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Internal shear layers in librating spherical shells: the case of attractors

Published online by Cambridge University Press:  23 October 2023

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 Université, CNRS, Centrale Marseille, IRPHE, F-13384 Marseille, France
Benjamin Favier
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, IRPHE, F-13384 Marseille, France
Michel Rieutord
Affiliation:
IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 avenue Édouard Belin, F-31400 Toulouse, France
Stéphane Le Dizès
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, IRPHE, F-13384 Marseille, France
*
†Email address for correspondence: jiyanghe123@gmail.com
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Abstract

Following our previous work on periodic ray paths (He et al., J. Fluid Mech., vol. 939, 2022, A3), we study asymptotically and numerically the structure of internal shear layers for very small Ekman numbers in a three-dimensional spherical shell and in a two-dimensional cylindrical annulus when the rays converge towards an attractor. We first show that the asymptotic solution obtained by propagating the self-similar solution generated at the critical latitude on the librating inner core describes the main features of the numerical solution. The internal shear layer structure and the scaling for its width and velocity amplitude in $E^{1/3}$ and $E^{1/12}$, respectively, are recovered. The amplitude of the asymptotic solution is shown to decrease to $E^{1/6}$ when it reaches the attractor, as is also observed numerically. However, some discrepancies are observed close to the particular attractors along which the phase of the wave beam remains constant. Another asymptotic solution close to those attractors is then constructed using the model of Ogilvie (J. Fluid Mech., vol. 543, 2005, pp. 19–44). The solution obtained for the velocity has an $O(E^{1/6})$ amplitude, but a self-similar structure different from the critical-latitude solution. It also depends on the Ekman pumping at the contact points of the attractor with the boundaries. We demonstrate that it reproduces correctly the numerical solution. Surprisingly, the numerical solution close to an attractor with phase shift (that is, an attractor touching the axis in three or two dimensions with a symmetric forcing) is also found to be $O(E^{1/6})$, but its amplitude is much weaker. However, its asymptotic structure remains a mystery.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Geophysical and Geological Flows: Rotating flows Boundary Layers: Boundary layer separation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 974 , 10 November 2023 , A3
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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