Self-Consistent Weak Turbulence Theory of Resonant Three-Wave Processes

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Abstract
Coupled kinetic equations for the density of plasmas Nkα(t) and spatially averaged distribution function fj(v, t) are presented in the electrostatic approximation for a weakly turbulent, unmagnetized plasma. Discrete particle collisions are neglected, and resonant three-wave processes (ωkα = ωk′β+ωk″γ, k = k′+k″) are assumed to provide the dominant collective interaction mechanism. By taking appropriate velocity moments of the kinetic equations for the particles, rate equations for the changes in (spatially averaged) mean velocity Vj(t) and kinetic energy relative to the mean Kj(t) for each plasma component are derived. The resulting rate equations describe the bulk features of the reaction of the particle distributions (i.e., acceleration and heating rates) to resonant three-wave interactions, and are applicable in both the nonlinearly stable and explosively unstable regimes. As an example corresponding to explosive instability, the self-consistent evolution of well-displaced counterstreaming ion beams in a hot electron background is examined. It is found that as the energy in the field fluctuations increases, the ion beams lose kinetic energy of mean motion, and the electrons and ions gain kinetic energy relative to the mean. This produces significant alterations in the background dielectric properties. Within the context of a simple model for the particle distributions, the region of k space for which the three-wave resonance conditions can be satisfied shrinks to zero volume, which quenches the nonlinear instability.