Waves of Topological Origin in Continuous Media*

The discovery that topology plays a role in the origin of certain boundary or interfacial waves was first made in the context of quantum matter, specifically the integer quantum Hall effect.  David Thouless and co-workers used linear response theory (Kubo formula) to show that the quantized conductivity was a manifestation of the integer-valued Chern number.   Topology also manifests in other forms of quantum matter such as topological insulators.  Because it is the wave-like nature of the Schrödinger equation that is relevant for topology, rather than the particle-like aspect, classical waves of topological origin were soon found in media such as optical and acoustic lattices and even mechanical systems such as coupled gyroscopes.  All these systems have in common an underlying lattice with a compact Brillouin zone in momentum or wavevector space that permits the Chern number to be well-defined.  Therefore, it came as a surprise that waves of topological origin also exist along the boundaries or interfaces of continuous media such as fluids and plasmas.  In retrospect we can see now that the coastal Kelvin wave discovered by William Thomson in 1879 was likely the first such wave of topological origin to be noticed.  Non-trivial topology has now been revealed in observations of inertia-gravity waves in Earth’s stratosphere, and waves of topological origin are predicted in magnetized plasmas and in the interiors of many stars.  What new examples of topological waves are waiting to be discovered?