From the geometry of Foucault pendulum to the topology of planetary waves

Antoine Venaille,Pierre Delplace

doi:10.5802/crphys.28

Abstract

La physique des isolants topologiques permet de comprendre et prédire l’existence d’ondes unidirectionnelles piégées le long d’un bord ou d’une interface. Nous décrivons dans cette revue comment ces idées peuvent être adaptées aux ondes géophysiques et astrophysiques. Nous traitons en particulier le cas des ondes équatoriales planétaires, qui met en lumière les rôles clés combinés de la rotation et de la sphéricité de la planète pour expliquer l’émergence d’ondes qui ne propagent leur énergie que vers l’est. Ces ingrédients minimaux sont précisément ceux mis en avant dans l’interprétation géométrique du pendule de Foucault. Nous discutons cet exemple classique de mécanique pour introduire les concepts d’holonomie et de fibré vectoriel que nous utilisons ensuite pour le calcul des propriétés topologiques des ondes équatoriales en eau peu profonde.

Highlights

As recalled by Michael Berry [1], the investigation of the topological properties of waves started during the “miraculous 1830s” with the discovery of their singularities: the singularity of the intensity explaining the emergence rainbows; the singularity of the phase, as amphidromic points1 discovered at that time in the North sea; and the singularity of the polarization whose theoretical prediction led to the observation of the conical refraction in optics
We review here the recent input of topological tools inherited from topological insulators to these geophysical and astrophysical waves
We propose to use the Foucault pendulum as a starting point to introduce key notions of geometrical properties induced by a rotating planet, and use these tools to address the topology of equatorial waves

Summary

The renewal of topological waves

As recalled by Michael Berry [1], the investigation of the topological properties of waves started during the “miraculous 1830s” with the discovery of their singularities: the singularity of the intensity explaining the emergence rainbows; the singularity of the phase, as amphidromic points discovered at that time in the North sea; and the singularity of the polarization whose theoretical prediction led to the observation of the conical refraction in optics. For the last ten years, it was realized that topological properties similar to that of the integer quantum Hall effect and of the recently discovered topological insulators could be engineered in metamaterials with classical waves of various kinds, from optics [3] to mechanics [4, 5] and acoustics [6, 7] These topological properties are related to phase singularities of the complex eigenstates of the system in a parameter (or reciprocal) space, and translate in real space as the existence of trapped boundary modes that can be used to guide energy, through the celebrated bulk-boundary correspondence [8].

Coriolis force on Earth

A geometrical look at the Foucault pendulum

From geometrical phases to topological numbers for waves

Topological equatorial waves

Conclusion and prospects