COLLECTIVE WAVE-PARTICLE INTERACTIONS

IN SOLAR TYPE IV RADIO SOURCES

JAN KUIJPERS

STELLINGEN

1. De voorstelling dat de door liet water veroorzaakte ribbels in het zand een soort

beeld /.ijn \an de watergohen mist. in tegenstelling niet Minnaert 's bewering, niet

altijd elke grond, lien bepaalde klasse stroomribhels heeft dezelfde golflengte als de

bii de vorming optredende /waartekrachts-waiergolven die bij gegeven stroomsnel-

heid stil staan.

Allen. J .R.L.. 1468. Current Ripple-.. Norili-Holland

l'ubl. C>.. Am-iicrvlam. p.34-3d

Miniuicrt, M . 1972. I)e natuurkunde van 'l vnie veld.

W.J. Thieme en Cie. Zutphen. p. 220-228.

2. De tijden-, solaire radiocontmua waargenomen fiber-uitbarstingen wij/en op hefvoorkomen van whistlergolven in de lage corona.

Uu proi'lM.hnt'1, hiK.fdsUik V.

3. De in Dwmgeloo op 20 december l q ? 4 . 12.34.32 s UT. waargenomen gebroken

fibers /;in te \erklarei door een kombinatie van de in dit proefschrift gesuggereerde

inechjtiisinen '.nor / eb rapaüonen en libei-uitbarstingen.

D n p r . u - l ' M - l i n l i . h o o f d s t u k k e n i l l • V .

4 Het optreden van breedbandige kortdurende tluxverminderingen in solaire t\pe IVdm radio-uitbarstingen kan worden verklaard door herhaalde injectie van bundelssnelle deeltjes in hel brongebied van üe radiostraling evenwijdig aan de rich1 ing vanh.et magnetische veld.

5. Bestaande modellen van l\ pc 111 radio-uitbarstingen gaan uit van een in voorwaartserichting sterk gepiekte bundel snelle deeltjes. Het is /eer de vraag of een aanzienlijkeanisotropie in de dc-eltjesverdeline. kan blijven bestaan tijdens de coherente opwek-king van Langmuirgolven.

Melfv.se. H.B.. W 4 . Solar. Pliys. 38 . 205.ZaiiM'v, X.X.. Niituik.iï. N.A. and Rapoport. V.O..1972. Solar Pl.yv 24 ,444

(•>. l'it polarisaticwaarnemingen van de solaire type 111 radio-uitbarsting op 10 maartl')73, 11.1').U3S UT. volgi een waarde voor de magnetische veldsterkte in hetbrongebied van 0,05 G bij een elektronendiehtheid van 1.1 . I09 cm"3.

De aanvankelijke evolutie van Je spectrale index van de R.Mitgenstraling en van üeinjectiesnelheid van Je eiv^rgienjke elektronen tijdens de zonncviam van 4 augustus1^72, b.2\ - 6.3') UT. wijst er op dat plasmaturbulentie een essentiële rol speelt bijhet versneliingsproces.

B r o w n . J .C . md H t n n j : . I ' , 1 ^ 7 5 . Astn>,'>!>\ v j . . te

s. De sneue deeltjes die verantwoordelijk /.i|ii vct-r de waargenomen radiovlammen inC\g X-3 kunnen, uitgaande van hei model van Peterson, worden gemaakt dooreenschokgolf tijdens zijn passage door een magnetische ties.

Peterson. ! AS.. 1°73. Nature 242. 1 ~.V

l). Instelling van universitaire toelatingsexamens naast liet bestaande eindexamen vanhet Voorbereidend Wetenschappelijk Onderwijs kan m het Nederlandse onderwijs-systeem niet meer inhouden dan een nieuwe regeling van een numerus fi:

10. De verdediging van het Nederlandse volk is meer gediend mei training van deburgers in otidermtinende activiteiten in geval van bezetting dan met het huidigebezit van duidelijk offensieve atoomwapens.

l\bert. T., 1971, Gevu ldl.>»e opwind. VVoher-.-Noordhoff, Gronmycn.de Lange, !!.. \Qn5. Het moderne ocrli't'isysteem ende vrede, H. Melissen, i

11. De bewapeningswedloop is een duidelijk voorbeeld van een ten koste \an het v.s-teem groeiende verstoring, een instabiliteit.

12. De brandslang- of Alfvéngolt'-instabiliteit kan zeer droog worden gedemonstreerdonder de kraan aan een waterstraaltje langs een hellend oppervlak met geringeadhesie.

J. Kuijpers 14 april 1975

COLLECTIVE WAVE-PARTICLE INTERACTIONS

IN SOLAR TYPE IV RADIO SOURCES

PROLiSCHRIM

TLR VI RKRUGlNc; VAN D! GRAAD VAN DOCTOR IN

m wtskfNDt: I N N A T I : I ' R W I - Ï [ - N S < MAPFI-.N A A N D I

RUKSl'MVi-RSiTt".IT II- I IR ICHT, OP Gl/.AG VAN

Dl R1CTOR MAC,Nil KTS PRO! . DR. SJ. GROi-NMAN,

VOI.t;i N'SBl S.LI'1TV-\N lUTCOI l.VCA VAN !;l KAMA

IN HIT OIM NBAAR M ^l RDI-DH'.l N Of MAXNDVG

14 APRIL 14/5 \)\ SNAMiUDXGSTi: 4.15 I LR

DOOK

JOANNES MARIA ELISABETH KU1JPERS

CJtBORKN OP I 1 SLPTIMB! R 194b IT KINDHOVEN

HRrKKI-RU I IINKW'JK Ü.V. I'TRl-VHT

PROMOTOREN: PROF. DR. H.G. VAN BUEREN

PROF. DR. M. KUPERUS

Wat baat dat u verstant soo wijs is en gheleertDat al de werelt dat venvondert acht en eert,En dat de Fame u ontsterfiijckheydt aan doetAls ghy des nachts alleen in't bedde slapen moet?

G.A. Brederode,Boertigh, Amoreus, en Aendaehtigh Groot Liedboeck, 1622.

Aan mijn ouders.aan Carla

CONTENTS

SAMENVATTING 9

L INTRODUCTION 111. Characteristics of stationary solar radio con tinua 112. Velocity space instabilities 14References 21

II. A COHERENT RADIATION MECHANISM FOR TYPE IV dm RADIOBURSTS 22

1. Introduction 222. Radiation from plasma waves 233. Gyro-synchrotron radiation 304. Conclusion 33References 33

III. A UNIFIED EXPLANATION OF SOLAR TYPE IV dm CONTINUA ANDZEBRA PATTERNS 35

1. Introduction 352. Instability mechanism responsible for continuum and zebra patterns 353. Observations in favour of the proposed mechanism 374. Determination of the magnetic field strength 435. Determination of the energetic particle density 446. Conclusion 44References 45

IV. A POSSIBLE GENERATING MECHANISM FOR INTERMEDIATE DRIFTBURSTS 46

V. GENERATION OF INTERMEDIATE DRIFT BURSTS IN SOLAR TYPE IVRADIO CONTINUA THROUGH COUPLING OF WHISTLER SOLITONSAND LANGMUIR WAVES 49

1. Introduction 492. Observations 513. Propagation characteristics of whistlers 524. Excitation 575. Coupling 606. Application to the type IV dm source region 647. Conclusion 69References 69Appendix 70

CURRICULUM VITAE 72

Prof. Dr. H.G. van Bueren en Prof. Dr. M. Kuperus hen ik zeer dankbaar voor deenthousiaste en kundige wijze waarop zij mij als student hebben ingeleid in deplasma-astrofysica en voor de vele inspirerende gesprekken tijdens mijn onderzoek. Dr. J.Rosenberg ben ik bijzonder erkentelijk voor zijn uitgebreide en waardevolle kritiek. Ikdank Dr. Ir. J. van Nieuwkoop, Ir. C. Slottje, Dr. T. de Groot, Dr. A.D. Fokker en deoverige teden van de radiogroep voor de vruchtbare discussies. Verder dank ik detechnische staf te Dwingeloo voor de kwaliteit van de waarnemingen en Mevr. L.H. deBruijn-Tappermann, Mevr. J.G. Odijk-Nijenhuis en Mevr. R. van de Klomp voor het type-en stencilwerk. Mej. C.F.M. Jansen en de heer Joh. Otten dank ik voor het opsporen vanti!.i9??_artike.1_en en de heren R. Staleman en J.A. Koopalvpor.de Verzorging vantekeningen en foto's.

De waarnemingen die de grondslag vormen van dit proefschrift werden mogelijkgemaakt door de Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek(ZWO).

SAMENVATTING

De buitenlagen van de zonneatmosfeer bestaan uit een ijl geïoniseerd gas, de corona.Tijdens een zonnevlatn komt binnen enkele minuten een grote hoeveelheid energie vrij inde vorm van zeer snelle deeltjes. Deze snelle deeltjes verstoren het rustige coronals plasmaen geven aanleiding tot een grote verscheidenheid van uitbarstingen van ladiostraling.

In dit proefschrift wordt een verklaring gegeven van zogenaamde stationaire type IVconünua zoals deze zijn waargenomen met de 60-kanaals radiospectrograaf, die thans teDwingeloo is opgesteld. Kenmerkend voörTdeze radiouitbarstingen is een aantalopvallende fijne structuren in een overigens geleidelijk in tijd en frequentie veranderendeachtergrond van straling. Vermoedelijk worden deze uitbarstingen veroorzaakt door deeniÊrgierijke elektronen die eerder, tijdens een zonnevlam, zijn versneld en daarna zijningesloten in een magnetische fles in het coronale plasma.

In hel inleidende hoofdstuk worden de •radiowaarnemingen van type IV uitbarstingenen de eigenschappen van de brongebieden gekenschetst. Tevens wordt ingegaan op denatuurkundige achtergrond van de micro-instabiliteiten die vermoedelijk een rol spelen bijde opwekking van deze radiostraling.

Hoofdstuk II handelt over de oorsprong van de continue straling. Wanneer Siielledeeltjes worden ingeschoten in een magnetische fles die aan beide uiteinden in defotosfeer verankerd is, ontstaat een anisotrope snelheidsverdeling van de deeltjes; eenzogenaamde verlieskegel treedt op. Ten gevolge van deze anisotropie kunnen Langmuir-golven worden opgewekt met een frequentie vrijwel gelijk aan de lokale plasmafrequentie.Omzetting van deze plasmagolvcn in elektromagnetische straling die het brengebied kanverlaten, is mogelijk door middel van geïnduceerde verstrooiing aan de ionen van hetthermische plasma. DU mechanisme wordt vergeleken met de uitzending van s>nchrotron-straling. Synchrotronstraling van zwak relativistische, deeltjes blijkt minder doeltreffend teworden opgewekt indien de plaatselijke plasmafrequentïe aanzienlijk groter is dan decyclotronfrequentie van de elektronen. De invloed van het magnetisch veld op deopwekking van Langmuirgolven blijft in hoofdstuk II buiten beschouwing.

• In hoofdstuk III gaan wij in op een gevolg van de aanwezigheid van een magnetischveld. De instabiliteit voor Langmuirgolven vindt plaats, ofwel in een aaneengeslotengebied in de bron bij de verschillende plasmaniveau's, ofwel slechts bij die niveau's waar:de plaatselijke piasmafrequentië een veelvoud is van de cyelotronfrequentie. In welke van?de hier genoemde vormen de instabiliteit optreedt, hangt af van de verhouding van hetaantal snelle deeltjes tot het aantal thermische deeltjes en van de verhouding tussenplasmafrequentie en cyclotronfrequentie. De waargenomen "zebra-patronen" en hetcontinue spectrum kunnen op deze wijze met hetzelfde mechanisme worden verklaard.

In hoofdstuk IV komen de "fiber-uitbarstingen" (intermediate drift bursts) ter sprake.Een "fiber" is een modulatie van de achtergrondstraling in de vorm van een emissierichelniet een absorptierichel aan de laagfrequente zijde. Zijn helling in het frequentie-tijdvlakligt in tussen die van de type III en de type II uitbarstingen. Juist deze eigenschappenleiden tot de suggestie dat een fiber wordt veroorzaakt door een verzameling vanwhistlergolven die op haar weg door het brongebied koppelt met ter plaatse aanwezigegolven van een frequentie ongeveer gelijk aan de plasmafrequentie.

In hoofdstuk V wordt dit mechanisme nader onderzocht. Als gevolg van verlieskegel-verdeiingen van de snelle deeltjes kunnen whistlers worden opgewekt aan de voctpuntenvan de magnetische boog. Zij kunnen zich voortplanten als "solitons" evenwijdig aan derichting van het magnetisch veld. Elektromagnetische straling kan ontstaan door dekoppeling vat) wlüulm met Lungniuirgolven mits de fasesnelheden van de Langminr-golven voldoende groot ^ijn en de richtingen van hun golsvectoren ongeveer samenvallenmet de richting van het magnetisch veld. Aan de hand van dit model is het mogelijk een--schatting te maken van de magnetische veldsterkte in het brongebied.

10

I. INTRODUCTION

This thesis contains four papers (chapters II - V) on the interpretation of stationary solartype. IV t .Jio continua and their fine structures. The observations have been made withthe 60 -channel;radio spectrograph at Dwingeloo on frequencies in between 160 and 320MHz (Van Nieuwkoop, 1971). Due to its high time resolution and its sensitivity thisinstrument is particularly suited to observe fine structures in radio bursts.

This chapter is meant to provide a general background to the subsequent papers. Insection 1 a brief review is given of the observations and the source conditions. In section2 we treat some gener/J features of the velocity space instabilities that are invoked for theexplanation of the observations. A general review on the occurrence of instabilities hi thesolar corona has been given by Rosenberg (1973).

1. Characteristics of Stationary Solar Radio Continua

1.1. OBSERVATIONS OF THE DYNAMIC SPECTROGRAMS

Solar continuum bursts in the "decimetre" wavelength region (200-1000 MHz) show ingeneral a large amount of fine stiuctures of durations of one tenth to a few secondssuperimposed on relatively smooth emissions over a broad band of frequencies("continua") which persist for a few to tens of minutes (Young et. ai., 1961: Thompsonand Maxwell, 1962; Kundu and Spencer, 1963). Some characteristic features of these finestructures have been mentioned by Kundu (1965), who has distinguished between1. fast-drift bursts. These occur in the frequency range 400-800 MHz, ofien in quick

succession, and are of very short duration (< 0.5 s). Their frequency coverageexceeds a value of 100 MHz and their frequency drift rates are-larger than 100 MHzs*1 in absolute value. A large proportion of these fast-drift bursts (> 20%) shows apositive Frequency drift rate in contrast with the well-known type III bursts. Also,

• generally the fast-drift bursts do not extend down to the metre band. Thereforethey cannot be considered as the high-frequency extension of the type III bursts

- (Kundu etal., 1961).2. sprays, consisting of a large number of fast-drift bursts of very short duration and

bandwidth (10 - 20 MHz) and occurring during one or two minutes. Sometimesthe bursts are associated with dark features of similar duration and bandwidth onthe low-frequency side- ("bursts in absorption"; Cf. the "broad-band short-livedreductions" in 5 below).

3. intermediate drift bursts or fiber bursts, the main observational characteristics ofwhich are described in chapter V of this thesis. Typical examples are shown in thefigure of chapter IV and in Figure 1 of chapter V.

A great number of fine structures have been observed in recent years with the Utrechtspectrograph in type IV continua between 160 and 320 MHz. Apart from the abovementioned fiber bursts Slottje (1972a, b) has distinguished between

11

4. zebra patterns, appearing as a modulation of the background continuum in theform of alternating absorption and emiss on stripes at different frequencies (seeFigures 1 4 of chapter III)

5. broad band short-lived redactions in intensity. We refer to chapter 11 for adescription (see the figure in chapter IV where they occur simultaneous'*.' with fiberbursts).

The generation of the continuum radiation is the subject of chapters II and III. Anexplanation for the zebra patterns is also described in chapter III and the intermediatedrift bursts are explored in chapters IV and V. The broad band sudden reductions arcnot dealt with in this thesis but they will be discussed by Benz and Kuijpers (1975) in aseparate paper.

1.2. SOURCE CHARACTERISTICS

The following general remarks on the physical conditions under which the type IV dmbursts are formed can be made:1. The decimetre-wavelength sources usually are situated near the flare region and at

the same place as the sources of the slowly varying component (the coronalcondensations) (Kundu, 1965). Their height is in between 0.06 - 0.3 R. (4 • 104 -2' 10s km) above the photosphere on observing frequencies of 1430, 340 and 200MHz (Krishnan and Mullaly, 1962; Kundu and Firor, 1961; Morimotc and Kai,1962). The available observations incicate no significant motion on frequencieshigher than 200 MHz (Morimoto and Kai, 1962). We confine ourselves to stationarycontinua on decimetre wavelengths and assume that the bursts originate from fastparticles that are trapped in a closed and stationary magnetic structure.

