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Non-Hermitian Chern number in rotating Rayleigh–Bénard convection

Published online by Cambridge University Press:  15 November 2024

Furu Zhang Open the ORCID record for Furu Zhang [Opens in a new window]  and
Jin-Han Xie Open the ORCID record for Jin-Han Xie [Opens in a new window]
Furu Zhang
Affiliation:
School of Science, Beijing Forestry University, Beijing 100083, PR China Department of Mechanics and Engineering Science at College of Engineering and State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, PR China
Jin-Han Xie*
Affiliation:
Department of Mechanics and Engineering Science at College of Engineering and State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: jinhanxie@pku.edu.cn
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Abstract

We show that rotating Rayleigh–Bénard convection, where a rotating fluid is heated from below, exhibits a non-Hermitian topological invariant. Recently, Favier & Knobloch (J. Fluid Mech., vol. 895, 2020, R1) hypothesized that the robust sidewall modes in rapidly rotating convection are topologically protected. By considering a Berry curvature defined in the complex wavenumber space, we reveal that the bulk states can be characterized by a non-zero integer Chern number, implying a potential topological origin of the edge modes based on the Atiyah–Patodi–Singer index theorem (Fukaya et al., Phys. Rev. D, vol. 96 2017, 125004; Yu et al., Nucl. Phys. B, vol. 916, 2017, pp. 550–566). The linearized eigenvalue problem is intrinsically non-Hermitian, therefore, the definition of Berry curvature generalizes that of the stably stratified problem. Moreover, the three-dimensional set-up naturally regularizes the eigenvector, avoiding the compactification problem in shallow water waves (Tauber et al., J. Fluid Mech., vol. 868, 2019, R2). Under the hydrostatic approximation, it recovers a two-dimensional analogue of the one which explains the topological origin of the equatorial Kelvin and Yanai waves (Delplace et al., Science, vol. 358, issue 6366, 2017, pp. 1075–1077). The non-zero Chern number relies only on rotation when the fluid is stratified, no matter whether it is stable or unstable. However, the neutrally stratified system does not support a topological invariant. In addition, we define a winding number to visualize the topological nature of the fluid. Our results represent a step forward for the topologically protected states in convection, but the bulk-boundary correspondence requires a further direct analysis for proof, and the robustness of the edge states under varying boundary conditions remains a question to be answered.

JFM classification

Geophysical and Geological Flows: Rotating flows Mathematical Foundations: Topological fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 999 , 25 November 2024 , A65
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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