- Electrothermal instability
__NOTOC__The

**electrothermal instability**(also known as the**ionization instability**or**Velikhov instability**in the literature) is a magnetohydrodynamic (MHD)instability appearing in magnetized non-thermal plasmas used in MHD converters. It was first theoretically discovered in 1962 and experimentally measured into aMHD generator in 1964 byEvgeny Velikhov . [*cite conference*] [

author = E.P. Velikhov

year = 1962

title = Hall instability of current-carrying slightly-ionized plasmas

conference = 1^{st}International Conference on MHD Electrical Power Generation, Paper 47

booktitle = Newcastle-upon-Tyne, England*cite conference*] [

author = E.P. Velikhov

coauthors = A.M. Dykhne

date = 13-18 July 1963

title = Plasma turbulence due to the ionization instability in a strong magnetic field

conference = 6^{th}International Conference on Ionization Phenomena in Gases

booktitle = Proceedings

editor = Paris, France

volume = 4*cite conference*]

author = E.P. Velikhov

coauthors = A.M. Dykhne, I.Ya Shipuk

year = 1965

title = Ionization instability of a plasma with hot electrons

conference = 7^{th}International Conference on Ionization Phenomena in Gases

booktitle = Belgrade, Yugoslavia**Physical explanation and characteristics**This instability is a

turbulence of the electron gas in a non-equilibrium plasma (i.e. where theelectron temperature T_{e}is greatly higher than the overall gas temperature T_{g}). It arises when amagnetic field powerful enough is applied in such a plasma, reaching a critical Hall parameter β_{cr}.Locally, the number of electrons and their temperature fluctuate (

electron density andthermal velocity ) as theelectric current and theelectric field .The Velikhov instability is a kind of ionization wave system, almost frozen in the two temperature gas. The reader can evidence such a stationary wave phenomenon just applying a transverse magnetic field with a permanent magnet on the low-pressure control gauge (

Geissler tube ) provided on vacuum pumps. In this little gas-discharge bulb a highvoltage electric potential is applied between twoelectrode s which generates anelectric 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**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 gyrofrequency Ω

_{e}and the electron-heavy particles collision frequency ν::$eta\; ,\; =\; ,\; frac\; \{Omega\_e\}\{\; u\}\; ,\; =\; ,\; frac\; \{e\; B\}\{m\_e\; u\}$

where: "e" is the electron charge (1.6 × 10

^{-19}coulomb ): "B" is the magnetic field (in teslas): m_{e}is the electron mass (0.9 × 10^{-30}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 theelectric field vector**E**. The two vectors**J**and**E**make the**Hall angle**θ which also gives the Hall parameter::$eta\; ,\; =\; ,\; an\; heta$

**Plasma conductivity and magnetic fields**In a non-equilibrium ionized gas with high Hall parameter,

Ohm's law ,:$mathbf\{J\}\; =\; sigmamathbf\{E\}$where "σ" is the

electrical conductivity (in siemens permetre ),is a matrix, because the electrical conductivity σ is a matrix:

:$sigma\; =\; sigma\_s\; egin\{Vmatrix\}\; dfrac\{1\}\{1+eta^2\}\; dfrac\{-eta\}\{1+eta^2\}\; \backslash \; dfrac\{eta\}\{1+eta^2\}\; dfrac\{1\}\{1+eta^2\}\; end\{Vmatrix\}$

σ

_{S}is the scalar electrical conductivity::$sigma\_s\; =\; frac\; \{n\_e\; e^2\}\{m\_e\; u\}$

where n

_{e}is the electron density (number of electrons per cubic meter).The current density

**J**has two components::$J\_\{parallel\}\; =\; frac\; \{n\_e\; e^2\}\{m\_e\; u\}\; frac\; \{E\}\{1+eta^2\}\; qquad\; ext\{and\}\; qquad\; J\_\{perp\}\; =\; frac\; \{-n\_e\; e^2\}\{m\_e\; u\}\; frac\; \{eta\; E\}\{1+eta^2\}$

Therefore

:$J\_\{perp\}\; =\; J\_\{parallel\}\; eta$

The Hall effect makes electrons "crabwalk".

When the magnetic field B is high, the Hall parameter β is also high, and $frac\; \{1\}\{1+eta^2\}\; ll\; 1$

Thus both conductivities

$sigma\_\{parallel\}\; approx\; frac\; \{sigma\_s\}\{eta^2\}\; qquad\; ext\{and\}\; qquad\; sigma\_\{perp\}\; approx\; frac\; \{sigma\_s\}\{eta\}$

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 stongest.

