Two-Stream Instability (1D PIC)

What you are seeing: two counter-streaming beams of electrons in a 1D plasma. The configuration is unstable: any tiny perturbation in density grows exponentially, the beams form rolling vortices in phase space ("electron holes"), then trap each other, eventually settling into a single thermal distribution. This is the canonical kinetic plasma instability.

The top plot is the (x, v) phase space, sampled by 10000 macro-particles, drawn with persistence so the electron-hole vortices leave trails. The middle strip is the density-mode spectrogram ( $|\hat\rho_k|$ for $k = 1\ldots 8$ versus time): mode 1 dominates the linear phase, harmonics appear at saturation. The bottom trace is $\log|\hat\rho_{k=1}|$ with the dashed analytic reference of slope $\gamma = \omega_p/(2\sqrt 2)\approx 0.354$ (plasma units): the measured slope tracks it in the linear regime, then the mode saturates. Default $v_0 = 0.6$ puts the fundamental near the peak-growth wavenumber.

Figure 1. 1D-1V particle-in-cell simulation of the two-stream instability. Method: NGP charge deposit, FFT Poisson solve, leapfrog particle push.

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WHAT TO TRY

Watch the two counter-streaming beams go unstable: a tiny density ripple grows exponentially and the beams roll up into phase-space vortices, the cat-eye structure of the two-stream instability.
Raise the beam velocity v_0: the instability growth rate and the vortex size change, since the unstable wavenumbers scale with the streaming speed.
Follow the field-energy diagnostic: it grows exponentially while the instability develops, then saturates as the beams thermalize and the vortices merge, the classic linear-then-nonlinear story.