A toy model of the instability in the equatorially trapped convectively coupled waves

The equatorial atmospheric variability shows a spectrum of significant peaks in the wave number-frequency domain. These peaks have been identified (Wheeler and Kiladis, 1999) with the equatorially trapped wave modes of rotating shallow water wave theory (Matsuno, 1966). We are attempting to address the observation that the various wave types (e.g. Kelvin, Rossby, etc.) and wave numbers show differing amplitudes. It is hypothesized that this is due to the (linear) stability of the atmosphere to the various wave types depending both upon the specific wave type and the wavenumber in question.

We use the simplified model of the convectively coupled waves in first two baroclinic vertical modes developed in Kuang 2007, extended to the equatorial beta plane. The convective parameterization for this model is based upon the quasi-equilibrium concept. The inclusion of tropospheric moisture as an input to the convection calculation to represent the effect of lateral entrainment of the (generally) drier tropospheric air into a rising convective cloud, controls the depth of convection.

The linear instability spectrum of the resulting coupled system is found by eigenvalue analysis. We use realistic model parameters estimated from cloud system resolving model (CSRM) studies of the convectively coupled waves. The instability analysis produces unstable waves with phase speeds, growth rates and structures (vertical and horizontal) that compare well with the results from CSRM simulations and observations.

The linear system shows peak unstable “Kelvin” waves around planetary wavenumber seven with peak growth rates of 0.09/day (e-folding time of eleven days). The system also shows unstable “Mixed Rossby Gravity” (MRG) and “Inertia-Gravity” waves with significant growth in the wavenumber range from negative ten to positive ten. The peak MRG growth rate is around one third that of the Kelvin wave and occurs at planetary wavenumber three.

This demonstrates that one aspect of the convectively coupled waves which cannot be captured without a beta plane (or some other representation of meridional variation) is the instability's dependence upon wave type. We believe this is related to the differing roles of divergent and rotational flows in the convective parameterization. We further conducted analytical investigation of simplified cases to explore the details of the mechanisms responsible for wave number and wave type selection in the instability spectrum.