Volume44A,number3 PHYSICSLETIERS 4 June1973

ELECTROSTATIC WAVES WITH FREQUENCIES ABOVE THEGYROFREQUENCY IN A PLASMA WITH A LOSS-CONE

V.!. KARPMAN, Ju.K. ALEKHIN, N.D. BORISOVandN.A. RJABOVA!zmiran, Moscow,142092, USSR

Received24 April 1973

Analytical andnumericalresultsarepresentedfor thepropagationandgenerationconditionsof supercyclotronic(L’ > WC) electrostaticwavespropagatingnearlyperpendicularlyto themagneticfield in a plasmawith a loss-conedistribution function.

We considerelectrostaticoscillationswith frequen- J~ciesabovetheelectrongyrofrequencyco,~(supercyclo-

*5tronic waves)in a plasmadescribedby the electron 15

distribution function of the“loss-cone”type £7

v~ t,~F(v11, ~) (~3/2w2u)_1uiexp(_—i- — —“\ . (1) t~

w21

The wavesunderconsiderationsare of specialinterest tsnow,in particular,after observationsof electrostaticoscillationswith frequenciesof the order(~c(m~)in L

11

spaceplasmas[1, 2] , where the distributionfunction \is likely to be of the loss-conetype.

Thedispersionequationcanbewritten in theform \2w2 (

�~1+—~-11+ E a~Z(a~) (2) _______________________________

k2u2 I, n=—~ LI 4 ~

x1 (1÷1 n’\ a n ii } Fig. 1. DispersioncurvesM(z;a) for differentvaluesq(q=______ — ~0 15, 45, etc.),a 10, ~ = 10. Two typesof hatchingq ,i—n)az — q(p—n) ~_jze zI~(z) mark regionsof resonantandnonresonantinstabilities.

wherew is theelectronplasmafrequency,j.t =

z = k~w2~/2w~,q = w2/u2,a~= a(jt—n),a = z>>~2 andz<< 1 (seebelow) andnumericallyin theandZ(a~)is theplasmadispersionfunction [3]. In intermediateregion. Thegeneralpictureof the disper-(2) onesupposesthat thez axis is directly alongthe sive curvesi.i = i.z(z;a)is shownin thefigures. Formagneticfield. z>> ~2 thesecurvesaredescribedasymptoticallyby

Thecasea = °° was alreadyconsideredin detail (e.g.ref. [4—6], whereotherreferencesaregiven also). z312= — ~ —v sin~O/ i~2p.~otjrn—

2 w~ (~/2cotIL1r+ 2a2sin2.LnSomeremarksrelatedto thecaseof nearlyperpendi-cularpropagation,i.e. 1 <<a<00, weregiven in [4]. (3)In the presentnotea completepictureof dispersioncurvesandgrowthratesfor wavesof largebut finite If q <<ma2 thisexpressiondescribesonly theuppera are obtained.Usingthefact that in this caseImZ is branchesof the dispersivecurves(forwhichp -~m—0exponentiallysmall, onemay neglectIm e in thecom- withz -+ °°). If q ~ ma2,(3) describesalso the lowerputationof the realpartof the frequency.After that branchesandtheir linking with theupperones(seeit is comparativelyeasyto solve(2) analyticallyfor e.g.the curvewithq = 90 in fig. 1).

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Volume44A, number3 PHYSICSLETTERS 4 June1973

t Zcr~2[4~~~ q

m~~im+l

~ (7)m.j

at which the two mensionedabovebranchesof the

t dispersivecurve link (in the caseq/ma2 << 1). The

linking in the regionz << 1 is notpossiblein the bandsm - wherem > p

0 (fig. 2).

Now let us considergrowthrates.Usingthe formu-la y = —~~lmc/Re(ae/ap)one obtainsfor z >> p

2 andz << I, respectively

rn-i —~ __________

ay= —w — ~ [(q+l)(p--n)—plexp[—a2(p—n)2j

Fig. 2. A generalview of the dispersioncurves~(z;a) in the fl~_

frequencybandcontainingtheupperhybrid frequencyp

0

andin theadjacentbands(mis integerpart of p0). x (cotP~[l + ~2 2q+3 1~2 sin

2pir INow let usconsiderthe regionz << 1. If one intro-

ducesp = m + ~ (wherem = 2, 3, ... and—1 <~<0) ~ [ ~1 +and supposesthat -~ ~ [ ~2 (I + 3cot2~~)]} (8)

~i<<l, l<<aIb~,q/ma<<a, (4) cr~ ~ (9)

then eq. (2) canbe reducedtoFrom (8) it follows that ‘y may be positive only for the

c0(p) — B + q m ÷1 1 = ~, (5) upperbranchin that partof it which is locatedin the

~~2j region ö <—1/2 (this is possibleonly if q ~ma2).

where From (9) onecanseethat atz << I an unstablebranch

= 1 — ~ (sin2O + cos2O is the lowerone(~<&~.r).Formulae(8, 9), however,arenot valid in the immediatevicinity of critical point~ whereac/ap = 0. It is easily seenthat an additionalun-

B ~ 1 sin20 stablebranchstartsfrom this point. Forexample,the2 ~2) -~~-—i expression(6) becomescomplexifz <Zcr, whichgives

the instability with 7m~ ~ ‘6cr’ (This type of in-stability is calledhereas“nonresonant”.)A similar

(c0(p) is the dielectric permeabilityof a cold plasma situationalsooccursfor largez. A generalpictureof

and0 is the anglebetweenwavevectorandmagnetic the instability regionsis presentedin fig. 1. Our theo-field). Due to the condition(4) one may put �0(p) ry is notapplicablein somevicinitiesof critical pointsc0(m).Thenfrom (5) it follows that (encircledin fig. 1) wherethe small imaginarypart

Bm2( / 2(m+l)e

0q ) of the dispersionfunction is important.An investiga-— — 1 ±1/i + . (6) tion without restrictionof nearlyperpendicularpro-— 2c~~ V Bm

4a2pagation(but with z >>p2) will be publishedin [7].

Now let p~be dimensionlessfrequencyof an electro-static wave of thegiven 0 in the cold plasma,i.e.

= 0. Thene0(m)< 0 ifm <p~~andviceversa. References

In the bandswherem <p0, the radical in (6) vanishes [l~ CF. Kennel, F.L. Scarf,R.W. FredricsandJ.H. McGehee,

at the“critical” point J. Geophys.Res.75 (1970)6136.

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Volume 44A, number3 PHYSICS LETTERS 4 June 1973

[2] H. Oya, J. Geophys.Res.75 (1970)4279. [5] J.A. Tataronis and F.W. Crawford, J. PlasmaPhys.4 (1970)(3] B.D. Fried and S.D. Conte,The plasma dispersion function 231,249.

(AcademicPress,N.Y. and L., 1961). (6] R.W. Fredrics, J. Geophys.Res.76 (1970)5344.[4] D.E. Baldwin, I.B. Bernstein andM.P.H. Weenink, Advan- [71V.!. Karpman, Ju.K. Alekhin, N.D. Borisov and

cesin plasmaphysics,VoL 3, eds.A. Simonand N.A. Rjabova, Cosmic Electrodynamics4 (1973),to beW.B. Thompson (N.Y. 1969),ch. i. published.

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