The Madden-Julian oscillation (MJO) is an equatorial planetary-scale circulation system coupled with a multiscale convective complex, and it moves eastward slowly (about 5 m s(-1)) with a horizontal quadrupole vortex and vertical rearward-tilted structure. The nature and role of scale interaction (SI) is one of the elusive aspects of the MJO dynamics. Here a prototype theoretical model is formulated to advance the current understanding of the nature of SI in MJO dynamics. The model integrates three essential physical elements: (a) large-scale equatorial wave dynamics driven by boundary layer frictional convergence instability (FCI), (b) effects of the upscale eddy momentum transfer (EMT) by vertically tilted synoptic systems resulting from boundary layer convergence and multicloud heating, and (c) interaction between planetary-scale wave motion and synoptic-scale systems (the eastward-propagating super cloud clusters and westward-propagating 2-day waves). It is shown that the EMT mechanism tends to yield a stationary mode with a quadrupole vortex structure (enhanced Rossby wave component), whereas the FCI yields a relatively fast eastward-moving and rearward-tilted Gill-like pattern (enhanced Kelvin wave response). The SI instability stems from corporative FCI or EMT mechanisms, and its property is a mixture of FCI and EMT modes. The properties of the unstable modes depend on the proportion of deep convective versus stratiform/congestus heating or the ratio of deep convective versus total amount of heating. With increasing stratiform/congestus heating, the FCI weakens while the EMT becomes more effective. A growing SI mode has a horizontal quadrupole vortex and rearward-tilted structure and prefers slow eastward propagation, which resembles the observed MJO. The FCI sets the rearward tilt and eastward propagation, while the EMT slows down the propagation speed. The theoretical results presented here point to the need to observe multicloud structure and vertical heating profiles within the MJO convective complex and to improve general circulation models' capability to reproduce correct partitioning of cloud amounts between deep convective and stratiform/congestus clouds. Limitations and future work are also discussed.
