On th Possblty of Gnratng Harmoncs of th Elctron Plasma Frquncy n th Solar Atmosphr du to Explosv Instablty n a Systm of Intrpntratng Elctron and Ion Flows V. V. Fomchv, S. M. Fanshtn, and G. P. Chrnov Pushkov Insttut of Trrstral Magntsm, Ionosphr, and Rado Wav Propagaton, Russan Acadmy of Scncs, Moscow, 142191 Russa -mal: gchrnov@zmran.ru Abstract An altrnatv mchansm s proposd for th gnraton of harmoncs of th lctron plasma frquncy du to th dvlopmnt of xplosv nstablty n a systm of ntrpntratng lctron and proton flows n th solar atmosphr. Th ffcncy of th nw mchansm n comparson wth th arlr dscussd mchansms nvolvng multstag procsss of nonlnar ntracton of wavs n plasma s dtrmnd. It s shown that th dvlopmnt of xplosv nstablty can lad to th xctaton of th scond and thrd harmoncs of th plasma frquncy wth comparabl ampltuds. 1. INTRODUCTION Solar rado-frquncy bursts n th mtr wavlngth band ar, as a rul, vry unstady, shorttrm, and ntns radatv phnomna wth an ffctv radaton tmpratur (T ff 10 10 K) much hghr than thtmpratur of th solar corona (Т 10 6 K). Prsntly, thy ar xpland by a plasma mchansm accordng to whch plasma wavs ar frst xctd at frquncs clos to th lctron plasma frquncy n th solar corona and, thn, ths wavs transform by on or anothr mchansm nto lctromagntc wavs, whch can lav th gnraton rgon [1]. Typcal xampls of such bursts ar typ-ii and typ-iii solar rado bursts. Typ-III rado bursts, whch ar rapdly drftng n frquncy (from hgh to low), ar gnratd by fast lctron flows propagatng at an avrag vlocty of ~c/3 (whr c s th spd of lght n vacuum) through th solar corona nto th ntrplantary spac up to th Earth s orbt. Snc th plasma dnsty monotoncally dcrass wth dstanc from th Sun, th fast lctrons propagatng n such amdum xct plasma wavs at succssvly lowr frquncs, whch causs a ngatv drft of th rado frquncy n such bursts. Typ-II rado bursts, whch ar slowly drftng n frquncy, ar assocatd wth th propagaton of shock wavs n th solar corona and ntrplantary spac [2 5]. Undr th condtons of th solar corona, whr collsons of plasma partcls ar nsgnfcant, th shock wavs ar collsonlss. Accordng to th thory of wak collsonlss shock wavs (wth Mach numbrs of M < 2), th front of a shock wav propagatng n th solar corona prpndcular to th magntc fld has an oscllatory structur,.., comprss a squnc of comprsson soltons [6]. Du to th nonunformty of th magntc fld n soltons, a rlatv drft of lctrons and ons n th plan of th shock front arss, th vlocty of whch dpnds 1
on th shock wav ntnsty. If th Mach numbr xcds th crtcal valu, M 1 3 cr (8π n κ T H ) 4 2 1/3 0 0, thn th rlatv drft vlocty of lctrons and ons xcds th lctron thrmal vlocty. Hr, n 0, H 0, and T ar th plasma dnsty, magntc fld, and lctron tmpratur ahad of th shock front, rspctvly. In ths cas, Bunman nstablty of plasma oscllatons dvlops wthn th shock front, whch lads to th gnraton of lctromagntc wavs. Th dynamc spctra (tm varaton n th frquncy spctra) of both typs of rado bursts hav a dvrs fn structur. A typcal fatur of ths spctra s thr harmonc structur,.., th prsnc n thm of smultanously drftng rado msson bands at frquncs clos to th lctron plasma frquncy and ts scond harmonc. Such a harmonc structur of rado bursts ndcats that nonlnar procsss n th solar atmosphr play an mportant rol n th mchansm of rado burst gnraton. In partcular, lctromagntc