The effect of the energy equation on the onset of parametric instabilities

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Balescu and Paiva-Veretennicoff (B and P) (1977) have used a kinetic model to describe the parametric instabilities driven by an external, oscillating electric field in an isotropic plasma. They obtained a dispersion relation which differs significantly from that of Nishikawa (N). They attributed this difference to the fact that N did not use the energy equation. The authors attempt to resolve this difference and have used a fluid model, including the energy equation. In part I they have used the perturbed density, velocity and temperature of the fluid as variables. In part II they have used the variables suggested by B and P: the perturbed density, momentum density and total energy density of the fluid. Both calculations lead to N's dispersion relation. The reason for the difference between the authors' second calculation and that of B and P is evidently related to the expansion of the non-linear momentum and energy variables.