Comptes Rendus
no. 2
From the geometry of Foucault pendulum to the topology of planetary waves
[De la géométrie du pendule de Foucault à la topologie des ondes planétaires]
Pierre Delplace1  ; Antoine Venaille1
1 Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Comptes Rendus. Physique, Volume 21 (2020) no. 2, pp. 165-175.

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.

The physics of topological insulators makes it possible to understand and predict the existence of unidirectional waves trapped along an edge or an interface. In this review, we describe how these ideas can be adapted to geophysical and astrophysical waves. We deal in particular with the case of planetary equatorial waves, which highlights the key interplay between rotation and sphericity of the planet, to explain the emergence of waves which propagate their energy only towards the East. These minimal ingredients are precisely those put forward in the geometric interpretation of the Foucault pendulum. We discuss this classic example of mechanics to introduce the concepts of holonomy and vector bundle which we then use to calculate the topological properties of equatorial shallow water waves.

Publié le :
DOI : 10.5802/crphys.28
Keywords: Waves, Coriolis force, Chern numbers, Geometric phase, Astrophysical and geophysical flows
Mot clés : Ondes, Force de Coriolis, Nombre de Chern, Phase géométrique, Flots géophysiques et astrophysiques
Pierre Delplace 1 ; Antoine Venaille 1

1 Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Pierre Delplace; Antoine Venaille. From the geometry of Foucault pendulum to the topology of planetary waves. Comptes Rendus. Physique, Volume 21 (2020) no. 2, pp. 165-175. doi : 10.5802/crphys.28. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.28/

[1] M. Berry Making waves in physics, Nature, Volume 403 (2000), p. 21 | DOI

[2] M. V. Berry; R. G. Chambers; M. D. Large; C. Upstill; J. C. Walmsley Wavefront dislocations in the Aharonov–Bohm effect and its water wave analogue, Eur. J. Phys., Volume 1 (1980), p. 154 | DOI | MR

[3] T. Ozawa; H. M. Price; A. Amo; N. Goldman; M. Hafezi; L. Lu; M. C. Rechtsman; D. Schuster; J. Simon; O. Zilberberg; I. Carusotto Topological photonics, Rev. Mod. Phys., Volume 91 (2019), 015006 | DOI | MR

[4] L. M. Nash; D. Kleckner; A. Read; V. Vitelli; A. M. Turner; W. Irvine Topological mechanics of gyroscopic metamaterials, Proc. Natl Acad. Sci. USA, Volume 112 (2015) no. 47, pp. 14495-14500 | DOI

[5] R. Süsstrunk; S. D. Huber Observation of phononic helical edge states in a mechanical topological insulator, Science, Volume 349 (2015) no. 6243, pp. 47-50 | DOI

[6] X. Zhang; M. Xiao; Y. Cheng; M.-H. Lu; J. Christensen Topological sound, Commun. Phys., Volume 1 (2018) no. 1, p. 97 | DOI

[7] A. Souslov; B. C. van Zuiden; D. Bartolo; V. Vitelli Topological sound in active-liquid metamaterials, Nat. Phys., Volume 13 (2017), p. 1091 | DOI

[8] Y. Hatsugai Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett., Volume 71 (1993), pp. 3697-3700 | DOI | MR | Zbl

[9] M. Berry The quantum phase, five years after, Geometric Phases on Physics (A. Shapere; F. Wilczek, eds.), World Scientific, Singapore, 1989, 7 pages

[10] P. Delplace; J. Marston; A. Venaille Topological origin of equatorial waves, Science, Volume 358 (2017) no. 6366, pp. 1075-1077 | DOI | MR | Zbl

[11] M. Perrot; P. Delplace; A. Venaille Topological transition in stratified fluids, Nat. Phys., Volume 15 (2019) no. 8, pp. 781-784 | DOI

[12] J. B. Parker; J. Marston; S. M. Tobias; Z. Zhu Topological gaseous plasmon polariton in realistic plasma, Phys. Rev. Lett., Volume 124 (2020) no. 19, 195001 | MR

[13] S. Shankar; M. Bowick; M. C. Marchetti Topological sound and flocking on curved surfaces, Phys. Rev. X, Volume 7 (2017) no. 3, 031039

[14] A. Souslov; K. Dasbiswas; M. Fruchart; S. Vaikuntanathan; V. Vitelli Topological waves in fluids with odd viscosity, Phys. Rev. Lett., Volume 122 (2019) no. 12, 128001 | DOI | MR

[15] G. K. Vallis Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, Cambridge, UK, 2017 | Zbl

[16] L. Foucault Démonstration physique du mouvement de rotation de la Terre au moyen du pendule, C. R. Séances Acad. Sci., Volume 32 (1851), p. 157

[17] J. Sommeria Foucault and the rotation of the Earth, C. R. Phys., Volume 18 (2017) no. 9–10, pp. 520-525 | DOI

[18] R. Snieder; C. Sens-Schönfelder; E. Ruigrok; K. Shiomi Seismic shear waves as Foucault pendulum, Geophys. Res. Lett., Volume 43 (2016) no. 6, pp. 2576-2581 | DOI

[19] M. V. Berry Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A, Volume 392 (1984), p. 45 | MR | Zbl

[20] B. Simon Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett., Volume 51 (1983), pp. 2167-2170 | DOI | MR

[21] G. Jotzu; M. Messer; R. Desbuquois; M. Lebrat; T. Uehlinger; D. Greif; T. Esslinger Experimental realization of the topological Haldane model with ultracold fermions, Nature, Volume 515 (2014), p. 237 | DOI

[22] M. Wimmer; H. M. Price; I. Carusotto; U. Peschel Experimental measurement of the Berry curvature from anomalous transport, Nat. Phys., Volume 13 (2017) no. 6, p. 545 | DOI

[23] N. Perez; P. Delplace; A. Venaille “Manifestation of Berry curvature in geophysical ray tracing”, preprint, arXiv:2010.05575 (2020)

[24] D. J. Thouless; M. Kohmoto; M. P. Nightingale; M. den Nijs Quantized hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., Volume 49 (1982), pp. 405-408 | DOI

[25] C. Tauber; P. Delplace; A. Venaille A bulk-interface correspondence for equatorial waves, J. Fluid Mech., Volume 868 (2019), R2 | DOI | MR | Zbl

[26] C. Tauber; P. Delplace; A. Venaille Anomalous bulk-edge correspondence in continuous media, Phys. Rev. Res., Volume 2 (2020) no. 1, 013147 | DOI

[27] G. M. Graf; C. Jud; H. Tauber “Topology in shallow-water waves: a violation of bulk-edge correspondence”, preprint, arXiv:2001.00439 (2020)

[28] S. Sakai; I. Lizawa; E. Aramaki (“GFD Online ressources”, https://www.gfd-dennou.org/)

[29] T. Matsuno Quasi-geostrophic motions in the equatorial area, J. Meteorol. Soc. Japan. Ser. II, Volume 44 (1966) no. 1, pp. 25-43 | DOI

[30] F. Faure “Manifestation of the topological index formula in quantum waves and geophysical waves”, preprint, arXiv:1901.10592 (2019)

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