The collisionless damping of shear Alfvén waves in the limit of low frequency and small but finite Larmor radius is discussed. Because the averaging of the wave electric field over the Larmor circle (the so-called finite cyclotron radius effect) creates a small difference in the transverse velocities of the ions and electrons, a longitudinal electric field appears in the Alfvén wave which is responsible for the transfer of the wave energy into the thermal motion of the “resonant” particles Vz = ω/kz. Numerical solutions (in the complex ω plane) of the dispersion relation for two wave modes have been obtained: the shear Alfvén mode and the least damped Fried and Gould ion acoustic mode. The normalized damping rate (Im ω)/(Re ω) has been computed for both waves, and it is shown that the damping factor for the Alfvén wave is maximal when the Alfvén velocity is equal to the real part of the phase velocity of the ion acoustic wave. Moreover, there exists a particular direction of propagation with respect to the magnetic field for which both the Alfvén and the least damped ion acoustic waves have the same Landau damping. The damping factor for the ion acoustic wave is a minimum for this particular mode and it is shown that at low values of the electron-to-ion temperature ratio Te/Ti (Te/Ti ≲ 2), the critical drift velocity necessary for instability occurs at a much lower velocity than is usually predicted.
The gyrokinetic differential equation for waves propagating in a hot collisionless current-carrying plasma is derived in cylindrical geometry. It is shown that the averaging of the wave electric field over the ion Larmor circle leads to a transcendental differential equation (d.e.) of infinite order in the radial derivative. This reduces to a d.e. of sixth order when the scale length of the plasma inhomogeneity Ln is greater than the ion gyroradius ρi by a factor (M/m)1/2, where M and m are, respectively, the ion and electron mass. This sixth-order d.e. describes the properties of the two (compressional and torsional) Alfvén modes and the ion acoustic mode. When Ln <(M/m)1/2ρi, the plasma can only support modes of the magnetokinetic (short wavelength) type. In the absence of a finite Larmor radius (FLR) effect, shear, and equilibrium current, we find that the correct equation to start with is the Hain and Lust [Z. Naturforsch. A 13, 936 (1958)] d.e. of second order that is singular at the Alfvén resonance layer (ARL). The ARL behaves in that case like a Budden absorption layer that traps the global Alfvén eigenmodes (GAEM) inside the plasma cavity where they are damped by transit time magnetic pumping (TTMP). The logarithmic singularity does not disappear with the introduction of the FLR effect in the Hain and Lust d.e., but only with the TTMP damping term. There is no mode conversion between the fast magnetosonic mode and the shear or magnetokinetic mode at the ARL or anywhere in the plasma. In the presence of shear and equilibrium current, the correct equation to use is a d.e. of fourth (or greater) order whose solutions descibe the shear Alfvén mode in the long wavelength limit or the magnetokinetic mode at shorter wavelengths. In the true magnetohydrodynamic (MHD) limit, both modes become degenerate. It is shown that the slow Alfvén eigenmodes are (almost) completely decoupled from the fast magnetosonic wave and therefore the growth rates show no dependence on the poloidal number m. The quicker the current density drops from the cylinder axis, the more unstable the modes are. This fourth-order d.e. is singular at the hybrid resonances, not at the ARL. It is therefore found that no normal mode solution can exist for linear shear Alfvén perturbations in the full FLR limit.
The propagation characteristics of nonpotential ion cyclotron waves traveling at very large angles to the unperturbed magnetic field are discussed. The wave dispersion relationship is derived in situations in which we can use a power series or an asymptotic expansion for the plasma dispersion function, namely, when the phase velocity is less than the electron thermal speed, or when the phase velocity is greater than the electron thermal speed. When the phase velocity is less than the electron thermal speed, it is shown that the analysis includes not only the electrostatic wave investigated by W. E. Drummond and M. N. Rosenbluth, but also a slightly damped nonpotential ion cyclotron mode for any ratio of the plasma pressure to magnetic pressure β. When the phase velocity is greater than the electron thermal speed only the nonpotential ion cyclotron modes exist. These waves have (E • k)/(E ⋏ k)≳1 near the cyclotron harmonics, i.e., at large values of kz (wave number parallel to the magnetic field). At smaller values of kz the ion cyclotron modes tend to be mostly longitudinal. Numerical solutions of the wave dispersion relationship are obtained, and application is made to some relevant space observations. It is shown that the fine structure of the proton whistlers recently observed by D. A. Gurnett and others is well explained by the propagation characteristics of ion (O+) harmonic waves through the ionosphere. Attention is also paid to the recent satellite observations of proton harmonic waves. Finally, the properties of these slow waves near the lower hybrid frequency are discussed.
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