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Zonal flows driven by libration in rotating spherical shells: the case of periodic characteristic paths

Part of: JFM Notebooks

Published online by Cambridge University Press:  02 January 2026

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

This work investigates the weakly nonlinear dynamics of internal shear layers and the mean zonal flow induced by the longitudinal libration of an inner core within a spherical shell. Building on the work of He et al. (2022 J. Fluid Mech., vol. 939, p. A3), which focused on linear dynamics, we adopt a similar set-up to explore the nonlinear regime using both asymptotic theory and numerical computations, with Ekman numbers as low as $E=10^{-10}$. A specific forcing frequency of $\widehat {\omega }=\sqrt {2}\widehat {\varOmega }$, where $\widehat {\varOmega }$ denotes the rotation rate, is introduced to generate a closed rectangular path of characteristics for the inertial wave beam generated at the critical latitude. Our approach extends previous results by Le Dizès (2020 J. Fluid Mech., vol. 899, p. A21) and reveals that nonlinear interactions are predominantly localised around regions where the wave beam reflects on the boundary. We derive specific scaling laws governing the nonlinear interactions: the width of the interaction region scales as $E^{1/3}$ and the amplitude of the resulting mean zonal flow scales as $E^{1/6}$ in general. However, near the rotation axis, where the singularity of the self-similar solution becomes more pronounced, the amplitude exhibits a scaling of $E^{-1/2}$. In addition, our study also examines the nonlinear interactions of beams that are governed by different scaling laws. Through comparison with numerical results, we validate the theoretical predictions of the asymptotic framework, observing good agreement as the Ekman number decreases.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Geophysical and Geological Flows: Rotating flows

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JFM Papers
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Journal of Fluid Mechanics , Volume 1026 , 10 January 2026 , A15
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© The Author(s), 2026. Published by Cambridge University Press

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