**Critical Hall parameter**The electrothermal instability occurs in a plasma at a (T

_{e}> T_{g}) regime when the Hall parameter is higher that a critical value β_{cr}.We have

:$f\; =\; frac\{left(\; frac\{delta\; mu\}\{mu\}\; ight)\}\{left(\; frac\{delta\; n\_e\}\{n\_e\}\; ight)\}$

where μ is the

electron mobility (in m^{2}/(V·s))and

:$s\; =\; frac\{2\; k\; T\_e^2\}\{E\_i;\; (T\_e\; -\; T\_g)\}\; imes\; frac\; \{1\}\{1\; +\; dfrac\{3\}\{2\}\; dfrac\{k;\; T\_e\}\{E\_i$

where "E

_{i}" is the ionization energy (inelectron volt s) and "k" theBoltzmann constant .The

**growth rate of the instability**is:$g\; =\; frac\{sigma\; E^2\}\{n\_e;\; left(\; E\_i\; +\; frac\{3\}\{2\}\; k;\; T\_e\; ight);\; left(\; 1\; +\; eta^2\; ight)\};\; (eta\; -\; eta\_\{cr\})$

And the

**critical Hall parameter**is:$eta\_\{cr\}\; =\; 1.935\; f\; +\; 0.065\; +\; s~$

The critical Hall parameter β

_{cr}greatly varies according to thedegree of ionization α ::$alpha\; =\; frac\{n\_i\}\{n\_n\}$

where n

_{i}is the ion density and n_{n}the neutral density (in particles per cubic metre).The electron-ion collision frequency ν

_{ei}is much greater than the electron-neutral collision frequency ν_{en}.Therefore with a weak energy degree of ionization α, the electron-ion collision frequency ν

_{ei}can equal the electron-neutral collision frequency ν_{en}.* For a

**weakly ionized gas**(non-Coulombian plasma, when ν_{ei}< ν_{en}): :$eta\_\{cr\}\; approx\; (s^2\; +\; 2s)^\{frac\{1\}\{2$* For a

**fully ionized gas**(Coulombian plasma, when ν_{ei}> ν_{en}): :$eta\_\{cr\}\; approx\; (2\; +\; s)$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**A two-temperature gas, globally cool but with hot electrons (T

_{e}>> T_{g}) is a key feature for practical MHD converters, because it allows the gas to reach sufficientelectrical conductivity while protecting materials from thermalablation . This idea was first introduced for MHD generators in the early 1960s byJack L. Kerrebrock [*Cite journal*] and

author = J.L. Kerrebrock

date = November 1, 1960

title = Non-equilibrium effects on conductivity and electrode heat transfer in ionized gases

journal = Technical Note #4

publisher = AFOSR-165

location = Guggenheim Jet Propulsion Center, Caltech, Pasadena, California.OSTI|4843920Alexander E. Sheindlin [*cite conference*] .

author = A.E. Sheindlin

coauthors = V.A. Batenin, E.I. Asinovsky

date = July 6, 1964

title = investigation of non-equilibrium ionization in a mixture of argon and potassium

conference = International symposium on magnetohydrodynamic electric power generation

booktitle = CONF-640701-102

editor = Paris, FranceOSTI|5024025But the unexpected large and quick drop of

current density due to the electrothermal instability ruined many MHD projects worldwide, while previous calculation envisaged energy conversion efficiencies over 60% with these devices. Whereas some studies were made about the instability by various researchers, [*cite conference*] [

author = A. Solbes

date = 24–30 July 1968

title = A quasi linear plane wave study of electrothermal instabilities

conference = 8^{th}International Conference on MHD Electrical Power Generation

booktitle = SM/107/26

editor = International Atomic Energy Agency, Warsaw, Poland*cite conference*] no real solution was found at that time. This prevented further developments of non-equilibrium MHD generators and compelled most engaged countries to cancel their MHD power plants programs and to retire completely from this research field in the early 1970s, because this technical problem was considered as an impassable stumbling block in these days.

author = A.H. Nelson

coauthors = M.G. Haines

date = 26–28 March 1969

title = Analysis of the nature and growth of electrothermal waves

conference = 10^{th}Symposium in Engineering Aspects of MHD

booktitle = Proceedings

editor = MIT, Cambridge, MA, USA

doi = 10.1088/0032-1028/11/10/003Nevertheless 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. [

