Wave topology brought to the coast

TL;DR

This paper reveals that coastal Kelvin waves are topologically protected modes, analogous to topological insulators, by linking shallow water wave dynamics to the Haldane model and using Chern numbers to predict wave behavior.

Contribution

It introduces a topological framework for understanding Kelvin waves in coastal dynamics, connecting ocean wave phenomena to condensed matter physics concepts.

Findings

Kelvin waves are topologically protected modes.

Chern numbers predict the number of wave modes crossing frequency gaps.

The topological approach is robust to boundary shape and model changes.

Abstract

Since the pioneering work of Kelvin on Laplace tidal equations, a zoology of trapped waves have been found in the context of coastal dynamics. Among them, the one originally computed by Kelvin plays a particular role, as it is an unidirectional mode filling a frequency gap between different wave bands. The existence of such Kelvin waves is robust to changes in the boundary shape and in changes of the underlying model for the coast. This suggests a topological interpretation that has yet up to now remained elusive. Here we rectify the situation, by taking advantage of a reformulation of the shallow water dynamics that highlights an analogy with the celebrated Haldane model in condensed matter physics. For any profile of bottom topography, the number of modes that transit from one wave band to another in the dispersion relation is predicted by computing a first Chern number describing the topology of complex eigenmodes in a dual, simpler wave problem.