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 tlie 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 tlie same location. However, each step of the resonant cascade occurs at a slightly different location so that the daughter wave can satisfy tlie local dispersion relation. In tlie resonance case the threshold power remains tlie same in each cascade. However, tlie mother Langmuir wave has to propagate downward to tlie resonant location of the daughter Langmuir wave for the next resonant cascade step. To compensate the propagation loss of tlie 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 tlie dependence of the threshold power on tlie step number N of tlie cascade for the nonresonant case is governed by v_e/v_i where v_e and v_i are the electron collision frequency and ion Landau damping rate, respectively. Tlie threshold power has a N² dependence for v_e« 2Nv _i and a √N dependence for v_e» 2JNv _i. It explains why the second cascade line hardly generates a third in early Trornso heating experiments and why more cascade steps can occur in Arecibo's heating experiments.