Farley–Buneman instability


The Farley–Buneman instability, or FB instability, is a microscopic plasma instability named after Donald T. Farley and Oscar Buneman. It is similar to the ionospheric Rayleigh-Taylor instability.
It occurs in collisional plasma with neutral component, and is driven by drift currents. It can be thought of as a modified two-stream instability arising from the difference in drifts of electrons and ions exceeding the ion acoustic speed. It occurs in collisional plasma with neutrals driven by drift current for two stream instability for unmagnetized plasma it becomes "Buneman instability".
It is present in the equatorial and polar ionospheric E-regions. In particular, it occurs in the equatorial electrojet due to the drift of electrons relative to ions, and also in the trails behind ablating meteoroids.
Since the FB fluctuations can scatter electromagnetic waves, the instability can be used to diagnose the state of ionosphere by the use of electromagnetic pulses.

Conditions

To derive the dispersion relation below, we make the following assumptions. First, quasi-neutrality is assumed. This is appropriate if we restrict ourselves to wavelengths longer than the Debye length. Second, the collision frequency between ions and background neutral particles is assumed to be much greater than the ion cyclotron frequency, allowing the ions to be treated as unmagnetized. Third, the collision frequency between electrons and background neutrals is assumed to be much less than the electron cyclotron frequency. Finally, we only analyze low frequency waves so that we can neglect electron inertia. Because the Buneman instability is electrostatic in nature, only electrostatic perturbations are considered.

Dispersion relation

We use linearized fluid equations for electrons and ions with Lorentz force and collisional terms. The equation of motion for each species is:
where
Note that electron inertia has been neglected, and that both species are assumed to have the same number density at every point in space.The collisional term describes the momentum loss frequency of each fluid due to collisions of charged particles with neutral particles in the plasma. We denote as the frequency of collisions between electrons and neutrals, and as the frequency of collisions between ions and neutrals. We also assume that all perturbed properties, such as species velocity, density, and the electric field, behave as plane waves. In other words, all physical quantities will behave as an exponential function of time and position :
This can lead to oscillations if the frequency is a real number, or to either exponential growth or exponential decay if is complex. If we assume that the ambient electric and magnetic fields are perpendicular to one another and only analyze waves propagating perpendicular to both of these fields, the dispersion relation takes the form of:
where is the drift and is the acoustic speed of ions. The coefficient described the combined effect of electron and ion collisions as well as their cyclotron frequencies and :

Growth rate

Solving the dispersion we arrive at frequency given as:
where describes the growth rate of the instability. For FB we have the following:

Buneman instability

The dispersion relation is
and the growth rate is