Convective organization and eastward propagating equatorial disturbances in a simple excitable system

We describe and illustrate a mechanism whereby convective aggregation and eastward propagating equatorial disturbances, which are similar in some respects to the Madden-Julian Oscillation (MJO), arise. We construct a simple explicit system consisting only of the shallow-water equations plus a humidity variable; moisture enters via evaporation from a wet surface, is transported by the flow and removed by condensation, thus producing an anomaly in the height field. For a broad range of parameters the system is excitable even when linearly stable, with condensation producing convergence and gravity waves that, acting together, trigger more condensation. On the equatorial beta-plane the convection first aggregates near the Equator, generating patterns related to those encountered in the Matsuno-Gill problem. However, the pattern is unstable and more convection is triggered on its eastern edge, leading to a self-sustaining precipitating disturbance that progresses eastward. The propagation is eastward because the warm, moist converging air from the east induced by the Matsuno-Gill pattern is more convectively unstable than the converging air from the west. The pattern is confined to a region within a few deformation radii of the Equator, as here the convection can create the convergence needed to organize itself into a self-sustaining pattern; thus, smaller values of the beta parameter give rise to a wider disturbance. Formation of the disturbance preferentially occurs where the surface is warmer, and sufficient time (typically a few tens of days) must pass before conditions arise that enable the disturbance to reform, a well-known characteristic of the MJO. The speed of the disturbance depends on the efficiency of evaporation and the heat released by condensation, and is typically a few metres per second, much less than the Kelvin wave speed.