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Internal shear layers in librating spherical shells: the case of periodic characteristic paths

Published online by Cambridge University Press:  23 March 2022

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] ,
Michel Rieutord Open the ORCID record for Michel Rieutord [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:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, F-13384Marseille, France
Benjamin Favier
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, F-13384Marseille, France
Michel Rieutord
Affiliation:
IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 avenue Édouard Belin, F-31400Toulouse, France
Stéphane Le Dizès
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, F-13384Marseille, France
*
†Email address for correspondence: he@irphe.univ-mrs.fr
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Abstract

Internal shear layers generated by the longitudinal libration of the inner core in a spherical shell rotating at a rate $\varOmega ^*$ are analysed asymptotically and numerically. The forcing frequency is chosen as $\sqrt {2}\varOmega ^*$ such that the layers issued from the inner core at the critical latitude in the form of concentrated conical beams draw a simple rectangular pattern in meridional cross-sections. The asymptotic structure of the internal shear layers is described by extending the self-similar solution known for open domains to closed domains where reflections on the boundaries occur. The periodic ray path ensures that the beams remain localised around it. Asymptotic solutions for both the main beam along the critical line and the weaker secondary beam perpendicular to it are obtained. The asymptotic predictions are compared with direct numerical results obtained for Ekman numbers as low as $E=10^{-10}$. The agreement between the asymptotic predictions and numerical results improves as the Ekman number decreases. The asymptotic scalings in $E^{1/12}$ and $E^{1/4}$ for the amplitudes of the main and secondary beams, respectively, are recovered numerically. Since the self-similar solution is singular on the axis, a new local asymptotic solution is derived close to the axis and is also validated numerically. This study demonstrates that, in the limit of vanishing Ekman numbers and for particular frequencies, the main features of the flow generated by a librating inner core are obtained by propagating through the spherical shell the self-similar solution generated by the singularity at the critical latitude on the inner core.

JFM classification

Boundary Layers: Boundary layer separation Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A3
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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