Farley–Buneman instability

The Farley–Buneman instability, or FB instability, is a microscopic plasma instability named after Donald T. Farley[1] and Oscar Buneman.[2] 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. In particular, it occurs in the equatorial electrojet due to the drift of electrons relative to ions,[3] and also in the trails behind ablating meteoroids.[4]

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.[3] Because the Buneman instability is electrostatic in nature, only electrostatic perturbations are considered.

Dispersion relation

We use linearized fluid equations (equation of motion, equation of continuity) for electrons and ions with Lorentz force and collisional terms. The equation of motion for each species is:

Electrons: ${\displaystyle 0=-en({\vec {E}}+{\vec {v}}_{e}\times {\vec {B}})-k_{b}T_{e}\nabla n-m_{e}n\nu _{en}{\vec {v}}_{e}}$

Ions: ${\displaystyle m_{i}n{dv_{i} \over dt}=en({\vec {E}}+{\vec {v}}_{i}\times {\vec {B}})-k_{b}T_{i}\nabla n-m_{i}n\nu _{in}{\vec {v}}_{i}}$

where

• ${\displaystyle m_{s}}$ is the mass of species ${\displaystyle s}$
• ${\displaystyle v_{s}}$ is the velocity of species ${\displaystyle s}$
• ${\displaystyle T_{s}}$ is the temperature of species ${\displaystyle s}$
• ${\displaystyle \nu _{sn}}$ is the frequency of collisions between species s and neutral particles
• ${\displaystyle e}$ is the charge of an electron
• ${\displaystyle n}$ is the electron number density
• ${\displaystyle k_{b}}$ is the Boltzmann Constant

Note that electron inertia has been neglected, and that both species are assumed to have the same number density at every point in space (${\displaystyle n_{i}=n_{e}=n}$).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 ${\displaystyle \nu _{en}}$ as the frequency of collisions between electrons and neutrals, and ${\displaystyle \nu _{in}}$ 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 ${\displaystyle f}$ will behave as an exponential function of time ${\displaystyle t}$ and position ${\displaystyle x}$ (where ${\displaystyle k}$ is the wave number) :

${\displaystyle f\sim \exp(-i\omega t+ikx)}$.

This can lead to oscillations if the frequency ${\displaystyle \omega }$ is a real number, or to either exponential growth or exponential decay if ${\displaystyle \omega }$ 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:

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

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

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

Growth rate

Solving the dispersion we arrive at frequency given as:

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

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

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

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

See also

• Plasma stability
• Plasma Instabilities
• List of plasma (physics) articles

References

1. Farley, D. T. (1963). "Two-Stream Plasma Instability as a Source of Irregularities in the Ionosphere". Physical Review Letters. 10 (7): 279–282. doi:10.1103/PhysRevLett.10.279.
2. Buneman, O. (1963). "Excitation of Field Aligned Sound Waves by Electron Streams". Physical Review Letters. 10 (7): 285–287. doi:10.1103/PhysRevLett.10.285.
3. Treumann, Rudolf A; Baumjohann, Wolfgang (1997). Advanced space plasma physics. World Scientific. ISBN 978-1-86094-026-2.
4. Oppenheim, Meers M.; Endt, Axel F. vom; Dyrud, Lars P. (October 2000). "Electrodynamics of meteor trail evolution in the equatorial E-region ionosphere". Geophysical Research Letters. 27 (19): 3173. doi:10.1029/1999GL000013.