  Topology is bringing new tools for the study of fluid waves. The existence of
unidirectional Yanai and Kelvin equatorial waves has been related to a
topological invariant, the Chern number, that describes the winding of
$f$-plane shallow water eigenmodes around band crossing points in parameter
space. In this previous study, the topological invariant was a property of the
interface between two hemispheres. Here we ask whether a topological index can
be assigned to each hemisphere. We show that this can be done if the shallow
water model in $f$-plane geometry is regularized by an additional odd-viscous
term. We then compute the spectrum of a shallow water model with a sharp
equator separating two flat hemispheres, and recover the Kelvin and Yanai waves
as two exponentially trapped waves along the equator, with all the other modes
delocalized into the bulk. This model provides an exactly solvable example of
bulk-interface correspondence in a flow with a sharp interface, and offers a
topological interpretation for some of the transition modes described by [Iga,
Journal of Fluid Mechanics 1995]. It also paves the way towards a topological
interpretation of coastal Kelvin waves along a boundary, and more generally, to
an understanding of bulk-boundary correspondence in continuous media.