2. The very appearance of multiple fine structures in these bursts in comparison withother types made Kundu (1965) suggest that the dm continua are closely related tothe particle accelerating region of the solar flare. Since in such a region magneticenergy is converted into particle energy partly in the form of heating and partly inthe form of stochastic acceleration, the ratio of the gas pressure to the magneticpressure is expected to be large there (of the order of unity). This idea is inaccordance with the results reached in chapter III.4.

3. Since the bursts originate near coronal condensations, one expects high (coional)temperatures, > 106K, in the source region in conjunction with more or lesschromospheric densities (10 !0 cm"3, corresponding to a plasmafrequency of 900'MHz) (Billings, 1966). Due to such a high temperature radiation around theplasmafrequency can escape without too much weakening, the collisional dampingrate of the electromagnetic waves being proportional to the electron—ion collisionfrequency v ~ i\T3n (n electron density, T temperature) (Wild et al., 1963).

4. The continuum sources observed on 340 MHz by Kundu and Firor (1961) had acharacteristic angular size of 3' (1.3 • 10s km). However the appearance of suddenreductions (Cf. Chapter II.1) as observed with the Utrecht spectrograph around 300

12

MHz, implies that at least a considerable fraction of the emission is generated in aregion smaller than 3'104km. This upper limit is derived from the abruptsimultaneous weakening of the burst over a bandwidth of 100 MHz by a factor oftwo or more within 0.1 s. Such a relatively small extent of the source favours ofcourse the appearance of fine structures. For example, according to the mechanismpresented in chapter III, zebra patterns result from the exclusive production ofLangmuir waves in those regions of the source where the ratio of the plasma-frequency to the cyclotron frequency is integer. Clearly the patterns are smoothedaway unless the variation in plasmafrequency along a separate harmonic surface isless than the average cyclotron frequency. To produce an observable structure ofintermediate drift bursts according to chapters FV and V, the passing whistler wavepacket should overlap the major part of a surface of fixed plasmafrequency in thesource region.

5. The observed hard (> 20 keV) X ray emission which often accompanies a solarflare indicates the production of electrons with energies between 10 100 keV(Kane and Anderson, 1970). From direct particle observations at \ AU the numberof electrons injected into the interplanetary medium is estimated to be of the order1033 electrons between 20 — 100 keV for electron events (Lin, 1974). In a largeproton event the number of non—relativistic electrons can be as large as 1036

electrons above 20 keV. Moreover, during proton events the electron spectra extendsmoothly with a power law dependence out to 10 MeV (Lin, 1974).

The energy loss time of a non-relativistic electron (of energy 20-100 keV)due to collisions with the thermal electrons of an ionized hydrogen plasma isaccording to Brown (1972) given by

1.47- 108E3 '2

t r . s. ( 1 ,

E is the kinetic energy of the fast electrons in keV and n is the thermal electrondensity cm"3. Assuming that the fast particles are injected instantaneously a icwsrlimit for their energies can be found from the burst duration. In the model adoptedthe source radiates at the plasmafrequency. At an observing frequency of 300 MHzthe energy loss time of 100 keV electrons is about 2 minutes. Thus the injectedelectrons must have higher energies unless the electrons are injected repeatedly.

6. Since the particle density increases inwards from the corona towards thephotosphere by a factor exceeding 106, the collision frequency between a fastparticle and the particles of the thermal plasma increases by the same factor.Therefore fast particles that have their mirror points close to the photospherevery rapidly lose their energy through collisions. Thus an instantaneous injectionof fast particles into a closed magnetic configuration evolves quickly into adistribution which is depleted in fast particles at each point inside a certain conewhich is called the local loss—cone. Such a particle distribution consisting of anisotropic "background" thermal plasma and a superposed "loss-cone" distributionof fast particles can lead to an instability. For instance the system can be unstable

13

to the generation of Langmuir waves (chapter 11.2.2, chapter III.2) and of whistlerwaves (chapter V.4.1).

2. Velocity Space Instabilities

Let us consider the stability ofhomogeneous, stationary and coiiisioniess piasmas. Theword "coiiisioniess" means that binary particle interactions are neglected; the particlesare supposed to interact only collectively through "smeared out" electromagnetic fields(Van Kampen and Felderhof, 1967). In this case the plasma is described by the Vlasovequation.

Dt ~ 0 ' U )

supplemented by Maxwell's equations. (Here fj(r,v,t) is the normalized distribution ofparticles of species j . A plasma is called stable if a small disturbance does not grow. In thehomogeneous case the occurrence of instability depends on the distribution functions ofthe constituting species and the instabilities are called micro— or velocity spaceinstabilities. We are primarily interested in instabilities that are generated by non-thermalparticle distributions and we do not consider the so—called parametric instabilities thatare driven by the presence of a large amount of electromagnetic energy. Then the plasmais unstable if a small disturbance causes a large transfer of kinetic energy of the particlesto energy in the electric and magnetic fields (Bekefi, 1966).

2.1. GENERAL INSTABILITY CRITERIA

If the distribution functions are isotropic and monotonically decreasing with increasingabsolute value of the velocity, the plasma is stable against all possible disturbances bothwith and without the presence of an external magnetic field (Rosenbluth.. 1964). Theproof follows from Liouville's theorem and the conservation of energy:

The total kinetic energy, . j }dmjV2fjd3vd3r, reaches a minimum value if thelargest value of fj occurs at v = 0 and if the distribution functions decreasemonotonically with the absolute value of the velocity. OR the other hand, for agiven isotropic and monotonically decreasing distribution function any deviationwithin the constraint of incompressible flow in (r,v)~ space results in an increase inthe kinetic energy. Since the total (kinetic and electromagnetic) energy is conservedthe electromagnetic energy will decrease. Since we have assumed that no excess inmagnetic energy will decrease. Since we have assumed that, no excess inelectromagnetic energy is present, the system is stable.

Thus a necessary condition for the occurrence of instability is a deviation from isotropyor, in the isotropic case, a positive radial velocity derivative. Both possibilities arecontained in the single condition that at some velocity v a positive partial derivative in

14

velocity space exists in some outward direction n, where outward mean:. n.*v>0. Thus thedomain of instability is ch.'iracteri/ed by beams, populations thai drift with respect toeach other, currents or loss cones.

To achieve instability the above condition is not always sufficient. Necessary diidsufficient criteria for instability can only be obtained in specific eases, for instance in theabsence of an external magnetic field a necessary and sufficient criterion for electrostaticwaves to grow exponentially has been formulated by pen rose {l%0). If, and only if, F(u)has a local minimum at a value u = um such thai

--OO

. dF/du - F(u F(u,,i)du = du • • '"- > 0 , (3)

u um -' u - u,,/,2Jm

electrostatic waves that grow exponentially in time exist. Here F(u) =

•j'^PJ/d3 v f,(v) 5 (e-v u)with e an arbitrary' unit vector and cjp j the angular

plasmafrequency of species j .To elucidate criterion (3) we consider the following examples (Mikhaiiovskii,

1974):1. A plasma which is composed only of arbitrary isotropic distributions f,(v) = fj(v)

for the various species is stable to electrostatic perturbations despite the occurrenceof positive derivatives of f (see Figure la). In this case dF(u)/du =

S2:rutOpj fj (u)<0 and F(u) does not attain a local minimum.] The above property Is used by Benz and Kuijpers (1975) to explain the

broad- band short lived reductions. If beams of fast particles are injected along themagnetic Held direction, an initially unstable "loss cone" distribution becomes lessanisotropic and the original instability can be quenched.

2. Anisotropy is not sufficient for electrostatic instabilities to occur. For instance, itcan be seen from Figure lb that in the case of bimaxwellian distribution functions(different temperatures in perpendicular directions) no local minimum of F(u)f»\i*!t iC*f lrn C%}\ THprf>fns"P tnp njacma ic ctill cfaKlg tO elsCtrOSt2tiC per turbat ions.

In this case the- plasma is in general not stable to electromagnetic perturba-tions. If also a magnetic field is present, whistler waves can be excited due to ananisotropic distribution of particles (Cf. chapter V.4.1).

3. In this example we consider an electron distribution that is cylindrically symmetricwith respect to the velocity, f(v) = f(vj, v//), where the symbol // denotes thecomponent parallel to the axis of symmetry and the symbol i denotes theperpendicular component. Moreover we use the approximation of infinitely heavyions. Then if the function / dw/ f(v| , v/i) has a single hum in v|, the system is;

stable to electrostatic waves in directions perpendicular to the axis of symmetry(see Figure 1c), as has been shown by Mikhaiiovskii (1974).

However, if apart from a hot anisotropic population also a cold neutralbackground plasma is present (see Figure Id), the function / dv,,f(v^, v,,) has alsoa maximum at r [ = 0 and the plasma can become unstable. This possibility is usedin chapter II to explain the generation of Langmuir waves in type IV dm sources.

15

*'-«)

Fig. 1. The particle distribution f(v) is sketched in velocity space for a number of cases at the lefttogether with the integrated function F(u) (see Eq. 3)) at the right: a) an isotropic distributionconsisting of a cold part and a hot shell, b) a bimaxwellian distribution, c) a ring distribution and d) aring distribution superimposed on a cold background. The direction of the unit vector e along whichthe velocity component u is measured is drawn in each case at the left.

16

In the presence of a homogeneous magnetic field an instability criterion for electrostaticwaves analogous to the Penrose criierion Ibui more complicated) has been derived byMikhailovskii (1974).

2.2. RESONANCES

In the presence of a magnetic field resonant wave panicle interactions exist in thefollowing situations:1. First we consider waves that propagate parallel to the magnetic field direction. If

the waves have a longitudinal electric field component, the wave field isindependent of time in the reference frame moving with a velocity cj/k,,, parallel tothe magnetic field direction. Then net energy exchange can occur with particles forwhich the relationco k ; /v ;, = 0 (4)is fulfilled (co is the wr.ve frequency., k is the wave number, v is the particle velocityand the suffix ,'i denotes the coordinate of any vector along the magnetic fielddirection).

If the waves have a circularly polarized transverse component of the field,resonance can occur if the gyrating particle "sees"' the wave field with a frequencyequal to its own cyclotron frequency and with the appropriate sense of rotation. Thusthe resonance condition isco •k,7v,,= ± OJC (5)with the appropriate sign dependent on the configuration.

2. Let us consider an obliquely propagating wave of which the fields van.' in time t andspace r as exp ( i(tot-k-r)). Then the wave acts upon a particle which movesunperturbed in the magnetic field, with a time dependenceexp | i (co k, ,v,) t - i(k|V^/toc) sin (coct) [ =

exp { - i ( w l . ^ t N u J t \ (6)

with J N the Bessel function of order N (Mikhailovskii, 1974) (the suffix 1 denotesthe component perpendicular to the magnetic field direction). Therefore some sortof resonance takes place under the general condition

co-kz/V/, = NGJC , (7)

where N can by any integer.

The resonance condition (7) can be used to know the reaction on the emitting particledistribution when waves of different kinds are emitted. If a particle interacting with afluctuation quantum of the turbulent background loses energy AE to the wave, theenergy loss is (Kennel and Engelmann, 1966)

AE = mvAv = - htok. (S)

17

The loss in parallel energy is

AE = mv;Av,, = - h k v,7.

Here Inh is Planck's constant and m is the mass o\~ the particle.Let us introduce the pitcii angle a of the particle defined as the angle between thevelocity and the magnetic field direction. Noting that v sina Ac* = cosa- Av-Av// onefinds from Eq. (8) and (9) tire relation

^ ( C O S a...k v /W). (10)v sina coses

Alter substitution of the resonance condition (7) one arrives at

sina cosa

From this relation we conclude that the emission of waves with frequencies that are smallin comparison with the cyclotron frequency is attended by strong pitch angle scatteringof the particles. Therefore we expect that a large anisotropy in the non thermalcomponent (e.g. a "loss- cone" distribution) gives rise to an instability for whistler waveswith W « C J C (Cf. chapter V.4). From Eq. (11) it follows that the emission of Langmuirwaves at the plasmafrequency involves mainly energy loss of the emitting particleswithout much pitch angle scattering(coce/a>pe « 1 ) .Therefore we expect that a "loss- cone" distribution of fast particles is unstable toLangmuir waves if 3f(v)/3v. is sufficiently large and positive in some finite yelocity range(Cf. section 2.3 below and chapters II.2.2 and III.2).

If the whistlers are generated by particles with energies that differ appreciably fromthe energies of the particles that emit Langmuir waves, we expect both wave modes to beexcited at the same time. In the case of whistlers the resonant particles have parallelenergies (Kennel and Petschek, 1966)

where n is the background density, x = a>/ajce and B is the magnetic field strength.For B = 3G, n = 1.1-109 cm"3 and x = O.I we find E,,« 1.5 keV. Thus for a loss-conewith cosa = 0.5 a typical value of the resonant particle energy is 6 keV. A large amount ofthe electrons accelerated during a flare has an energy well above this value (section 1.2.5).If the Langmuir waves are generated predominantly by these energetic particles bothinstabilities can proceed at the same time.

A distinction is usually made between hydrodynamic and kinetic instabilities. Bothclasses arise merely from different approaches to the dispersion relation. In thehydrodynamic approach the system consists of different subsets of "cold" particles, eachwith a mean velocity Vj and a small dispersion vjj around it such that

lw-k -Vj l»kv T j , (13a)

18

or <n the presence of a magnetic field and vanishing Vj

Ico Nu;c i » k vTj. (13b)

Here k is real and OJ is complex. Then one can neglect the contribution of the poles in thedispersion relation and use the appropriate expansion of the factor (cJ-k/,v/,-Noi;c)"

1.Then the growing solutions that appear are called hydrodynamic instabilities.

If the spread in velocities is too large for the inequalities (13) to be fulfilled thecontribution of the poles in the dispersion relation cannot be neglected and resonantwave -particle interactions have to be taken into account. The hydrodynamic instabilitieschange into the so—called kinetic instabilities.

Somewhat confusingly the term resonant is also used in connection with hydrodyna-mic instabilities. When in two clearly defined subsystems separate waves exist with thesame phase velocity, a resonant hydrodynamic instability can arise. A necessary conditionis that one of the waves carries negative small signal energy. If the difference in total(kinetic and electromagnetic) energy of a plasma with and without the presence of asmall-amplitude wave is negative (which can occur if the system has a directed motion),the wave is called a negative energy wave, if such a wave grows, energy is released fromone subsystem and can be supplied to the positive energy wave of the second subsystem.For example, for the case of a low- density beam directed along the magnetic fielddirection and a background plasma the beam modes (suffix b) are characterized by(Briggs, 1964)

oj-kvb = ±cOpt,T beam plasma wave, (14a)and OJ kvfc == ±coct;, cyclotron wave. (14b)

Both the slow beam plasma wave and the slow beam cyclotron wave (minus signs) arenegative energy waves. Instability arises when one of these beam waves has the samephase velocity as any wave of the background plasma. (In the case of coupling of thecircularly polarized slow cyclotron beam wave with a cyclotron wave of the backgroundplasma both waves should have of course also the same sense of polarization in the samereference frame).

Summarizing this section we remark that the existence of some sort of resonance betweenthe particles of the non-thermal distribution and a wave mode of the (thermal) plasmafavours a strong interaction.

2.3 HIGH-FREQUENCY LOSS-CONE INSTABILITIES

The occurrence of an instability to Langmuir waves propagating perpendicular to themagnetic Field direction plays an essential role in the explanation of the continuumradiation and the zebra patterns (chapters II and III). Therefore we list here withoutdiscussion the electrostatic instabilities that can occur around the upper hybrid frequency

19

^ ^ p e + toc,V-) in the case of a cold background plasma and a superimposedloss-cone distribution of fast electrons (Cf. Figure Id). We assume that in the sourceregion ujee/ojpe « 1 so that the upper hybrid frequency nearly equals the plasmafrequency. Further we assume that the fractional density, •'-, of the fast particles is small(< v « l ) . In the case of perpendicular propagation the resonance condition (7) reduces tou)=Ncjce. Since Langmuir waves can propagate in the background plasma in a directionperpendicular to the magnetic field direction around the upper hybrid frequency, weexpect that the fast particles can excite Langmuir waves most easily if copi,=NcoCl..

This high frequency instability has been treated both in the hydrodynamicapproximation where the plasma is composed of well-separated hot and cold particledistributions (Cf. bq. (13)) and in the kinetic formalism (Pearlstein et. al., 1966:Mikhailovskii, 1974: Shimizu et. ai., 1974). The hydrodynarruc instabilities are thefollowing (Shimizu et. al., 1974):1. If ujpe =« Nojce and if the fast particles have a loss-cone distribution so that 8f(v)

/3v'|>0 in some finite velocity range for the combined distribution function, aresonant instability exists for nearly perpendicular wave vectors with(k,., / k)2 < 0.1 ('-'/N)'2- The maximum growth rate is attained at k,•,.• = 0 and isgiven by

7 * 0.2 (wpewceA)*. (15)

This is the instability described by Pearlstein et. al. (1966), which is used inchapter III.

