Resonant interactions between planetary waves

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Abstract
In contrast to surface gravity waves, planetary waves can interact resonantly at the second order. Hence triplets of planetary waves may occur which are in resonance with each other. For simplicity the situation is studied first on a β plane. The geometrical conditions for two waves to form a triplet with a given third wave are determined, and so also is the rate of energy transfer. Some conservation theorems are proved. The analysis allows for a non-zero horizontal divergence of the motion. For waves which cover a complete sphere it is shown that the resonant interactions between three different harmonic components take place, if at all, in the neighbourhoods of two latitude circles, situated symmetrically north and south of the equator.

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