Electron-ion two-stream instability in anisotropic isothermal plasma

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Electron-ion two-stream instability in anisotropic isothermal plasma

Sh. M. Khalil, B.F. Mohamed

To cite this version:

Sh. M. Khalil, B.F. Mohamed. Electron-ion two-stream instability in anisotropic isothermal plasma.

Journal de Physique, 1988, 49 (3), pp.451-455. �10.1051/jphys:01988004903045100�. �jpa-00210715�

Electron-ion two-stream instability ⁱⁿ anisotropic isothermal plasma

Sh. M. Khalil (1, *) ^{and B. F. Mohamed} (2)

(1) International Centre for Theoretical Physics, Trieste, Italy

(2) Nuclear Research Centre, ^Plasma Physics ^{Unit, Atomic} Energy Authority, Cairo, Egypt

(Requ le 19 août 1986, révisé le 29 juillet 1987, accept6 le 9 octobre 1987)

Résumé.

L’excitation linéaire de l’instabilité à deux faisceaux électron-ion, ^dans

plasma ^isotherme (Tc

Ti), magnétisé ^et

uniforme, ^{est étudiée} analytiquement. ^Des expressions pour la fréquence, ^{le taux}

de croissance et les conditions de l’instabilité sont obtenues quand la vitesse de dérive dépasse légèrement ^le

seuil

vitesse de l’instabilité. La non-uniformité de la densité ^et l’existence d’un champ magnétique ^externe

constant peuvent stabiliser l’instabilité.

Abstract.

The linear excitation of electron-ion two-stream instability excited in inhomogeneous, magnetized

isothermal plasma (Te

Ti) ^is investigated analytically. Expressions ^{for the} frequency, growth ^{rate and}

conditions of instability

obtained when the current velocity slightly ^{exceeds the} instability ^threshold velocity. Inhomogeneity ⁱⁿ density and existence of external static magnetic field may stabilize the instability.

Classification

Physics ^Abstracts

52.55K

52.50D

1. Introduction.

Since the pioneering ^{work of} Buneman [1], ^the

electron-ion two-stream instability has been the

subject ^of extensive studies [2-4]. ^This type ^{of insta-}

bility is excited hydrodynamically ^{when the current} velocity ^u (relative velocity ^{due to} the motion of electrons and ions, ^u

_{Uc -} ui) considerably ^exceeds

a certain threshold value (u > ucr oc V Te, ^where V T c

(Te/me)ll2 is the electron thermal velocity) [5, 6]. ^{In this case the} development ^{of the} instability

involves a displacement ^{of the} plasma regions ^and

results in the variation of the spatial configuration ^of

the plasma. On the other hand, ^near the threshold when AU

u - Ucr Ucr, ^or when waves receive energy

due to Cherenkov resonance

from a

small group of ^resonance particles, ^this instability

becomes kinetic [2, ^3, 7]. ^{In this case, the} phase

velocities of the waves is less by ^{an order of} magnitude than the thermal velocities of the partic-

les. The amplification of oscillations is due to the difference in the character motion of the various groups of particles.

(*) Permanent address : Nuclear Research Centre, ^Plas-

ma

Physics ^{Unit., Atomic} Energy Authority, Cairo, Egypt.

The electron-ion two-stream is found to be the

reason for some rapid ^turbulent heating ^of plasma [5, 6, 8, 9]. ^{Besides, it is of} interest due to its

possible connection with the thermal conductivity ^of

laser-heated plasmas [10, 11] ^and plasma heating

with relativistic electron beam return current [12].

El-Naggar et ^al. [3] studied in detail the linear and

quasi-linear stages of Buneman instability ^{in a homo-}

geneous isotropic ^{hot ion} plasma (Ti > T, ). ^The instability ^{in this case} will lead to an excitation of

negative energy waves, and have a threshold charac- ter, u,, az V Tc ^This instability ^{was also} investigated

in the linear and nonlinear approximations by ^Hus-

sein et al. [2] ^{for a} plasma placed ^{in an} external static

magnetic ^{field and a} temperature Ti > Te.