2. If cope * Nwce and if the distribution of fast particles is sufficiently anisotropic sothat Ti/T// > 4.5 (N/^)2 3 , a resonant instability exists for oblique wave vectors.For non—thermal distributions one can roughly use the effective temperaturesdefined by KBT^,// = /d3vmv|,/,f(v) (KQ is Boltzmann's constant). The maximumgrowth rate is attained for kj = 0 and given by

7!*_0.4JWpiwCJ.<i)Ifl. (16)

candidate for the origin of zebra patterns in type IV Bursts due to the severe requirementon the anisotropy. In chapter III we suggest that the first instability gives rise to the zebrapatterns, since in the inhomogeneous corona this instability occurs at the levels where

p

To understand the absence of a high-frequency instability at the intermediatelevels we look for the possibility of a non—resonant instability (GJ = cape) at the levelswhere tope/a;ce = N-0.5. From the work of Shimizu et. al. (1974) one can conclude thatsuch a non-resonant hydrodynamic instability only exists (for perpendicularly propaga-ting Langmuir waves) if a>pe

A /cj c e > n \J2JA, where we have used ajp e /cjCe» 1 a s

above. This condition roughly coincides with the criterion for the appearance ofinstability throughout the source in the analysis of Pearlstein et. al. (1966) as has beenused in chapter III.

20

References

Bekefi. G. 1966, Radiation Processes in Plasmas. John Wiley and Sons, Inc., New York. Ch. 9.Benz, A.O. and Kuijpeis, J. 1975, to be published.Billings, D.L. 1966. A Guide to the Solar Corona, Academic Press, New ^ urk, Ch. H D.Briggs, R.J. 1964, Election-Stream Interaction with Plasmas, M.I.T. PTCSV Cambridce. Massachusetts

Ch. 3.Brown, J.C. 1972, Solar Phys. 25, 158.l'ane, S.R. and Anderson, K.A. 1970, Astrophys. J. 162. 1003.FC.mnel, C.F. and Engelrnann. F. 1966, Phys. Fluids 9. 2377.Ke.inel, C.F. and Petschek. ILK. 1966, J. Gcophys. Res. 71, 1.Kriinnan, T. and Muilaly, R.F. 1962, Australian J. Phys. 15, 86.Kundu, M.R. 1965, Solar Radio Astronomy, Interscience Publishers, New York, Ch. 8, 11.Kundu, MR; and Firor, J.W. 1961 Astrophys. J. 134, 389.Kundu, M:R , Roberts, J.A., Spencer, C.L. and Kuiper, J.W. 1961, Astrophys. J. 133, 255.Kundu, M.R. and Spencer. C.L. 1963, Astrophys. J. 137, 572.Lin, R.P. 1974, Space Sci. Rev. 16, 189.Mikhailovskii, A.B. 1974, Theory of Plasma Instabilities, I, Consultants Bureau, New York, Ch.

2,10,1!.Morimoto, M. and Kai, K. 1962, J. Phys. Soc. Japan 17, Suppl. A-1I, 220.Pcarlstein, L.D., Rosenbluth, M.N. and Chang, D.B. 1966, Phyt. Fluids 9, 953.Penrose, O. 1960, Phys. Fluids 3, 258.Rosenberg, J. 1973, Instabilities^ the Solar Corona Utrecht University (Thesis).Rosenbluth, M;N. •1964, in M-N. Rosenbluth (ed.), Teoria dei Plasmi, Rendiconti deila Scuola

Internazionale di Fisica="Enrico Fermi", XXVCorso, p. 137.Shimizu, K-, Todorbki, J. arid Sato, M. 1974, J. Phys. Soc. Japan 37. 460.Slottje, C. 1972a, Solai Phys. 25, 210.Slottje, C. 1972b, in A. Abrairii (ed.), Proc. of the Second Meeting of the Committee of European

Solar Radio Astronomers, Trieste, p. 88.Thompson, A.R. and Maxwell, A. 1962, Astrophys. J. 136, 546.Van Kampen, N.G. and Felderhof, B.U. 1967, Theoretical Methods in Plasma Physics, North^Holland

Publ. Cy., Amsterdam, I, XI.Van Nicuwkoop, J. 1971. A Multi-Channel Solar Radio Spectrograph, Utrecht University (Thesis).Wild, J.P., Smerd, S.F. and Weiss, A.A. 1963, Ann. Rev. Astron. Astrophys. 1, 291.

21

II. A COHERENT RADIATION MECHANISM

FOR TYPE IV dm RADIO BURSTS

JAN KUUPERS*

Stcrrekundig Instiiuut, Utrecht, The Netherlands

(Received 28 December, 1973; in revised form 21 February, 1974)

Abstract. An interpretation is presented of the decimetric type IV continuum with tine structure onMarch 6, 1972:and of the corresponding source region^ in terms of Cerenkov plasma radiation andalternatively of synchrotron radiation, both in case of coherent and incoherent generation. If themagnetic field strengthvin the source region is a few gauss, in a stationary situation a loss cone insta-bility develops which generates electron plasma waves coherently. The amount of energetic electronsrequired for consecutive induced scattering of the plasma waves at the thermal ions into electro-magnetic waves is less than in case of synchrotron radiation. It is concluded that the former mechanismprovides the explanation of type IV continua with fine structure such as intermediate drift bursts andsudden reductions of the continuum level.

1. IntroductionA type IV continuum..which occurred in^thedrn-band on March 6, 1972 between1150 UT and 1300 UTihas ;beenTexten^repeat briefly the principal observational details and subsequent requirements fora model for dm type iV continua with similar fine structure.

The continuum radiation consisted of at least three sub-bursts of 10 to 15 minduration and 'decay times' of about 5 min. In these continua mainly two kinds offine structures were apparent: intensity reductions and the so called fiber bursts. Theintensity reductions are broad band features lasting 0.1-2 s and occurring simulta-neously over 80-100 MHz within 0.1 s; the reduction amounted to a factor 2 to 3.Such a structured continuum has been observed on several other occasions withthe Utrecht spectrograph. It sometimes shows 'zebra patterns' (Slottje, 1972a, b).

The projected source of the March 6 emission, as observed with a one-dimensionalinterferometer, remained quasistationary (Lantos-Jarry, 1972). We assume that theradiation was caused by fast particles injected into a closed magnetic structure afterand during the flare of 1117 UT. The flare was associated with a proton event and itis therefore plausible that the injected electrons had relativistic energies, in excess of300 keV but mostly less than 1 MeV (Lin, 1970). We use therefore low energy particlesin explaining the continuum.

By far the strongest radio component was located near the optical flare region.Magnetograms indicate that the source was probably situated in a magnetic fieldpointing towards the Sun. Under the assumption that the radiation did not changeits sense of polarization on its path through the corona, the sense of polarization in

* This work was sponsored by the Netherlands Organization for Pure Research (Z.W.O.).

Solar Physics 36 (1974) 157-169. All Rights ReservedCopyright © 1974 by D. Reidel Publishing Company, Dordrecht-Holland

22

the source region was rigluhanded, i.e. had the same sense of rotation as electrons ina magnetic field.

From the observed abrupt changes in intensity, one finds as the upper limit of thesize of the responsible source 3 x I04 km. The maximum flux at 240 MHz amountedat least to 1C~16 erg cm"2 Hz"1 s"1 around 1223 UT (Droge, 1972). However duringthis maximum there were no reduction features of the continuum level by a factorof two or more and we cannot be sure that this emission was produced by the samesmall region. Strong reductions occurred at 1205 UT around 234 MHz, the 'missingfl^x' being 6.5.x I0" l 8 ergcm- 2 Hz"1 s~' (HHI, Kriiger, 1972). If we take for theeffective emitting surface 7x 108 km2, the effective temperature of the burst is atleast 10t0K. This strongly suggests a coherent emission mechanism.

The burst was absent at !69 and 530 MHz. The emission v,as concentrated roughlybetween 240 and 300-350 MHz and it could be tracediup to 410 MHz. From theinferred lower limit on the exten* in frequency, 100 MHz, and the upper limit :nsize it follows that the emission at the different frequencies cannot originate at theircorresponding plasma levels in the corona in case of density distributions exceeding5 limes the Newkirk values (Newkirk, 1967); the 200 MHz level corresponds to aheight of 0.15 -/?o in a coronal model with density 5 times the Newkirk values.

As regards the operating mechanism we discuss three possibilities:(i) induced conversion into electromagnetic waves of coherently generated

Cerenkovrplasma Waves,(ii) coherent synchrotron radiation in the presence of a background plasma due to

a population inversion of the fast particles,(iii) synchrotron radiation due to phase coherence of bunched particles.For the magnetic field strength we adopt values of 1-8 G. Values of 3-8 G have been

deduced from the observed zebra patterns in related continua (Rosenberg, 1972;Chiuderi et ai, 1973), and, perhaps more important in this case, many intermediatedrift or fiber bursts are present, each individual burst having a mean frequencyextent of 1-3 MHz. which leads to values of the magnetic field in the source regionof 1-3 G (Kuijpers, 1973).

2. Radiation from Plasma Waves

For any wave mode an effective temperature, reff, is defined according to its mono-chromatic intensity

8ir3c2

in the case where the direction {6, <p) of the ray and the wave vector coincide andhoi <§ kTcSr. Here /; = cfc/ttx k the wave vector, a) the corresponding angular frequency ofthe particular mode, c the velocity of light in vacuo, K and 2kh the Bolizmann andPlanck constants. In thermal equilibrium the effective temperature equals the truetemperature.

"""•"'.' " •' ' 2 3

Furthermore a number density, N(k), of plasmons of any particular mode isdefined in wave vector space according to

h.TcfT(k) - A'(k)/ico(k).

"Again this relation coincides with the appropriate quantummechanical definilion inthermal equilibrium.

2.1. INCOHERENT GENERATION • - r ,

Considering the case where the transverse waves are generated |byelectron plasmawaves, the effective temperature of the former imposeS"a lowerliminipon the effectivetemperature of the latter in the following way.

The number density of transverse quanta can never exceed the instantaneousdensity of the longitudinal quanta in the case of conversion processes for which eachingoing longitudinal plasma wave-quantum corresponds to one outgoing electro-magnetic quantum (e.g. scattering at particles or coalescence of waves of differentkinds) and situations such that the period of emission of transverse waves from thesource equals the time interval during which the longitudinal waves are generated.This is certainly true on time scales as large as the characteristic duration of thecontinuum outbursts, which is about ten minutes. If moreover.the ratio of the fre-quencies of the outgoing and incoming wave is two or less, the required temperaturefor the electron plasma waves should exceed half the temperature of the transversewaves, 5 x 109 K, with the proviso that the occupied volume in wave vector spacefor both wave modes is about equal. If on the other hand a single particle mechanismis governing the (Cerenkov) emission and absorption of plasma Waves, or in otherwords if the waves are generated incoherently, their effective temperature cannotexceed the kinetic energy of the relevant particles divided by the Boltzmann constant(Melrose, 1970a). Therefore for 300 keV particles the (extreme)' upper limit of theobserved radiation temperature is 4x 109K, (hardly the observed value), unless theplasma waves are generated coherently.

2.2. COHERENT GENERATION

Will the circumstances lead to an unstable situation in which coherent generationof waves can take place which grow proportionally to the amount of waves present?When a beam of fast particles is injected into a magnetic arch, loss cone distributionswill be set ap for the fast particles after a few trapping periods, the loss cones beingdetermined by the magnetic structure and the destructive influence of collisions.

It should be mentioned that loss cones can give rise to Harris instabilities atmultiples of the cyclotron frequencies (Harris, 1969; Baldwin et al., 1969). This kindof instability is invoked by Mollwo (1971, 197?.) in the explanation of a.o. type IVdm bursts. However, essentially he assumes a rather high value of the magneticfield stiength (o^/cOp^l), contrary to the case considered here.

For weak magnetic fields, characterized by c%/cop«l, the dispersion relation for24

electron plasma waves is

Q)2 = oip + o),2, sin2 3 + 3fr2r2..

!) is the angle between the wave vector and the field direction, o)p is the electron plasmafrequency, wn the nonrelativistic gyrofrcquency of the electrons and vle = (KTchnc)

il2

the thermal velocity of the electrons of the background plasma and Tc and mc theirtemperature and mass.

For a particular wave the resonant particles are those having velocity componentsin the wave vector direction equal to the phase velocity of the wave.

The effect of the magnetic field on the generation of plasma waves with wavenumber A' will be negligible whenever a>;//a>p<^l and AT,,^ 1 where rH denotes theLarmor radius of the resonant particle. For Cerenkov emission, a ' = k v , the twoconditions coi ,cide. Then in the source region, where a;p/a>,,> 10. coherent excitationof plasma waves will lake place when the instability criterion for the nonmagnetic case

cF- > 0du u=oi,k

is fulfilled, where F(u)=$f(\)5(u-k-\-lk)d3\ and/(v) is the velocity distributionfunction for the electrons (Krall and Trivelpiece, 1973). Consequently for loss conedistributions an instability may be expected for wave vectors non-parallel to themagnetic field.

For simplicity we proceed from a beam of fast particles with a density in velocityspace which is constant within a certain sphere of radius 1O10 cm s~J (correspondingto 30keV e'ectrons) at the time of injection. In a static magnetic field the resultantequilibrium density distributions will be constant along the flux tube and independentof the velocity in the accessible regions in velocity space, i.e. outside the loss cone(Roederer, 1970). The half-aperture a, of the loss cone is given at every point by

sin2 a 1U = r7"»

where Hc is the value of the magnetic field at the 'level' where collisions dominate.Performing the integration of the distribution function over two directions in velocity-space, one parallel, the other perpendicular to the magnetic field, and taking thederivative with respect to the third velocity component we find indeed that thedistribution is unstable against the excitation of plasma waves. Taking into accountthat in the instability criterion the total distribution function should be consideredincluding in this case an isotropic distribution of electrons with a temperature of106K an instability can occur only for waves that are not too heavily Landau-damped on the thermal background, say with phase velocities larger than 3 vte, i.e.1.2 x 109 cm s~'. In our example plasma waves are generated coherently if loss coneshave half-apertures larger than 7° and up to 80°, for velocities up to 5x 109 cms"1 .We conclude therefore that coherent generation of plasma waves in a directionpreferably perpendicular to the magnetic field (the upper hybrid wave) will proceed

25

quite naturally from an injection of fast particles into a closed flux lube with a weakmagnetic field up to heights where the field strength has decreased to 10 ~2 times thecritical field Hc.

2.3. SPONTANEOUS CONVERSION - - - - - - - . . _ _ _ _ _ _ _

As for the conversion of plasma waves into electromagnetic waxes coupling can occurdue to the existence of large scale gradients in regions where the approximation ofgeometrical optics 1? violated. The efficiency is about 3 x !0~B at 300 MHz for a densityscale height of 10'° cm, a temperature of ID6K .and. a broad angular spectrum ofplasma waves with a solid angle of 2K steradians and phase velocities of 1010 cms" 1

(Zheleznyakov, 1970).Coupling (nonlinear) of plasma waves with sound waves is not likely to be important

either since there is no reason to assume that ion-acoustic waves are present to anextent larger than the thermal noise level, because the ratio of the electron to the iontemperature is certainly not much larger than unity and therefore their damping rateis high (Fried and Gould, 1961).

In principle in a magneloplasma other low frequency wave; modes, such as Alfvenand magnetoacoustic waves, can be present and may be of importance here. Partic-ularly the observed fine structure can be taken as evidence for their existence, e.g.whistlers for the fiber structure (Kuijpers, 1973} and very low frequency waves forthe sudden reductions in a way analogous to the explanation of pulsating structureby Rosenberg (1970). We will not make an ad hoc assumption about their importancein relation to the continuum radiation and exclude them. So, scattering at particlesis the only remaining mechanism.

With the scattering process the particle orbits are perturbed by the field of theincoming (plasma) wave and the associated current density acts as the source for theoutgoing (electromagnetic) wave, in general one has to include the reaction of thesurrounding plasma on the perturbed particle motion, which also generates a current.While in the first case the current is proportional to the electric field of the incomingwa\e and is known as Thomsc - >r Compton scattering, in the second case the current(for weak fields) is proportional to the product of the shielding field and that of theincoming wave and is known as nonlinear scattering. The latter effect counteractsthe former because the shielding gives rise to a polarization cloud with a chargeexcess o fa sign opposite to that of the inner particle while both oscillate not inde-pendently in the field of the incoming wave. In general the degree of cancelling dependson the mass of the particle and its velocity.

If no external magnetic field is present, the scattering process is governed by theconservation relation

(j>, - w, = (k, - k,)-v.

Here co, and wl denote the angular frequency of the transverse respectively plasmawave, kr and k, their wave vectors and v the velocity of the scattering particle. In theweak field case, cyH/cyp<§l, considered here, this relation is also valid.26

ft?

