Counter-propagating electron beams drive exponential electrostatic wave growth. Explore dispersion relations, growth rates, SMILEI PIC data, simulation box design, and live phase-space evolution — all interactive.
Our goal — detect and study the two-stream instability (TSI) in real NASA spacecraft data, and validate our theoretical predictions (Equations 18–21) against direct measurements. This page has two parts: the Theory Laboratory (sections 01–07) with your exact interactive equations, SMILEI PIC data, and phase-space animation — and the MMS Observatory (sections 08+) with live MMS-1 data, theory validation, ML prediction, 6-month history, and unusual event detection.
Four signatures searched: electron holes (bipolar E∥ pulses), bump-on-tail df/dv > 0, TSI→Weibel causal chain, and Langmuir wave enhancement near f_pe.
Two beams of electrons travel in opposite directions at ±v₀. Each carries half the total density. A tiny density perturbation δn·cos(kx) seeds the instability.
When the resonance condition ω/k ≈ v₀ is met, particles and electric waves exchange energy coherently. Small perturbations amplify exponentially at rate γ.
Growth stops when particles become trapped in wave potential wells — "cat's eye" vortices in x–pₓ phase space. Beam energy converts to wave energy, then heat.
| Parameter | Value | Meaning for TSI |
|---|
B magnitude in nT, subsampled for display. Fluctuations δB/B₀ represent the turbulent component. Spectral index α is fitted from the PSD in the inertial range 0.01–2 Hz. Kolmogorov (1941) predicts α = −5/3; Iroshnikov-Kraichnan (1965) predicts α = −3/2.
Field direction reveals magnetospheric region. Rapid rotations = boundary crossings.
Log-log PSD. Amber = Kolmogorov −5/3. Purple = IK −3/2. Steeper slope = kinetic dissipation.
The measured electron bulk speed V_e becomes v₀ = V_e/c. The resonant wavenumber k_res = ω_pe/v₀ (beam-wave resonance: k·v₀ = ω_pe). Growth rates are computed across k = 0.1×k_res to 3×k_res and compared to the measured plasma state.
Growth rate curves from Eq.18–21 using the MMS-measured beam velocity. The red dashed vertical line marks k_res = ω_pe/v₀ (resonance condition). If γ_max > 0, the instability is theoretically active under current conditions.
Bipolar pulses (positive spike then negative, or reverse) are electron hole signatures. Amber dashed lines = ±detection threshold.
Full E magnitude. Large spikes coinciding with bipolar events confirm hole passage. Background level tracks general wave activity (Langmuir, whistler, ion acoustic).
Pink dashed line = 300 km/s beam threshold. High speed = directional beam = TSI driver present. The 2025 measurement shows Ve = 488 km/s — well above threshold.
Green = n_e (left axis). Blue = f_pe = 8980×√(n_e) in kHz (right axis). Higher f_pe → faster TSI growth. The 2025 data shows Ne = 7.53 cm⁻³, f_pe = 86.7 kHz.
| Quantity | Value | Interpretation |
|---|
Blue = T_par (parallel). Amber = T_perp (perpendicular). When T_perp persistently > T_par, Weibel is driven. When T_par > T_perp, firehose instability is possible.
Purple = ratio. Red dashed = stability boundary (ratio = 1). Above 1 = Weibel active. Below 1 = firehose possible (as seen in 2025 data with ratio ~0.85).
Peak at positive lag τ > 0 → anisotropy follows beam (correct causal order for TSI→Weibel). Peak at τ = 0 → coincident. Peak at τ < 0 → anisotropy preceded beam (reverse causality). Detection threshold: C = 0.2 (amber dashed).
Log-log PSD. Amber vertical = f_pe. Enhanced power near f_pe = Langmuir waves active. Note: f_pe ~86 kHz in 2025 data is above EDP slow-survey Nyquist (~16 Hz) — proxy only.
Measured spectral index α vs Kolmogorov (−5/3) and IK (−3/2). The 2025 measurement gives α = −1.796 — slightly steeper than IK, suggesting enhanced sub-ion scale dissipation consistent with kinetic Alfvén wave damping.
Each bar shows how strongly that feature contributes to the TSI prediction. Full bar = maximum theoretical value. The weighted sum determines the final probability.
Cross-validation: if ML probability and rule-based score agree, we have higher confidence. If they disagree, it reveals which features are driving the discrepancy.
Radar chart showing the 5 feature contributions to the current prediction. Larger area = more features pointing toward active TSI.
Each workflow run appends one record. The database stores up to 720 records (~180 days). Because MMS data has ~90 day latency, the history covers approximately 90–270 days before today. These plots reveal how plasma conditions evolve as MMS-1 orbits through different magnetospheric regions — plasmasphere, plasma sheet, magnetosheath, reconnection sites.
Large values = inner magnetosphere; small = magnetotail or boundary layers.
Amber dashed = Kolmogorov −5/3. Purple dashed = IK −3/2. Steeper = kinetic dissipation.
Values above 300 km/s (pink dashed) = beam present = TSI conditions.
Above dashed line = Weibel drive. Below = firehose. Track how often anisotropy follows beam events.
High counts = active TSI saturation. Correlate with V_e spikes to test causality.
Both estimates plotted for cross-validation. Agreement = high confidence. Divergence reveals feature importance.