*cite conference*] [

author = J.P. Petit

coauthors = J. Valensi, J.P. Caressa

date = 24–30 July 1968

title = Theoretical and experimental study in shock tube of non-equilibrium phenomena in a closed-cycle MHD generator

conference = 8^{th}International Conference on MHD Electrical Power Generation

booktitle = Proceedings

editor = International Atomic Energy Agency, Warsaw, Poland

volume = 2

pages = 745–750*cite journal*] The stability is given by a

author = J.P. Petit

coauthors = J. Valensi

date = September 1, 1969

title = Growth rate of electrothermal instability and critical Hall parameter in closed-cycle MHD generators when the electron mobility is variable

journal = Comptes rendus de l'Académie des sciences

publisher = French Academy of Sciences

location = Paris

issue = 269

pages = 365–367**quick transition to "fully ionized" conditions**(fast enough to overtake the growth rate of the electrothermal instability) where the Hall parameter decreases cause of the collision frequency rising, below its critical value which is then about 2. Stable operation with several megawatts in power output had been experimentally achieved as from 1967 with high electron temperature. [*cite conference*] [

author = J.P. Petit

coauthors = J. Valensi, J.P. Caressa

date = 24–30 July 1968

title = Electrical characteristics of a converter using as a conversion fluid a binary mix of rare gases with non-equilibrium ionization

conference = 8^{th}International Conference on MHD Electrical Power Generation

booktitle = Proceedings

editor = International Atomic Energy Agency, Warsaw, Poland

volume = 3*cite journal*] [

author = J.P. Petit

coauthors = J. Valensi, D. Dufresnes, J.P. Caressa

date = January 27, 1969

title = Electrical characteristics of a Faraday linear generator using a binary mix of rare gases, with non-equilibrium ionization

journal = Comptes rendus de l'Académie des sciences

publisher = French Academy of Sciences

location = Paris

volume = 268

issue = A

pages = 245–247*cite journal*] But this electrothermal control does not allow to decrease T

author = S. Hatori

coauthors = S. Shioda

year = 1974

month = March

title = Stabilization of Ionization Instability in an MHD Generator

journal = Journal of the Physical Society of Japan

volume = 36

issue = 3

publisher = Tokyo Institute of Technology, Yokohama, Japan

pages = 920–920

doi = 10.1143/JPSJ.36.920_{g}low enough for long duration conditions (thermal ablation) so such a solution is not practical for any industrial energy conversion.Another idea to control the instability would be to increase non-thermal ionisation rate thanks to 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 under its critical value along these paths. But this concept has never been tested experimentally.In the 1970s and more recently, some researchers tried to master the instability thanks to

**oscillating fields**. Oscillations of the electric field or of an additional RF electromagnetic field locally modify the Hall parameter. [*cite journal*] [

author = G.I. Shapiro

coauthors = A.H. Nelsone

date = April 12, 1978

title = Stabilization of ionization instability in a variable electric field

journal = Pis'ma v Zhurnal Tekhnicheskoi Fiziki

volume = 4

issue = 12

publisher = Akademiia Nauk SSSR, Institut Problem Mekhaniki, Moscow, USSR

pages = 393–396*cite journal*]

author = T. Murakami

coauthors = Y. Okuno, H. Yamasaki

year = 2005

month = May

month = December

title = Dynamic stabilization of the electrothermal instability

journal = Applied Physics Letters

volume = 86

issue = 19

publisher = Tokyo Institute of Technology, Yokohama, Japan

pages = 191502–191502.3

doi = 10.1063/1.1926410Finally, a solution has been found in the early 1980s to annihilate completely the electrothermal instability within MHD converters, thanks to

**non-homogeneous magnetic fields**. A strong magnetic field implies a high Hall parameter, therefore a low electrical conductivity in the medium. So the idea is to make some "paths" linking an 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 is risen. Experiments where streamers do not present any inhomogeneity has been obtained with this method. [*cite journal*] [

author = J.P. Petit

coauthors = M. Billiotte

date = April 27, 1981

title = Method for eliminating the Velikhov instability

journal = Comptes-rendus de l'Académie des Sciences

publisher = French Academy of Sciences

location = Paris

pages = 158–161*cite conference*] This effect, strongly nonlinear, was unexpected but led to a very effective system for streamer guidance.

author = J.P. Petit

year = 1983

month = September

title = Cancellation of the Velikhov instability by magnetic confinment

conference = 8^{th}International Conference on MHD Electrical Power Generation

booktitle = Moscow, Russia

url = http://www.mhdprospects.com/pdf/cancellation_of_the_velikhov_instability_by_magnetic_confinment.pdf

format = PDFBut 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 metJean-Pierre Petit in 1983 at the 9^{th}MHD International conference in Moscow, made the following comment to the inventor of the magnetic stabilization method:

cquote

"You bring the cure, but the patient already died..."However it should be noted that this electrothermal stablization by magnetic confinment, if 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 ).**See also***

Magnetohydrodynamics

*MHD generator

*Evgeny Velikhov **External links*** M. Mitchner, C.H. Kruger Jr., [

*http://navier.stanford.edu/PIG/C4_S10.pdf Two-temperature ionization instability*] : Chapter 4 (MHD) - Section 10, pp. 230–241. From the plasma physics course book [*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**

*Wikimedia Foundation.
2010.*

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