Abstract
he stability of planetary waves in a two-layer fluid on a channel beta-plane is discussed. The lowest mode of baroclinic waves is shown to be always unstable in the sense of triad resonance instability. The lowest mode of barotropic waves, however, is stable if the zonal wavenumber is smaller than the critical value ηc. The transition corresponds to the occurrence of resonating phenomenon between the group velocity of the primary wave and the phase speed of long waves. This phenomenon is important as a mechanism generating relatively strong zonal flow. The coupled evolution equations which govern the phenomenon are derived. Also derived is the evolution equation which governs the modulation for a barotropic wave whose zonal wavenumber is smaller than ηc.
These evolution equations have exact solutions. The stability of exact plane wave solutions is examined and related to the long wave resonance instability and the sideband instability reported by Plumb (1977). The exact solutions of a solitary type, which seem to be final states after above instabilities, are also obtained and presented as planetary solitons.
