Contrastreaming Instability in a One‐Dimensional Finite‐Length System

Two independent and oppositely directed electron beams with equal densities and speeds are considered to interpenetrate one another in a finite‐length system with a stationary neutralizing ion background. The nonlinear development of the resulting electrostatic waves is followed both in time and space by numerical simulation. The results include the time‐dependent behavior of the particle trajectories governed by the equation W = ½mv2−∣ qϕ0(t) ∣ coskmaxzW=12mv2−∣qϕ0(t)∣coskmaxz. In the cold beam case a hysteresis‐like loop in phase space is observed to develop in the transition from the laminar to a turbulent state. The vortices formed in velocity space are found to be absorbed at the right‐hand boundary of the finite‐length system and coalesce into one another for all warm beam cases. The total electric field energy saturates at approximately 16% of the total system energy. This result is three times greater than for a one‐dimensional system with periodic boundary conditions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70642/2/PFLDAS-15-1-166-1.pd