Fundamentals of Plasma Physics 2025, 25 - Streaming Instabilities
Start from the electrostatic dispersion relation for zero-temperature plasma
using ωp for the normal mode of cold plasma. An oscillation persists indefinitely, because no dissipation occurs. It requires external driving, as no free energy is available to start it simultaneously. For an instability to emerge in situations where each particle species has the same equilibrium streaming velocity uσ0 the linearized e.o.m., continuity equation, and Poisson's equations factor into the frequency expression, and changes the dispersion relation to use ωDoppler = ω - k uσ0. At equal positron/electron densities streaming past each other with equal and opposite velocities (more theoretical, than practical), the positron plasma frequency and electron plasma frequency are the same (derived from their mass), and the dispersion relation reduces to a quartic which solves for the wavelength
When the electrons stream through a background of stationary neutralizing ions, the dispersion relation reduces differently and more easily
Fast growing instabilities occur for k u0 ≅ ωpe. Using the plasma dispersion function Z(α) introduced by the Landau problem,
For the pole corresponding to the fastest growing mode at well below the real axis, said mode would be highly damped. This is disallowed by the correspondence between Vlasov and fluid models predicting similar waves in the regime. The Vlasov solution can have poles only slightly below the real axis. The ξ integration divides into three sections -∞ < α - δ < α + δ < ∞. The principle parts is the sum of the straight line segments in the first and third segment for δ → 0, and the semicircle is half a residue as per functional analysis for such integrals. We assume |α| >> 1 for the adiabatic limit and |α| < < 1 for isothermal fluids.
To avoid the ill-defined integral featured in the Landau method, a shortcut is offered by the Plemelj formula
