A circularly polarized, shear Alfvén wave of finite amplitude is shown to become rapidly unstable as its frequency ω1 approaches ion cyclotron resonance Ω. A particular mode of decay is into two perturbations consisting of a shear Alfvén wave of slightly lower frequency and an acoustic wave of much lower frequency. The decay rate is proportional to S½/(1 − ω1/Ω) where S is the power flux.
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J. D. Jukes (1965) studied this question.