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KAZUE TAKAHASHI RESPONSE OF ENERGETIC PARTICLES TO MAGNETOSPHERIC ULTRA-LOW-FREQUENCY WAVES Ultra-low-frequency (ULF) waves in the space physics context are the lowest-frequency plasma waves propagating in the Earth's magnetosphere. Although the propagation of the waves is explained in mag­ netohydrodynamic (MHD) theory, excitation of the waves involves not only MHD but also kinetic pro­ cesses. In this article, we describe how in situ magnetic field and particle measurements can be used to distinguish the basic properties of ULF waves, including the azimuthal wave number of the standing wave structure and the propagation direction. Examples are taken from observations with the AMPTE/CCE spacecraft. INTRODUCTION A long time ago it was realized that the geomagnetic perimentally determined. This provides a quantitative ba­ field as observed on the ground exhibits small-amplitude sis for testing various theories of plasma instabilities. oscillations with a period of a few minutes. 1 Since then, In this article we describe how energetic particle data the phenomenon has attracted many researchers and has combined with magnetic field data can be used to under­ been called by several different names, including geo­ stand the properties of ULF waves. In some cases, the magnetic micropulsations, magnetic pulsations, ultra­ particles can be treated separately from the bulk of the low-frequency (ULF) pulsations (i.e., frequencies lower plasma, and can be used as test particles probing the spa­ than 3 Hz), or ULF waves. In 1954, Dungey2 predicted tial and temporal structure of ULF pulsations. In that that the oscillations originate in the magnetosphere as case, the type of wave-particle interactions critically de­ magnetohydrodynamic (MHD) waves. As soon as satel­ pends on the spatial structure of ULF waves. We show lite measurements of the Earth's magnetic fields became examples of wave-particle interactions from data ac­ available, this theoretical prediction was confirmed. 3 quired by the Active Magnetospheric Particle Tracer Ex­ Nowadays, magnetometers on scientific satellites routine­ plorers/Charge Composition Explorer (AMPTE/ CCE) ly detect ULF pulsations in the magnetosphere. spacecraft and demonstrate that from single spacecraft While the propagation of ULF waves in the magneto­ measurements of particle fluxes we can infer the spatial sphere can be explained in terms of MHO, the mechan­ structure of ULF waves. The organization of this article isms for exciting the waves are not fully understood. is as follows: In the following section, the general theory Thus a great deal of effort is still being expended, both of ULF waves and the behavior of energetic particles are in observation and in theory, to identify the excitation outlined. Then the magnetic field and particle experi­ mechanisms. One class of pulsations can be generated ments on AMPTE/ CCE are described. Finally, examples from external sources such as the Kelvin-Helmholtz in­ of ion flux oscillations are illustrated. stability on the magnetopause,4,5 and other class of pul­ STANDING WAVES AND MOTION sations can be generated by plasma instabilities in the 6 OF ENERGETIC PARTICLES magnetosphere. ,7 However, the basic distinguishing properties of the waves (such as wave number and prop­ Magnetohydrodynamics describes the fluid-like be­ agation direction) are not easily determined from obser­ havior of a plasma at frequencies lower than the ion cy­ vation. One experimental difficulty is that ULF waves are clotron frequency, which is about 1 Hz or higher in the of large scale, often comparable to the dimension of the Earth's magnetosphere. Magnetohydrodynamics allows magnetosphere itself, while spacecraft measurements are propagation of both compressional waves, called fast­ made at one point or perhaps at a half-dozen points in mode waves, and transverse waves, called Alfven waves. the entire magnetosphere. The fast-mode waves propagate across the ambient geo­ One major scientific motivation for studying magneto­ magnetic field and transfer wave energy between differ­ spheric ULF waves is that a detailed comparison is pos­ ent magnetic shells. The Alfven waves, in contrast, prop­ sible between plasma theory and observation in the mag­ agate along geomagnetic field lines. Because the iono­ netospheric environment. Unlike laboratory plasmas, the sphere is a good reflector for Alfven waves that are in­ magnetospheric plasma is free of walls or disturbances cident from the magnetosphere, the energy of plasma from diagnostic tools (spacecraft). In addition, the wave and field perturbations propagating in the Alfven mode frequency can be very low, so that particle distributions is reflected back and forth between the northern and in phase space at different oscillation phases can be ex- southern ends of geomagnetic field lines. As a conse- Johns Hopkins APL Technical Digest, Volume 11 , Numbers 3 and 4 (1990) 255 K. Takahashi quence, ULF disturbances tend to establish standing N Alfven waves, as was predicted by Dungey.2 The waves are in a sense analogous to the vibration of a string fixed at both ends. In MHD, the restoring force (tension) is provided by the Maxwell stress of the geomagnetic field. The two lowest-order modes of the standing waves are illustrated in Figure 1. The fundamental mode has an antinode of field line displacement at the equator. Because the transverse magnetic field perturbation, S DB 1. , is proportional to the local tilt angle of the dis­ placed field line, bB 1. has a node at the equator. For Symmetric wave Antisymmetric wave the second harmonic, the field displacement is null at (fundamental) (second harmonic) the equator, while bB 1. has an antinode there. The fre­ N quency of a standing Alfven wave is given approximately I by the fundamental frequency fA = (Jds / VA) - I, or by I its harmonic, where VA is the Alfven velocity and ds I is the line element along a geomagnetic field line, and -+- E the integral is taken over a return path from the southern I end to the northern end of the field line. The Alfven I velocity is given by VA = B/(4'Trp) Yz , where B is the I I field line intensity and p is the plasma mass density. S Therefore, the standing wave frequency, fA, depends Figure 1. Schematic of standing waves on a geomagnetic field on both Band p, as well as on the length of the geo­ line. For simplicity, the curved geomagnetic field line is magnetic field line. It must be mentioned that waves ex­ represented by a straight field string with ends fixed. The fun­ cited in a hot plasma often exhibit a strong compres­ damental mode has maximum displacement at the equator. The sional component. In that case, the waves cannot be a second harmonic has null displacement at the equator. pure Alfven mode. There is strong coupling, however, between compressional waves and Alfven waves, and the wave reflection at the ionosphere leads to standing struc­ where M is the particle mass, v is the particle velocity, tures even for the mixed-mode waves. 8 and q is the electric charge. In the absence of temporal Very clear evidence for the standing waves has been variations in E and B, particle motion in a dipole geo­ obtained from dynamic spectra of magnetic field mea­ magnetic field is composed of gyromotion, bounce mo­ surements from Earth-orbiting satellites. 9 Example tion, and drift motion of the guiding center, as illustrated spectra from four consecutive inbound passes of AMPTEI in Figure 3. The circular motion of particles about the CCE are shown in Figure 2. Power spectral density rela­ magnetic field that arises from the magnetic part of the tive to the background, which is defined by a best-fit Lorenz force qv x B is called the gyromotion. The peri­ second-order polynomial to the original spectrum, is dis­ od of the gyromotion (called the gyroperiod), 27rM/qB, played by color, with blue, green, yellow, and red indi­ does not depend on energy for nonrelativistic particles. cating increasingly high power levels. Each color-coded For a magnetic field of 100 nT, which is typical at geo­ spectrum shows a frequency structure of the azimuthal stationary orbit, the proton gyroperiod is 0.7 s. The guid­ magnetic field oscillation as a function of geocentric dis­ ing center is the instantaneous center of the gyromotion, tance, L, measured in units of Earth radii (R E ). The and its bounce and drift motions arise from the dipole clearly visible red traces correspond to the fundamental configuration of the geomagnetic field. The former arises and the third-harmonic standing Alfven waves on the because the magnetic field lines converging at higher lati­ local geomagnetic field line. The frequency decrease with tudes push particles back to the equator. The latter arises L is caused by a rapid decrease of magnetic field mag­ because the curvature of the field lines gives a centrifugal nitude with distance. force for the guiding center and also because the radial In addition to the magnetic pulsation, an electric field gradient of B leads to displacement of the guiding center oscillation given by E = - V x B accompanies an at consecutive gyrations. The speed of the guiding center Alfven wave, where E is the electric field, V is the plasma motions depends on particle energy. For example, the bulk velocity, and B is the magnetic field. Because mea­ bounce period and the drift period (the time interval that surements of ULF electric fields are more difficult than it takes a guiding center to complete an azimuthal rota­ magnetic field measurements, the presence of a ULF wave tion about the Earth) at geostationary altitude are is usually determined by magnetic field experiments. -120 sand -8 h for 10-keV protons and -40 sand Many magnetic pulsations accompany particle flux os­ - 50 min for l00-keV protons.

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