Resonant interactions between unstable and neutral baroclinic waves in a continuous model of the atmosphere

Resonant interactions between a finite-amplitude marginally unstable wave and two neutral baroclinic waves are investigated in a quasi-geostrophic, continuous, infinite depth model on the beta-plane channel. Since the neutral waves are characterized by total horizontal wavenumbers less than that of the unstable wave, the kinematic resonant conditions can only be satisfied when the meridional scale of the unstable wave is sufficiently small. Employing asymptotic methods, equations are obtained governing the temporal evolution of the amplitudes and phases of the triad. The neutral waves are scaled smaller in magnitude than the unstable wave as a result of an asymptotic imbalance between the advective non-linearity and instability. Numerical solutions of the amplitude equations reveal a separation between the instability and interaction time scales. The neutral wave amplitudes vacillate on the longer interaction time scale. The unstable wave amplitude vacillates locally on the instability time scale while its envelope vacillates on the interaction time scale in direct proportion to the initial neutral wave amplitudes.