On the resonant interaction of planetary waves in a two-layer model with extreme wave forcing. I. The shape of the amplitude

The weak nonlinear propagation of three planetary waves is investigated in a quasi-geostrophic two-layer model. Nonlinear amplitude equations are derived and discussed. In the presence of stationary forcing either a stable state of oscillation or two stable and an unstable one exist dependent on the amplitude of a neutral wave and the forcing amplitude. The rate of instability is determined by the forcing amplitude, by the forcing frequency and by the amplitude of a neutral wave. For the boundary case where the exchange of energy between the basic wind and the triad is a direct function of the amplitudes of the waves it is shown that without forcing the evolution of the amplitudes can be divided into four categories. These categories characterize a different degree of interaction between the waves of the triad.