1^ If plasma waves are converted into electromagnetic radiation through scattering

at particles of the (thermal) background plasma, a predominant role will be playedby the ions (Tsytovich, 1970). The reason is that (nonlinear) scattering at the polar-ization clouds of the ions (far more important than the (linear or Compton) scatteringat the ions themselves) dominates the net effect of scattering on the electrons Wherethe two contributions are equal to first order and cancel each other. Because|k,-vl<?|k,-v| the frequency difference between the outgoing and incoming wavessatisfies A(o<k,vn where r(l- denotes the thermal velocity of the ions. Further onaccount of Landau damping the range of possible phase velocities of the plasmawaves is restricted to'•.totjk[>vtc, where o)t~a)p. Therefore Aa)<(opvtilvl.e~2.3x

<f I0"2iop. Consequently the radiation would originate at the respective plasma levels;4 in the corona, apart from the bandwidth of the original plasma waves due to theI finite temperature of the sustaining plasma, e.g. <0.2cup. The efficiency for the con-r^ version of plasma waves into transverse waves through spontaneous scattering on§ thermakdensity-fluctuations (the. pplarizaiiQnr=clouds-ofkther-ions)iis-ofathe-orderj§ 4 x 10"5 far a scatteringiregion with linear dimensions: of 3 x 109 cm, a density of| 1.1 x 109 c m " 3 arid a temperature of 10^ K (Zheleznyakov, 1970).| The waves can scatter also on the fast particles of the beam. For relativistic electrons|g the scattering is essentially Cdmpton scattering (Kaplan and Tsytovich, 1969) and|j will dominate the non-linear scattering at the polarization cloud, in contrast withIJ scattering at the thermal electrons. Actually these fast electrons have left their polar-It ization cloud trailing behind and consequently they emit (Cerenkov) plasma waves.I The frequency difference between the transverse and longitudinal waves will be at-| most (jjil(aj,lktv—l) when the transverse wave and the particle are moving in the§ same direction, four times the plasma frequency for 300 keV electrons. However,:| due to the small fraction of fast particles, the contribution of this process to spon-j taneous scattering will be less important compared to thaLpf _the_ thgriMl ions

qjf " (Zheleznyakov, 1970). " " " " " ^ 7 7 ^ ~ ^ 7 ^ - r — - — - - . : Trr--_—^.

% 2.4. INDUCED SCATTERING

If Apparently the spontaneous conversion efficiency for the different processes is rather| | low being less than 10"4. However, the conversion of longitudjnal to transverse3 waves can be stimulated by transverse waves already present. Precisely through such=§ an induced-conversion the temperature of the produced waves can eventually reach~| the temperature of the original waves. The kinetic wave equation for scattering canIf in the case of homogeneity be written as (Tsytovich, 1970)

IIi +.

d3p d3k'1W

27

The indices l and / denote the transverse respectively plasma waves. % the particlespecies, p the momentum and /j(p) the distribution function in momentum spaceand u!,'(p, k, k') the probability per unit time that a panicle of species a andmomentum p scatters a plasma wave with wave vector k' into an electromagneticwave with wave vector k. The first term on the right hand side describes the in-duced scattering, the other two the spontaneous contributions.

The formula shows that the conversion rate for induced scattering of plasma wavesoiYthe thermal ions will be positive for conversion towards lower frequencies, sincefor any isotropic distribution the kernel will be proportional to (to,-a),) Cfjdp.

At the same time we see that induced Compton scattering at the fast electronstowards higher frequencies can only take place for a population with a positive radialderivative in velocity space. Actually, according to Kaplan and Tsytovich (1969).the fast particles should also have an anisotropic distribution. However, even in oursimple example, the induced scattering off the thermal ions towards lower frequenciesis positive while the induced Compton scattering off the fast, particles is zero, due tothe independence of the fast particle distribution on momentum. Consequently wereject Compton scattering off the fast electrons as a possible generating mechanismof the continuum radiation and will proceed to investigate the requirements for inducedscattering off the thermal ions.

According to Tsytovich (i970), the characteristic time for this process is less thanor equal to

for the case of v'ph/v,c4.(9 m-Jm,.)*12 which is certainly fulfilled here. Wl is the totalenergy density of the plasma waves. v'ph their phase velocity. The requirements forinduced scattering into transverse waves reduce to

T.VL igr < 3 x 10" crnand

where i^r is the group velocity of the transverse waves, TCOI, the effective collisiontime for the electrons and ions of the background plasma, the latter being a measureof the damping of the transverse wa\'es.

The first condition imposes a lower limit of 10"' erg cm"3 on the energy densityof the plasma waves, taking the vacuum value for the group velocity of the transversewaves, 7x 109cm s ' 1 for the phase velocity of the longitudinal waves, IO6K forthe temperature of the background plasma and 1.1 x 10q cm"3 for its density corre-sponding to a plasma frequency of 2n x 300 MHz. Choosing a value of 300 keV for theenergy of the electrons and assuming that they loose about one third of their energyas plasma waves within 103 s (the maximum duration of a sub-burst), then we arriveat a lower limit of 6 x 104 cm"3 on their density.

To satisfy the second condition the derived number density of fast particles should28

be as large as 6 x 10" cm \ the collision time being (Zheleznyakov, 1970)

Tu,ll*:7--'i:.(5.5«eln(1047"2''\H:''))

a: 1.4 x 10 2 s

for the same environment as before.Such a densitv corresponds to a value for the fraction of the background plasma

of 5x 10~4. It should be noted that for the mentioned densities and energies theweak turbulence condition, Wx 4.nkT, necessary for the validity of the kinetic waveequation, is easily satisfied. Fora sphere of radius 1.5 x 10" cm the required numberof fast electrons amounts to 1034.

The most plausible geometric configuration in which amplification will take placeis one in which the transverse waves move perpendicular to the density gradient,more cr less coinciding with a direction perpendicular to the magnetic field. Forin this case they will not enter those regions of the source where most of the plasmawaves have frequencies, lower than the incoming transverse waves. In these regionsinduced absorption would prevail and weaken the transverse waves.

2.5. POLARIZATION

As regards the polarization of the outgoing radiation, conversion into the extra-ordinary mode, for which the smallest frequency is 0.5 coH above the local plasmafrequency, is possible in case that the frequency of the original plasma wave is highenough. This requires a local bandwidth of the plasma waves of 0.05-0.02 wp for aratio uiplu)H m 10-30.

Although the emission of the extraordinary mode would be easy enough, a pre-dominance of the extraordinary over the ordinary mode is not to be expected becauseof collisional influence. In a weak magnetic field (u)H/wp<g 1) the absorption coefficientfor ihe two modes is

K = !± f f (?!iL_ __

for parallel propagation (Zheleznyakov, 1970). Here vcff denotes the effective numberof collisions and n the index of refraction. Consequently the influence of collisions onthe two different modes is not essentially different.

However, the converting mechanism itself favour? the production of waves of theextraordinary mode in the following way. When a particle with velocity v scatters ata plasma wave with wave vector k, the current produced is proportional tok,~(k,-v/c) v/cxk, for the thermal ions (Gailitis and Tsytovich, 1964). Due to thenature of the loss cone instability the plasma waves will travel preferably perpen-dicular to the magnetic field. Therefore the generating current will also be perpen-dicular to the magnetic field. The power radiated into a particular wave mode equalsthe work done by the electric field E of the outgoing wave on the current and istherefore proportional to E-k,. If the transverse waves are moving perpendicular tothe external magnetic field the electric field for the ordinary mode is parallel to the

29

magnetic field and that for the extraordinary mode perpendicular lo it (Krall andTrivelpiece, 197?). Consequently, only waves of the extraordinary mode would beproduced. Therefore the suggested mechanism is also consistent wiih the observedpolarization.

2.6. SUDDEN REDUCTIONS

If the continuum is indeed generated by the coherent excitation of plasma waves byfast particles and the consecutive induced conversion lo electromagnetic waves throughscattering off the thermal ions, a natural explanation of the reductions would beprovided by sudden absence of the conditions required for induced scattering orcoherent excitation. The first will hoppen when the nonlinear transfer time. T V / ,becomes too large, which happens when the energy density of the plasma wavesbecomes too small. In its turn the energy density of the plasma waves will drop ifthe coherent excitation stops. Maybe this could be achieved by low frequency wavestravelling through the source region, changing the velocity space distribution of thefast electrons adiabatically.

We conclude that coherent generation of plasma waves is likely to occur in a closedflux tube after an injection of fast particles. Besides efficient conversion into electro-magnetic radiation through induced scattering at the thermal ions easily occurs underthe constraints from the observed continuum.

3. Gyro-Synchrotron Radiation

The movement of the particles perpendicular to the magnetic field certainly favoursthe emission of synchrotron radiation.

3.1. INCOHERENT GENERATION

Firstly we investigate the incoherent emission. In case of weak magnetic fields, whenojHjtop<^ 1, k{ru > 1 and consequently one can neglect the influence of the magneticfield on the Cerenkov emission of electron plasma waves, the amount of energy fed bya single particle into (incoherent) plasma waves exceeds the synchrotron losses byseveral orders of magnitude (Zheleznyakov, 1970). Therefore if the conversion efficiencyfor plasma waves is larger than one percent, which means that the conversion shouldproceed in an induced way, the synchrotron contribution will be far less importantthan that of the Cerenkov plasma waves: even more so when the plasma waves aregenerated coherently.

Jf the continuum is generated by the synchrotron mechanism in an incoherentway, the reductions would have to be caused by sudden changes of the magnetic field.From the amount of flux absent during the intensity reductions we can get an idea ofthe number of electrons involved, using the results of Ramaty and Lingenfelter, (1968).Again we consider the case of a weak magnetic field, with mildly relativistic electrons.For a magnetic field strength of 3.7 G (a value allowing us to use their calculations),30

the observed intensity maximum at 240 MH/ (23 /„) agrees with the speclral maxi-mum of an isotropic velocity distribution o|" electrons of the form

" ( V ) - t 2 for l ^ H w i t h y = [ l - ( r 2 r ) J - 1 2 .

superposed on the coronal background plasma with 1.5 /"/, /"r = 0.2 i / is the frequency).This implies that the source is situated high in the corona at a plasma level of SO MHz,The sense of polarization agrees with the observed one. since mildly relativisticelectrons radiate predominantly in I he extraordinary mode.

Let us take as a characteristic vaiue for the observed flux density of the incoherentradiation, originating in a source of a diameter less than 3x 10" cm. a value of6.5x10 IK erg cm : H/ ' s ~ \ Then ! .4x lO 3 4 electrons with a Lorentz factor••-2.1 I are required, which corresponds to a density of fast particles in the trappingregion of !0"em \ more than 10 "2 times the background number density. Thisnun:her density is larger than thai required for induced conversion and the electronsare of higher energy. It is true that for a magnetic field strength as large as 9 G (themaximum value considered here) the required number density would he much less,as low as a few times IO4cm \ Bui one should realize that in the case of a localplasma frequency of 80 MHz the critical density of fast particles required for theinduced scattering of ph;-,ma waves into electromagnetic waves, reduces to a vaiueof the same order. Therefore we abandon the incoherent synchrotron mechanism asthe origin of the observed radiation.

3.2 C;>HERI-\I CilNI.RMlON

Let us now investigate the conditions foi the coherent emission of synchrotronradiation. In this case the observed reduction* would arise through sudden violation ofthe conditions for amplification. The residual radiation svould then be due to inco-herent idiation originating throughout the whole flux tube. There are two ways inwhich t;;e coherent excitation can be achieved, either through a population inversionor an amsoiropy in velocity space (Wild ct <//., 19631 followed by stimulated emission,or by bunching of particles (Takakura, 1956).

3.3. POPULATION INVERSION OR ANISOTROPY IN VFI.OCITY SPA< f-

For the first way of producing stimulated emission the underlying physical picture isthe following. In the presence of a background plasma of sufficiently high density, anelectromagnetic wave can cause grouping of the radiating particles through itslongitudinal electric component. Such an effect could not occur in vacuo!

For simplicity we consider a monoenergetic and isotropic system of relativisticelectrons., situated in a homogeneous background plasma. Then the maser effect ispossible if the following conditions are met (Kaplan. 19661:

30H4R31

5.3 x 10 "* n}'2

H

where », is the density o\' fast particles. //_. that oi' the background plasma and R theradius of ihe emitting region

The radiation would be centered around

1 8 n.

For the burst considered /„. —240 MHz , so n , , / ! ^ 1.3 * 10". Given a Reid strength of

3.7 G the electron densi ty is about 5 v if) cm \ cor responding to a plasma frequency

of 66 MHz. The requi rements for the fast electrons a re : yS-IO and / ; v > l 0 ' \ and

these again are much m o r e stringent than for the discussed C e r e n k m mechanism.

Therefore this mechanism is ruled out.

3.4. BLNCHED PAR11C1 ir.S

Coherent generation could also occur in vacuo if the radiating electrons are movingin littie lumps (Takakura, 1956). Then effectively the radiating system consists ofparticles with a mass and charge s times higher than the corresponding electronquantities, if.? particles are moving together within a volume of wavelength dimension.

However, without a continually working organizing mechanism maintaining therequired phase coherence of the electron motion, density fluctuations on the scale ofa wavelength will be destroyed very quickly. Already owing to the thermal velocitydispersion this happens within 10" s for a wavelength of 100cm. The bunchingmight be achieved in a plasma by whistler and electromagnetic cyclotron waves witha frequency close to vjn, ascending in the solar corona and generated in the form ofAlfven waves at the basis, or by electrostatic cyclotron (Bernstein) waves with fre-qMencies-W.hichiarei-a--muhiple-o^

Owing to radiation damping, an electron radiates ina'-"bandwidth•'

(2eW\<5n = 2 , = 5 x 10"' Hz at 300 MHz.

\ 3me" /

Consequently, for a group of 5 particles the bandwidth will be

— 1

.'line3 ' S -

If the absorptions are caused by destruction of the phase relation, sudden reducti m

features would appear in the continuum with a bandwidth of 5 x I0" 5 s Hz around300 MHz. For the observed bandwidth of 100 MHz the number of particles per groupwould have to be tremendously large, 2x I013, corresponding to a monoenergetic andphaserelated density of fast particles of 2 x 107 cm"3 . So the absorption cannot beexplained by this mechanism.32

Therefore, synchrotron radiation cannot be considered as an alternative to theCerenkov mechanism, since the number density required in the case of the formermechanism already implies the existence of the latter.

4. Conclusion

We have looked for an interpretation of the continuum radiation on the assumptionthai the magnetic field in the source region is weak (a),, w ^ I), evidence for which isprovided by the work on fine structure in similar continua. From the reductionfeatures an upper limit of the size of the source region was deduced. As a naturalconsequence of the trapping of the faM particles in a stationary flux tube, a kind ofloss cone instability developes generating electron plasma waves in a coherent way.Future interferometric observations of these continua can confirm this suggestion:presumably a double structure refers to the case of a large magnetic arch with losscones in the highest parts too small for an instability to develop.

The conditions for consecutive induced scattering are less stringent than for anexplanation in terms of synchrotron radiation, either coherently or incoherently.Therefore we propose Cerenkov wave generation and consecutive induced scatteringoff the thermal ions as the generating mechanism of dm type IV continua like the oneof March 6, 1972.

Acknowledgement

The author wishes to thank the members of the Utrecht radiogroup for their stim-ulating discussions.

References

Bald vin, D. E.. Bernstein, I B.. and Weenink. M. P. H.: 1969. in A. Simon and \V. B. Thompson(ed-> '. A(hu/iia in Plasinu Phmcs 3, lniersciencc Pjhlisheis, New York,

t hiudcn, <_"., Giachetti. R.. and Rosenberg, H : 1973. Solar Phyif337125".'Droge. F.: 1972, private communication.

••Fried, B. D. and Gould. \V. R.- 1961, Phys Fluids 4. 139.Gailitis, ^. and Tsyiovich, V. N.: 1964, SOL Phys. JETP W. 1165.Harris, E. C : 1969, in D. G. Wentzel and I). A. Tidman (eds.), Plasma Instabilities in Astrophysics,

Gordon and Breach, New York, p. 59.Kaplan, S. A.: 1966, Astrophysics 2, 221.Kaplan, S.-A. and Tsytovich, V. N.: 1969. Soviet Phys. Usp. 12, 42.Krall, N. A. and Trivelpiece, A. W.: 1973, Principles of Plasma Physics. McGraw-Hill Co. New

York, Ch: 8,9.Kruger, A.: 1972, private communication.Kuijpers, j . : 1973, in J. Delannoy and F. Poumeyrol (eds ), Proc. of the Thini Meeting of the CESRA,

Floirac, p. 130.Lantos-Jarry, M. F.; 1972. private communication.Lin, R. P.: 1970, Solar Phys. 12, 266.Mangeney, A.: 1972, in K. Schindler (ed.). Cosmic Plasma Physics. Plenum Press, New York, p. 185,Melrose, D. B.; 1970a, Australian J. Phys. 23, 871.Melrose, D. B.: 1970b, Australian J. Phys. 23, 885.Mollwo, L.: 197!, Solan Phys. 19, 128.Mollvvo, L.: 1973, Solar Phys. 30, 497.Newkirk, G.: 1967, Ann. Rev. Astron. Astrophys. 5, 213.

