Poincaré inertio-gravity modes described by the shallow water equations in a rotating frame have nontrivial topology, providing a perspective on the origin of equatorially trapped Kelvin and Yanai waves. We investigate the topology of rotating shallow water equations and continuously stratified primitive equations with and without background shear flow. Continuously stratified fluids support waves that are analogous to the edge modes of weak three-dimensional topological insulators. Background shear flow not only breaks the Hermiticity and homogeneity of the system but also leads to instabilities. By introducing a gauge-invariant winding number, we show that singularities in the phase of the Poincaré waves of the unforced shallow-water equations and primitive equations persist in the presence of both horizontal and vertical shear flows. Thus, the bulk Poincaré bands have nontrivial topology and we expect and confirm the persistence of the equatorial waves in the presence of shear along the equator where the Coriolis parameter $f$ changes signs.

• Received 19 February 2022
• Accepted 31 August 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033191

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Published by the American Physical Society