Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-25T00:39:16.828Z Has data issue: false hasContentIssue false
  • English
  • Français

A bulk-interface correspondence for equatorial waves

Published online by Cambridge University Press:  10 April 2019

C. Tauber Open the ORCID record for C. Tauber [Opens in a new window] ,
P. Delplace  and
A. Venaille Open the ORCID record for A. Venaille [Opens in a new window]
C. Tauber*
Affiliation:
Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, CH-8093 Zürich, Switzerland
P. Delplace
Affiliation:
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
A. Venaille
Affiliation:
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
*
†Email address for correspondence: tauberc@phys.ethz.ch
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Geophysical and Geological Flows: Shallow water flows Mathematical Foundations: Topological fluid dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 868 , 10 June 2019 , R2
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avron, J. E. 1998 Odd viscosity. J. Stat. Phys. 92 (3–4), 543–557.Google Scholar
Bal, G.2018 Continuous bulk and interface description of topological insulators. arXiv:1808.07908.Google Scholar
Banerjee, D., Souslov, A., Abanov, A. G & Vitelli, V. 2017 Odd viscosity in chiral active fluids. Nat. Commun. 8 (1), 1573.Google Scholar
Dedalus, Project2016 http://ascl.net/1603.015, http://dedalus-project.org.Google Scholar
Delplace, P., Marston, J. B. & Venaille, A. 2017 Topological origin of equatorial waves. Science 358 (6366), 1075–1077.Google Scholar
Faure, F.2019 Manifestation of the topological index formula in quantum waves and geophysical waves. arXiv:1901.10592.Google Scholar
Faure, F. & Zhilinskii, B. 2000 Topological Chern indices in molecular spectra. Phys. Rev. Lett. 85 (5), 960–963.Google Scholar
Graf, G. M. & Porta, M. 2013 Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324 (3), 851–895.Google Scholar
Hatsugai, Y. 1993 Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71 (22), 3697–3700.Google Scholar
Iga, K. 1995 Transition modes of rotating shallow water waves in a channel. J. Fluid Mech. 294, 367–390.Google Scholar
Matsuno, T. 1966 Quasi-geostrophic motions in the equatorial area. J. Meteorol. Soc. Japan Ser. II 44 (1), 25–43.Google Scholar
Nakahara, M. 2003 Geometry, Topology and Physics. CRC Press.Google Scholar
Perrot, M., Delplace, P. & Venaille, A.2018 Topological transition in stratified atmospheres. arXiv:1810.03328.Google Scholar
Shankar, S., Bowick, M. J. & Marchetti, M. C. 2017 Topological sound and flocking on curved surfaces. Phys. Rev. X 7 (3), 031039.Google Scholar
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.Google Scholar
Souslov, A., van Zuiden, B. C., Bartolo, D. & Vitelli, V. 2017 Topological sound in active-liquid metamaterials. Nat. Phys. 13 (11), 1091–1094.Google Scholar
Vallis, G. K. 2017 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Volovik, G. E. 1988 Analogue of quantum Hall effect in a superfluid 3 He film. Zh. Eksp. Teor. Fiz. 94 (9), 123–137.Google Scholar
Wiegmann, P. B. 2013 Hydrodynamics of Euler incompressible fluid and the fractional quantum Hall effect. Phys. Rev. B 88 (24), 241305.Google Scholar
Yang, Z., Gao, F., Shi, X., Lin, X., Gao, Z., Chong, Y. & Zhang, B. 2015 Topological acoustics. Phys. Rev. Lett. 114 (11), 114301.Google Scholar
Supplementary material: File

Tauber et al. supplementary material

Tauber et al. supplementary material 1

Download Tauber et al. supplementary material(File)
File 259.3 KB