential rate expression is intended for small ${\text{kλ}}_D$ and weak damping. Large wave amplitudes or non-Maxwellian velocity tails can produce behavior, including growth or plateau formation, that this model does not capture. Explicit Euler stepping also requires a sufficiently small $\mathrm{\Delta t}$; a coarse step can create numerical artifacts rather than a faithful damped envelope. The displayed wave is one-dimensional, so oblique propagation and mode coupling are omitted. The image therefore should be read as a trace with a calculated envelope, not as a spatially calibrated measurement of $x$.

Suggested extensions for Landau damping studies

A more complete Landau damping study could solve the Vlasov–Poisson system with a particle-in-cell or continuum method, animate phase-space vortices, or add ion Landau damping for ion-acoustic waves. External electric fields could be used to explore inverse Landau damping and wave growth. A phase-space overlay could identify the resonant velocity range, while adaptive stepping could reduce work when damping is slow. Those additions would change the physical model substantially; this page deliberately retains a small linear Langmuir-wave visualization based on the rate shown above.

References and related plasma tools

For derivations of collisionless Landau damping, see Landau’s original paper (J. Phys. USSR 10, 25, 1946) or textbooks such as Nicholson’s Introduction to Plasma Theory. This simulator complements our Lorentz Force Calculator, the Mean Free Path Calculator, and the Point Charge Field Calculator, which address other plasma and electromagnetic quantities.

How to use this Landau damping calculator

1. Enter ne as electron density in m⁻³.
2. Enter Te as electron temperature in kelvin.
3. Enter k as the Langmuir-wave wavenumber in m⁻¹.
4. Press Play to animate the calculated damping, then reset and vary one plasma parameter at a time to compare its effect on the rate.