Parametric Instabilities of Ion Cyclotron Waves in a Plasma

An externally applied large amplitude circularly polarized ion cyclotron wave of long wavelength propagating along the static magnetic field can drive instabilities of ion cyclotron waves and ion acoustic waves. The dispersion relation is first derived using a kinetic equation approach. Later, it is shown that the same relation can be derived on the basis of a “$\mathrm{K\Delta X}$ ” analysis. The instabilities can be classified into three classes: ${\mathrm{|\hspace{0.17em}k\hspace{0.17em}|⋙k}}_0$ corresponds to the usual purely growing instability; ${\mathrm{|\hspace{0.17em}k\hspace{0.17em}|\hspace{0.17em}\sim k}}_0$ corresponds to the decay instability, and ${\mathrm{|\hspace{0.17em}k\hspace{0.17em}|\hspace{0.17em}\gg k}}_0$ corresponds to slightly oscillatory instability which is actually the finite $k_0$ modification of the purely growing instability. The dependence of both, the threshold and maximum growth rates, on various parameters is discussed. The relevance of the results to the anomalous ion heating for fusion research is also pointed out.