Multiscale asymptotics for the Skeleton of the Madden-Julian Oscillation and Tropical–Extratropical Interactions

Anew model is derived and analyzed for tropical–extratropical interactions involving the Madden– Julian oscillation (MJO). The model combines (i) the tropical dynamics of the MJO and equatorial baroclinic waves and (ii) the dynamics of barotropic Rossby waves with significant extratropical structure, and the combined system has a conserved energy. The method of multiscale asymptotics is applied to systematically derive a system of ordinary differential equations (ODEs) for three-wave resonant interactions. Two novel features are (i) a degenerate auxiliary problem with overdetermined equations due to a compatibility condition (meridional geostrophic balance) and (ii) cubic self-interaction terms that are not typically found in threewave resonance ODEs. Several examples illustrate applications to MJO initiation and termination, including cases of (i) the MJO, equatorial baroclinic Rossbywaves, and barotropic Rossbywaves interacting, and (ii) the MJO, baroclinic Kelvinwaves, and barotropic Rossbywaves interacting. Resonance with the Kelvinwave is not possible here if only dry variables are considered, but it occurs in the moist model here through interactions with water vapor and convective activity.