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We study the two-dimensional rotating shallow-water model describing Earth'soceanic layers. It is formally analogue to a Schr\"odinger equation where thetools from topological insulators are relevant. Once regularized at small scaleby an odd-viscous term, such a model has a well-defined bulk topological index.However, in presence of a sharp boundary, the number of edge modes depends onthe boundary condition, showing an explicit violation of the bulk-edgecorrespondence. We study a continuous family of boundary conditions with a richphase diagram, and explain the origin of this mismatch. Our approach relies onscattering theory and Levinson's theorem. The latter does not apply at infinitemomentum because of the analytic structure of the scattering amplitude there,ultimately responsible for the violation.

Topology in shallow-water waves: a violation of bulk-edge correspondence