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The Madden–Julian oscillation (MJO) is an equatorial eastward moving system with a planetary-scale coupled Kelvin–Rossby wave structure. The equatorial waves and their interaction with convection are expected to play an important role in MJO dynamics. Using the trio-interaction model for essential MJO dynamics, this study investigates the importance of dynamic feedback that includes wave feedback (WF) and boundary layer convergence feedback (BLCF), by comparing the moisture mode (MM) that contains only moisture feedback (MF) and cloud-radiative feedback (CRF), with the dynamic moisture mode (DMM) that includes additional WF and BLCF. It is shown that the dynamic feedbacks fundamentally change the properties of the MJO mode. For the MM, the MF alone yields a damping and quasi-stationary mode on wavenumber 2–4. The CRF can destabilize the MM, but it cannot produce planetary wave selection. By including the dynamic feedbacks (WF and BLCF), the resultant DMM is an unstable mode with a preferred planetary scale, which moves eastward slowly, yielding a 30–90-day period. The dynamic feedbacks produce the planetary scale selection of the DMM through generating more eddy available moist static energy on the longer wavelengths. The WF can significantly change the structure of the MM and links the propagation of the DMM to the Kelvin and Rossby wave components, with stronger Kelvin (Rossby) wave favoring faster (slower) propagation. The BLCF enhances the Kelvin wave component on the longer wavelengths, changing the horizontal structures and accelerating the eastward propagation. The WF relates the dispersion feature of the DMM to the properties of the Kelvin and Rossby waves. Since the Kelvin-wave (Rossby-wave) frequency increases (decreases) with increasing wavenumber, their coupling in the DMM yields a quasi-constant frequency at the planetary scales (wavenumber 1–3).

Dynamic moisture mode versus moisture mode in MJO dynamics: importance of the wave feedback and boundary layer convergence feedback