View source for Electrothermal instability

{{Distinguish|text = the [[magnetoacoustic]] instability discovered by the same author}}
{{Use dmy dates|date=April 2021}}
__NOTOC__
The '''electrothermal instability''' (also known as '''[[ionization instability]]''', '''non-equilibrium instability''' or '''Velikhov instability''' in the literature) is a [[magnetohydrodynamics|magnetohydrodynamic]] (MHD) [[instability]] appearing in [[Plasma (physics)#Magnetization|magnetized]] [[Plasma (physics)#Thermal vs. nonthermal (cold) plasmas|non-thermal]] [[Plasma (physics)|plasmas]] used in [[MHD converter]]s. It was first theoretically discovered in 1962 and experimentally measured into a [[MHD generator]] in 1963 by [[Evgeny Velikhov]].<ref name="Velikhov 1962">{{cite conference
 | last1=Velikhov |first1=E. P.
 | year = 1962
 | title = Hall instability of current carrying slightly ionized plasmas
 | conference = 1st International Conference on MHD Electrical Power Generation
 | location = Newcastle upon Tyne, England
 | article=  Paper 47
 | page= 135
}}</ref><ref name="Velikhov 1963">{{cite conference
 | last1=Velikhov |first1=E. P.
 | last2=Dykhne |first2=A. M.
 | date = 8–13 July 1963
 | title = Plasma turbulence due to the ionization instability in a strong magnetic field
 | conference = 6th International Conference on Phenomena in Ionized Gases
 | book-title = Proceedings
 | location = Paris, France
 | volume = 4
 | page = 511
 | bibcode = 1963pig4.conf..511V
}}</ref><ref name="Velikhov 1965">{{cite conference
 | last1=Velikhov |first1=E. P.
 | last2=Dykhne |first2=A. M.
 | last3=Shipuk |first3=I. Ya
 | year = 1965
 | title = Ionization instability of a plasma with hot electrons
 | conference = 7th International Conference on Ionization Phenomena in Gases
 | location = Belgrade, Yugoslavia
 | url = https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670016557.pdf
}}</ref>

{{blockquote
 |"This paper shows that it is possible to assert sufficiently specifically that the ionization instability is the number one problem for the utilization of a plasma with hot electrons."|author=Dr. Evgeny Velikov|source=at the 7th International Conference on Ionization Phenomena in Gases, Belgrade, Yugoslavia (1965).<ref name="Velikhov 1965" />
}}

== Physical explanation and characteristics ==
[[File:electrothermal instability evolution.gif|thumb|490px|Evolution of the electrothermal instability in a Faraday MHD converter. Electric current lines.]]
This instability is a [[turbulence]] of the [[Free electron model|electron gas]] in a [[Thermal equilibrium|non-equilibrium]] plasma (i.e. where the [[electron temperature]] T<sub>e</sub> is greatly higher than the overall gas temperature T<sub>g</sub>). It arises when a [[magnetic field]] powerful enough is applied in such a plasma, reaching a critical Hall parameter β<sub>cr</sub>.

Locally, the number of [[electron]]s and their temperature fluctuate ([[electron density]] and [[thermal velocity]]) as the [[electric current]] and the [[electric field]].

The Velikhov instability is a kind of ionization wave system, almost frozen in the two temperature gas. The reader can evidence such a [[Standing wave|stationary wave]] phenomenon just applying a transverse magnetic field with a [[magnet|permanent magnet]] on the low-pressure control gauge ([[Geissler tube]]) provided on vacuum pumps. In this little [[Gas-discharge lamp|gas-discharge bulb]] a high [[voltage]] [[electric potential]] is applied between two [[electrode]]s which generates an [[electric glow discharge]] (pinkish for air) when the pressure has become low enough. When the transverse magnetic field is applied on the bulb, some oblique grooves appear in the plasma, typical of the electrothermal instability.

The electrothermal instability occurs extremely quickly, in a few microseconds. The plasma becomes non-homogeneous, transformed into alternating layers of high free electron and poor free electron densities. Visually the plasma appears stratified, as a "pile of plates".