radaton at frquncs clos to th plasma frquncy was attrbutd to th scattrng of plasma wavs by dnsty fluctuatons (or to thr ntracton wth on-acoustc wavs), whl lctromagntc radaton at frquncs clos to th scond harmonc of th plasma frquncy was xpland by th ntracton btwn plasma wavs xctd by a fast lctron flow and oppostly drctd plasma wavs arsng du to thr stmulatd scattrng of xctd plasma wavs by ons or thr dcay nstablty. Hghr ordr nonlnar procsss wr nvolvd n [1, 5, 7] to xplan th gnraton of typ-iii rado bursts at th thrd harmonc of th lctron plasma frquncy. In partcular, n [7], gnraton of lctromagntc radaton at th thrd harmonc was xpland n two ways: () as a rsult of th mrgng of thr plasma wavs (procss l 1 + l 2 + l 3 t) or () by th combnaton ntracton of two plasma wavs (procss l 1 + l 2 t II ) and th subsqunt ntracton of th rsultng lctromagntc radaton wth a plasma wav (procss l 3 + t II t III ). Hr, l and t rfr to plasma and lctromagntc wavs, rspctvly. Estmats of th ffcncs of ths two procsss vdnc n favor of th twostp mchansm of thrd-harmonc gnraton. Ths procsss gv a satsfactory xplanaton f th thrd harmonc gnratd du to th thrd stpws procss of mrgng s much wakr than th frst two. Howvr, accordng to obsrvatons, th thrd harmonc can b comparabl n th ntnsty wth th frst two. In ths cas, th apparanc of th thrd-harmonc radaton wth an ntnsty comparabl to th ntnsty of th scond harmonc rqurs th prsnc of ntns plasma wavs,.., plasma turbulnc should b strong. It s worth notng that, n som cass, th dynamc spctra of typ-ii rado bursts ndcat th prsnc of radaton at harmoncs wth numbrs n = 3 and 4 (and, possbly, 5) [7 9], th ntnsts of th hghr harmoncs bng narly th sam. Snc 2
th ffcncs of th abov mchansms sharply dcras whn passng to th hghr harmoncs of th plasma frquncy, th mchansm of gnraton of hghr harmoncs n solar rado msson rmans unclar. On possbl soluton to ths problm can b th dvlopmnt of xplosv nstablty n th nonlnar ntracton of thr typs of wavs n th radaton sourc. In [10], a mchansm was consdrd accordng to whch harmoncs of th lctron plasma frquncy wr gnratd as a rsult of th dvlopmnt of xplosv nstablty n plasma pntratd by an lctron flow. It was shown that ths mchansm maks t possbl to xplan th mrgnc of hghr harmoncs wth comparabl ntnsts n th solar rado msson undr th condtons xstng n th sourcs of typ-iii rado burst. In ths papr, w consdr th possblty of xctaton of lctromagntc radaton at harmoncs of th lctron plasma frquncy du to th dvlopmnt of xplosv nstablty n a systm of ntrpntratng lctron and on flows. As notd abov, such a stuaton can occur at th front of a collsonlss shock wav du to th gradnt drft of lctrons and ons, whl th plasma wavs xctd du to nstablty can b a sourc of typ-ii rado bursts. Condtons for th dvlopmnt and saturaton of xplosv nstablty ar rvald. Th ampltuds of harmoncs of th lctron plasma frquncy for th solar plasma condtons ar stmatd qualtatvly. Such a mchansm for th gnraton of rado msson at th thrd and hghr harmoncs of th lctron plasma frquncy n typ-ii solar rado bursts has not bn consdrd arlr. 2. BASIC EQUATIONS AND SYNCHRONISM CONDITIONS As was pontd out n th prvous scton, du to th rgular nonunformty of th magntc fld at th front of a wak (M < 2) collsonlss shock wav propagatng n th solar corona prpndcular to th magntc fld, a rlatv drft of lctrons and ons n th plan of th front taks plac. Undr th condton М > М cr, th vlocty of th rlatv drft xcds th lctron thrmal vlocty, whch lads to th dvlopmnt of Bunman nstablty and xctaton of plasma oscllatons nsd th shock front. Th systm of on-dmnsonal ( / x 0) quashydrodynamc quatons dscrbng th moton of wakly rlatvstc lctrons and postv ons (protons) n th slf-consstnt lctrc fld has th form [11]: 3