Experimentally, ^{it has been observed} that, ^{when a}

turbulent heating pulse ^is applied ^{to a} very well confined tokamak discharge, remarkably ^efficient heating results. The heating mechanism may be attributed to the onset of the electron-ion two- stream instability [13].

Besides, ⁱⁿ investigating ^the equilibrium properties

of a low-density single-ended Q-Machines with isothermal plasma Ti

Te, ^this instability ^{has been}

identified as a major ^factor disturbing the assumed

equilibrium [14].

In the present ^{work we} shall consider the problem

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004903045100

452

treated in [2 ^and 3], ^{but under new} considerations :

i) isothermal plasma Ti

Te, ^and ii) inhomogeneous magnetized plasma. The external magnetic ^{field is}

uniform and directed along ^the z-direction (i.e., Ho

ez Ho), ^{and the} inhomogeneity ^{due to the} gradient ⁱⁿ plasma density, which is taken along ^the x-direction, i.e., perpendicular ^{to the} magnetic ^field.

The excitation of the instability ^{will be considered}

in the kinetic case at current velocity insignificantly

exceeds the threshold velocity ^{of the} instability (Au

^{u -} Ucr Ucr). The smallness of the parameter

AU/Ucr ^{is used to find an} analytical solution of the

dispersion equation ^{for waves.}

2. Boundaries of instability.

The general ^{form of the} dispersion relation describes the electrostatic oscillations in a one-dimensional

inhomogeneous magnetized plasma ^with particles

have a Maxwellian velocity distribution function is [15] :

where a indicates the type ^of charged particles (e ^for

electrons and i for ions), _{úJ P a}

(4 7Te 2 nO/ m a )1/2 ^is

the plasma frequency, VT. = (Ta/ma)l12 is the ther- mal velocity, W (za ) ^{is the} probability integral (or Kramp function) ^with complex argument given by

P a ^{is the} inhomogeneity factor, given by :

and A (J.L a) ^{is the} magnetic ^field effect, given by :

CJJ Ca = (eHo/ ma ^{c ) is the cyclotron} frequency, p a is the Larmor (gyration) ^{radius, and} I (JL a) ^{is the}

modified Bessel function.

It is easy ^to check that the relation ue/gj .c 1, ^is

valid for isothermal plasma (Ti

Te ).

For electrons, equation (1) is valid under the condition (oi, KV Te) W ce. ^{In this case we can}

consider that the oscillation wavelength is consider-

ably higher than the electrons’ Larmor radius with thermal velocity V Te, ^{this implies that K 1. p e 1.}

For ions, ^{we consider a} plasma ^{with a} finite ion Larmor radius, i. e. , ^a perturbation ^with K1 pi > ^1.

This is satisfied under the condition that (p i/ a ) >

Me ^1/2 _(a is the inhomogeneity scale) ^{and that} Mi

KV Ti > ^{wci (i. e. ,} the ion thermal motion is more

effective than the Larmor gyration ^{and the condition} Im ú) :> ú) ci ^{need not to be} satisfied). Generally speaking, ^we shall consider the ions to be weakly magnetized, ^{while electrons are} strongly magnetized.

Accordingly, ^{we have for} A (JL a) :

Let us also denote :

and under the condition of weak inhomogeneity,

we can write the operator t,, for both electrons and ions as :

The dispersion equation (1) ^{will now have the} simple

form (tri

Te) :

where w’ is a complex frequency,

To find the limit of instability, ^{we set:}

and hence,

up indicates the current velocity parallel ^{to the wave}

vector k (ull

^{u cos 0, where 0 is the} angle ^between

u and k).

From (2), ^{we can obtain the} following equations

(by separating ^{real and} imaginary parts) :

Combining (3) ^and (4), ^and taking ^{into account the}

range ^of frequencies I ku I ^{« w’, we} get :

where,

The function 1Jf (zo i) is characterized _{as ; at} Zo i -> 0,

W(zoi)= 1, ^and at zoi,.>l ;

From (4), ^{we can} obtain the following expression ^for

the current velocity ul :

where

and from (6), ^{we can} obtain the minimum value of up (minm ul|

ucr) ^{at which the} instability ^{starts to}

take place :

Relation (7) ^{is valid at}

The threshold ucr ^{in the case of a} homogeneous unmagnetized plasma ^with Ti

T, ^is given by [14] :

Comparing (7) ^and (8), ^{we see that this} type ^of instabilities has a lower threshold value in the presence of external static magnetic ^{field and in-} homogeneity ⁱⁿ density, ^{and the} instability ^takes place earlier than in the case of isotropic plasma.

