JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. A4, PAGES 4345-4351, APRIL 1, 1986 Coupling of Global Magnetospheric MHD Eigenmodes to Field Line Resonances MARGARET G. KIVELSON Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los An•ieles DAVID J. SOUTHWOOD Institute of Geophysics and Planetary Physics, University of California, Los Angeles Department of Physics, Imperial College of Science and Technology, London, United Kingdom Many features of the magnetospheric hydromagnetic wave spectrum are fully understood, but a remaining puzzle is why selected field line resonances appear to be excited by a broadband source. In this paper, we propose that the preferential resonances correspond to global eigenmodes of the mag- netospheric cavity. In order to understand the behavior of ultralow frequency waves in the terrestrial magnetosphere (geomagnetic pulsations), we describe the structure of global eigenmodes in the hy- dromagnetic box model of an inhomogeneous plasma. Global modes are large-scale modes whose frequencies match field line resonance frequencies somewhere in the system. The coupling between localized field line resonance effects and the large-scale mode leads to damping of the latter. In the box model the eigenmode equation yields a spectrum of discrete eigenfrequencies. For each eigenfrequency, the equation contains singular points where the field line resonance occurs. We base much of our discussion on earlier works which have used the same equation in other contexts. We describe solutions and then use the results to outline features of the modes that may be observable. 1. INTRODUCTION Magnetohydrodynamic wave phenomena in the terrestrial magnetosphere can be monitored by spacecraft instruments, ground magnetometer arrays, ground radars [Cummings et al., 1969; Samson, 1972; Walker et al., 1982] and in other ways. One can use the measurements as diagnostics of the mag- netospheric system, but also one may examine the physics of the waves themselves. The propagation anisotropies of the wave modes and the inhomogeneities of the system combine to provide some curious effects, most familiarly, the field line resonance effect [Southwood, 1974, 1975a; Chen and Hase- gawa, 1974a, b]. The latter papers explained how ultralow frequency (ULF) hydromagnetic wave energy entering the magnetosphere on high-latitude flux tubes can tunnel to lo- cations where the frequency matches the Alfv•n guided mode standing wave frequency and give rise to a distinctive spatial phase and amplitude structure. Both Southwood and Chen/Hasegawa worked in a limiting case equivalent to imposing a fairly rapid phase variation across the field (large azimuthal wave numbers, m). There were sound reasons for such an assumption, but it should be noted that some subsequent work [e.g., Green, 1976; Olson and Ros- toker, 1978; Mier-Jedrzejowicz and Southwood, 1979; H. Gough, private communication, 1985] has shown that many pulsation signals reveal relatively small east-west phase vari- ation. These and other shortcomings of the theory led Kivelson and Southwood [1985] to propose that there was a class of ULF eigenmodes possible in an inhomogeneous bounded magnetized plasma system such as the terrestrial mag- netosphere whose behavior had not been fully investigated by theorists. The modes in question, not restricted to the large m limit, have the form of global compressional eigenmodes Copyright 1986 by the American Geophysical Union. Paper number 5A8873. 0148-0227/86/005A-8873505.00 whose frequency is quantized by the radial structure of the magnetosphere. Evidence for a global compressional wave with m = 0 and frequency established by the radial inhomoge- neity of the outer magnetosphere had been presented pre- viously by Kivelson et al. [1984]. For m :• 0, the modes are damped through an irreversible coupling to an Alfv6n mode localized to a magnetic shell with the same resonant fre- quency. The physics of the modes is closely allied to that of the damped surface eigenmode investigated by Chen and ttase- gawa [1974b] among others. In this paper we show that if we use the simple "hydromagnetic box" model plasma, the equa- tion governing such modes is the same as that used in several other areas of research, and we call on results derived for problems in ionospheric radio propagation, laser fusion, and RF (radio frequency) laboratory plasma heating. 2. THE INHOMOGENEOUS "Box" MODEL In this section we reintroduce Southwood's [1974] simple model of a nonuniform plasma in which cold plasma hy- dromagnetic mode coupling occurs and we outline the form of a solution of the coupled equations. The magnetic field Bz is uniform, but the mass density, p(x), varies with x. Boundaries (corresponding to ionospheres) are placed at z = +[. If the boundaries are perfectly reflecting, wave fields must have standing structure in the z direction, and allowed parallel wave numbers are quantized (k = mr/2[). If the boundaries are weakly absorptive, the parallel wave numbers are complex, but the real parts are still quantized as above [Newton et al., 1978' Ellis and Southwood, 1983]. Boundaries in x and y need to be considered. We impose periodic boundary conditions in y, causing ,•, the wave number in y, to be quantized. In section 5, we discuss the implications of this assumed periodic structure. The presence of boundaries at large and small x is central to our discussion. The boundaries correspond to the magnetopause and the plas- mapause or equatorial ionosphere in application to the mag- netosphere. We shall take one at x = c and the others at x=a>>c. 43454346 KIVELSON AND SOUTHWOOD' COUPLED GLOBAL MODES, RESONANT WAVES In the system thus described, the x and y components of the cold plasma momentum equation combined with the three components of the frozen-in field equation give /,toPO) k2)• x 1 db: • - B dx (1) iXb: ( 'uøpw2 k 2 •y- (2) b: = --iX•yB -- B d'-•- (3) The (linearized) wave field components •,, •, representing field line displacement, and the compressional wave magnetic field b: have been assumed to vary as standard functions of the form exp (iXy -- icot)[exp (ikz) ___ exp (-ikz)] (4) with y, z, t. Except where explicitly noted, only the x variation of the amplitudes