VOL. 80, NO. 19 JOURNAL OF GEOPHYSICAL RESEARCH JULY 1, 1975 A Contribution to the Theory of the Electrostatic Half-Harmonic Electron Gyrofrequency Waves in the Magnetosphere M. ASHOUR-ABDALLA AND G. CHANTEUR Groupe de Recherches Ionosph•riques, Centre National d'Etudes des T•l•communications Issy-les-Moulineaux, France R. PELLAT Equipede Recherche Associke au CentreNational de la Recherche Scientifique Ecole Polytechnique, Paris, France The effectof differentplasmaparameters on the stabilityof the half-harmonic electron gyrofrequency wavesis investigated. By making somesimplifyingassumptions about the cold plasmaand usinga ring distribution for the warm component the role of the differentparameters is demonstrated. A maximum value of the ratio of the cold to hot plasma densityabovewhich we cannot get instability is derived,and this ratio increases with the temperature anisotropy. Calculations of the growthratesshow that thereis a sharp peak when 0 = tan-x kz/kll • 70o-80ø. 1. INTRODUCTION Observations of wavesin the magnetosphere have until re- cently been limited to electromagnetic waves due to experi- mental difficulties.Satellite Ogo 5 carried the first broad band electric field experiment to operate successfully beyond the plasmapause [Kennel et al., 1970].Analysis of the electric field data showed that there frequentlyexisted noiseabout the half harmonicof the electron gyrofrequency o•/•2 = n + (1/2), the (3/2)•2 being the most frequently detected.The waves have largeamplitudes, typicallyin the range 1-10 mY/m, but • 100- m¾/m bursts have been observed during a substormexpan- sion phase[Scarfet al., 1973].Thesehalf-harmonicemissions are typically narrow band (a few hundred hertz at a few kilo- hertz) and exhibit uniform spectral densityacross their band- width [Coroniti et al., 1971]. These emissions have no ap- parent local time preference and have been observed in the region4 < L < 10 at low geomagnetic latitudes.They also oc- cur at high geomagnetic latitudes(up to at least 50 ø ) but are weaker and more sporadic in appearance there [Scarf et al., 1973]. Sometimes chorus is observed simultaneouslywith thesewaves [Scarf et al., 1973]. The large electric field observedimplies that these waves play an important role in the energization and precipitation of particles.Thus thesewaves are likely to be important in de- termining the structure of the outer zoneplasma. The studyof the theory of these electrostatic waves should lead us hope- fully to understand under what magnetospheric conditions thesewaves are observed. There have been two different ap- proaches: Oya [1972] has considered a wave-waveprocess as being the possible generating mechanism; othershave studied the lineardispersion relationfor electrostatic waves [Fredricks, 1971; Younget al., 1973]. We will adopt the latter approach here.So by making some simplifying assumptions about the cold plasma temperature we find an easy way to demonstratethe effect of different parameters such as anisotropyof the distribution function, plasma density, etc., on the half-harmonic instability. In sec- tion 2 we studythe linear dispersion relation, defineour nota- tion, and discuss someof the assumptions made. In section 3 Copyright¸ 1975 by the AmericanGeophysical Union. we study the effect of different parameters suchas the ratio of the cold plasma density to the hot plasmadensityand the tem- perature anisotropyof the distribution function on the curves of marginal stability. Next we computethe linear growth rates and seeat what propagationangle the growth rate maximizes (section 4). 2. LINEAR DISPERSION RELATION The half-harmonic wavesare thought to be electrostatic in nature, no magnetic component having been observed to within the sensitivityof the magnetic loop sensors (<1 mT) [Fredricks, 1975]. Thus we can make use of the general dis- persion relation for electrostaticwaves as given by Harris [1959]: o• • *• f Gdv,) dv = 1 q- -• ,,.•_• w* --k•vtl -- n• where G,•(v,) =2. n-v• Or. -{" k,, • J,• [--•'-]o. do.t = o o•* complex frequency, equal to k wave vector; k• component of the wave vector parallel to the static magnetic field; k• component of the wave vector perpendicular to the static magnetic field; •2 electron gyrofrequency; %, plasmafrequency, equal to (4•rne•'/rn)X/•'; vz, v, perpendicularand parallel particle velocities,re- spectively; Jn Bessel function of the first kind. The function fis the normalized electrondistribution func- tion (protons have been neglected, since we are dealingwith frequencies much higher than the ion gyrofrequency).For small growth rates • << o•, (1) can be expandedas follows: 0 O(w*, •')= O(w, •') -.[- i• • O(w, •) = 0 2775