Cascade of the parametric decay instability in ionospheric heating experiments

A cascade of Langmuir waves excited by the O‐mode radio waves in ionospheric heating experiments via the parametric decay instability (PDI) is studied. The threshold powers for each resonant and nonresonant cascade step are determined. The nonresonant cascade of mother Langmuir waves proceeds at the same location. However, each step of the resonant cascade occurs at a slightly different location so that the daughter wave can satisfy the local dispersion relation. In the resonance case the threshold power remains the same in each cascade. However, the mother Langmuir wave has to propagate downward to the resonant location of the daughter Langmuir wave for the next resonant cascade step. To compensate the propagation loss of the mother Langmuir wave, a large power ratio (∼10 dB) between two consecutive cascade lines in the observed HF wave enhanced plasma lines is required. The threshold powers for the nonresonant cascade steps are found to increase progressively as the cascade process proceeds. It is shown that the dependence of the threshold power on the step number N of the cascade for the nonresonant case is governed by ve/vi, where ve and vi, are the electron collision frequency and ion Landau damping rate, respectively. The threshold power has a N2 dependence for ve ≫ 2Nvi and a √N dependence for ve ≪ 2Nvi. It explains why the second cascade line hardly generates a third in early Tromso heating experiments and why more cascade steps can occur in Arecibo's heating experiments.