Weibel Instability Growth Rate Calculator

Introduction to estimating Weibel filamentation growth

Modeling a Weibel or current-filamentation case starts with a practical question: given a beam density, a background density, and a Lorentz factor, how fast should a transverse magnetic perturbation grow? This calculator provides that first-pass estimate. It checks that the beam density is positive and below the background density, then reports the maximum growth rate γ_max, the e-folding time τ, and the corresponding fastest-growing wavenumber k_max.

A quick estimate is valuable because the answer changes significantly with the density ratio. In the cold-beam picture used here, growth does not rise linearly with n_b. It follows the square root of the beam-to-background ratio after the background electron plasma frequency has been calculated. The tool is therefore useful for screening simulation inputs, checking hand calculations, and deciding whether a proposed time window resolves the early linear phase.

This estimate is deliberately narrower than a general kinetic plasma solver. It describes an idealized early-time instability rather than nonlinear filament merging, saturation, collisions, or a complete electromagnetic dispersion relation. The result is best treated as a scale estimate that helps organize a more detailed analysis.

What the Weibel current-filamentation calculator solves

The Weibel calculator estimates the early exponential growth of a beam-driven current-filamentation mode in a denser background plasma. A weaker beam gives a smaller density ratio and slower growth. Increasing the beam Lorentz factor reduces the estimated rate through the relativistic denominator. The seed field determines where the displayed magnetic trace starts, but it does not determine the calculated linear growth rate.

The three headline results describe different views of the same instability. The growth rate γ_max, measured in s⁻¹, indicates how quickly the amplitude changes. Its reciprocal τ is the time required for an ideal linear perturbation to grow by a factor of e. The wavenumber k_max, measured in m⁻¹, identifies a characteristic spatial scale for the fastest-growing mode under this simplified relation.

How to use the Weibel instability growth-rate calculator

The form combines physical plasma parameters with numerical display settings. Enter all densities in m⁻³, magnetic field in tesla, and times in seconds. Press Play to generate and animate the trace, Pause to stop it temporarily, Reset to return to the first sample, or CSV to download the generated time series.

1. Enter n_b as the density of the beam or other current-carrying population that drives filamentation.
2. Enter n₀ as the background electron density. For this model, n_b must remain below n₀.
3. Set γ₀ to the initial Lorentz factor of the drifting beam. The accepted minimum is 1.
4. Set B₀ to the seed magnetic-field amplitude used at the beginning of the trace.
5. Choose Δt, the interval between numerical samples, and T, the total displayed duration.
6. Run the calculation and compare γ_max, τ, k_max, and the shape of the field history.

When comparing scenarios, change one physical input at a time. Holding n₀, γ₀, Δt, and T fixed while varying n_b, for example, makes the square-root density dependence much easier to recognize. The CSV export is useful for preserving those controlled sweeps.

Choosing beam, background, seed-field, and time inputs

The beam density n_b represents the population supplying the unstable current, while n₀ establishes the background electron plasma frequency. Their ratio is dimensionless. Because the implemented approximation assumes a tenuous beam, the form rejects cases where n_b is equal to or greater than n₀. A rejected case is not proof that the physical plasma is stable; it only lies outside the intended input regime of this calculator.

The Lorentz factor γ₀ describes relativistic beam motion. Larger values reduce the estimated growth rate when the densities are unchanged. B₀ is different: it scales the initial amplitude of the plotted field but does not appear in the growth-rate formula. If B₀ is exactly zero, the equation dB/dt = γ_maxB remains at zero, so use a small nonzero seed when you want a visible exponential trace.

The numerical controls do not introduce new physics. Δt controls the spacing of output samples and T controls the time span. A useful display window covers several e-folding times without extending so far that exponential values exceed a meaningful linear-regime scale. A smaller step can make the graph smoother, but it cannot compensate for missing temperature, geometry, or nonlinear saturation physics.

