Nonlinear Stability of the Extraordinary Wave in a Plasma

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Abstract
The linear theory predicts that certain anisotropic velocity distributions will produce unstable extraordinary waves. The development of these unstable waves in the nonlinear regime is investigated and their final amplitudes are estimated. The analysis is restricted to infinite homogeneous plasmas where the background distribution and the wave energy density may be considered as slowly varying functions of time. A set of nonlinear integro‐differential equations, which describe the evolution of the system, are derived and discussed. For the physically interesting case where the wave frequency is much less than the cyclotron frequency, the basic behavior of the system and the characteristics of the equilibrium solutions may be estimated without computing the transient behavior of the system. For this case, it is shown that the main diffusion of the background distribution occurs along the v = const lines, in which v is the magnitude of the velocity. Since these lines are also constant energy lines, very little energy is transferred to the waves during the diffusion process. A Maxwellian plasma which has been disturbed by injecting a stream of electrons into it is considered as an example.

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