=== Hall effect in plasmas ===
{{Unreferenced section|date=April 2021}}
The [[Hall effect]] in ionized gases has nothing to do with the Hall effect in solids (where the '''Hall parameter''' is always very inferior to unity). In a plasma, the Hall parameter can take any value.

The Hall parameter β in a plasma is the ratio between the electron [[Gyroradius|gyrofrequency]] Ω<sub>e</sub> and the electron-heavy particles collision frequency ν:

:<math>
\beta \, = \, \frac {\Omega_e}{\nu} \, = \, \frac {e\ B}{m_e\ \nu}
</math>

where
: ''e'' is the [[Elementary charge|electron charge]] (1.6 × 10<sup>−19</sup> [[coulomb]])
: ''B'' is the magnetic field (in [[tesla (unit)|teslas]])
: m<sub>e</sub> is the electron mass (0.9 × 10<sup>−30</sup> kg)

The Hall parameter value increases with the magnetic field strength.

Physically, when the Hall parameter is low, the trajectories of electrons between two encounters with heavy particles (neutral or ion) are almost linear. But if the Hall parameter is high, the electron movements are highly curved. The [[current density]] vector '''J''' is no more colinear with the [[electric field]] vector '''E'''. The two vectors '''J''' and '''E''' make the '''Hall angle''' θ which also gives the Hall parameter:

:<math>
\ \beta \, = \, \tan \theta
</math>

=== Plasma conductivity and magnetic fields ===
{{Unreferenced section|date=April 2021}}
In a non-equilibrium ionized gas with high Hall parameter, [[Ohm's law]],

:<math>
\mathbf{J} = \sigma\mathbf{E}
</math>
where ''σ'' is the [[electrical conductivity]] (in [[siemens (unit)|siemens]] per metre),

is a [[Matrix (mathematics)|matrix]], because the electrical conductivity σ is a matrix{{dubious|reason=What dimension, 2x2, 3x3 ?|date=April 2021}}:

:<math>
\sigma = \sigma_s \begin{Vmatrix} \dfrac{1}{1+\beta^2} & \dfrac{-\beta}{1+\beta^2} \\ \dfrac{\beta}{1+\beta^2} & \dfrac{1}{1+\beta^2} \end{Vmatrix}
</math>

σ<sub>S</sub> is the scalar electrical conductivity:

:<math>
\sigma_s = \frac {n_e\ e^2}{m_e\ \nu}
</math>

where n<sub>e</sub> is the electron density (number of electrons per cubic meter).

The current density '''J''' has two components:

:<math>
J_{\parallel} = \frac {n_e\ e^2}{m_e\ \nu} \ \frac {E}{1+\beta^2} \qquad \text{and} \qquad J_{\perp} = \frac {-n_e\ e^2}{m_e\ \nu} \ \frac {\beta\ E}{1+\beta^2}
</math>

Therefore,

:<math>
J_{\perp} = J_{\parallel}\ \beta
</math>

The Hall effect makes electrons "crabwalk".

When the magnetic field B is high, the Hall parameter β is also high, and <math>\frac {1}{1+\beta^2} \ll 1</math>

Thus both conductivities

<math>
\sigma_{\parallel} \approx \frac {\sigma_s}{\beta^2} \qquad \text{and} \qquad \sigma_{\perp} \approx \frac {\sigma_s}{\beta}
</math>

become weak, therefore the electric current cannot flow in these areas. This explains why the electron current density is weak where the magnetic field is the strongest.

=== Critical Hall parameter ===
{{Unreferenced section|date=April 2021}}
The electrothermal instability occurs in a plasma at a (T<sub>e</sub> > T<sub>g</sub>) regime when the Hall parameter is higher than a critical value β<sub>cr</sub>.

We have

:<math>
f = \frac{\left( \frac{\delta \mu}{\mu} \right)}{\left( \frac{\delta n_e}{n_e} \right)}
</math>

where μ is the [[electron mobility]] (in m<sup>2</sup>/([[volt|V]]·[[second|s]]))

and

:<math>
s = \frac{2\ k\ T_e^2}{E_i\; (T_e - T_g)} \times \frac {1}{1 + \dfrac{3}{2}\ \dfrac{k\; T_e}{E_i}}
</math>

where ''E<sub>i</sub>'' is the [[Ionization potential|ionization energy]] (in [[electron volt]]s) and ''k'' the [[Boltzmann constant]].

