- Two stream instability
The

**two-stream instability**is a very commoninstability inplasma physics . It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species (ions andelectrons ) can have different drift velocities. The energy from the particles can lead toplasma wave excitation.**Dispersion relation**Consider a cold, uniform, and unmagnetized plasma, where ions are stationary and the electrons have velocity $mathbf\{V\}\_0$, that is, the reference frame is moving with the ion stream. Let the electrostatic waves be of the form:

$mathbf\{E\}\_1\; =\; xi\_1\; exp\; [i(kx\; -\; omega\; t)]\; mathbf\{hat\{x$

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 $partial\_t\; ightarrow\; -iomega$, $abla\; ightarrow\; ik$ we can get the following expression:

$1\; =\; omega\_\{pe\}^2\; left\; [frac\{m\_e/m\_i\}\{omega^2\}\; +\; frac\{1\}\{(omega\; -\; kv\_0)^2\}\; ight]$

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

$omega\_j\; =\; omega\_j^R\; +\; igamma\_j$

If the imaginary part ($Im(omega\_j)$) is zero, then the solutions represent all the possible modes, and there is no temporal wave growth or damping at all:

$mathbf\{E\}\; =\; xi\; exp\; [i(kx\; -\; omega\; t)]\; mathbf\{hat\{x$

If $Im(omega\_j)\; e\; 0$, that is, any of the roots are complex, they will occur in complex conjugate pairs. Substituting in the expression for electrostatic waves leads to:

$mathbf\{E\}\; =\; xi\; exp\; [i(kx\; -\; omega\_j^R\; t)]\; exp\; [gamma\; t]\; mathbf\{hat\{x$

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

**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 $v\_\{ph\}$ 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 particle's 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 ($v\; >\; v\_\{ph\}$) 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.**Bibliography***Bittencourt, J. A. "Fundamentals of Plasma Physics", Third Ed. 2004 Springer-Verlag, New York.

*Chen, Francis F. "Introduction to Plasma Physics and Controlled Fusion". Second Ed., 1984 Plenum Press, New York.

*Nicholson, D. R. "Introduction to Plasma Theory". 1983 John Wiley & Sons, New York.

*Tsurutani, B., and Lakhina, G. "Some basic concepts of wave-particle interactions in collisionless plasmas". Reviews of Geophysics 35(4), p.491-502

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Two-stream instability**— The two stream instability is a very common instability in plasma physics. It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species (ions and electrons) can have different… … Wikipedia**Instability**— [ Hydrodynamics simulation of the Rayleigh–Taylor instability [Shengtai Li, Hui Li Parallel AMR Code for Compressible MHD or HD Equations (Los Alamos National Laboratory) [http://math.lanl.gov/Research/Highlights/amrmhd.shtml] ] ] Instability in… … Wikipedia**Rayleigh–Taylor instability**— The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities, which occurs when the lighter fluid is pushing the heavier fluid. [citation… … Wikipedia**Weibel instability**— The Weibel instability is a plasma instability present in homogeneous or nearly homogeneous electromagnetic plasmas which possess an anisotropy in momentum (velocity) space. In the linear limit the instability causes exponential growth of… … Wikipedia**двухпотоковая неустойчивость в ионосфере**— Неустойчивость, генерируемая в плазме при наличии в ней двух потоков с разными скоростями. [ГОСТ 25645.113 84] Тематики ионосфера земли Обобщающие термины волны и неустойчивости ионосферной плазмы EN two stream instability FR instabilité biflux … Справочник технического переводчика**National Party (South Africa)**— National Party of South Africa Nasionale Party van Suid Afrika The National Party s final logo The National Party s Flag … Wikipedia**Jicamarca Radio Observatory**— Infobox Telescope bgcolour = name = Jicamarca Radio Observatory caption = organization = [http://geo.igp.gob.pe Instituto Geofisico del Peru] ,Cornell University, National Science Foundation location = coord|11|57|05|S|76|52|27.5|W| wavelength =… … Wikipedia**climate**— /kluy mit/, n. 1. the composite or generally prevailing weather conditions of a region, as temperature, air pressure, humidity, precipitation, sunshine, cloudiness, and winds, throughout the year, averaged over a series of years. 2. a region or… … Universalium**Economic Affairs**— ▪ 2006 Introduction In 2005 rising U.S. deficits, tight monetary policies, and higher oil prices triggered by hurricane damage in the Gulf of Mexico were moderating influences on the world economy and on U.S. stock markets, but some other… … Universalium**china**— /chuy neuh/, n. 1. a translucent ceramic material, biscuit fired at a high temperature, its glaze fired at a low temperature. 2. any porcelain ware. 3. plates, cups, saucers, etc., collectively. 4. figurines made of porcelain or ceramic material … Universalium