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The properties of the instability of combined gravity-inertial-Rossby waveson a β-plane are investigated. The wave-energy exchange equation showsthat there is an exchange of energy with the background stratified medium.The energy source driving the instability lies in the background enthalpyreleased by the gravitational buoyancy force. It is shown that if the phase speed of the westward propagating lowfrequency-long wavelength Rossby wave exceeds the Poincaré-Kelvin (or"equivalent" shallow water) wave speed, instability arises from the mergingof Rossby and Poincaré modes. There are two key parameters in thisinstability condition; namely, the equatorial/rotational Mach (or Froude)number M and the latitude θ0 of the β-plane. In general wavesequatorward of a critical latitude for given M can be driven unstable, withcorresponding growth rates of the order of a day or so. Although theseconclusions may only be safely drawn for short wavelengths corresponding to aJWKB wave packet propagating internally and located far from boundaries,nevertheless such a local instability may play a significant role inatmosphere-ocean dynamics

Instability of combined gravity-inertial-Rossby waves in atmospheres and oceans