Volume148,number5 PHYSICS LETFERS A 20August 1990

Electronbeamacousticand magneto-acousticmodesin amagnetizedbeam—plasmasystem

V.K. Jam ‘and N.L. Tsintsadze2InternationalCentrefor TheoreticalPhysics,Trieste,Italy

Received4August 1989;acceptedfor publication12 June1990Communicatedby R.C.Davidson

Thedispersionrelationcharacterizingelectrostaticmodesin a coldmagnetizedplasmapenetratedby a hot electronbeamissolvedanalytically.In thelimit k2A~~ 1 wherekis thewavenumberof themodeand~b is theDebyelengthof beamelectrons,twonewmodes,viz, theelectronacousticandthemagneto-acousticmode,arefoundto exist.Expressionsfor thegrowthrateandthresholdbeamenergyrequiredfor theonsetof thekineticinstabilitiesofthesemodesareobtained.

Electrostaticinstabilitiesin beam—plasmasystemshavebeenstudiedextensivelyoverthe lastcoupleof dec-adesdueto their relevanceto bothlaboratoryandspaceplasmaconfigurations[1—101.It is well-knownthata magnetizedplasmasustainsa variety of electrostaticmodeson ion andelectrontime scales.For example,Trivelpiece—GouldmodeandBernsteinmodewavesexist for k~p~~< 1 andk~p~~ 1 respectively,wherek

1is the transversewavenumberandPe is the electronLarmor radius. In caseof cold ions and in the limitk22~.~ 1, where~ is the Debyelengthof plasmaelectrons,ion-acousticmodesareobserved.Variousmodes

inplasmacanbedrivenunstablevia Cerenkovor cyclotroninteractionsbyadrifting coldor hot electronbeamwhichactsasafreeenergysourcein abeam—plasmasystem.In spiteof voluminousliteratureavailableto dateon theexcitationof electrostaticmodeinstabilities,modesin the limit k2A~~ 1, where~‘b is the Debyelengthfor beamelectrons,haveescapedthe attentionof theorists.In thisLetterwethereforeaddressourselvesto thestudyof electrostaticmodesin the limit k2~~< 1 andinvestigatethe effect of the beamon their dispersioncharacteristics.

We consideracoldbut finite temperatureplasmain amagneticfield penetratedby adrifting electronbeamwith finite thermalspread.The dispersionrelationfor electrostaticmodesin suchasystemcanbe written as[2]

l+�p+�bO, (1)

where

= 1 + ~ [~+ ~, ~iz(w_fl~~)i(k2±V~)exp(~.. (2)

2w~bI” w—k1 V0 ,jw—k1Vo—nw,~’\(k~v~~,\( k~v~’\]

1+ ‘9 b i 2 ,exp~—~, 2 11’ ( ),‘,~ Vtb L r9Vtb ~ \ l9Vft~ / \ / \

wherew, w~,w~,co~are the wave,plasma,beam—plasmaandelectroncyclotronfrequenciesrespectively;k~,k1, k are the parallel,perpendicularandtotal wavenumbersrespectively;V0, v~,Vtb arebeamdrift velocity,

Permanentaddress:PhysicsGroup,Schoolof EnvironmentalSciences,JawaharLalNehruUniversity,NewDelhi 110067,India.2 Permanentaddress:InstituteofPhysics,AcademyofSciencesof theUSSR,GuramishviliStreet6, Tibiisi 38007,USSR.

0375-9601/90/S03.50© 1990 — ElsevierSciencePublishersB.V. (North-Holland) 269

Volume 148,numberS PHYSICSLETTERSA 20August 1990

plasmaandbeamelectronthermalvelocitiesrespectively.Z is the plasmadispersionfunction [8] andI~isthe modifiedBesselfunction of nth order.

