Two stream instability

Dispersion relation

Consider a cold, uniform, and unmagnetized plasma, where ions are stationary and the electrons have velocity 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 ^

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 , ∇ → i k we can get the following expression:

1 = ω p e 2 [ m e / m i ω 2 + 1 ( ω − k v 0 ) 2 ] ,

which represents the dispersion relation for longitudinal waves, and represents a quartic equation in ω . The roots can be expressed in the form:

ω j = ω j R + i γ j

If the imaginary part ( I m ( ω j ) ) is zero, then the solutions represent all the possible modes, and there is no temporal wave growth or damping at all:

E = ξ exp ⁡ [ i ( k x − ω t ) ] x ^

If I m ( ω j ) ≠ 0 , that is, any of the roots are complex, they will occur in complex conjugate pairs. Substituting in the expression for electrostatic waves leads to:

E = ξ exp ⁡ [ i ( k x − ω j R t ) ] exp ⁡ [ γ t ] x ^

Because of the second exponential function at the right, the temporal dynamics of the wave amplitude depends strongly on the parameter γ ; if γ < 0 , then the waves will be exponentially damped; on the other hand, if γ > 0 , then the waves are unstable and will grow at an exponential rate.

Wave–particle interactions

The two stream instability can be thought of as the inverse of Landau damping, where the existence of a greater number of particles that move slower than the wave phase velocity v p h as compared with those that move faster, leads to an energy transfer from the wave to the particles. In the case of the two stream instability, when an electron stream is injected to the plasma, the particles' velocity distribution function has a "bump" on its "tail". If a wave has phase velocity in the region where the slope is positive, there is a greater number of faster particles ( v > v p h ) than slower particles, and so there is a greater amount of energy being transferred from the fast particles to the wave, giving rise to exponential wave growth.