Nonlinear interaction of ion cyclotron waves with energetic protons in the Earth magnetosphere

Investigated is the interaction of the quasi-monochromatic ionic-cyclotron wave with energy protons captured in a magnetic trap, in particular, in the Earth's magnetosphere. Determined are the paths of the protons captured by the magnetic trap in the presence of an ionic-cyclotron wave of small amplitude. A function has been found to quantify the distribution of particles in the field of a constant-amplitude wave. The nonlinear decrement of the wave damping is calculated. Evaluated is the level of stabilization of the cyclotron instability in single-mode conditions. The results are employed for interpreting satellite observations of ionic-cyclotron oscillations of the Pc1-Pc2 range produced by the cyclotron instability of energy protons of the circular current in the Earth's magnetosphere. Ionic-cyclotron waves recorded on an Earth's satellite fall into two classes. The first one comprises observations on a geostationary orbit L=6.7 (L is the McIlvain's parameter), and the second class - measurements on satellites with lower orbits. In the former case, the linear increment γsub(l) of the cyclotron instability must exceed the characteristic bounce-frequency Ωsub(b) of protons of the circular current, in the latter case, the condition γsub(b)<Ωsub(b) is fulfilled