A perturbation theory and a kinetic theory are developed to investigate the novel physics of relativistic ion cyclotron instabilities. The existence of the instabilities is determined by the normalized mass deficits per nucleon of fast and slow ions (δmf and δms, respectively), and by their Lorentz factors (γf and γs, respectively); while the ion bunching is caused by the relativistic variation of ion mass. If δmf - δms - γf + γs > 0, only a quadratic instability can occur at high cyclotron harmonics of the fast ion in the lower-hybrid frequency regime and above; the threshold on the harmonic number is determined by the dielectric constant of the slow ion. The peak growth rate is higher at the harmonics just above the threshold. If it is negative, both a cubic instability (or instead a coupled quadratic instability if the resonant slow ion cyclotron harmonic is the first harmonic) and the high harmonic quadratic instability can be driven. The cubic instability is due to the harmonic interaction of fast and slow ion cyclotron motions with the wave frequency in between. This introduces a novel instability concept, namely, a two-streaming process in gyrospace. Thus, the cubic instability is also called a two-gyro-stream instability even without beams in real space in contrast to conventional two-stream instability. Both theories show that, as compared to the conventional axial phase bunching mechanism, the importance of the inclusion of the relativistic mass variation effect (and the gyro-bunching mechanism) depends on the phase velocity of the wave along the external magnetic field, and is not related to the Lorentz factors (or kinetic energies); that is, if ω/kz > c (e.g., kz = 0), the relativistic gyro-bunching mechanism always dominates. While the importance of this study in fundamental plasma physics is emphasized here, some issues (e.g., nonlinear saturation, wave polarization, and nonuniform magnetic field) related to its application are also discussed.
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