The '''growth rate of the instability''' is

:<math>
g = \frac{\sigma\ E^2}{n_e\; \left( E_i + \frac{3}{2} k\; T_e \right)\; \left( 1 + \beta^2 \right)}\; (\beta - \beta_{cr})
</math>

And the '''critical Hall parameter''' is

:<math>
\beta_{cr} = 1.935 f + 0.065 + s
~</math>

The critical Hall parameter β<sub>cr</sub> greatly varies according to the [[degree of ionization]] α :

:<math>
\alpha = \frac{n_i}{n_n}
</math>

where n<sub>i</sub> is the ion density and n<sub>n</sub> the neutral density (in particles per cubic metre).

The electron-ion collision frequency ν<sub>ei</sub> is much greater than the electron-neutral collision frequency ν<sub>en</sub>.

Therefore, with a weak energy degree of ionization α, the electron-ion collision frequency ν<sub>ei</sub> can equal the electron-neutral collision frequency ν<sub>en</sub>.

* For a '''weakly ionized gas''' (non-Coulombian plasma, when ν<sub>ei</sub> < ν<sub>en</sub> ):
:<math>
\beta_{cr} \approx (s^2 + 2s)^{\frac{1}{2}}
</math>

* For a '''fully ionized gas''' (Coulombian plasma, when ν<sub>ei</sub> > ν<sub>en</sub> ):
:<math>
\beta_{cr} \approx (2 + s)
</math>

NB: The term "fully ionized gas", introduced by [[Lyman Spitzer]], does not mean the degree of ionization is unity, but only that the plasma is Coulomb-collision dominated, which can correspond to a degree of ionization as low as 0.01%.

== Technical problems and solutions ==
{{Primary sources|section|date=May 2021}}
A two-temperature gas, globally cool but with hot electrons (T<sub>e</sub> >> T<sub>g</sub>) is a key feature for practical MHD converters, because it allows the gas to reach sufficient [[electrical conductivity]] while protecting materials from [[heat|thermal]] [[ablation]]. This idea was first introduced for MHD generators in the early 1960s by [[Jack L. Kerrebrock]]<ref>{{Cite journal
 | last1 = Kerrebrock |first1 = J. L.
 | date = 1 November 1960
 | title = Non-equilibrium effects on conductivity and electrode heat transfer in ionized gases
 | journal=Technical Note No.&nbsp;4
 |osti=4843920
}}</ref><ref>{{cite journal
 |last1   = Kerrebrock
 |first1  = J. L.
 |date    = June 1964
 |title   = Nonequilibrium ionization due to electron heating: I. Theory
 |journal = AIAA Journal
 |volume  = 2
 |issue   = 6
 |pages   = 1072–1080
 |url     = http://ayuba.fr/pdf/kerrebrock1974a.pdf
 |doi     = 10.2514/3.2496
 |bibcode = 1964AIAAJ...2.1072K
}}{{dead link|date=August 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{cite journal
 |last1   = Kerrebrock
 |first1  = J. L.
 |last2   = Hoffman
 |first2  = M. A.
 |date    = June 1964
 |title   = Nonequilibrium ionization due to electron heating: II. Experiments
 |journal = AIAA Journal
 |volume  = 2
 |issue   = 6
 |pages   = 1080–1087
 |url     = http://ayuba.fr/pdf/kerrebrock1974b.pdf
 |doi     = 10.2514/3.2497
 |bibcode = 1964AIAAJ...2.1080H
}}{{dead link|date=August 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> and [[Alexander E. Sheindlin]].<ref>{{cite conference
 | last1 = Sheindlin |first1 = A. E.
 | last2 = Batenin |first2 = V. A.
 | last3 = Asinovsky |first3 = E. I.
 | date = 6 July 1964
 | title = Investigation of non-equilibrium ionization in a mixture of argon and potassium
 | conference = International symposium on magnetohydrodynamic electric power generation
 | book-title = CONF-640701-102
 | location =  Paris, France
|osti=5024025
}}</ref>

