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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 is present in the equatorial and polar ionospheric E-regions. Since the FB fluctuations can scatter the electromagnetic waves, the instability can be used to diagnose the state of ionosphere by the use of electromagnetic pulses.

Dispersion relation

We use linearized fluid equations (equation of motion, equation of continuity) for electrons and ions with Lorenz force and collisional term. This additional term describes the collisions of charged particles with neutral particles in the plasma by frequency of collisions. We denote $\nu _{en}$ as the frequency of collisions between electrons and neutrals, and $\nu _{in}$ as the frequency of collisions between ions and neutrals. We also assume exponential type of behaviour, meaning that all physical quantities $f$ will behave as an exponential function of time $t$ and position $x$ (where $k$ is the wave number) :

$f\sim \exp(i\omega t+ikx)$.

This can lead to oscillations if the frequency $\omega$ is a real number, or to either exponential growth or exponential decay if $\omega$ is complex. Dispersion relation takes the form of:

$\omega \left(1+i\psi _{0}{\frac {\omega -i\nu _{in}}{\nu _{in}}}\right)=kv_{E}+i\psi _{0}{\frac {k^{2}c_{i}^{2}}{\nu _{in}}}$,

where $v_{E}$ is the $E\times B$ drift and $c_{i}$ is the acoustic speed of ions. The coefficient $\psi _{0}$ described the combined effect of electron and ion collisions as well as their cyclotron frequencies $\Omega _{i}$ and $\Omega _{e}$:

$\psi _{0}={\frac {\nu _{in}\nu _{en}}{\Omega _{i}\Omega _{e}}}$

Growth rate

Solving the dispersion we arrive at freuqency given as:

$\omega =\omega _{r}+i\gamma$,

where $\gamma$ describes the growth rate of the instability. For FB we have the following:

$\omega _{r}={\frac {kv_{E}}{1+\psi _{0}}}$

$\gamma ={\frac {\psi _{0}}{\nu _{in}}}{\frac {w_{r}^{2}-k^{2}c_{i}^{2}}{1+\psi _{0}}}$.

References

• Treumann, Rudolf A; Baumjohann, Wolfgang (1997). Advanced space plasma physics. World Scientific. ISBN 1-86094-026-9.