Two-stream instability

The two-stream instability is a very common instability in plasma physics. It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species (ions and electrons) can have different drift velocities. The energy from the particles can lead to plasma wave excitation. The two-stream instability is a very common instability in plasma physics. It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species (ions and electrons) can have different drift velocities. The energy from the particles can lead to plasma wave excitation. Two-stream instability can arise from the case of two cold beams, in which no particles are resonant with the wave, or from two hot beams, in which there exist particles from one or both beams which are resonant with the wave. Two-stream instability is known in various limiting cases as beam-plasma instability, beam instability, or bump-on-tail instability. Consider a cold, uniform, and unmagnetized plasma, where ions are stationary and the electrons have velocity V 0 {displaystyle mathbf {V} _{0}} , that is, the reference frame is moving with the ion stream. Let the electrostatic waves be of the form: E 1 = ξ 1 exp ⁡ [ i ( k x − ω t ) ] x ^ {displaystyle mathbf {E} _{1}=xi _{1}expmathbf {hat {x}} } Applying linearization techniques to the equation of motions for both species, to the equation of continuity, and Poisson's equation, and introducing the spatial and temporal harmonic operators ∂ t → − i ω {displaystyle partial _{t} ightarrow -iomega } , ∇ → i k {displaystyle abla ightarrow ik} we can get the following expression: 1 = ω p e 2 [ m e / m i ω 2 + 1 ( ω − k v 0 ) 2 ] , {displaystyle 1=omega _{pe}^{2}left,} which represents the dispersion relation for longitudinal waves, and represents a quartic equation in ω {displaystyle omega } . The roots can be expressed in the form: ω j = ω j R + i γ j {displaystyle omega _{j}=omega _{j}^{R}+igamma _{j}}