New Multiscale Models and Self-Similarity in Tropical Convection

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Abstract
One of the unexplained striking features of tropical convection is the observed statistical self-similarity in clusters, superclusters, and intraseasonal oscillations through complex multiscale processes ranging from the mesoscales to the equatorial synoptic scales to the intraseasonal/planetary scales. Here new multispatial-scale, multitime-scale, simplified asymptotic models are derived systematically from the equatorial primitive equations on the range of scales from mesoscale to equatorial synoptic to planetary/intraseasonal, which provide a useful analytic framework for addressing these issues. New mesoscale equatorial synoptic dynamical (MESD) models and balanced MESD (BMESD) models are developed for the multitime, multispace interaction from mesoscales to equatorial synoptic scales; new multitime versions of the intraseasonal planetary equatorial synoptic dynamics (IPESD) models are developed for multiple spatiotemporal interactions on equatorial synoptic scales and planetary scales. The mathematical character derived below for all these simplified models explicitly demonstrates that the main nonlinear interactions across scales are quasi-linear where eddy flux divergences of momentum and temperature from nonlinear advection from the smaller-scale spatiotemporal flows as well as mean source effects accumulate in time and drive the waves on the successively larger spatiotemporal scales. Furthermore, these processes that transfer energy to the next larger, longer, spatiotemporal scales are self-similar in a suitable sense established here. On the other hand, the larger scales set the environment for this transport through processes such as mean advection of the smaller scales.