Cascade of the parametric decay instability in ionospheric heating experiments

A cascade of Langmuir waves excited by tile 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 tile resonant cascade occurs at a slightly different location so that tile daughter wave can satisfy the local dispersion relation. In tile resonance case the threshold power remains the same in each cascade. However, tile 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 (similar to 10 dB) between two consecutive cascade lines in the observed HF wave enhanced plasma lines is required. Tile threshold powers for the nonresonant cascade steps are found to increase progressively as the cascade process proceeds. It is shown that tile dependence of tile threshold power on the step number N of the cascade for the nonresonant case is governed by nu (e)/nu (i), where nu (e) and nu (i) are the electron collision frequency and ion Landau damping rate, respectively. The threshold power has a N-2 dependence for nu (e) >> 2N nu (i) and a rootN dependence for nu (e) << 2N nu (i). It explains why the second cascade line hardly generates a third in early Tromso heating experiments and why more cascade steps dan occur in Arecibo's heating experiments.