Two-stream and bump-on-tail instabilities

Consider two equal-density electron beams with velocities plus and minus U, their total density defining the electron plasma frequency. Immobile ions neutralize the total electron charge. Each beam contributes half the total squared plasma frequency in its Doppler-shifted cold susceptibility. This factor of one half matters: assigning the full density to both beams double counts the electrons and shifts the instability threshold. The electrostatic model neglects transverse electromagnetic instabilities of the same drifting system.

Combining the two rational terms gives a quadratic equation in frequency squared. Introduce a dimensionless squared wavenumber parameter so that the lower branch can be inspected directly. For a nonzero parameter below one, that branch is negative, giving one exponentially growing root and one decaying root. At exactly zero wavenumber the original rational equation has degeneracies, so the limiting low-frequency branch should not be mistaken for an additional finite-frequency physical mode at k = 0.

Solving the quadratic yields two values of normalized frequency squared. For the unstable branch choose the square root with positive imaginary frequency under exp(ikx − iωt), which gives temporal growth. This is an exact result within the cold two-beam model, not a weak-damping perturbation of a smooth Maxwellian. Finite beam spread changes the dispersion and may suppress the instability, so the cold threshold cannot be used unchanged for an arbitrary warm distribution.

A warm distribution can have two maxima and a minimum between them without being unstable at every wavenumber. The full dielectric response or a suitable Penrose/Nyquist test determines whether a growing root exists. A necessary shape feature and a sufficient spectral condition are different statements. This distinction is especially useful when a velocity histogram has noisy positive slopes: not every small local rise in sampled data proves an exponentially growing collective mode.

For a weak bump on a broad background, a Langmuir branch can remain close to its original real frequency while the distribution slope at phase velocity changes sign. On a positive-energy branch with positive real dielectric frequency derivative, a positive total resonant slope gives growth instead of damping. Use the total distribution: a rising beam contribution can be outweighed by a falling bulk contribution. Other species and collisions can also supply competing damping.

Linear growth cannot continue indefinitely without changing its own assumptions. Growing fields trap resonant particles, modify the velocity distribution, and can couple to other modes. A plateau can reduce resonant drive, but the saturation amplitude and late-time state depend on coherence, spectrum, boundaries, and collisions. A linear growth rate is therefore a prediction for the early interval of a specified equilibrium, not a universal estimate of the final turbulent transport or a proof of complete relaxation to a Maxwellian.