E ρ ρ = 4 π ρ ρ ; +V 0 + N = μ ( ρ ) ; x t x x x ρ ρ +V 0 + N = μ ( ρ ); t x x x V + E = μ t x m x 0 υ μ E +V E = μ +. t x m x m c 2 0 2 (1) Hr, x s th coordnat n th plan of th shock front, along whch th currnt flows; E s th lctrc fld; m and m ar th masss of an lctron and on, rspctvly; V 0 and V 0 ar th qulbrum vlocts of ons and lctrons; N and N ar th lctron and on qulbrum dnsts; ʋ, ʋ, ρ, and ρ ar th dvatons of th lctron and on vlocts and dnsts from thr qulbrum valus; c s th spd of lght n vacuum; s th lctron charg; and th paramtr μ~ ρ / N ~ ρ / N ~ υ / V 0 ~ υ / V ( μ << 1 ) s ntroducd to dnot th smallnss of th rght-hand sds of Eqs. (1). Systm of quatons (1) contans quadratc and cubc nonlnarts. Not that, n th quaton for υ, w tak nto account wak lctron rlatvsm, whch, as wll b shown blow, stablzs th xploson. Lt us consdr wav procsss n whch th varabl quantts dpnd on th coordnat and tm as ~xp(ωt kx), whr ω and k ar th crcular frquncy and wavnumbr of th wav prturbaton, rspctvly. Aftr substtutng prturbatons n such a form nto systm of quatons (1), w obtan th followng dsprson rlaton dscrbng th normal mods of th plasma systm undr study: 1 ω ( ω+kv ) ω ( ω kv ) = 0, (2) 2-2 2 2 0 0 0 0 2 2 whr ω 0 4 N / m and ω 4 / 2 2 0 N m ar th on and lctron plasma frquncs, rspctvly. Lt us consdr th rsonanc thr-wav ntracton of normal wavs n ths plasma systm. In ths ntracton, th followng laws of consrvaton of nrgy and momntum should b satsfd [12]: ω3 ω1 ω 2, k3 k1 k2. (3) Usng dsprson rlaton (2) and condtons (3), w obtan th paramtrs of th rsonant trplt ω3 ω2 ω 0, ω2 = nω0 ω 0, k3 ~ n 0/ V0, k2 ~ k3, 1 ~ 0, k ~ / V, 1 0 0 whr ω 1 s th Bunman mod and ω 2 and ω 3 ar th bam mods (wth n bng th mod (4) 4
numbr). Havng dtrmnd th nrgs of th ntractng mods, w fnd that a wav wth th frquncy ω 3 (th thrd plasma frquncy harmonc) has a ngatv nrgy, whl th two othrs (ω 1 and ω 2 ) hav postv nrgs;.., th systm admts an xplosv soluton, whch s stablzd by th cubc nonlnarty n th last of Eqs. (1). 3. ANALYSIS OF EQUATIONS FOR THE WAVE AMPLITUDES W rprsnt all varabls n Eqs. (1) n th form ~a j xp(ω j t k j x) and xpand th nonlnar trms n Eqs. (1) n a Taylor srs up to cubc trms (th cubc trms ar rlatd to th dpndnc of th mass of a movng lctron on th vlocty V 0 ). Thn, usng th standard tchnqu [13, 14], w obtan rducd quatons for th slowly varyng complx wav ampltuds a j (μt, μx), (E xj = a j (μt, μx)xp(ω j t k j x), j = 1, 2, 3). Hr, w consdr th on-dmnsonal rsonanc ntracton of wavs ( / x 0, / y = 0, / z = 0). Followng th procdur usd n [15], w obtan whr a t 3 3 a v 3 gr3 3 1 2 3 3 x j 1 a t v 2,1 2,1 gr2,1 2,1 3 1,2 2 j σ a a a α a, a x σ a a*, σ, σ 2n, α, 1 4 2,3 3 2 2 mv0 β mv0 4m c ω0 V c 0 β, V0, V0 3 (6) (5) and v gr1,2,3 ar th group vlocts of th mods wth n = 1, 2, and 3. Accordng to formulas (6), th sgns of th coffcnts σ 1,2,3 ar th sam; thrfor, th wav ampltuds can ncras smultanously. Ths mans