Assumingw/k11v~,(w ±w~) /k11v1>> 0 for weakdamping,andin thelimit k~v~/2w~<<1, eq. (2) reducesto

21.2 21.2(Op __________

~ w2k2 — (w2—w~)k2

+ i ~ ~ {ex~[— ()2] + exp[ — (w_w~)2] + exp[ — (w+w~)2]} (4)

Wefurtherassumek~v~/2w~~ 1 andconsideronly the Cerenkovinteractionterm in eq. (3). We, there-fore, expandZ( (w—k

11V0)/kllvtb) in the small argumentapproximationw— k11 V0 0 andwrite eq. (3) in thefollowing form,

~ ~*{l+i ,1—w—k1v0 exp[_(°)~/t1Vo)2]} (5)

Vtb V~

1, Vtb

Neglectingthe imaginarytermsandassumingk2A~~*z1 whereAb ( Vtb/.~J~(Opb) is theDebyelength of the

beamelectrons,the dispersionrelation (1) simplifies to

F(wk)—’ ~P!EL w~k~ ~ (6)— — (O2k2 — (w2—w~)k2 k2v?b —

The solutionsof eq. (6) are

= ~ [ (w~+k2v~nP/nb)±~~/(w~+k2v~nP/nb)2—4k~v~a~nP/nb]. (7)

When

(~+k2v?bnP)2 4k~v~,w~n~

w÷andw_ canbe written in the following form,

k2v~n~ (8)

— k~v~np/nb 9— 1 + (nP/nb)k2V~,/w~

It may be notedthat the solutionsrepresentedby w~and w,. are similar to the dispersionrelationscor-respondingto megneto-acousticandion-acousticmodesexceptthat the ion inertia hasbeenreplacedby thebeamelectroninertia andthe plasmaelectrontemperatureby the beamelectrontemperature.We thereforetermthemodescharacterizedby w andw~asbeamelectronacousticandmagneto-acousticmodesrespectively.

To studythe stability of thesemodeswe write eq. (1) as

w~k~ ~

w2k2 (w2—w~)k2 k2v~

= — i ~ ~ {exp[ — (w)2] + exp[ — (w_wc )2] + exp[ — (w±coc)2]}

—i ,1r~2w~(o—k1V0) exp[_(~~V0)

2]. (10)

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Volume148,number5 PHYSICS LETFERSA 20August 1990

In theabsenceof thetermson therighthandside,thesolutionsofeq. (10) arew=w~,i.e.,F(co1)=0.Now

let (Obecomplex,W=COr+~Wj.Expandingthe left handsidearoundWr,

we obtainthe following expressionfor the growth rate,

= — ~ ((Or {exp[_(~)2]±:xp[_(0~)2]+exp[_(~0~)2]}

_ flt, C0rk1 V0 J’ (Wr’4’i V0\ ]‘\1. 3 ex~—~, ill,

I Vtb L \ “1 ~ / ji

where1.2 1.2

x=~~-+~~3 ((O2.(O2)2’

ThethresholdbeamenergyV~is obtainedby puttingCo, = 0 in eq. (11),

~ ~ (i + ~ {exp[_(~)2]+exp[_(~)~0c)2]}).

The contributiondue to the term exp{— [(w+w~)/k,v~]2} is very small andhenceis neglected.Thus for

beamenergy V0> VOT, the modeswill becomeunstable.

Theauthorsaregratefulto the InternationalCentreforTheoreticalPhysicsfor providinghospitalityduringthecourseof thiswork. Oneof us (V.K.J.) would like to thankProfessorP.K. Kaw, Institutefor PlasmaRe-searchBhat, Gandhinagar,Indiaforhelpful discussions.

References

[I] T.M. O’Neil andJ.H.Malmberg,Phys.Fluids11(1968)1974.[2]M.Seidl,Phys.Fluids13 (1970)966.[3] S.A. Self,MM. ShoucriandF.W.Crawford,J.AppI. Phys.42 (1971)704.[4] J.A.TataronisandF.W.Crawford,J.PlasmaPhys.4 (1970)249.5] A.B. Mikhailovskii, Theoryofplasmainstabilities,Vol. 1 (ConsultantsBureau,NewYork, 1974).

[6] A.L Akhiezer,IA. Akhiezer, R.V. Polovin, A.G. SitenkoandK.N. Stepanov,Plasmaelectrodynamics,Vol. I (Pergamon,Oxford,1975).

[7] J.E.AllenandA.D.R. Phelps,Rep.Prog.Phys.40 (1977)1305.[81B.D. FriedandS.D. Conte,Theplasmadispersionfunction (AcademicPress,NewYork, 1961).[91K. MizunoandS.Tanaka,Phys.fluids 17 (1974)156.

[10] V.K. Jam andPJ.Christiansen,PlasmaPhys.Contr.Fusion26 (1984)613.

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