Weibel Instability Growth Simulator

1. Real-world phenomenon

Plasmas that stream faster in one direction than another do not remain quiescent. Tiny transverse magnetic perturbations deflect the charged particles, generating current filaments that in turn amplify the field. This positive feedback is the Weibel, or current-filamentation, instability. It converts particle anisotropy into magnetic turbulence and appears in environments ranging from laser-driven experiments to gamma-ray-burst shocks. Whereas a traditional calculator simply outputs the maximum growth rate $\gamma$_max, the simulator above integrates the resulting exponential growth of magnetic field amplitude. Watching the field line shoot upward conveys how rapidly anisotropy can seed magnetism.

The instability matters because magnetic fields mediate energy exchange in collisionless plasmas. In astrophysical shocks, Weibel-generated fields scatter particles and enable Fermi acceleration. In the laboratory, uncontrolled filamentation can spoil beam quality or ignite undesired hot spots. Understanding the timescale of magnetic amplification helps diagnose and tame these systems.

2. Variables and assumptions

The form collects the beam density $\mathrm{n\_b}$, background density $\mathrm{n\_0}$, and beam Lorentz factor $\mathrm{\gamma \_0}$. The calculation assumes symmetric counter-streaming electron beams in an initially unmagnetized plasma. The density ratio $\alpha =\frac{\mathrm{n\_b}}{\mathrm{n\_0}}$ must be less than unity for the background to dominate. An initial seed magnetic field $\mathrm{B\_0}$ provides the starting amplitude for growth. All values are in SI units and inputs are validated to prevent negative or non-finite entries. The time step $\mathrm{\Delta t}$ is clamped between 10⁻¹⁵ and 10⁻⁶ seconds and the simulation runs for total time $T$.

3. Core equations

The electron plasma frequency for the background is

$\omega$_p=n_0e2ε_0m_e.

The maximum temporal growth rate is

$\gamma$_max=ω_pαγ_0.

The fastest-growing wavenumber is $k$_max=ω_pcαγ_0. For the animation we integrate $\frac{\mathrm{dB}}{\mathrm{dt}}=\gamma$_maxB using numerical methods.

4. Numerical scheme

Although the growth equation admits an exponential solution, the simulator uses a fourth-order Runge–Kutta method to highlight numerical integration. Each step evaluates four slopes and combines them in the classic RK4 weighted average. Magnetic energy density $U=\frac{B^2}{2}$ (in units with $\mathrm{\mu \_0}=\mathrm{4\pi \times 10^\{-7\}}$) scales the striped energy bar. Stability demands a time step significantly smaller than the e-folding time $\tau =\frac{1}{\gamma }$_max.

5. Worked example tied to the simulation

Take a beam with density 10²⁰ m⁻³ streaming through a background of 10²¹ m⁻³ at Lorentz factor 2. The density ratio is 0.1. Plugging these numbers yields a plasma frequency of about 5.6×10¹² s⁻¹ and a growth rate of roughly 1.8×10¹² s⁻¹, corresponding to an e-folding time near 5.6×10⁻¹³ s. With a seed field of 10⁻⁹ T, pressing Play shows the curve on the canvas climb steeply; the energy bar rapidly approaches its maximum as the field multiplies by tens of thousands within a few nanoseconds. Doubling γ₀ to 4 while keeping densities fixed slows the growth, a change immediately visible in the shallower slope and the caption's updated summaries.

6. Comparison table

The table contrasts the baseline parameters with two variations.

Scenario	n_b (m⁻³)	γ₀	γ_max (s⁻¹)	τ (ps)
Baseline	1e20	2	1.8e12	0.56
Lower beam density	5e19	2	1.3e12	0.77
Higher Lorentz factor	1e20	4	1.3e12	0.77

Both reducing beam density and increasing γ₀ reduce the growth rate in this cold-beam model. Enter these values in the form to see the animation update and confirm the table.

7. How to read the animation

The canvas plots magnetic field amplitude versus time. The horizontal axis spans the total simulation time, while the vertical axis auto-scales to the peak field encountered. A magenta curve traces B(t); a red dot marks the current state. Beneath the plot, the striped magenta bar represents magnetic energy density relative to the maximum value. The caption and hidden summary list time, field strength, and growth rate for accessibility. Keyboard users can press the space bar while the canvas is focused to toggle play and pause.

8. Limitations

The model assumes cold, symmetric electron beams and neglects thermal effects, ion dynamics, and pre-existing magnetic fields. Real plasmas may saturate due to nonlinear coupling, filament merging, or external fields not captured here. The RK4 integration demonstrates numerical technique but is unnecessary given the analytic solution; nevertheless, using a finite step introduces truncation error and illustrates stability concerns when Δt becomes too large.

9. Extensions

Extensions could include modeling saturation by adding a nonlinear term, coupling the magnetic field to particle momentum, or allowing multiple spatial modes. Another path is to animate the wavenumber spectrum instead of a single mode. Because the script is self-contained, curious students can modify it to explore these scenarios or adapt it for classroom demonstrations of plasma instabilities.

10. References and related tools

Key references: E. Weibel, "Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution," Physical Review Letters 2, 83 (1959). For modern treatments see M. Yoon and C. Rhee, Reviews of Modern Plasma Physics 5, 10 (2021). Investigate related topics with the Plasma Frequency Calculator, the Zeeman Effect Calculator, or relativistic shifts via the Relativistic Doppler Shift Calculator.