that, n th systm undr consdraton, xplosv nstablty can dvlop. For th convnnc of solvng Eq. (5), w rprsnt th ampltuds a j n th form a j = u j (σ m σ n ) 1/2 (j = 1, 2, 3). As a rsult, for wavs wth th sam ntal condtons u 3 (0) = u 2 (0) = u 1 (0), w obtan a soluton to systm of quatons (5) for wav ampltuds n th spatally homognous rgm ( / x = 0, / t 0). Th maxmum ampltud of th mod wth th frquncy ~nω 0 (n = 1, 2, 3) can asly b obtand from Eq. (5), σ σ a, (7) max 1 2 n αn whr α s th coffcnt n th trm wth cubc nonlnarty n Eq. (5). It s obtand by avragng th cubc trm n th fourth of Eqs. (1). Snc 3 1,2, th xprsson only for α 3 s gvn n formulas (6). 5
Th charactrstc xploson tm s t* = 1/u 0, whr u 0 s th ntal normalzd wav ampltud, whch s rlatd to а j (0) as u 1 / 2 j a j (σmσ n). Hnc, w hav t*(ω 3) σ2σ 1 a3(0), whr a 3(0) s th ntal nos ampltud. 4. QUALITATIVE ESTIMATES Lt us stmat th ampltuds of a solton at th frquncy nω 0 (n = 1, 2, 3) for solar plasma wth N ~ 10 8 cm 3 (ω 0 ~ 10 9 s 1 ) at V 0 ~ c/3 and a constant magntc fld of H 0 ~ 1 G, whch corrsponds to an lctron gyrofrquncy of ω H ~ 2 10 7 s 1 (.., ω 0 ωh ). It s asy to fnd from xprsson (7) that a n mc V,.., max 2 2 3 4 ω0 0β max a 3 ~ 70 V/cm, max a 2 ~ 56 V/ cm, and max a 1 ~ 40 V/cm. Th closnss of th valus of а 1,2,3 n spt of th rlatonshp α3 α 1,2 s du to th fact that th nqualty 1 σ 2, 3 σ s smultanously satsfd (s formulas (6)). Takng th nos valu a N ~ 10 3 α as th ntal valu of а 3 (0), w fnd that th max 3 dvlopmnt tm of xploson s t* = 1/u 0 ~ 20 μs. Comparng th ampltuds of th plasma frquncy harmoncs (n = 1, 2, 3) obtand n [9] as a rsult of th dvlopmnt of xplosv nstablty n plasma pntratd by a proton flow wth th ampltuds of th harmoncs gnratd by ntrpntratng on and lctron flows, w fnd that th lattr ar hghr. Thus, n th prsnt papr, an altrnatv mchansm for th gnraton of harmoncs of th lctron plasma frquncy by ntrpntratng on and lctron flows has bn rvald. 5. CONCLUSIONS Th abov-dscussd stuaton wth th gnraton of lctromagntc radaton n a systm of ntrpntratng lctron and proton flows s drctly rlatd to th gnraton of typ-ii solar rado bursts. At prsnt, accordng to th most dvlopd and rcognzd modl, typ-ii rado bursts ar gnratd by collsonlss shock wavs xctd n th solar corona thr at th fronts of coronal mass jctons or aftr solar bursts. In any cas, at th front of a collsonlss shock wav propagatng prpndcular to th xtrnal magntc fld, Bunman nstablty dvlops du to th gradnt drft of lctrons and protons, whch can lad to th gnraton of rado wavs at th frst and scond harmoncs of th lctron plasma frquncy [2 5]. It s also not xcludd that som typ-ii bursts ar assocatd wth shock wavs propagatng along th magntc fld n th solar corona. In ths cas, modfd Bunman nstablty can dvlop, whch lads to th gnraton of plasma wavs at th lowr hybrd frquncy. 6