• Units: use m⁻³ for both densities, tesla for B₀, and seconds for Δt and T.
• Density regime: keep 0 < n_b/n₀ < 1 for the implemented cold, tenuous-beam estimate.
• Time resolution: choose Δt substantially shorter than τ if individual early-time samples matter.
• Comparison practice: preserve the same units and numerical window across a parameter sweep.

Formulas for the Weibel growth rate, e-folding time, and wavenumber

The calculation begins with the background electron plasma frequency. It then scales that frequency by the square root of the beam fraction divided by the beam Lorentz factor. The constants are the elementary charge e, electron mass m_e, vacuum permittivity ε₀, and speed of light c.

The magnetic history is advanced with the linear equation dB/dt = γ_maxB using a fourth-order Runge–Kutta step. Its ideal solution is B(t) = B₀e^{γmax t}. This exponential law is appropriate only during the linear stage. Real filaments eventually alter the distribution and magnetic geometry, causing saturation or a transition to nonlinear evolution.

Worked example: a 10% beam-density fraction

Using the default values, n_b = 1.0×10²⁰ m⁻³, n₀ = 1.0×10²¹ m⁻³, γ₀ = 2, and B₀ = 1.0×10⁻⁹ T. The beam fraction is 0.1. With the physical constants in the script, the estimated maximum rate is approximately 3.99×10¹¹ s⁻¹. The corresponding e-folding time is approximately 2.51×10⁻¹² s, while k_max is approximately 1.33×10³ m⁻¹.

That e-folding time is much shorter than the default total window of 5×10⁻⁹ s. The ideal linear equation therefore covers many e-folds and can produce an extremely steep trace. For a closer inspection of the onset, reduce T to a modest multiple of τ and choose Δt well below τ. Doing so changes the displayed window, not γ_max itself.

Suppose n_b is reduced by 20% while n₀ and γ₀ remain fixed. Because the rate depends on the square root of n_b, γ_max falls to roughly 3.57×10¹¹ s⁻¹ rather than falling by a full 20%. Conversely, raising n_b by 20% gives approximately 4.37×10¹¹ s⁻¹. This illustrates why a density sweep is more informative than assuming a linear response.

How to interpret the Weibel output and growth trace

A larger γ_max means faster ideal exponential amplification, while a smaller τ expresses the same result in the time domain. k_max offers a characteristic inverse length scale. If a wavelength is needed for comparison, the conventional conversion is λ = 2π/k, subject to the assumptions behind the selected dispersion estimate.

Read the numerical result together with the plot. A line that climbs almost vertically usually means the selected duration spans many e-folding times. It does not imply that a real plasma would continue growing indefinitely. In an experiment or particle-in-cell calculation, saturation, finite temperature, finite system size, return currents, and evolving particle distributions can become decisive before the end of that ideal trace.

The progress indicator reports the currently displayed field relative to the largest value generated for the trace. It is a visual animation aid, not an efficiency, probability, or physical saturation fraction. Likewise, the energy-density column in the CSV is calculated from B²/(2μ₀); it inherits the same idealized field evolution.

Limitations of this cold-beam Weibel estimate

This calculator is intended for rapid scale checks rather than definitive plasma predictions. It uses a compact beam-driven relation and does not solve the full kinetic dispersion relation. Temperature anisotropy, velocity spread, ion dynamics, oblique modes, collisions, return-current structure, boundaries, and multidimensional geometry can all change the dominant growth rate or mode.

• Linear stage only: the trace does not model magnetic trapping, filament coalescence, depletion, or saturation.
• Cold-beam restriction: thermal spread and anisotropic pressure are absent from the implemented rate.
• Ideal geometry: the k_max relation is a characteristic estimate rather than a complete geometry-dependent spectrum.
• Classical inputs: the calculator assumes the stated SI quantities are physically compatible and does not infer unit conversions.
• Numerical range: a very long window can push exponential values beyond useful floating-point or linear-regime scales.

For experiment design or publication work, compare this estimate with an appropriate dispersion solver, kinetic simulation, and authoritative plasma-physics references. Its practical value lies in making the leading density and Lorentz-factor dependence transparent before more expensive modeling begins.