Equatorial waves in the tropics possess meridional trapping structures, whereas disturbances associated with the midlatitude westerlies in the atmosphere exhibit vertical trapping structures through the coupling and organization of upper- and lower-layer eddies. Despite this common dynamical feature, the similarities between these two classes of waves have received little attention in previous studies. Our research group initially began its work motivated by equatorial waves. The theoretical understanding developed for trapped wave structures has subsequently been extended to baroclinically unstable disturbances in midlatitude jets, leading to new approaches that connect geophysical fluid dynamics with climate diagnostics. The wave-energy flux diagnostics developed by Aiki et al. (2017 PEPS, 2026 JAS) provide a continuous and dynamically consistent framework for evaluating wave-energy transfer pathways from the tropics to the midlatitudes, including baroclinically unstable waves embedded in the midlatitude westerlies. The associated energy flux vectors can be interpreted in terms of group velocity, allowing quantitative diagnosis of wave-energy pathways. This framework provides a powerful tool for investigating the generation, transfer, and dissipation of wave energy in both the atmosphere and ocean, without geographical restrictions and without limitations imposed by the strength of the background shear.
Shallow water models provide a fundamental framework for studying large-scale dynamics in the atmosphere and ocean. The present model incorporates online diagnostics of wave-energy fluxes during model integration, allowing energy pathways and group velocities to be evaluated without additional post-processing. These diagnostics connect wave dynamics with large-scale energy transport, providing a useful bridge between geophysical fluid dynamics and climate dynamics.