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Topology of shallow-water waves on a rotating sphere

Published online by Cambridge University Press:  24 January 2025

Nicolas Perez Open the ORCID record for Nicolas Perez [Opens in a new window] ,
Armand Leclerc Open the ORCID record for Armand Leclerc [Opens in a new window] ,
Guillaume Laibe Open the ORCID record for Guillaume Laibe [Opens in a new window]  and
Pierre Delplace Open the ORCID record for Pierre Delplace [Opens in a new window]
Nicolas Perez*
Affiliation:
Centre de Recherche Astrophysique de Lyon (UMR CNRS 5574), Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, F-69230 Saint-Genis-Laval, France
Armand Leclerc
Affiliation:
Centre de Recherche Astrophysique de Lyon (UMR CNRS 5574), Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, F-69230 Saint-Genis-Laval, France
Guillaume Laibe
Affiliation:
Centre de Recherche Astrophysique de Lyon (UMR CNRS 5574), Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, F-69230 Saint-Genis-Laval, France
Pierre Delplace
Affiliation:
Laboratoire de Physique (UMR CNRS 5672), Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, F-69342 Lyon, France
*
†Email address for correspondence: [email protected]
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Abstract

Topological properties of the spectrum of shallow-water waves on a rotating spherical body are established. Particular attention is paid to spectral flow, i.e. the modes whose frequencies transit between the Rossby and inertia–gravity wavebands as the zonal wavenumber is varied. Organising the modes according to the number of zeros of their meridional velocity, we conclude that the net number of modes transiting between the shallow-water wavebands on the sphere is null, in contrast to the Matsuno spectrum. This difference can be explained by a miscount of zeros under the $\beta$-plane approximation. We corroborate this result with the analysis of Delplace et al. (Science, vol. 358, 2017, pp. 1075–1077) by showing that the curved metric discloses a pair of degeneracy points in the Weyl symbol of the wave operator, non-existent under the $\beta$-plane approximation, each of them bearing a Chern number of $-1$.

JFM classification

Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Waves in rotating fluids Mathematical Foundations: Topological fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1003 , 25 January 2025 , A35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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