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A bulk-interface correspondence for equatorial waves

  • C. Tauber (a1), P. Delplace (a2) and A. Venaille (a2)

Abstract

Topology is introducing new tools for the study of fluid waves. The existence of unidirectional Yanai and Kelvin equatorial waves has been related to a topological invariant, the Chern number, that describes the winding of $f$ -plane shallow water eigenmodes around band-crossing points in parameter space. In this previous study, the topological invariant was a property of the interface between two hemispheres. Here we ask whether a topological index can be assigned to each hemisphere. We show that this can be done if the shallow water model in the $f$ -plane geometry is regularized by an additional odd-viscosity term. We then compute the spectrum of a shallow water model with a sharp equator separating two flat hemispheres, and recover the Kelvin and Yanai waves as two exponentially trapped waves along the equator, with all the other modes delocalized into the bulk. This model provides an exactly solvable example of bulk-interface correspondence in a flow with a sharp interface, and offers a topological interpretation for some of the transition modes described by Iga (J. Fluid Mech., vol. 294, 1995, pp. 367–390). It also paves the way towards a topological interpretation of coastal Kelvin waves along a boundary and, more generally, to an understanding of bulk-boundary correspondence in continuous media.

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Corresponding author

Email address for correspondence: tauberc@phys.ethz.ch

References

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Avron, J. E. 1998 Odd viscosity. J. Stat. Phys. 92 (3–4), 543557.
Bal, G.2018 Continuous bulk and interface description of topological insulators. arXiv:1808.07908.
Banerjee, D., Souslov, A., Abanov, A. G & Vitelli, V. 2017 Odd viscosity in chiral active fluids. Nat. Commun. 8 (1), 1573.
Delplace, P., Marston, J. B. & Venaille, A. 2017 Topological origin of equatorial waves. Science 358 (6366), 10751077.
Faure, F.2019 Manifestation of the topological index formula in quantum waves and geophysical waves. arXiv:1901.10592.
Faure, F. & Zhilinskii, B. 2000 Topological Chern indices in molecular spectra. Phys. Rev. Lett. 85 (5), 960963.
Graf, G. M. & Porta, M. 2013 Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324 (3), 851895.
Hatsugai, Y. 1993 Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71 (22), 36973700.
Iga, K. 1995 Transition modes of rotating shallow water waves in a channel. J. Fluid Mech. 294, 367390.
Matsuno, T. 1966 Quasi-geostrophic motions in the equatorial area. J. Meteorol. Soc. Japan Ser. II 44 (1), 2543.
Nakahara, M. 2003 Geometry, Topology and Physics. CRC Press.
Perrot, M., Delplace, P. & Venaille, A.2018 Topological transition in stratified atmospheres. arXiv:1810.03328.
Shankar, S., Bowick, M. J. & Marchetti, M. C. 2017 Topological sound and flocking on curved surfaces. Phys. Rev. X 7 (3), 031039.
Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. 2019 Topological waves in fluids with odd viscosity. Phys. Rev. Lett. (submitted). arXiv:1802.09649.
Souslov, A., van Zuiden, B. C., Bartolo, D. & Vitelli, V. 2017 Topological sound in active-liquid metamaterials. Nat. Phys. 13 (11), 10911094.
Vallis, G. K. 2017 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.
Volovik, G. E. 1988 Analogue of quantum Hall effect in a superfluid 3 He film. Zh. Eksp. Teor. Fiz. 94 (9), 123137.
Wiegmann, P. B. 2013 Hydrodynamics of Euler incompressible fluid and the fractional quantum Hall effect. Phys. Rev. B 88 (24), 241305.
Yang, Z., Gao, F., Shi, X., Lin, X., Gao, Z., Chong, Y. & Zhang, B. 2015 Topological acoustics. Phys. Rev. Lett. 114 (11), 114301.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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