But the unexpected large and quick drop of [[current density]] due to electrothermal instability ruined many MHD projects worldwide, while previous calculations had envisaged [[Energy conversion efficiency|energy conversion efficiencies]] over 60% with these devices. Although some studies were made into the instability by various researchers,<ref>{{cite conference
 | last1 = Solbes | first1 = A.
 | date = 24–30 July 1968
 | title = A quasi linear plane wave study of electrothermal instabilities
 | conference=Electricity from MHD: Proceedings of a Symposium on magnetohydrodynamic electrical power generation
 | book-title = SM/107/26
 | location = Warsaw, Poland
 | volume = I
 | publisher = [[International Atomic Energy Agency]]
}}</ref><ref>{{cite conference
 | last1 = Nelson | first1 = A. H.
 | last2 = Haines | first2 = M. G.
 | date = 26–28 March 1969
 | title = Analysis of the nature and growth of electrothermal waves
 | conference = 10th Symposium in Engineering Aspects of MHD
 | book-title = Proceedings
 | location =  MIT, Cambridge, MA, USA
 | url = http://ayuba.fr/pdf/nelson1969.pdf
 | doi = 10.1088/0032-1028/11/10/003
| bibcode = 1969PlPh...11..811N}}</ref> no real solution was found at that time. This prevented further developments of non-equilibrium MHD generators and caused most engaged countries to cancel their MHD [[Power station|power plant]] programs and to retire completely from this field of research in the early 1970s, because this technical problem was then considered to be an impassable stumbling block.

Nevertheless, experimental studies about the growth rate of the electrothermal instability and the critical conditions showed that a stability region still exists for high electron temperatures.<ref name="Petit 1968">{{cite conference
 |last1=Petit |first1=J.-P.
 |last2=Caressa |first2=J.-P.
 |last3=Valensi |first3=J.
 |date=24–30 July 1968
 |title=Etude théorique et expérimentale, en tube à choc, des phénomènes accompagnant la mise hors d'équilibre dans un générateur MHD en cycle fermé
 |trans-title=Theoretical and experimental study, using a shock tube, of the phenomena accompanying equilibrium breakdown in a closed-cycle MHD generator
 |language=fr
 |conference=Electricity from MHD: Proceedings of a Symposium on magnetohydrodynamic electrical power generation
 |location=Warsaw, Poland
 |volume=II
 |publisher=International Atomic Energy Agency
 |pages=745–750
 |url=http://ayuba.fr/pdf/petit1968.pdf
}}</ref> The stability is gained by a '''quick transition to "fully ionized" conditions''' (fast enough to overtake the growth rate of the electrothermal instability) where the Hall parameter decreases because of the collision frequency rising, below its critical value which is then about 2. Stable operation with several megawatts of power output had been experimentally achieved as from 1967 with high electron temperature.<ref name="Petit 1969a">{{cite journal
 | last1= Petit |first1 = J.-P.
 | last2= Valensi | first2= J.
 | last3= Dufresne | first3= D.
 | last4= Caressa | first4= J.-P.
 | date = 27 January 1969
 | title = Caractéristiques électriques d'un générateur linéaire de Faraday utilisant un mélange binaire de gaz rares, avec ionisation hors d'équilibre
 | trans-title = Electrical characteristics of a linear Faraday generator using a binary mix of rare gases, with non-equilibrium ionization
 | language = fr
 | journal = [[Comptes rendus de l'Académie des sciences]]
 | series = Série A
 | issue = 268
 | pages = 245–247
 | url = http://ayuba.fr/pdf/cras_mhd_1969a.pdf
}}</ref><ref name="Petit 1969b">{{cite journal
 |last1=Petit |first1=J.-P.
 |date=14 April 1969
 |title=Performances théoriques d'un générateur du type de Faraday avec ionisation hors d'équilibre dans le gaz de conversion
 |trans-title=Theoretical performances of a Faraday-type generator with non-equilibrium ionization in the conversion gas
 |language=fr
 |volume=268
 |series=Série A
 |pages=835–838
 |url=http://ayuba.fr/pdf/cras_mhd_1969b.pdf
}}</ref><ref name="Petit 1969c">{{cite journal
 |last1=Petit |first1=J.-P.
 |date=21 April 1969
 |title=Instabilité de régime dans un générateur de Hall, avec ionisation hors d'équilibre
 |trans-title=Rate instability in a Hall generator with non-equilibrium ionization
 |language=fr
 |volume=268
 |series=Série A
 |pages=906–909
 |url=http://ayuba.fr/pdf/cras_mhd_1969c.pdf
}}</ref><ref name="Petit 1969d">{{cite journal
 | last1= Petit |first1 = J.-P.
 | last2= Valensi | first2= J.
 | date = 1 September 1969
 | title = Taux de croissance de l'instabilité électrothermique et paramètre de Hall critique dans les générateurs linéaires à cycle fermé lorsque la mobilité électronique est variable
 | trans-title = Growth rate of electrothermal instability and critical Hall parameter in closed-cycle MHD generators when the electron mobility is variable
 | language = fr
 | journal = Comptes Rendus de l'Académie des Sciences
 | series = Série A
 | issue = 269
 | pages = 365–367
 | url = http://ayuba.fr/pdf/cras_mhd_1969d.pdf
}}</ref><ref>{{cite journal
 | last1=Hatori | first1=S.
 | last2=Shioda | first2=S.
 |date=March 1974
 | title = Stabilization of Ionization Instability in an MHD Generator
 | journal=Journal of the Physical Society of Japan
 | volume = 36
 | issue = 3
 | page = 920
 | url = http://ayuba.fr/pdf/hatori1974.pdf
 | doi = 10.1143/JPSJ.36.920
 | bibcode = 1974JPSJ...36..920H }}</ref> But this electrothermal control cannot provide an adequate decrease of T<sub>g</sub> over long durations (to avoid thermal ablation), so such a solution is not practical for industrial energy conversion.