33

Ramaiy, R. and Lingenfelter. R. E.: 1968. Solar Phys. 5, 531.Roederer, J. G.: 1970. Dynamics of Geomagnetic-ally Trapped Radiation. Springer-Verlag, Berlin,

Ch. IV.Rosenberg, H.: 1970. Asiron. Astrophyx. •», 15" .Rosenberg, H.: W2, Solar Ptiys.25,'\$S.Slottje. C : 1972a, in A. Abrami (ed). Proc. ol the Second Met ting o/ the CLSRA. Trieste, p. SK.Sloltje. C : 1972b, Solar Phys. 25, 210.Slottje. C. !c'74, to he publishedTakakura. T.: 1956. Pnbl Astron. Sot Japan 8. 182.Tsytcnich, V. N.- |970. \onhnear L(lecf> in Plautni, Plenum Press, New York. C'h. VI, VH, VI11.Wild. J. P., Smeid, S. 1-., and Weiss. A A. l * \ Ann. Rev. Auron. A.urophys. !, 291.Zhele/nyakov. V. V.: 197Q, Radio Emh\ian ot the Sun <.md Plane;*. Pcrgamon Press, Oxford, §25, 26.

34

III. A UNIFIED EXPLANATION OF SOLAR TYPE IV dm CONTINUAAND ZEBRA PATTERNS

Summary. Both type IV dm continuum radiation and zebra patterns ian hau their origin in plasmawave-, at the upper hybrid frequency, which arc excited b\ a loss com* distribution oftast electronssuperposed on the thermal background 11" the inverse fractional densit) of last eli\ (runs surpasses UKratio of the electron plasma:frciiueiu> to the electron tulotron Iruqueiuy, instability only L-XISIS atthose places in the corona.where,this u t io is integer, thus giving rise to zebra patterns.

Sonic observations in favour' ofUhe present explanation arc discussed. Upper limit-, to tlu maunetn.field strength andthefastiiarticle density in the source aie derived from ihe observations

Key words. Solat corona - loss cone instability - type. IV cnt inua — magnetic iield.

1. Introduction

Zebra patterns constitute a characteristic microstructurc in type IV din annum:! (Slottjc.il>72a. b). Zheleziiyakov and Zlotnik i l l '71) explained these emission ridges throughgyro-resonance absorption of plasma waves at the levels where the plasma frequency andthe electron ••••cyclotron frequency are harmonically related; however this: mechanism iseffective only at rather small harmonic numbers (< )0). Afterwards the ridges have beenexplained through coalescence of electrostatic electron cyclotron (Bernstein ) waveswith the upper hybrid into electromagnetic waves (Rosenberg 1972; Chiuderi et al..1 c'73). On the other hand, an explanation of the continuum radiation can be provided byscattered electron plasma (Langmuir ) waves around the upper hybrid frequency whichare generated through a loss cone instability preferably perpendicular to the ambientmagnetic field (Kuijpers. 1974). The last two mechanisms cannot consistently merge intoone explanation of the combined phenomenon. Indeed, the zebra radiation should beproduced in a small region of the corona (the upper liybrid frequency must not vary morethan a fraction of the electron cyclotron frequency over the source region for the patternto exist at all), while the continuum radiation originates essentialh at the different

. plasma leveU in the corona.

Here an attempt is made to reconcile this contradiction on thr basis of a modifiedgeneration of the zebra pattern. In section 2 a unified mechanism is presented for theproduction of continuum radiation and zebra microstructure. In section 3 a fewobservations are described which can naturally be explained within the framework of thismechanism. In section 4 the determination of the magnetic field strength is discussed andin section 5 an upper limit to the number density of the energetic particles in the sourceregion is .obtained.

2. instability Mechanism Responsible for Continuum and Zebra Patterns

We consider a stationary magnetic arch in the corona which at each point is filled with a"cold" isotropic background component (with a temperature of 106 K) and a superposedloss- cone distribution of injected fast electrons (> 30 keV). The loss-cone distribution

35

is effected by the influence of collisions on trapped fast particles in the loweratmosphere, that are initially injected isotropicully at some point into the arch. We areinterested primarily in unstable electrostatic waves that move perpendicular to theambient magnetic field.

If the background distribution is sufficiently cold, so that kvx / GL\. « 1 and ifmoreover the reverse inequality is valid for the fast panicles, we can use the analysis ofPearlstcin et al. (1966). Here k is the wave number, vi the respective mean velocityperpendicular to the magnetic field and wc the angular electron cyclotron frequency. It isassumed that relativistic effects can be neglected. Thus for their results to be applicable tothe above particle distribution at a plasma frequency of 200 MHz. the magnetic fieldstrength is required to be within the limits 3 - 72 G(3.9 • 10"2 « w c / w p « 1). Here wehave used k~u>p/vp[lase with vphase * 10l ° cms"1 and have substituted (KT/'m)1^ for thecharacteristic perpendicular velocity of the background particles and IO10 cms"1 (30keV)as a lower limit to the velocity of the loss-cone particles (K Boltzmann's constant;T temperature; m electron mass; u;p angular election plasma frequency for the totaldensity).

Previously adopted values of the magnetic field strength in,thesource region are 1 8G5(Rosenberg^l972^K:uiipers, 1974). Consequently the ratio of the electron cyclotronfrequency ( 3 - 23 MHz) to the plasma frequency(^200: MHz) is small.

Then an instability exists for electrostatic waves perpendicular to the magnetic field.The character of the instability is determined by the value of coy H A / < c> where A is thefractional density of the fast particles (nfast/(nfast + n b a c k g r o u n d ) ) and U>UH = (wp

2 +(JJC

2 ):/l == Wp is the upper hybrid frequency. Two possibilities for instability arise:I. If UOH'VWC > 2/7T, waves are unstable at the upper hybrid frequency with a linear

growth rate 0.5 coy^-II. If uijH^/fJc •*"•• 2/TT- instability only exists if the ratio OJIJH/OOC is integer and larger

than one. In this case the unstable frequency again is the upper hybrid and thelinear growth rate is (coy H OJC<-/27r)1/2.

If the e—folding growth time of the instability is smaller than one half of the electron"cyclotron period, we expect the occurrence of the instability not to be influenced by themagnetic field. Substituting the above expressions for the growth rates one consistentlyarrives at the above stated criterion.

Consequently, if the magnetic field strength is small enough (and the fast particiedensity sufficiently high), at every level in the source region plasma waves will be presentaround the local upper hybrid frequency (radiation mode I). On the contrary, if themagnetic field strength is sufficiently high and the fast particle density low, plasma waveswill only be excited at those levels in the source region where OJJJH/^C is integer(radiation mode II). As the conversion of the plasma waves into electromagnetic wavesproceeds by scattering on the ions of the cold background, one observes at the eartheither a continuum (mode I) or a zebra pattern (mode IJ), if not smeared out by theeffects of an inhomogeneous magnetic field strength along the separate surfaces cop/coc = N.

36

3. Observations in Favour of the Proposed Mechanism

Some observed features which so far remained puzzling, can be explained readily withinthe context of the proposed mechanism:1. It has been observed by Slottje (1972 b), that the ridges in the zebra patterns are not

always equidistant in frequency at fixed times, e.g,, March 2, 1970, 14.17 - 14.20UT and August 14, 1970, 36.58.42s UT (Fig. 1). Bernstein waves at the higherharmonics are presumably generated by resonance with directly unstable lowerharmonics (Rozenberg. 1974). Therefore, if the ridges were produced by Bernsteinwaves at a fixed position, their frequencies would differ by the cyclotron frequencyat that point and equidistancy would be expected. In our model the ridges areproduced at different places where, in general, the magnetic field and thus theridge separation will vary.

2. On August 14, 1970 at 16.58 - 17.00 UT (Fig. 1) a zebra pattern was observed inthe frequency band 160 - 200 MHz-of which-.the ridges drifted towards higherfrequencies. After 16.58.16s UT, at which time the mean frequency separationamounted to 3 MHz, 67 - 7i ridges entered the observing frequency band of thespectrograph. In Rosenberg's picture all these ridges would have to originate at thesame region and fulfill the relation to = u>UH + NCJC for successive integers N.which apparently is not possible for the given numbers (o; = 200 MHz, tor = 3 MHz,N>67) .

3. The zebra patterns are causally related to the source of the continuum in whichthey are embedded, since the observed emission between the ridges is considerablyreduced in comparison with the continuum, e.g. June 29, 1971, 16.52.00s UT (Fig.2). In the proposed model this lack of emission naturally occurs if the sourcechanges over from mode I into mode II.

4. On the low frequency side of a zebra pattern often a frequency continuum ispresent, e.g. March 2. 1970, 14.19 UT, in Slottje (1972a). This radiation wouldnaturally arise in the higher regions in the corona where the magnetic field issufficiently weak, so that the source is radiating in mode 1.

5. Sometimes in the course of a zebra event one ridge is disappearing, while theneighbours approach each other and continue, e.g. August 14,1970,16.43.07s UT(Fig. 3) and June 29, 1971, 1652.02s;UT(Fig.2)f -Another intriguing observation which (if a zebra pattern at all) cannot be explainedby Rosenberg's mechanism is the observation of March 21, 1969, 13.38.40s UT(Fig. 4). The phenomenon consists of a series of boomerang—like ridges of whichthe direction in the frequency—time plane coincides with the arrow of time.Both phenomena can be explained in our model if more than one distinct surfaceexists with cjp/cjc = N for the same N in the source region. In illustration, this occursfor example if the field strength has a power—law dependence on height and theplasma frequency decreases exponentially with height in the corona (Fig. 5). Whenthe field is being compressed, the plasma frequencies at which cop/cjc = N for thesame N approach each other. Eventually the ridges disappear successively in time.

37

16.58.00 UT 16.58.20 UT

-160 MHz

- 2 0 0 MHz

- 16 58 40 UT - - 16.59.00 UT

-160 MHz

-200MHz

16.59 20 UT 16.59 40 UT

-160 MHz

-200 MHz

17.00.00 UT 1700.00 UT

-160 MHz

- 2 0 0 MHz

Fig. 1. Zebra pattern on March 2, 1970 observed with the Utrecht spectrograph. The figure shows theflux variations ( < 3 S with respecf to a floating zerolevel; range ± 1.7 dB) and consists of 60 channelswith a width of 0.9 M Hz each.

38

16.51.50 UT

-213 MHz

21- 253 MHz'-213 MHz

-253 MHz

16.52 05 UTPig. 2. June 29, 1971. The upper part shows the flux \ .uui inns (see legend ,il I ig 1). 'Iho lou i r p j i i slums the normal tlux measurements(threshold 10 2 2 W m"3 H / " ' : sensitivity 15 (IB; time constant 0.02s). Here the 60 channels ire divided into three groups which iarc slightlymistimed with respect to each other thus giving rise l<> apparent misconneetinns in flu zebu pa iun i . ' •!'' "!:1 : it) j:pt

r ii"^uf1 i

-160 MHz

-200 MHz

16.43.00 UT 16 43 20 UT

Fig. 3. August 14. 1970; flux variations.llj

-160MHz

-320 MHz

13.38.30 UT 13.39.00 UT

n t <'

Fig. 4. March 21, 1969; flux variations.

li, Ir I

o

pol-'ig. 5. The frequencies with cop = NOJC are the solutions of the equation 3 cOp/ojpO = 1.5Nojixj/fwpo i &i CJp/cjpO j) in the case that the electron cyclotron frequency varies with height Ii(measured in units of 7 . 104 km) as OJC = GJCO h"1 and the plasma frequency as OJp = oJpO exp ( -h/2). The left-hand side corresponds to the straight line and the right-hand side is drawn for N =24-44, where we have used a value of coco = O)p0/51 so tnat at h = 1 the plasma frequency is 220MHz and the magnetic field strength 4.2 G.

At compression of the field the curved lines rise and the cross-sections approach each otherand disappear successively,

42

4. Determination of the Magnetic Field Strength

We note thai the (requena separation of two neighbouring ridges in .1 /ehia pattern onl\remains the local electron-cyclotron frequency a , in the picture n\ Rusenberj (]L>72), ifthe magnetic field is not varying ovei the source

In I he ease that the harmonic number increases with increasing frequency an upperlimit to the magnetic field strength can be found. Then the frequency separation betweenneighbouring ridges corresponds to

wN+1 - u;N = N (GJC N + 1 • coc N ) + G;C N + ] > L0(. N + , (1)

on the assumption that thn magnetic field strength decreases with hciaht.The ridges of the zebra pattern on August 14, 1970 (Fig. 1) drifted monotonically

towards higher frequencies. Around 16.59.22s UT two ridges are merging. Therefore (Cf.section 3.5) the harmonic number has an extreme for these ridges.

Concerning the magneto hydrodynamic stability of the source region we note thatsince the beginning of the zebra pattern up to 16.59.22s UT at least 46 ridges can becounted on the high-frequency side of the disappearing ridge. Assuming that there are nosecondary reversals in harmonic number dependence of these ridges, we find that at leastfor one ridge N = 46. Consequently the ratio of the gas pressure to the magnetic pressurei3 takes a value of

a 2n KT ,top,-> 4 K T , .0 = ^TT7~ = (—E * — - = 1.4, n\

B /STT coc me K~'

where we have neglected the pressure of the energetic particle population and T = 10s K(B magnetic Held strength; n background electron density: c velocity of light). Since j3 > 1the source structure is to be expected unstable (Rosenberg. 1973) and the observed driftpresumably corresponds to a reconfiguration of the source region towards a stable state.that is u ^ l ^ of tile source (Tow frequencies)where stabilizingline-=^tynigeffectsare less-important.

t!ienv:s|nce-;a-'ndge-is-disdppeariniv"'te"-"ablive extreme in harmonic number is likely tobe a maximum so that the disappearance signifies a lowering of j3. Apparently in this casewe can use Eq. (1) to determine an upper limit to the field strength from the ridgeseparation on the low frequency side of the extremum (3.8 MHz at 16.59.22s UT) of 1.4Ga t 180 MHz. This field strength corresponds to (3 = 1.7 (N = 50), of the same order ofmagnitude as the previously derived value.Finally we mention that the above evolution of the source region may consist of astretching of the photospherically anchored magnetic loop in which particle energy istransferred to the magnetic field.

43

5. Determination of the Energetic Particle Density

The transition from zebra to continuum radiation can be caused by a loss of homogercemalong the various harmonic surfaces, that is. if for fixed harmonic numbers the plasmafrequency varies more than a value equal to the cyclotron frequency

Another potential cause lies in the transition from radiation mode I . into 11.characterized by u>p-''u\. = 2,'ir Consequent^, one then can determine the fractionaldensity of fast particles at the moment of transition for the different ridges if theharmonic numbers are known. For the example of August 14. 1970 (N = 50) thefractional fast particle density is of the order of IO"2. Indeed the corresponding densitywell exceeds the number density required for induced scattering of the plasma waves intoelectromagnetic waves (Kuiipers, 1974).

Finally we mention that in regime I the energetic particle pressure leads to an extracontribution

0 f « t = g ^ ) ^ . % ( 7 _ D > ^ E ^ ( 7 - 1), ( 3 )

where 7 is the Lorentz factor of the energetic particles. For the August 14 event thiswould lead to the value 0 f a s t =B 4 for 30 keV particles. In that case the evolution of thesource will be determined mainly by the energetic particles.

6. Conclusion

Both the continuum radiation and the zebra patterns in solar type IV dm bunts can beexplained on the basis of the instability described by Pearlstein et al. (1966). Thezebra-like patterns then result from the electrostatic instability at the surfaces in thesource region with integer ratios of CJP/COC if the fraction of fast particles is small andy p / w c j s large,;A eoRtm^case if the inhomogeneity of the magnetic field makes the plasma frequency vary morethan~ a • value e^ual to the average "electron cyclotron7 frequency along the differentharmonic surfaces.

Adopting the above explanation it is possible to determine in certain cases themagnetic field structure and evolution of the source and the energetic particle density.

Acknowledgement

1 would like to thank Dr. J. Rosenberg, Professors H.G. van Bueren and M. Kuperus, andDr. C, Slottje for their useful discussions and providing the observations.

I am indebted to the Dutch Organization for the Advancement of Pure Research(Z.W.O.) for its financial support of the observations.

44

References

Chiudcri, C , Giachetti, R., and Rosenberu. IF. 1<P3. Solar Phys 33 T>5Kuijpers. .!. 1974, Solar Phys. 36, 157. ' - •• •Pearkteiri, L.D., Rosen biuth, M.N.. and Cliaim, D.B. 1966 Plus Fluids 9 953Rownberg.J. 1972, Solar Phys. 25, 18S.Ro-.oiibfrg. i. 1973v:instabilities in the Solar Corona. Thesis. TtrcJit I'liivcrsity. p. 12.Rosenberg, J. 1974, privatecommunication.Slottjc, C. 1972a, Solar Phys. 25, 210.Slottje, C. 1972b, Proceedings of the Second Meetinp of tin- Committee of F.urnpean Solar Radio

Asironomers, Id. A. Abrami. Trieste, p. 88.Zhekv.nyakov, V.V. and Zlotnik. ILYu. 1971. Solar Phyv. 20. 85.

45

IV. A POSSIBLE GENERATING MECHANISM FOR INTERMEDIATEDRIFT BURSTS*

Jan KuijpersAstronomical Institute, Univcrsi'v of Utrecht, The Netherlands

Intermediate drift bursts (1) or fiberbursts can be seen on dynamic mdiospectra of theNERA -spectrograph in type lV-like continua in the range 200 320 MHz. Anindividual fiber consists of an emission ridge, typically 1.5 MHz wide, with an adjoiningabsorption edge at the low frequency side of the same width. The drift rate in theconsidered frequency band generally lies in the range from 2 in 10 Mil/ s"1 ,occasionally showing a positive drift (see figure).

