JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. A12, PAGES 23,685-23,693, DECEMBER 1, 1994 A model for the generation of obliquely propagating ULF waves near the magnetic equator K. G. McClements, R. O. Dendy, and C. N. Lashmore-Davies United Kingdom Atomic Energy Authority Government Division, Fusion, Culham, England Abstract. Severalspacecraft, notably OGO 3, IMP 6, Hawkeye1, and GEOS 1 and 2, have detected harmonically structured waveemissions in the ULF rangeat low magnetic latitudes between L values of 4 and 8. These waves propagate at large angles with respect to the magnetic field, and, in the case of GEOS 1, appearto be associated with ringlike proton distributions. It is demonstrated that the excitationof obliquelypropagating fast Alfv6n waves by suchdistributions can account for many of the observed featuresof har- monicallystructured ULF emissions. Linear instability can occur at, or slightly below, the proton cyclotron frequency and all its harmonics.The maximum growth rates of some harmonics, notably the fundamental,occur at propagation angles which are oblique rather than strictly perpendicular. Because of magneticfield gradients,waves which propagateapproximately azimuthally and within a few degrees of the perpendiculardi- rectionare likely to be amplified to the highest levels. Certain features of the ULF data, suchas the finite bandwidth of the harmonics, can be explained only by allowing the wave vector to have a finite parallel component. 1. Introduction Wave data from the equatorial plasmasphere showing har- monically structured emissions in the ULF range have been obtained from various spacecraft since1970 [Russell et al. 1970; Gurnett, 1976; Perraut et al., 1982]. Suchwaves may be responsible for heating the bulk equatorial plasma from a few eV up to several hundred eV [Curtis, 1985],and it is thereforeimportant to understand the physicalprocesses re- sponsible for producing them. It is generally accepted that the waves are amplified by an instability driven by non- thermal ions, and particle data from Explorer45 [Fritz and Spjeldvik, 1979], Active Magnetospheric Particle Tracer Ex- plorers/Charge Composition Explorer (AMPTE/CCE)[Lui et al., 1990] and CRRES indicate the presence at equato- rial latitudes of anisotropic ions with energies above 50 keV. Proton energy distributions which exhibit population inver- sion,i.e., are nonmonotonic, at lowerenergies (~ 10 keV) have also been detected by instruments on the GEOS 1 [Per- raut et al., 1982]and AMPTE/CCE spacecraft [Kistler et al., 1989]. Perhaps because of the relative scarcity of observational data, there has been little theoretical work in this area. Gul'elmi et al. [1975] considered the excitation of per- pendicularlypropagatingelectromagnetic waves at high cy- clotron harmonics, due to the presence of a proton distri- bution peaked at a nonzero perpendicular speed. Curtis and Wu [1979]examined the caseof obliquepropagation at arbitrary frequencies, and, on the basis of the Explorer 45 particle data available at that time [Fritz and Spjeld- vik, 1979], concluded that cyclotron harmonic waves would be strongly damped except for propagation angles lying ex- trexnely close to 90 ø. Perraut et al. [1982] studied the same instability, using parameters inferred from the later GEOS 1 data. They restricted their analysis to strictly perpen- Published in 1994 by the American Geophysical Union. Paper number 94JA01979. dicular propagation, and, representingthe energetic proton distribution as a cold velocity-space ring, found numerically that the GEOS 1 particle data implied hnear instabihty at multiple proton cyclotron harmonics. The case of perpen- dicular propagation was studied more fully by McClements and Dendy [1993], who obtained instabilitythreshold crite- ria in a warm uniform plasma. Nekrasov [1988] considered the growth of cyclotron harmonic wavespropagating across a longitudinally inhomogeneous magnetic field. Our aim in this paper is to reexamine the case of obhque propagation. Sucha reexamination is prompted by the wave and particledata published by Perraut et al. [1982]and Laakso et al. [1990], whichwas not available to Curtis and Wu [1979]. The relevant features of this data set are sum- marized in section 2. In sections 3 and 4, drawing on the analysis of Dendy et al. [1994],we obtain a perturbative analytical solution of the dispersion relation for obhquely propagating fast Alfv6n waves. This solution is compared with exact numerical solutions for the case of equatorial magnetospheric parameters in section 5, and in section 6 we discussthe conditions under which obhquely propagat- ing fast Alfv6n waves can be convectively amplified to the observed levels. Our results are discussed in the context of previous theoretical work in section 7. 2. ULF Waves and Energetic Protons Figure1, takenfrom Perraut et al. [1982], shows an exam- ple of harmonically-structured wave emissionin the equato- rial radiation belts. This is a magnetic field power spectrum from GEOS 1, in which two componentsof the wave mag- netic field are shown: Bz denotes the component which is aligned with the ambient magnetic field, and Ba denotes the right circularly polarized component. The direction of the ambient field (whichmay differ from the directionper- pendicular to the equatorial plane) was determined using the dc magnetometer experiment S-331. There are two harmonically related, narrowband features, the harmonic 23,685