Topology of rotating stratified fluids with and without background shear flow

Poincar\'e-gravity modes described by the shallow water equations in a rotating frame have non-trivial topology, providing a new 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 in the presence of a background sinusoidal shear flow. The introduction of a background shear flow not only breaks the Hermiticity and homogeneity of the system but also leads to instabilities. We show that singularities in the phase of the Poincar\'e waves of the unforced shallow-water equations and primitive equations persist in the presence of shear. Thus the bulk Poincar\'e bands have non-trivial 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 sign.