Ths wavs can hat th plasma at th shock wav front. At suffcntly larg Mach numbrs, fast lctrons njctd ahad of th front can xct cohrnt plasma wavs, th scattrng of whch lads to typ-ii rado bursts. Th prsnc of bands at th thrd and hghr plasma frquncy harmoncs n th dynamc spctra of typ-ii rado bursts rqurs an addtonal analyss of mchansms for th gnraton of solar rado bursts. In [10], a mchansm for th gnraton of harmoncs of th lctron plasma frquncy du to th dvlopmnt of xplosv nstablty n plasma pntratd by an lctron flow was consdrd; t was notd hat ths mchansm mght b rsponsbl for th spcfc faturs of typ-iii rado bursts. In th prsnt papr, w hav analyzd th mchansm for th xctaton of plasma frquncy harmoncs n typ-ii bursts. Th mchansm basd on xplosv nstablty n a systm of ntrpntratng lctron and proton flows, whch actually xst at th front of a collsonlss shock wav, turns out to b mor ffcnt than th altrnatv mchansms of harmonc gnraton n qulbrum plasma or mchansms of thr-wav ntracton (s [7]). Th rason for ths s that, n th mchansm proposd hr, mods wth dffrnt n ar xctd smultanously. In addton, th ampltuds of th lctrc fld harmoncs provdd by ths mchansm (.., upon th ntracton of lctron flows wth plasma) xcd thr typcal valus n sourcs of typ-iii rado bursts by mor than on ordr of magntud du to th contrbuton of on flows to th xctaton of plasma turbulnc. Condtons for th dvlopmnt of xplosv nstablty and ts saturaton du to th nonlnar frquncy shft causd by th dpndnc of th lctron mass on vlocty hav bn dtrmnd. Qualtatv stmats show that th ampltuds of dffrnt plasma frquncy harmoncs gnratd n th solar plasma ar comparabl. Th rsults of ths study can b usd to ntrprt typ-ii rado bursts gnratd n th solar corona. In addton, ntns lctromagntc fluxs gnratd n th cours of xplosv nstablty can b usd to ntrprt strong lctromagntc radaton mttd from th atmosphrs of stars and pulsars [16]. To conclud, t should b notd that, hr, w dd not tak nto account th lnar nstablty that can ars durng Bunman nstablty. As was shown n [17], th lnar nstablty s supprssd by strong RF flds;.., n our cas, th powrful RF fld arsng n th cours of xplosv nstablty compltly supprsss th lnar nstablty. 17-02-00308. ACKNOWLEDGMENTS Ths work was supportd by th Russan Foundaton for Basc Rsarch, projct no. 7
REFERENCES 1. V. V. Zhlznyakov, Rado Emsson of th Sun and Plants (Nauka, Moskow, 1964; Nw York, Prgamon, 1970). 2. S. B. Pkl nr and M. L. Gntsburg, Sov. Phys. Astron. 7, 639 (1963). 3. V. V. Zhlznyakov, Sov. Phys. Astron. 9, 191 (1965). 4. V. V. Zatsv, Sov. Phys. Astron. 9, 572 (1965). 5. V. V. Zatsv, Radophys. Quant. Elctron. 20, 951 (1977). 6. R. Z. Sagdv, n Rvws of Plasma Physcs, Ed. by M. A. Lontovch (Consultants Burau, Nw York, 1968), Vol. 4, p. 23. 7. V. V. Zhlznyakov and E. Ya. Zlotnk, Solar Phys. 36, 443 (1974). 8. B. Klm, A. Krugr, V. V. Fomchv, I. M. Chrtok, R. V. Gorgutsa, and H. W. Urbarz, Prprnt No. PRE-ZIAP 89-07 (Zntralnsttut fϋr Astrophysk, Potsdam-Bablsbrg, 1989). 9. I. M. Chrtok, V. V. Fomchv, R. V. Gorgutsa, and A. K. Markv, Astron. Nachr. 311, 55 (1990). 10. V. V. Fomchv, S. M. Fanshtn, and G. P. Chrnov, Plasma Phys. Rp. 39, 387 (2013). 11. V. L. Gnzburg, Th Propagaton of Elctromagntc Wavs n Plasmas (Nauka, Moscow, 1967; Prgamon, Oxford, 1970). 12. V. N. Tsytovch, Nonlnar Effcts n Plasmas (Nauka, Moscow, 1967; Plnum, Nw York, 1970). 13. J. Wland and H. Wlhlmsson, Cohrnt Nonlnar Intracton of Wavs n Plasmas (Prgamon, Oxford, 1976). 14. A. V. Gaponov, L. A. Ostrovsk, and M. I. Rabnovch, Radophys. Quant. Elctron. 13, 121 (1970). 15. Hgh-Frquncy Plasma Hatng, Ed. by A. G. Ltvak (IPF AN SSSR, Gork, 1983; AIP, Nw York, 1992). 16. V. V. Zhlznyakov, Radaton n Astrophyscal Plasmas (Yanus-K, Moscow, 1997; Kluwr, Dordrcht, 1996). 17. Yu. M. Alv and V. P. Sln, Sov. Phys. JETP 21, 601 (1965). 8