Instabilities of finite-amplitude oscillations of a plasma in a magnetic field in the presence of trapped particles

A study is made of the instabilities of a plane monochromatic electrostatic wave of finite amplitude in a constant magnetic field due to particles trapped by the electric field of this wave under conditions of Cherenkov resonance or cyclotron resonances. If the particles are trapped near a centre on the phase plane with the transverse or longitudinal velocity and the helical phase as coordinates in the case n not=0 or on the phase plane with the longitudinal velocity and the helical phase as coordinates in the case n=0, and if the particles have a small spread with respect to the velocities and helical phase, there arise coherent instabilities excited by all the trapped particles. If the wave vector k of the unstable oscillations is not near k0, there arises an instability of the two-stream type. If k approximately=k0, there arises a parametric satellite instability. In both cases, the maximal growth rate is gamma max approximately=(n1/n0)1/3 Omega 0, where n1 is the number of trapped particles in unit volume and n0 is the density of the plasma.