As the generating mechanism of this phenomenon we propose nonlinear coupling inthe source region between radiation and passing wave packets of the whistlertypetravelling parallel to the magnetic Held.

Because of the unsettled nature of the associated continua two possibilities exist.First the radiation may be synchrotron radiation from relativistic electrons, escaping

predominantly in the extra ordinary mode at a level above the local plasma frequency. Apassing whistler (with frequency wh) can couple nonlinearly at its instantaneous positionto the transverse radiation (with frequency t) with small group velocity (e.g. near thecorresponding reflection levels). During the coupling t + wh ->• t', the original synchrotronradiation is upconverted in frequency, giving rise to the observed frequency profile.

Secondly, the continuum, radiation may be due to scattering of Cercnkov plasmaradiation (C) on particles or ion acoustic waves (p). The frequency of the scatteredtransverse wave is essentially the frequency of the plasma wave. If a whistler packet ispresent the plasma waves can couple with the whistler (6 + wh •* t). also giving rise to theobserved frequency dependence. **

In both cases the frequency separation of the emission and absorption ridges in a fibercorresponds to the frequency of the whistler. The last attains its maximum group velocityaround one third of the electron gyrofrequency (2). If we take this value as thecharacteristic frequency of the whistler, the magnetic field strength in the source region isapproximately 2 Gauss.

Whistlers can propagate in the form of discrete wave packets, solitons. with velocitiesbetween 21 to 31 times the Alfven speed, depending upon their amplitudes (3).

The measured drift rate of the fibers, - 2 to -10 MHz s'1, corresponds to 4500 km s'1

in an enhanced coronal model (lOx Newkirk). Indeed this agrees with the soliton speedfor a field strength of 2 Gauss and a density of 109 cm ~3, 4 - 6 • 103 km s'1 .

* reprinted from Prof. Third Meeting Committee European Solar Radio Astron.. J. Delannoy and F.Poumeyrol (eds.), Floirac, 1973, p. 130.

** Detailed calcinations of the coupling coefficients are in progress.

46

MHz

315

12.11.35 U.T. 12.11 45 U.T.March 6, 1972, The s t ruc tu re of the f ibers is clearly visible on the sensi-tive channels in the middle. (Lower part : normal channels, upper pa r t : c i r -cular polarization.)

We did noi explain how the whistlers are generated. Multiple evidence exists that theyan be edited b\ an amsotropic high velocih particle distribution with pieferrcd

\~elocitio perpendicular to the magnetic Held (4, 5), eg in the mirror points.MtenutiveK they can be excited by MUD-disturbances.

References

1 Young C W Spencer, C.L., Moreton. C.E. and Roberts, J.A.: 1961, Asttophys. J. 133, 243.2. Formisano. V. and Kennel. C.F.: 1969, J. Plasma Physics 3, 55.3. Tidman, D.A. and Krall, N.A.: 1971, Shock waves in collisionlcss plasmas. Cb. 4.2 v\Viley~Inter-

sciencc, New York).4. Kennel, C: 1966, Physics Fluids 9, 2190.5. Kai Fona Lee: 197K J. Plasma Physics 6, 449.

48

V. GENERATION OF INTERMEDIATE DRIFT BURSTS IN SOLARTYPE IV RADIO CONTINUA THROUGH COUPLING OFWHISTLER SOLITONS AND LANGMUIR WAVES

Abstract. I k possible generation of intermediate drift bursts m t \ |x IV dm mntiiuia throuul.u-.upling between uhistler \ m w , traveling along the magnetic field, ami Langrnuir waves, excited in aloss i.one instability in the soune region, is elaborated. We investigate tlie generati.in. propagationand coupling of whistlers.

It is shown that tlie superposition of an isntropic bai-kpound plasma of 106 K and a loss- tonedistribution of fast electrons is unstable for whistler waves if the loss-cone aperture 2a is sufficientlylarge (see 0^ 4); a typical vaiuc. of the excited frequencies is 0.l0Jce (.OJCL. is tlie angular electroncyclotron frequency).

The whistlers can travel upwards through the source region of the continuum in the form of solitonswith a velocity of 21.5 v^ (y.\ is the Alfven velocity).

Coupling of the whistlers with Langmuir waves into escaping electromagnetic waves can lead t>> theobserved intermediate drift bursts, if the Langmuir waves have, phase velocities around the velocity oflight.

In our model the instantaneous bandwidth of the fibers corresponds to a fmniency oi' 0.1 —0.4'Cx>ce and leads to estimates'of the. magnetic field strength in tlie source region. These estimates arc-in good agreement with those derived from the observed drift rate, corresponding to liL5 v^. if we usea simple hydrostatic density model.

1. Introduction

Intermediate drift bursts or "fiber" bursts (see Figure 1) have been observed in solar typeIV dm continua by Young et a!. (1961), Slottje (1972a) and Etganiy (1973): their driftrates are intermediate between type III and type II drift rates. Typically, a fiber burstconsists of two adjoining depression and enhancement ridges relative to the surroundingcontinuum with the emission feature on the high-frequency side. Both the drift r^te andthe ridge separation led to the suggestion (Kuijpers, 1*373) that fiber bursts arc generatedb\ whistler wave packets._ traveling through the source region ami coupling with thelocally excited Langmuir waves, which presumably cause the observed continuumemission (Kuijpers, 1974). - -

In section 2 the observations are reviewed. Section 3 deals with the nature and thepropagation properties of whistler waves. The excitation of whistlers is the subject ofsection 4. Tlie possible coalescence of whistlers with Langmuir waves into transversewaves and the efficiency of this process are determined in section 5. Finally, in section 6we argue that the model might apply to the situation in a type IV source region. In thispaper we neglect the influence of collisions.

We assume that the source originates from fast electrons, injected into a stationarymagnetic arch configuration and that the fast particles are, at each point, distributedoutside a local loss—cone thus giving rise to an instability for Langmuir waves.Subsequently the transverse waves are produced by induced scattering on the thermalions (Kuijpers, 1974).

49

12.30.50 UT 12 31.00 UT

- 160 MHz*^173"N234

- -315

Fig. 1. Example of intermediate drift ^bursts observed with the 60-channei Utrecht spectrograph on March 6, 1972. The figure shows the- fluxVariations (< 3 s with respect to a floating zcrolevei; range ± ! .7 dB). F.ach channel has a width of 0.9 MH?..

2. Observations

The observational characteristics cf intermediate drift bursts arc summarized in table 1.

TABLE I

Observed characteristics of intermediate drift bursts

Observing bandwidth (MHz) 950 - 500 (1 )

instantaneous bandwidth (MHz) 10

Single frequency duration (s) 0.2 0.6

Frequency drift (MHz s"' ) -10 to -50 (+19 to+ 25

Frequency extent ofsingle fibers (MHz) (20*) 50 - 150 (300*)

3 4 0 - 31Oc3)

1.75

0.25

iG*) -9.5

300*)

320

1.5 -

0.2-

-10

•- 200< 2>

- 3

0.4

t o - 2+6 to+ 2

(5*) 50 - 100

175 145(3)

0.5

0.25

-3.5

* the values between brackets are rarely observed.(1) Young eta!., 1961.(2) Slottje, 1972a, b; here the listed values refer to emission and absorptk>7i ridges separately.(3) Elgar0y, 1973.

1. In most cases the fibers appear as fine structures within a continuum burst whichlasts 5 - 5 0 min, although the fibers are sometimes observed as emission ndgesbefore or after the continuum (Young el al., 1961) Often the fibers cluster in timewith (sometimes nearly constant) intervals of a few seconds or much less betweensuccessive fibers and are vury similar within one group.

2. The absorption edge is typically on the low frequency side; according to Young etal. (1961) and Elgar^y (1973) the absorption edge is at the high frequency side insome cases, but with the Utrecht 60-channel spectrograph no convincing examplethereof has been recorded (Slottje> 1974).

3. The absorption and emission ridges begin, and end simultaneously (Young et al.,1961; Slottje, 1972b).

4. The majority of fibers have a negative frequency drift rate.GenerallS' the absolute value of the drift rate of an individual fiber decreases withdecreasing frequency. Also the drift rates and instantaneous bandwidths of fibersin different frequency bands tend to decrease with decreasing frequency: for thesingle frequency duration this is less clear (Cf. Table I). From the observed drift

51

5.-

rates Young et al. (1961) and Elgar^y (1973) deduced velocities for the excitingagent less than the electron thermal velocity under assumption that the emissionoriginates at the local plasmafrequency.

The emission ridges and the surrounding continuum are strongly circularlypolarized and in the same sense (Slottje, 1974).

3. Propagation Characteristics of Whistlers*

3.1. WHISTLER WAVES IN A COLD PLASMA

Whistlers are transverse waves with frequencies below the electron cyclotron frequency. Ina cold plasma and for frequencies well above the ion cyclotron frequency the dispersion

k(crn-i)

id"1

i d 2

i d 3

i

• • • • ' : \

i

i

t

/

/

-

0 0.25 0.50 075X =

Fig. 2. Relation of the real part of the frequency with the wave number for whistlers in an isotropicMaxwelJian plasma with T = 106 K, tope/27r = 300 MHz and cope/coce = 30 (curve 1), resp. 10 (curve2). The curves end at -7/co c e « 0 . 1 . The dashed curve indicates the coid dispersion relation. Thevalues of k. are in cm"1.

* See also the appendix for the meaning of the symbols.

52

relation for whistler waves propagating parallel to the magnetic field is (Kennel andPetschek. 1966):

<jj2 O j (CJ c e ~ Gj) ' - - - _ _ . . _ _ )

where co is the angular wave frequency, k the wave number, cjpe the angular electronplasmafrequency and coce the (positive valued) electron cyclotron frequency (see Figure2). Since in thecorona cjpC/oJce » 1. the phase velocity vp can be written as

V = V I YC] _ V l'^ « C (1 \P Ae l -v •'• J > - ^ v i v-,;

where x = w/coce, vA e = B / (4vr nc me) is the electron Alfven velocity, which istypically a velocity intermediate between the usual Alfven velocity and the velocity oflight. Consequently, it follows from Maxwell's law, that the wave character is dominantlymagnetic

__. yp

where E^ and Bjj are respectively the electric and magnetic wave amplitudes,perpendicular to the wave vector. (We use the Gaussian system of units in this paper).

3.2. INTERACTION WITH PARTICLES IN A FINITE TEMPERATURE PLASMA

For parallel propagation the whistler is purely transverse and circularly polarized in thesame sense as a gyrating electron. Therefore, the wave can interact with a particle of givenkind, if in the reference frame of the gyration centre of the particle the wave frequencyequals the corresponding cyclotron frequency and if in the same frame the wave has thecorrect polarization (cyclotron resonance). Thus, for electrons the resonance velocityV// parallel to the ambient field direction is deteimined b y s s s ^ T ^ ^ ^ ^

- : (3)

and for ions

, (4)

where the velocity^component V// = (v. B ) / 3 and k,, = (k.B) / B.Sin^TaiiclJfdirlf to>Eq; (3) arid (4^ the values of the particle resonant energies at

frequencies ajjpve Q. I £ope are much smaller for electrons than for ions, the damping in aMaxwelliari" plasma will be determined mainly by the electrons. In this case the dispersionrelation for parallel propagation is (Scharer and Trivelpiece, 1967):

53

, 2 1.2C K= 1 + —

k vu. V :- t .

where v l (,: =KrjTL. / me and Z t"--Z(i^ t) is the plasma dispersion lunUion (Fiicd ,nul

Conte, 1961) with argument \|it:= (oj-oJc e) / (k y t e \ /2) . For large values of the real partof the argument ^ r e l a t i o n (5) reduces to"the"cold plasma dispersion relation (1). ingeneral, a solution of F_q. (5) with real k is complex: CJ = tur + i7. with 7 < 0corresponding to damping. If the damping is sufficiently weak, that is if I 7 I « a;r and] 7 • « t o v . L . tor and ! cor Re Z I » I 7 Im Z i. one can easily Miivt icidiiun {5} for cj r

and 7. From Figure 3 it can be seen, ihat under corona! circumstances(Te = 106 K, w p e / 2n = 300 MHz, u)pc I <JQCC = 30 10) and in the absence of aloss—cone distribution the role of the cyclotron damping is so important that thepropagating whistlers will not occur above 0.5 w c c . On the other hand Figure 2 showsthat the relation between the real part of the frequency and the wave number is hardlyaffected in the accessible frequency regime ( y / tuce < 0.1).

Fig. 3. Cyclotron damping of parallel whistlers in an isotropic Maxwcllian plasma witlfT^lO6 K,GJpC/27T = 300 MHz and Cx^e/coce = 30 (curve \). rep. 10 (curve 2). The effect of an extra isotropichot component (nn = 10"3, V2 vj, - 101 ° cm s"') is showi shown by the dashed curves.

3.3. OBLIQUE WAVE VECTORS

In the case of propagation at an angle d to the magnetic field direction the "cold"dispersion relation is (Kennel and Scarf, 1969)

VP = vAe [ x (cos -3 - x) ]X/l. (6)

From this relation one can derive the well-known property of whistlers that the

54

direction of the group velocity ( — = kr^+ - —; ' denotes a unit vector) is confined to adk dk k ou

small cone with aperture 2amax around the field direction at the lower frequencies. (SeeTable ll:Cf. Siix, ll7<,2).

TABLE II

Confinement of whistlers to the magnetic field direction

a ( 1 ) a ( 2 )

"max "max0 15.8' 0°0.58 J3.3° 35°

<1) local maximum for lower k-valucs(2) value for k ^ ° ° (cos i9 -» x).

Since in the oblique case whistlers have an electric field component parallel to themagnetic field direction (Allis et al.. 19b3), such waves can also interact with particlesthroughGherenkpv—resonance, provided that to = k,,v,7. -They'are-therefore,subject toLandau damping if the velocity distribution is monotonically decreasing when integratedover the velocity components perpendicular to the magnetic field (see for a numericalexpression page 21 in Kaplan and Tsytovich (1973)). Moreover, now higher harmoniccyclotron interactions are possible at

u> k.v , = m OJCC (7)

for any entire positive number m, apart from the already mentioned fundamentalcyclotron damping at m - 1 (Cf. Eq. (3)). Therefore in practice the occurring wave vectordirections will certainly fall well within the cone given by cos § = x and the alignment ofthe group velocity with the magnetic field direction will be even better than suggested h\

-Table II.

The inhomogeneity of the plasma can influence the wave vector directions withrespect to the magnetic field and consequently also the damping rate. Whistlers willrefract towards the magnetic field direction during their passage through theinhomogeneous corona if the phase velocity at a given frequency or (from Eq. (6)) thelocal Alfveri velocity decreases along their path. On the other hand, in the reversesituation, a whistler essentially refracts away from the field direction over a distanceAr = 7T (2 sin d d ?n v^ / dr)"1 and will be damped.

3.4. SOLITONS

The absolute value of the group velocity of parallel whistlers in the allowed frequency

55

range (x < 0.5 ; Cf. section 3.2.) follows from Eq. (6) as

vG = 2 v A , l x ( l -xV1 | - . («)

Since the group velocity equals the phase velocity at x = U.5, one can expect stationarypulse like solutions roughly at

x * 0 ; 5 (vG * 0 , 5 v A e * 21.5 vA) . "••-'••• "(?)

in the absence of dissipation isolated whistler wave packets can occur in the form ofsolitons if the nonlinear effects are balanced by dispersion. For the case that the electionsobey the isothermal and isotropic equation of state and the ions are cold, Kakutani et al.(1967) showed with a fluid approximation and the assumption of quasi—neutrality, thatparallel propagating whistler solitons can exist for the following range of Alfvenic Machnumbers Mr\

m L + - + Hk<2 , (10)

<

( M s - I ) 2 2MA2

with Ms the acoustic Mach number with respect to a speed of sound equal to(Kg Te / (mc •:- m,))1^ so that Ms

2 = 2 M^2 / 0 where 0 is the ratio between kinetic andmagnetic pressure: j3 = S nj KR TJ / (B2 / 8rr). Since m; / mc » 1 and, in our case, ft <1 ,vvefind Ms

2 » 1 and we can use the results for a cold plasma (Kakutani, 1966;Tidman andKrall, 1971) for the soliton velocity v:

when for the last part of the ec;ualiiy we assume that the maximum wave field amplitudeB., < B (Cf. Eq. (9)) -t^^ :: : - ~+^- : - • _ ^ H ^ - : ,^ ., =•_•.,:: .The characteristic width of these pulses is (Tidman and-Krall. 1971)

arid the passage time r is given by

— ~ Af ~«Wce"ci) 2B-

Summarizing briefly the last section, we have found that whistlers can propagate slightlydamped under coronal circumstances at frequencies below 0.5 wce. The group velocitiesare directed around the field within a small cone. Whistlers focus towards the magneticfield direction if the Alfven velocity decreases along the path but refract away from it in

56

the reverse situation. Finally, propagation in the form of discrete wave packets or solitonsalong the field is possible with a velocity of 21.5 vA and characteristic values of theinvolved frequencies in between 0.25 and 0.5 toCL,.