Another idea to control the instability is to increase the non-thermal ionisation rate by using a [[laser]] which would act like a guidance system for streamers between electrodes, increasing the electron density and the conductivity, therefore lowering the Hall parameter to below its critical value along these paths. But this concept has never been tested experimentally.<ref name="petit_thesis">{{cite thesis
 | last1 = Petit | first1 = J.-P.
 | type = Doctor of Science thesis
 | date = 10 March 1972
 | title = Applications de la théorie cinétique des gaz à la physique des plasmas et à la dynamique des galaxies
 | trans-title = Applications of the kinetic theory of gases to plasma physics and galactic dynamics
 | chapter = 5: Magnetohydrodynamique
 | language = fr
 | publisher = [[University of Provence]]
 | id = CNRS#6717
 | chapter-url = http://ayuba.fr/pdf/petit_thesis.pdf
 | pages = 172–195
}}</ref>

In the 1970s and more recently, some researchers tried to master the instability with '''oscillating fields'''. Oscillations of the electric field or of an additional RF electromagnetic field locally modify the Hall parameter.<ref>{{cite journal
 | last1 = Shapiro |first1 = G. I.
 | last2 = Nelson |first2 = A. H.
 | date = 12 April 1978
 | title = Stabilization of ionization instability in a variable electric field
 | journal=Pis'ma V Zhurnal Tekhnischeskoi Fiziki
 | volume = 4
 | issue = 12
 | pages = 393–396
|bibcode = 1978PZhTF...4..393S
 }}</ref><ref>{{cite journal
 | last1 = Murakami |first1 = T.
 | last2 = Okuno |first2 = Y.
 | last3 = Yamasaki | first3 = H.
 | date = December 2005
 | title = Suppression of ionization instability in a magnetohydrodynamic plasma by coupling with a radio-frequency electromagnetic field
 | journal=Applied Physics Letters
 | volume = 86
 | issue = 19
 | pages = 191502–191502.3
 | url = http://ayuba.fr/pdf/murakami2005.pdf
 | doi = 10.1063/1.1926410
| bibcode = 2005ApPhL..86s1502M }}</ref>

