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Internal shear layers generated by a vertically oscillating cylinder in unbounded and bounded rotating fluids

Part of: JFM Notebooks

Published online by Cambridge University Press:  22 July 2025

Jiyang He Open the ORCID record for Jiyang He [Opens in a new window] ,
Benjamin Favier Open the ORCID record for Benjamin Favier [Opens in a new window]  and
Stéphane Le Dizès Open the ORCID record for Stéphane Le Dizès [Opens in a new window]
Jiyang He*
Affiliation:
Department of Ocean Science, The Hong Kong University of Science and Technology, Hong Kong, PR China Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France
Benjamin Favier
Affiliation:
Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France
Stéphane Le Dizès
Affiliation:
Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France
*
Corresponding author: Jiyang He, jiyanghe123@gmail.com
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Abstract

In rotating fluids, the viscous smoothing of inviscid singular inertial waves leads to the formation of internal shear layers. In previous works, we analysed the internal shear layers excited by a viscous forcing (longitudinal libration) in a spherical shell geometry (He et al., 2022 J. Fluid Mech. 939, A3; He et al., 2023 J. Fluid Mech. 974, A3). We now consider the stronger inviscid forcing corresponding to the vertical oscillation of the inner boundary. We limit our analysis to two-dimensional geometries but examine three different configurations: freely propagating wave beams in an unbounded domain and two wave patterns (a periodic orbit and an attractor) in a cylindrical shell geometry. The asymptotic structures of the internal shear layers are assumed to follow the similarity solution of Moore & Saffman (1969 Phil. Trans. R. Soc. Lond. A, 264, 597–634) in the small viscous limit. The two undefined parameters of the similarity solution (singularity strength and amplitude) are derived by asymptotically matching the similarity solution with the inviscid solution. For each case, the derivation of the latter is achieved either through separation of variables combined with analytical continuation or the method of characteristics. Global inviscid solutions, when obtained, closely match numerical solutions for small Ekman numbers far from the critical lines, while viscous asymptotic solutions show excellent performance near those lines. The amplitude scalings of the internal shear layers excited by an inviscid forcing are found to be divergent as the Ekman number $E$ decreases, specifically $O(E^{-1/6})$ for the critical-point singularity and $O(E^{-1/3})$ for attractors, in contrast to the convergent scalings found for a viscous forcing.

JFM classification

Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids Boundary Layers: Boundary layer separation

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JFM Papers
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Journal of Fluid Mechanics , Volume 1015 , 25 July 2025 , A38
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© The Author(s), 2025. Published by Cambridge University Press

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