4. Excitation

4,1 , LINEAR INSTABILITY

As has been shown by Bnce (1964) the energy change of a particle upon emission orabsorption of whistler waves satisfies

A\ML v , , O J C C •

A ^ = I - V = ± - £ - - (I4)

where -1W is the change in total particle kinetic energy and AW'i = mvj_ Avj_ is the changein transverse energy only. Fos the last part of Eq. (14) we have used k , = k, Eq (3) forthe electrons and Eq. (4) for the ions. From Eq. (14) it follows that the transverse energyof an emitting electron (AW<0) decreases. Consequently, anisotropic electron distribu-tions with an excess of average transverse energy in comparison with the average parallelenergy are expected to be unstable for the generation of whistler waves through cyclotronresonance. We therefore investigate the eventual generation of whistlers from a loss-conetype distribution of fast electrons superposed on the thermal coronal plasma.

For simplicity we make the approximation that the fast electron distribution fh isMaxwellian outside the loss- cone with half-aperture a

f|,(v) = n h (27rv h2 ) ' 3 / 2 exp (-v2 / 2 v h

2 )

for ! arc tan (vj / w,,) \ > a (15)

and f]j (v) = 0 otherwise. It should be noted that this distribution function is normalizedto n^ cos a, where ni, is the fractional fast particle density for a zero degree loss—cone.-Then the dispersion rclatioiffor paral lerpTqp^et al., 1972) ;:""': ^ " " ^ -1—^ : ^ - ^ - ^ ^ ^ ^ ^ - . i v - - . i - - e - - ^ : -

c2k2

Here Zh = Z(^h) w»th i|/h = {co - <x)ce) / (kvj, y/l cos a) , Cf. Eq. (5), and n^ is assumedto be much smaller than unity. From Eq. (16) the growth rate 7 is derived for the regionwhere 17! « c o r and b l « !c<jce - a>r | as in section 3.2 and is

7 = v t v h v h ^ c o r "

57

Re Z t 2 3 k c o j n h R e Z] , 2 n j , t a n 2 a . . . 3 n j , t a n 2 a i ^ 2 R e Z ^v t ^ p e 2 v h v h v h

In deriving Eq. (17) we used the cold dispersion relation Hq. (1) for the real part ot the

frequency. Examples o( the growth (damping) rate are shown in Figure 4 us a function of

frequency for a number o( different cases. Their global behaviour is readily undershmd

Irom the tullowmu simple leasomng With a characteristic velociU ol the unstable

Fig. 4. flie_ vertical axis indicates the whistler growtHfa^^^^both; in umtl of_the_electron "cy.^^qn-.fre.qt«My..jrte:iiuinbered curves correspond to the parametervalues nh = I0"5, \ /2v t = 5.5 • 10s cm s"1 , ^ 2 vh = 10 '° cm s"1 and

curve sectt uce

1234567

1.011.1248158

30

10

The dashed curves present part of the damping rates (absolute value of 7/OJCL.); the dashed curve at theleft is common to cases 1-6, the other corresponds to case 7.

58

p a r t i c l e s v = V|, c o s o . cine c a n d e r i v e a r e l a t i o n b e t w e e n t h e e n e r g y o f d i e ' " h o t "p . I I i i c l e s . t h e l o s s c o n e a i i ^ l c o a n d t h e u n s t a b l e f r e q u e n c i e s f r o m F.q. ( 2 ) a n d ( 3 )

j t L .u j t L )V|, c o s a ' v , , = c , . U S )

t O p. ' X J

First, since the interaction at the higher frequencies (x * 1 ) takes place with particles ofMii.il! parallel velocities (Fq (18)). where the distribution is dominated by the (stable)background Maxweihan. the whistlers are expected to be unstable below a certainfrequency in agreement with Figure 4Secondly, from Eq. (18) we expect that the growth rate becomes larger and the unstablefrequencies shift to higher values when the loss-cone widens, in agreement with thebehaviour of curves ' to 6 in Figure 4.Finally, when the ratio ujpc/wc i , increases. Fq. (18) indicates that the instability shifts tohigher frequencies relative to the cyclotron frequency, again in agreement with Figure 4(curves ? and 7).

We conclude, that the configuration of a closed magnetic arch, where fast panicleshave loss-cone distributions (ii|, - 10"1) m an otherwise i.sotropic background plasma,can lead to the generation of whistlers. The growth rate is very sensitive to the degree ofanisoiropy and since the value of sec a varies much more than the ratio tu p e ' u ; c c alongthe source tube, we infer from Figure 4 that the excitation of whistler waves isconcentrated in the lower parts (sec a>4) and in the frequency range 0.1 - 0.2 coL.efavourable for the development of solitons. When the whistlers travel upwards intoregions with decreasing field strength, their frequency rises relative to the local cyclotronfrequency. Therefore, eventually they become thermalized upon entering regions withsmall loss-cone angles where the growth rate becomes negative.

4.2. Ql'ASILINEAR DEVELOPMENT

We assume that non linear wave-wave interactions are not..of .importance for .the. actualdevelopment of the instability in comparison with the effect of the excited whistlers onthe fast particles themselves (quasilinear diffusion). Then the^ffiarin^ffect'4:f6ir-~these-low- frequency waves is scattering of the particles into the loss-cone with a characteristicisotropization time (Kennel and Petschek. 1%6)

(Aa)2 ^ vj_ v _B_2

7<H ~ D a ^ 3 2 t^7~ Bk2 K }

where Do is the pitch angle diffusion coefficient and B^2 / 8JT the wave magnetic energydensity per unit wave number. We have taken Acs equal to 7r/4 and used a pitch angle ofTT/3 for the original particle with velocity v. Now we expect that after the onset of theinstability the waves continue to grow until their energy density is so high that thequasilinear relaxation time rq j for the electrons becomes comparable with the growthtime of the waves (27)"1. For the values 7 = 10'3 CJCC, v= 101 ° cm s'1, coce/27r = 10 MHz

59

we find a value of 0.035 erg cm"2 us an upper limit for the maximum wave energydensity per unit wave number. With the above growth rate taken It) be constant the wavewill reach tins level from its thermal equilibrium level given byB k

2 / STI- * 4ir k2 KB T * 4.4 • I0*12 erg cm'2 (k ^ 0.05 cm"1 ; Cf. Figure 2) after atime

* 23/(27)=* 1.8 * 10"4 s.

When a whistler of fixed frequency travels upwards, the ratio co/ojt.e increases and,therefore, the parallel energy of the resonant electrons decreases (Hq. (18)). On the otherhand, the parallel energy of a particle that resonates with the wave at a given altitudeincreases when it travels upwards due to the adiabatic invariance of the flux through theparticle's orbit (Jackson, 1962). Consequently, the wave and the particle can interactonly over a very limited distance. We consider the simplified picture in which both thewave growth and the scattering of particles is confined to one and the same region o\eiwhich the loss cone parameter sec a varies from infinity to VlO (Cf. Figure 4). Withinthis distance L the magnetic field strength varies 10% and we take L = 10' km. Thepassage time of a whistler through the unstable region is

"pass = L / vG * 0.2 s,

where we have used VQ = 21.5 vA (Cf. section 3.4) and cjp e /cj c e = 30.

Finally we estimate the bounce time of the fast particles in the magnetic loop to be

•* bounce ~" ' s-

Then since r m a x « T pass*^ 7 " bounce- t n e instability evolves in the following way.First the whistlers reach their maximum amplitudes and scatter the fast particles in theunstable region into the loss-cones. Subsequently these particles escape within 0.01 stowards the lower atmosphere where they lose their energy. The excited whistlers travelupwards and leave the unstable region after a time T p a s s = 0.2 s. Meanwhile the unstableregion is refilled by particles from above. However, as long as a major part of the whistlersis present in the unstable region, these particles are scattered immediately into theloss—cones and enhance the wave amplitudes slightly. Only after the disappearance of thewhistlers from the scattering region renewed generation of whistlers can take place. Thiswill happen if r p a s s « T b 0 U n c e -

We therefore expect the whistlers to be generated in a periodic manner with timeintervals exceeding r paii « 0.2 s.

5. Coupling

5.1. ADMISSIBILITY

We are interested in a thiee-wave coupling process: a whistler wave couples with aLangmuir wave to produce a transverse wave above the plasma frequency. Both in therandom phase approximation and in the case of fixed phases, the necessary conditions forcoupling are (Tsytovich, 1970)

60

ami (20i

which can be considered as conservation of the total energy and momentum of thecoalescing ptasmons. Here the index w indicates the whistler, C the Langmuir wave, t theescaping transverse wave and GJJ is the frequency related to the wave vector k, throughthe linear dispersion relation for wave mode j .

To solve Eq. (20) for a range of relevant values of ojg, cow and suitable directions ofthe different wave vectors, we proceed along the line described by lloijer andW'ilhclmsson (1970). I 'sing the notation OJ+ ^ u\v+u>t;,k+ = ku + k^ and adopting a certainvalue lor ky and t?vv (I3J is the angle of the wave vector ot kind j with the magnetic fielddirection), we examine graphically whether the curve formed by the pairs (k+,u;+)intersects the appropriate dispersion curves kt(cj+) for a sufficiently large interval ol dx

so that we can be sure that i?+ is equal to t9t at the point of intersection for some i9t- Weuse the whistler dispersion curve for parallel propagation as drawn in Figure 2. To obtainthe dispersion curves for the Langmiiir waves and for the two escaping transverse modeswe solve the well-known dispersion equation derived in the continuum approximationand for infinitely heavy ions (Denisse and Delcroix, 1961)

{1 a V - X H ( l - n 2 X I + Y L ) - XI j ( l -n2)( l Y L ) - X » =

= Y T2 ( l - n 2 ) ( i - V - X ) (21)

for wave numbers at given frequencies and a given direction d with respect to themagnetic field (see Figure 5). Here n = ck/o; is the index of refraction, Y = cjce/oj, X =ojp^/uj2, \ \ - Y cos i9. Yj = Y sin d and a2 = 3 v t e

2 /c2 . For the coronal conditions T= 105 K, cjp e /ojc e = 30, the dispersion curves so derived are plotted in Figure 5 forvarious angles &. Since for cjp c /ajc e = 30 the whistlers are strongly damped by thethermal plasma above CO/CJC6 = 0.37 (Cf. section 3.2.), the maximum allowable wavenumber is 0.056 cm*1. Since moreover k+ > kg - kw, it can be seen in Figure 5 thatcoupling can only take place if the longitudinal wave number is small enough, that is ifthe frequency of the Langmuir wave is very close to the plasma frequency. For the valueskg * 0.06 and #g = 0°, 10° and 20° the three curves k+(co+) for various whistlerfrequencies up to o)/cocc = 0.37 are drawn in Figure 5. Because the wlustler frequenciescannot exceed a value of 0.5 cocc under coronal conditions, clearly only the ordinaryelectromagnetic wave can be produced and this only then if the angle betweenlongitudinal and whistler waves is rather small (< 10°) (see Figure 5).

The amount of energy radiated in a particular mode equals the work jk (" } ' Efc done bythe electric field of the outgoing electromagnetic wave E^ on the nonhneaily (bycoalescing whistler and longitudinal wave) generated current, j k ^ - Considerinp, only theresponse of the electrons in a cold and magnetized plasma, we arrive, after Fouriertransformation and expansion of the fluid variables in the wave electric fields in a way

61

Mem"')

10

Fig. 5. Solution of the dispersion relation for Langmuir and transverse em waves for variouspropagation angles $ and frequencies in between Gjpg - 0.5 OJCC and Wpe + 0.5 COre. The escapingelectromagnetic branches are marked with symbols O (ordinary) and X (extraordinary). The dottedcurves are examples of the curves k+(0J+) produced from parallel (t?w

= '80°) whistlers with variablewave number up to x =0.37 {JiLO^ > - 0 . 1 ) and one fixed Langmuir wave with k£ = 0.06 and i5g = 0°and, resp., 10° and 20°. The values of k are in cm"1.

analogous to Tsytovich (1970) for an isotropic plasma, at the following result for thenonlinear current:

(2):_ ^pe2e- r d3k, &wx d3 k2

8 CJ c

5 (k - kj - k2)

me

\ -J - (I + A , ) : E k l (k2 - A) : k(E k ,

, ( E k l - A 2 : E k 2 ) + — B , : E k2) + • k,

(22)

Here the indices 1 and 2 refer to the coalescing modes while the outgoing wave isindicated without indices; the index kj is an abbreviation of (kj.Wj), I is the identity and

62

1 1ihe operators A and B lake the following form on the basis (-/^<x + iyl. — (x IV). z)with z parallel to the constant external magnetic field:

/ ( X j - 1 ) " ' 0 0 \Aj = I 0 • fxj + 1)"' " 0 I.

\ 0 0 0 /

0x(x+ir'

0

where, as before x = GJ/GJC. If the external magnetic field vanishes, the operators A and Bvanish also, and Eq. (22) reduces to the result (2.41) in Tsytovich (1970).

In the case of parallel propagating whistlers (k.A = (0.0,ktt ) and E k w = (0, E|<w, 0))and parallel .Langmuir waves (kg = ..(0,0,ke).sand,,Ejc6! = (O.OJE^)),no transverse waves ofUie ofdin^yjfrri^Opposite-tothat ofthe" whistler) are generated, as follows from inspection of the explicitform'-'nf (jkt " kt) ••(using Eq. (22)).. At the same time, it follows that waves in theordinary mode can be produced from parallel whistlers and oblique Langmuir waves,provided the conservation relations (20) are satisfied.

Combining the results of the previous paragraphs we conclude that the coalescence ofparallel whistlers with Langmuir waves can only lead to radiation in the ordinary modeand in directions non—parallel to the magnetic field.

5.2. EFFICIENCY

To estimate the coupling efficiency of a Langmuir wave and a whistler with theappropriate wave vectors (Cf, previous section) into an electromagnetic wave, we use thegrowth relation derived for isotropic spectra of the coalescing mod-"- and averaged overthe angles of the outgoing waves (Kaplan and Tsytovich, 1973, p. 284)

L^IEL y/2—gp2 - W k t ! ^ k * U 3.1 • 10"' w 4 Wk« erg cm'2 s'1 (23)

withHkxv f kg sk-Witjis^the total wave energy density per unit wave number for mode j .We have used in the right hand part of the equality ojpe/2rr = 300 MHz, top e /cj c e = 30arid k = Wpe/c.

In the present case we have to compare Eq. (23) with the production rate ofelectromagnetic waves from induced scattering of the Langmuir waves on the thermalions (Kuijpers, 1974). This scattering process is described by Tsytovich (1970, Eq. (8.61))

63

9 t(24)

Owms to tlie explicit dependence on \\\\ o\ the nulil hand pan ihe total tonver:iionefiicienc\ depends sen.Mtively on the detailed geometry o! the source legion, l'licicture acoinpan&on with Eq. (23> becomes ruther arbitrary. On the other hand the amount of

"radiation produced according to Eq. (23) can be compared direcih with the observedflux values. Tills will be done m >ection 6.2.

If the continuum radiation is the result of induced scattering of th-j Langmuir waveson panicles, harmonic generation by coalescing Liingmuir waves is. of minor importancein comparison with it and therefore also with the whistler coupling. The angle- averagedgeneration rate of first harmonic radiation from isotropie longitudinal waves with phasevelocities smaller than the velocity of light is (Kaplan and Tsytovieh. 1973, p- 284)

a t 5 nc mc c5 Ak, •'

Comparing Eq. (23) with Eq. (25) the ratio between the growth rates is found to be ofthe order

(26)

for ilkg = kg, Ak{ = ojpe/c and on the assumption that the Langmuir waves have the samewave numbers in both reactions. Apparently, the coalescence of Langmuir waves withwhistler waves is more efficient than the mutual coalescence of Langmuir waves intoelectromagnetic waves under coronal conditions where cjpe/d;ce is typically of order 30,

6. Application to the Type IV dm Source Region

6.1. LANGMUIR WAVE SPECTRUM

As we have stressed earlier (section 5.1), coupling of longitudinal waves with whistlers canonly lead to an observable structure in dynamic radiospectra if a substantial fraction ofthe Langmuir waves is concentrated around small wave numbers (i.e. high phasevelocities). Since the Langmuir waves are excited by fast particles, initially their phasevelocities are smaller than the velocity of light. But we know that in a Maxwellian plasmainduced scattering of longitudinal waves into longitudinal waves leads to a degradation ofthe Langmuir wave spectrum towards lower frequencies and higher phase velocities(Kaplan and Tsytovich, 1973). Moreover, in a magnetic field this induced scatteringresults in a "condensation" of the Langmuir waves parallel to the magnetic field (Kaplan

64

and Tsytovich, 1973) due to the angle dependence of the approximate dispersion relationfor the weak field case

GJ3 = cjpe2 + 3k2 vu. : + CL>CC2 sin2 t?. (27)

This effect occurs for phase velocities exceeding \/3 v.e w p e /c j c c s= 2.6 • 1010 cm s*!,and is therefore important in our case where the coupling takes place with Langmuir waveswith phase velocities of the order of the velocity of light.