In the case of magnetized ^{hot ion} plasma (Ti > Te) [2], ^{the threshold value is} given by :

which is higher ^{than that for} isothermal plasma.

This shows that by increasing the electrons tem-

perature ^{over that} of ions will decrease the threshold

velocity ^{and the} instability ^still appears faster in isothermal plasma.

3. Kinetic growing ^of instability.

As already ^known [5], this branch of oscillations is

hydrodynamic ^{when the current} velocity ^{u must}

exceed a certain threshold value ucr where ucr

slightly exceeds the electron thermal velocity V Te.

Besides, ^the theory proved that, ^near the threshold when AU

u - Ucr UCP this instability ^becomes

kinetic [2, 3].

Let us now find a solution for equation (2) ^at

y K > 0 corresponding ^{to the} growth ^of oscillations,

and we shall make use of the critical values (at

0 = 0, _wo=- 2 B 2 B ; _K0 ^{and u}

^{ucr)’ Besides, we set}

w’ as a complex frequency ^near the boundaries of the instability, i. e. , ^{we set :}

Am is a small complex addition which represents ^the

frequency ^and growth ^{rate of the} instability :

Also, it is convenient to set :

The values of Au , o 2, ^{and Aw are small} enough ^{to be}

used for expanding, ⁱⁿ Taylor’s series, ^{the terms in} (2). ^These considerations will lead to the appearance of the small parameters I åZe I ^« zo e aid [ Azi [ « ^{zo h}

where

Accordingly, ^{we can derive from the} dispersion equation (2) ^the following algebraic equations :

and

454

where

and

The solutions of (9) ^and (10) give ^the frequency ^and

the growth ^rate of the electron-ion two-stream

instability ^{as :}

where

It is clear from (11) ^and (12) ^{that this} instability ^is

excited in our case, under the conditions of super-

criticality, within the narrow wave vector range defined by ^the inequality :

The waves that grow ^most rapidly ^{are those with}

cos 9 = 1 and K2 p 2 - 1.

₂ ^{In this case w - K W ma,,}

and ’Y K -> ’Y max ^are the maximum frequency ^and

maximum growth ^{rate of the} instability, given by :

1l1 includes the dependence ^{of the} growth ^{rate on}

both the density gradient and the external magnetic

field. It is also clear that for different values of :It the factor R/l ⁺ :It 2 is less than unit, ^{and if the} magnetic ^{field is} increased, ^the growth ^{rate de-}

creases.

In the case of a homogeneous, isotropic plasma [16], ymax is given by :

where

From (15) ^and (16) ^{we see that the} growth ^{rate is less}

in an inhomogeneous plasma placed ^{in an external}

static magnetic ^field. Hence, in the linear stage ^of the instability, ^the magnetic ^{field and} inhomogeneity

in density, ^while reducing ^the growth ^{rate to some}

extent, ^{are unable to} inhibit the instability ^effec- tively.

In addition, ^{for a} strongly nonisothermal _mag- netized plasma ^with Ti > Te [2], the electron-ion two-stream instability is excited with maximum

growth ^rate given by :

It is easy ^to notice that the growth ^rate (17) ^{is less}

than that for the case T;

Te.

As a conclusion, ^the temperature ^ratio TilT,, ^is

not only ^of great importance ^to determine the

instability type, but also of importance ^{as a stabiliz-} ing factor. It is shown that increasing ^{the ion}

temperature ^over that of electrons will lead to a

reduction of the growth of the electron-ion two- stream instability.

In addition, it is shown that the inhomogeneity ⁱⁿ density and the existence of an external static

magnetic field may stabilize this instability ^{in the}

linear stage.

