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A new model is derived and analyzed for tropical-extratropical interactionsinvolving the Madden-Julian oscillation (MJO). The model combines (i) thetropical dynamics of the MJO and equatorial baroclinic waves and (ii) thedynamics of barotropic Rossby waves with significant extratropical structure,and the combined system has a conserved energy. The method of multiscaleasymptotics is applied to systematically derive a system of ordinarydifferential equations (ODEs) for three-wave resonant interactions. Two novelfeatures are (i) a degenerate auxiliary problem with overdetermined equationsdue to a compatibility condition (meridional geostrophic balance) and (ii)cubic self-interaction terms that are not typically found in three-waveresonance ODEs. Several examples illustrate applications to MJO initiation andtermination, including cases of (i) the MJO, equatorial baroclinic Rossbywaves, and barotropic Rossby waves interacting, and (ii) the MJO, baroclinicKelvin waves, and barotropic Rossby waves interacting. Resonance with theKelvin wave is not possible here if only dry variables are considered, but itoccurs in the moist model here through interactions with water vapor andconvective activity.

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