To find the actual shape of the longitudinal wave spectrum we compare the inducedscattering rate of longitudinal into longitudinal waves on one hand with that of longitudinalinto transverse waves on the other hand. These processes arc described by (Kaplan andTsytovich, 1973)

3wj = _ * ^ wkj awjj3t 108 neiriiVtc4 9 kg ^ '

and

3t 108 n em;v t

Consequently the ratio of the transfer rates is W g / Wjj-t, if we neglect Wk| toco rnparison• • • . . . • , . £ . . • . . :-.. ; o t - . , • • . .•' .;?..-:',.._.^*£-.;}.,-:.;=i:'j::. ,

with kj; 3 Wjcg"7 3 kg. Since W ^ g » WjftJ- the Langmuir waves will be concentrated, at

high phase velocities and, from the above discussion, along the magnetic field direction.

6.2. DIMENSIONS AND ENERGETICS OF THE WHISTLER WAVE PACKET

Pursuing the suggestion of the soliton—like appearance of the whistlers with a length andenergy, content that is uniquely determined by the maximum amplitude of the wavepacket, we can derive a regime for the maximum amplitude for which observablestructure in the radio signals can result.

At each instant the wave packet must extend all over that part of the source where theexcited Langmuir waves of one frequency are concentrated. Assuming that the Langmuirwaves lie within a frequency band (cjpe,o3pe + Aco) with

2

corresponding to phase velocities exceeding 30 v te (Cf. Eq. (27)), the soliton should coverthe same range of plasmafrequencies in the solar corona. For an isothermal corona inhydrostatic equilibrium the dependence of the density n on the distance from the solarcentre R (in solar radii) is

^ (30)

65

\vhc-e T5 is the temperature in. units of I06 K (Cf. Van de Hulst, 1950). From thisrelation we then find a value of order 100 km for the spatial extent of the soliton at 300MHz (R~=l). Then, from Eq. (12), the relative amplitude of the wave magnetic field is

10"4.Similarly, the lateral extent of the soliton must be of the order of the lateral source

dimension, say 10"*km.We can now compare the amount of radiation produced by the coalescing whistlers,

and Langmuir waves with the ouserveJ tlux values by means ot F.q. (23). For the whistlerenergy density per unit wave number we use

for Bw/Bo = 0.7 • 10"4, Bo = 3G, Akw = ojpe/c.If the continuum radiation is the resultof induced scattering of Langmuir waves on ions, the Langmuir wave energy density mustexceed 10"4 erg cm"3 (Kuijpers, 1974). Using Akg^6Jpe/(30 vte)we find'Skf ~ 6,2 -10'4

erg cm"2. If we assume that the total production rate of Wj-jL is simply given bymultiplication of the right hand part of Eq, (23) with the soliton volume7T 2.5 • 1024 cm3 and that, moreover, the transverse radiation is not weakened b>collisions and leaves the source at die same rale as it is-produced. the observed flux at adistance of "1 AU is 1.5 • \Q~6 erg cm"2 s"1 cm, or, in frequency measure, 5 • 10"'7 ergcm"2 s"1 Hz"1 - This value well exceeds the flux variations of 6.5 • 10"'8 ergcm"2 s"1 Hz"1 that have been observed in the type IV outburst of March 6, 1972 duringthe appearance of "sudden reductions" (Heinrich-Hertz-Institut, Berlin-Adlershof; Cf.Kuijpers, 1974). We have assumed that the source is optically thin for the couplingprocess and we have neglected the. decay of a transverse wave into a whistler and aLangmuir wave. Let us define an effective temperature for wave mode j according to

KBTk j=Wkj , (31)

where WjJ is the wave energy density per unit volume in wave vector space. Then theeffective temperature of the transverse waves that are produced by the coalescence ofwhistlers and Langmuir waves satisfies (Melrose, 1970)

For the above situation one canderive Tkg= 4.1 - 1013 K and Tkw « 1.2 • 101 ° K.Thenfrom Eq. (32) we find Tk t < 7 • 1011 K for ww - 0 5 coce and (xipeluice. = 30. The abovecalculated flux corresponds to T^t ~ 6.5 • 1012 K and the observed flux corresponds toTk =» 8.5 • 101 ' K. Consequently we conclude that the adopted source model is opticallythick for the coupling process and gives rise to a flux variation of the order of theobserved value.

66

6.3. EXPLANATION OF THE OBSERVATIONS

1. The association of intermediate drift bursts with conlinua naturally results from thesuggested mechanism, in which Langmuir waves are an essential ingredient, sinceLangmuir waves can produce a structureless continuum on their own (Kuijpers1974).

2. If at each instant the whistler wave packet extends over that part of the sourcewhere the buigmuir waves of one frequency are present (Cf. section 6.2.) and if theLangmuii waves are concentrated around wave vectors such that the conservationrelations (20) can be fulfilled (Cf. section "6.1.), coupling with the Langmuir wavescan lead to an observable fiux enhancement at a frequency u>pc + CJW relative tothe ..mbient continuum (section 6.2.).

I!", moreover, the whistler energy density is sufficiently, large, the energydensity of Langmuir waves decreases sufficiently by the coupling to affect the rateof induced scattering into radiation at ojp e significantly (Cf. Eq. (24)). Then thisleads to a depression of the observed fiux at a>pe.

Then the result of the coupling of whistler waves, that travel along themagnetic field in the form of a discrete wave packet, with the Langmuir wavesclearly is a structure that drifts in the frequency-time plane and having aninstantaneous frequency profile consisting of a depression relative to the ambientcontinuum on the low frequency side arid an enhancement shifted to higherfrequencies over a .distance equal to the dominant whistler frequency. Moreover,the depression—enhancement structure should begin (and end) suddenly, inagreement with the observations.

3. Since the coupling mechanism is independent of the induced scattering mechanism,in the; sense that the latter needs a minimum amplification length in the source(KuijperSj 1974) in contrast with the former, one expects to observe sometimes.only emission ridges, in conformity with a few observations by Young et al. (1961).

4. Due to the damping of waves with oblique wave vectors (section 3.3.), we expect tosee fibers ia a particular outburst predominantly with a positive ("reverse") or,alternatively, with a negative ("normal") drift rate depending on whether the localAlfven velocity increases with height or not. In the case where the Alfven velocitydecreases with height (negative drift rates) one expects the drift rate to decreasewith decreasing frequency, since, first, the local Alfven velocity diminishes and,secondly, the projected velocity along the density gradient decreases as the packetreaches the_ top of the magnetic arch. Occasionally, one expects to see fibersbending from negative to positive drift rates (Cf. Figure 1).

5. Since the magnetic field strength decreases with increasing height, the ridgeseparation should decrease for successive fibers centered around decreasingfrequencies, in agreement with the observations (Table I).

67

6. Since the whistler instability is faster at the "lower parts of the source where theloss-cone is large (section 4.1.), one expects to observe predominantly fibers withnegative drift rates.

•7. The occurrence of intermediate drift bursts in clusters of similar fibers can beexplained by the effect of quasilinear diffusion in piich angle and the finite extentof the whistler unstable region, (section 4.2.)

S. The field strength is most directly determined from the instantaneous ridgesepaiation. From the observed values (section 2) we have determined values of themagnetic field strength under the assumption that the whistler frequency is inbetween 0.1 - 0.4 wcc (see Table III).

On the other hand the observed drift rates allow to find a value for the fieldstrength by means of the coronal density model(30) and the relationship v *•• 21.5 v^(see Table III). Here we have neglected projection effects. From Table III it followsthat the latter values of the field strengths are in good agreement with the former.

TABLEIII

Magnetic field strength from observed intermediate drift bursts

plasma frequency (MHz) magnetic field strength (G)(1) (2)

900320160

91.40.45

- 3 6- 11- 1 .8

1530.66

(1) determined from instantaneous ridge separation assuming that A to = 0.1 — 0.4 tOgg.(2) determined from observed drift rates assuming v = 21.5 v^ and using relation (30) with TV

= 1 and R = 1. For the drift rates at 900, 320 and 160 MHz we have used, respectively,-50 , -10 and -3.5 MHz s"1.

9. [f the Langmuir waves are concentrated in a narrow band of width < 10'2 c j p e

above the local plasmafrequency (section 6.1.), the escaping radiation is purely inthe ordinary mode both in the case of coupling with whistlers (section 5.1.) and inthe case of induced scattering by the thermal particles, since the frequency of theoutgoing wave does not exceed a value of a>pe'+ 0.5 o)ce-

We mention that the result for the dominant sense of polarization is thereverse of that given by Kuijpers (1974) concerning the generation of thecontinuum, where the bandwidth of the longitudinal waves was assumed to bemuch larger (« 0.1 ojpe).

The result that the radiation produced by induced scattering as wel! as thatproduced bycoalescingwaves has the same sense of polarization is in agreement withthe observations with the Utrecht spectrograph.

68

7. Conclusion

We have found that the fiber phenomenon can be explained by coupling of whistler wavepackets with Langmuir waves in the source region of the continuum under the conditionthat the Langmuir "turbulence is concentrated at phase velocities approaching the velocityof light. The whistlers can travel as solitons along the magnetic field with a velocity of21.5 vA. The whistlers, like the Langmuir waves, can be excited by loss-conedistributions of fast electrons superimposed upon a thermal background for the assumedsource- conditions^ but; the whistlers originate preferably at lower heights than theLangmuir waves. Their pulse-like way of emission can be explained by quasi-linear pitchangle scattering of the anisotropic fast particle distribution into the loss-cone with aresultant quenching of the instability. Only after the whistlers have left the unstableregion, anisotropic loss- cone distributions can be built up again by the fast particles thattravel downwards into the unstable region.

The observed frequency separation of the absorption and emission ridges correspondsto a fraction of 0.1 - 0.4 of the local electron cyclotron frequency and leads to estimatesof the magnetic field strength in the source region. On the other hand the observed driftrate should correspond to the component of the propagation velocity along the densitygradient and also leads to estimates of the field strength under the assumption that thewhistlers propagate as solitons. The latter estimates are :n good agreement with theformer if one uses a simple hydrostatic equilibirum model. The observed intermediatedrift bursts with positive drift are naturally explained by whistlers travelling downwardsthrough the source region.

Acknowledgments

I thank Professors H.G. van Bueren and M. Kuperus, Dr. J. Rosenberg, Dr. C. Slottje, Dr.A.O. Benz, Dr. T. de Groot and Dr. A.D. Fokker for the valuable discussions andcriticism.

I am much indebted to the Dutch Organization of Pure Research (Z.W.O.) for theobservations with the spectrograph at Dwingeloo.

References

Allis, W.P., Buchsbaum, S.J. and Bcrs, A : 1963, Waves in Anisotropic Plasmas, M.I.T. Press,Cambridge, § 4.2.

Bricc, N.: 1964, J. Gcophys. Res. 69, 4515.Denisse, J.F. and Delcroix, J.L.: 1961, Thcorie des Ondes dans les Plasmas, Dunod, Paris, p. 77.E!gar0y, (p.: 1973, in J. Dclannoy and F. Poumeyiol (eds.), Proc. 3rd Meeting CESRA,

Bordeaux-Floirac, p. 174. .:--•:-..-.—:.::•Fried, B.D. and Come. S.D.: 1961, The Plasma Dispersion Function, Academic Press, New York.Hoijer, S. and Wilhelmsson, H.: 1970, Plasma Phys. 12f585.Jackson, J.D.: 1962, Classical Electrodynamics, John Wiley and Sons, Inc., New York. p. 421.Kakutani, T.: 1966, J. Phys. Soc. Japan 21, 385.Kakutani, T., Kawahara, T. and Taniuti, T.: 1967, J. Phys. Soc. Japan 23, 1138.

69

Kaplan, S.A. and Tsytovich. V.N.: 1973, Plasma Astrophysics, Pergamon Press. Oxford, p. "21, 43,283. 284.

Kennel. C-F. and Petschek. H.F.: 1966, J. Gcophys. Res. 71,1.Kennel. C.F. and Scarf, I-'.L.: 1969, in J.O. Thomas and B.J. Landmark ieds.), Plasma Waves m Space

and in the Laboratory, University Press, l.dinbugh. vol. 1, p. 451.Kuijpers, J.- 1973. in J. Dcfannoy and F. Poumeyrol, Proc. 3rd Meeting Cl SRA, Bordeaux 1'loirac. p.

130.Kuijpers, J.: 1974, Solar Phys. 36, 157.Mcltose, D.B.: 1970, Australian J. Phys. 23, 871.Ossakow, S.L., Ott, E. and Haber, I.: 1972, Phys. Fluids 15, 2314.Seharer.J.E.: 1967, Phys. Fluids 10, 652.Scharer, J.E. and Trivelpiece, A.W.: 1967, l'hys. Fluids 10, 591.Slottje,:C.::lS72a^Solar Phys;:25; 210;;.Slottje. C:?7l972b,yinrAi-EMangeriey:: (ed;), Proc. Summer School Plasma Physics and Solai

Radidastronorriyv Meudon, p. 245.SlottjerC: 1974, personal communication.Stix, T.H.: 1962, The "Theory of Plasma Waves, McGraw Hill Book Cy., New York, § 3-4.Tidman. D.A. and Krall. N.A.: 1971, Shock Waves in Collninnless Plasmas. Wiley - Interscicnce, New

York, § § 3.3.e. 4.2.Tsytovich, V.N.: 1970, Nonlinear Effects in Plasma. Plenum Press, New York. Ch. 11, HI, V, Y1H.Van de Hulst, H.C.: 1950, Bull. Astron. Inst. Neth. 11,150.Young, C.W., Spencer, C.L., Moreton, G.T. and Roberts. J A.: 1961. Astrophys. J. 133, 243.

Appendix

LIST OF SYMBOLS TO CHAPTER V

a = 3%v t e / c

a loss-cone half-aperture, particle pitch angle

& = Pgas / Pmag ratio of kinetic to magnetic pressure

B magnetic field vector

c velocity of light

DQ pitch angle diffusion coefficient

E electric field vector

: _ e _ . _ , „ electron charge in absolute value

fc h cold and, resp., hot electron distribution function

j ••—•-: .-e------- electric currentKg Boltzmann constant

L length of the whistler generating region

rtij particle mass

M A Alfvenic Mach number

Ms Acoustic Mach ^ number

n = ck / OJ index of refraction

n; particle density

nj, fractional fast particle density

cjj ,k; angular frequency, resp., wave vector

A angle between wave vector of kind j and magnetic field

70

k+ = kv + k(

0Jr-7 real. resp.. imaginary frequency part (7 > 0 : growth)

u;pj = (4rr iije" • in,) " plasmafrequencv<ol(

= eB / m, c cyclutron frequencyr height in the corona, lengthR distance fromsolai centreTj temperatureIk, effective temperature of wave mode j near kjrql quasilinear relaxation time•"max maximum wave growth timerpa^s passage time of whistlers through unstable regionr bounce bounce time of trapped particlesv particle velocityVU = ( K B T, : m,) " velocity spread of thermal particlesvh velocity spread of hot electron populationVA = B ,/(47r ne mj)' : Alfven velocityv.\e

= B / (4TT i\, n\,) : electron Alfven velocityvp = co / k phase velocityVG - b <JJ ' 9 k group velocityW «• particle kinetic energyWJ = / Wkj dkj wave energy density

wave energy density per unit wavenumberwave energy density per unit volume ofwave vector spacewk;

X

XY

v LYT

0/,i h

Zt

l O / COCl.

= t o p c2 / to2

= toct. / to

Y CO:. I?

= Y sin d= ico - coce) h

(co - coce)= Z(^t) plasma dispersion function

INDICES

/ / , 1 component,-parallel, resp., perpendicular to themagnetic field

e electronsi ionsw whistlers? Langmuir wavest electromagnetic waves above the electron piasmafrequencyk (k,co)

71

CURRICULUM VITA»-

J. Ku niH-i > 'Acid ueborcu '.'i1 I 1 september ll|4(> le 1 iudlm\en Hoi diploma (i'. IIIII.IMUIH

H behaalde hij in juni l'-'M aan iici CiViniuisiuin Auuustinianum te Hindhoven. -Lfli

september ll'fo4 begon hij /.ijn studie in de wis-, natuur- en sterrenkunde aan dc=.

Ri|ksuniversiteii te IHieclii. waar hij in mei il)(vS het kandidaai.sevainei) allcude. Zljïif

afstudeertsnder/oek verricht te hij in de werkgroep Laboratorium Astroiysica onder de-

stimulerende leidinu van l'iot. Dr. M. kupen i \ liet lioctuiaal exaruen k'üde iüj ai in

februari I U 7 I niet het hoofdvak theoretische sterrenkunde en de bijvakken theoretische

natuurkunde en wiskunde. Sindsdien is hij verbonden als wetenschappelijk medewerker

aan de S ienewachi van de Rijksuniversiteit te l ' i recht m de werkgroepen l'Iasma

Astrol\.sicy en Kadioasironomie van de Zon.

72