Finally, a solution has been found in the early 1980s to completely remove the electrothermal instability within MHD converters, by means of '''non-homogeneous magnetic fields'''. A strong magnetic field implies a high Hall parameter, and therefore a low electrical conductivity in the medium. So the idea is to create some "paths" linking one electrode to the other, ''where the magnetic field is locally attenuated''. Then the electric current tends to flow in these low B-field paths as thin plasma cords or ''streamers'', where the electron density and temperature increase. The plasma becomes locally Coulombian, and the local Hall parameter value falls, while its critical threshold rises. Experiments where streamers do not present any inhomogeneity have been obtained with this method.<ref>{{cite journal
 | last1= Petit |first1= J.-P.
 | last2= Billiotte |first2= M.
 | date = 4 May 1981
 | title = Méthode pour supprimer l'instabilité de Velikhov
 | trans-title = Method for suppressing the Velikhov instability
 | language = fr
 | journal = Comptes Rendus de l'Académie des Sciences
 | series = Série II
 | volume = 292
 | pages = 1115–1118
 | url = http://ayuba.fr/pdf/cras_mhd_1981.pdf
}}</ref><ref name="Vilnius 2009">{{cite journal
 | last1= Petit | first1= J.-P.
 | last2= Geffray | first2= J.
 | date=June 2009
 | title = Non equilibrium plasma instabilities
 | journal= Acta Physica Polonica A
 | publisher=Institute of Physics of the Polish Academy of Sciences
 | volume = 115
 | issue = 6
 | pages = 1170–1173
 | doi= 10.12693/APhysPolA.115.1170
 | bibcode= 2009AcPPA.115.1170P
 | url = http://przyrbwn.icm.edu.pl/APP/PDF/115/a115z675.pdf
}}</ref><ref name="Prague 2012">{{Cite journal
 |last1=Petit |first1=J.-P.
 |last2=Doré |first2=J.-C.
 |title=Velikhov electrothermal instability cancellation by a modification of electrical conductivity value in a streamer by magnetic confinement
 |year=2013
 |journal=Acta Polytechnica
 |volume=53
 |issue=2
 |pages=219–222
 |doi=10.14311/1765
 |url=https://ojs.cvut.cz/ojs/index.php/ap/article/view/1765/1597
|doi-access=free
 |hdl=10467/67041
 |hdl-access=free
 }}</ref> This effect, strongly [[Nonlinear system|nonlinear]], was unexpected but led to a very effective system for streamer guidance.

But this last working solution was discovered too late, 10 years after all the international effort about MHD power generation had been abandoned in most nations. [[Vladimir S. Golubev]], coworker of Evgeny Velikhov, who met Jean-Pierre Petit in 1983 at the 9th MHD International conference in Moscow, made the following comment{{citation needed|date=April 2021}} to the inventor of the magnetic stabilization method:

{{center|
{{cquote
 |''You bring the cure, but the patient already died...''
}}
}}

However, this electrothermal stabilization by magnetic confinement, although found too late for the development of MHD power plants, might be of interest for future applications of MHD to aerodynamics (magnetoplasma-aerodynamics for [[hypersonic flight]]).<ref name="Bremen 2009">{{Cite conference
 |last1=Petit |first1=J.-P.
 |last2=Geffray |first2=J.
 |last3=David |first3=F.
 |title=MHD Hypersonic Flow Control for Aerospace Applications
 |date=October 2009
 |conference=16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference (HyTASP)
 |location=Bremen, Germany
 |publisher=[[American Institute of Aeronautics and Astronautics]]
 |url=https://www.researchgate.net/publication/270820169
 |doi=10.2514/6.2009-7348
}}</ref>

== See also ==

* [[Magnetohydrodynamics]]
* [[MHD generator]]
* [[Evgeny Velikhov]]

== External links ==
* M. Mitchner, C.H. Kruger Jr., [https://web.archive.org/web/20070609185654/http://navier.stanford.edu/PIG/C4_S10.pdf Two-temperature ionization instability]: Chapter 4 (MHD) – Section 10, pp.&nbsp;230–241. From the plasma physics course book [https://web.archive.org/web/20080905113821/http://navier.stanford.edu/PIG/PIGdefault.html Partially Ionized Gases], [[John Wiley & Sons]], 1973 (reprint 1992), Mechanical Engineering Department, [[Stanford University]], CA, USA. {{ISBN|0-471-61172-7}}

== References ==

{{reflist}}

[[Category:Plasma instabilities]]