As experimental application, ^we mention here two examples. Firstly, in connection with fusion devices, ^{interest has} developed ^{in the effects of} instabilities, ^{and in} plasma heating, ^for T;

Te ^and

u -az VT.* ^{Under these} conditions, ^the strong ^ion Landau damping prevents ^the excitation of ion- sound waves. The kinetic electron-ion two-stream

instability ^{is excited and causes for} example ^en-

hanced resistivity, heating ^and perpendicular ^thermal conductivity. ^These phenomena ^are observed in Q- device plasma, ^{where the} plasma ^is produced ^{with a} temperature Ti

Te [14, 17].

Secondly, ^our results may be best applicable ^to

toroidal experiments ^{in which the electron drift} velocity ^{is somehow boosted so} that it is larger ^than

the thermal velocity. ^{This is satisfied under the}

condition that an extremely ^{short electric field} pulse

is applied ^{to the} isothermal plasma (i.e., ^when pulse

duration is shorter than the characteristic growth

time for the electron-ion two-stream instability) [18].

Acknowledgments.

One of the authors (Sh. ^M. Khalil) would like to

thank Prof. Abdus Salam, ^the International Atomic

Energy Agency and UNESCO for hospitality ^{at the}

International Centre for Theoretical Physics, Trieste,

where part ^{of the work was} done. Thanks are also

due to Prof. Adel El-Nadi (faculty ^of Engineering,

Cairo University) and Prof. M. Hassan (ICTP) ^for

valuable discussions and advice.

References

[1] BUNEMAN, O., Phys. Rev. Lett. 1 (1958) ^8.

[2] ^{HUSSEIN, A.} M., KHALIL, Sh. M. and SIZONENKO, V. L., ^Plasma Phys. ²⁰ (1978) ^545.

[3] EL-NAGGAR, ^I. A., KHALIL, Sh. M. and SIZONENKO, V. L., ^Plasma Phys. ²⁰ (1978) ^75.

[4] ISHIHARA, O., HIROSE, ^{A. and} LANGDON, ^A. B., Phys. Rev. Lett. 44 (1980) ^1404.

[5] SIZONENKO, V. L. and STEPANOV, ^K. N., ^Plasma Phys. ¹³ (1971) ^1033.

[6] ^{MANTEI, T.} D., DOVEIL, ^{F. and} GRESILLON, D., Plasma Phys. ¹⁸ (1976) ^705.

[7] ^{RANYUK, A. I. and} SIZONENKO, ^V. L., Sov. Phys. -

Tech. Phys. ¹⁹ (1974) ^705.

[8] HAMBERGER, S. M. and FRIEDMAN, M., Phys. ^Rev.

Lett. 21 (1968) ^674.

[9] SIZONENKO, V. L., ^Sov. Phys. - ^Tech. Phys. ¹⁹ (1975) ^1260.

[10] ^{DRUMMOND, W. E.} and ROSEBLUTH, ^M. N., Phys.

Fluids 5 (1962) ^1507.

[11] LOMINADZE, ^{D. G. and} STEPANOV, ^K. N., JETP 34

(1964) ^1823.

[12] BUNEMAN, O., Phys. ^{Rev. Lett. 10} (1963) ^285.

[13] TOI, K., HIRAKI, N., NAKAMURA, K., MITARAI, O., KAWAI, ^{Y. and} ITOH, S., Nucl. Fusion 20 (1980)

1169.

[14] ^{KUHN, S., Plasma} Phys. ²³ (1981) ^881.

[15] MIKHAILOVSKY, ^A. B., Theory of Plasma Instabilities

(Consultants ^{Bureau, New} York) (1974) ^{Vol. 2.}

[16] ^{KHALIL, Sh.} M., AMEIN, W. H. and MOHAMED, B. F., ^Acta Phys. Pol., ^{to be} published (1987).

[17] ^{MOTLEY, R. W.,} Q. ^Machines (Academic Press)

1975.

[18] ^PAULSON, J. D. and HIROSE, A., Phys. ^{Canada 38}

(1980) ^33.