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The Farley–Buneman instability, or FB instability, is a
microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale betwe ...
plasma instability In plasma physics, plasma stability concerns the stability properties of a plasma in equilibrium and its behavior under small perturbations. The stability of the system determines if the perturbations will grow, oscillate, or be damped out. It ...
named after Donald T. Farley and Oscar Buneman. It is similar to the
ionospheric The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays ...
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
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s and
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
s 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
equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...
ial and polar
ionospheric The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays ...
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 In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ran ...
, the
instability In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior. ...
can be used to diagnose the state of
ionosphere The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays ...
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 (
equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
, equation of continuity) for
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s and
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
s with
Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
and collisional terms. The equation of motion for each species is: : Electrons: 0 = -e n \left(\mathbf + \mathbf_e \times \mathbf\right) - k_\text T_e \nabla n - m_e n \nu_ \mathbf_e : Ions: m_i n \frac = e n \left(\mathbf + \mathbf_i \times \mathbf\right) - k_\text T_i \nabla n - m_i n \nu_ \mathbf_i where * m_s is the mass of species s * v_s is the velocity of species s * T_s is the temperature of species s * \nu_ is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
of collisions between species s and neutral particles * e is the charge of an electron * n is the electron number density * k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
Note that electron inertia has been neglected, and that both species are assumed to have the same number density at every point in space (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 \nu_ as the frequency of collisions between electrons and neutrals, and \nu_ 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 f will behave as an exponential function of time t and position x (where k is the
wave number In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of r ...
): f \sim \exp. This can lead to
oscillations Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
if the
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
\omega is a
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, or to either
exponential growth Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
or
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppe ...
if \omega is
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
. 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: \omega\left( 1 + i \psi_0 \frac\right) = k v_E + i \psi_0 \frac , 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.


Growth rate

Solving the dispersion we arrive at frequency 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 \gamma =\frac \frac.


Buneman instability

The dispersion relation is 1 - \frac - \frac = 0 and the growth rate is \gamma = \sqrt \omega_p ^


See also

* Plasma stability *
Plasma Instabilities In plasma physics, plasma stability concerns the stability properties of a plasma in equilibrium and its behavior under small perturbations. The stability of the system determines if the perturbations will grow, oscillate, or be damped out. It ...


References

{{DEFAULTSORT:Farley-Buneman instability Plasma instabilities