A Toy Model of the Instability in the Equatorially Trapped Convectively Coupled Waves on the Equatorial Beta Plane

The equatorial atmospheric variability shows a spectrum of significant peaks in the wavenumber-frequency domain. These peaks have been identified with the equatorially trapped wave modes of rotating shallow water wave theory. This paper addresses the observation that the various wave types (e.g., Kelvin, Rossby, etc.) and wavenumbers show differing signal strength relative to a red background. It is hypothesized that this may be due to variations in the linear stability of the atmosphere in response to the various wave types depending on both the specific wave type and the wavenumber. A simple model of the convectively coupled waves on the equatorial beta plane is constructed to identify processes that contribute to this dependence. The linear instability spectrum of the resulting coupled system is evaluated by eigenvalue analysis. This analysis shows unstable waves with phase speeds, growth rates, and structures (vertical and horizontal) that are broadly consistent with the results from observations. The linear system, with an idealized single intertropical convergence zone (ITCZ) as a mean state, shows peak unstable Kelvin waves around zonal wavenumber 7 with peak growth rates of similar to 0.08 day(-1) (e-folding time of similar to 13 days). The system also shows unstable mixed Rossby-gravity (MRG) and inertio-gravity waves with significant growth in the zonal wavenumber range from -15 (negative indicates westward phase speed) to +10 (positive indicates eastward phase speed). The peak MRG n = 0 eastward inertio-gravity wave (EIG) growth rate is around one-third that of the Kelvin wave and occurs at zonal wavenumber 3. The Rossby waves in this system are stable, and the Madden-Julian oscillation is not observed. Within this model, it is shown that in addition to the effect of the ITCZ configuration, the differing instabilities of the different wave modes are also related to their different efficiency in converting input energy into divergent flow. This energy conversion efficiency difference is suggested as an additional factor that helps to shape the observed wave spectrum.