Kinetics of ions during the development of parametric instability of   intensive Langmuir waves in plasma
  Nonlinear regimes of one-dimensional parametric instabilities of long-wave plasma waves are considered for the cases when the average plasma field energy density is less (Zakharov's model) or greater (Silin's model) than the plasma thermal energy density. The hybrid models presented in this paper treat the electrons as a fluid by way of an equation for the high-frequency wave and treat the ions kinetically with a super-particle simulation. This makes possible consideration of non-resonant particles trapped by high-frequency oscillations and the influence of trajectory crossing on the development of the parametric instability. The evolution of ion energy distribution is studied. It is shown that after saturation of the instability, the ion kinetic energy density normalized to the initial field energy density is of the order of the ratio of linear growth rate to the plasma frequency, for the case when the initial field energy far exceeds the plasma thermal energy. In this case, the ion energy distribution is different from the Maxwellian.In the opposite case of hot plasma, the ions acquire a part of initial field energy, which is approximately equal to the half of ratio of initial Langmuir field energy to the plasma thermal energy. At this, the ion kinetic energy distribution is close to the Maxwellian, and it is reasonably to speak about ion temperature. The crossing of ion trajectories in the surrounding of density cavities is a reason of instability quenching in both cases. 
A.V. Kirichok, V.M. Kuklin, A.V. Pryjmak, A.G. Zagorodny
physics.plasm-ph