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Libration-induced mean flow in a spherical shell

Published online by Cambridge University Press:  08 February 2013

Alban Sauret  and
Stéphane Le Dizès
Alban Sauret
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS, and Aix-Marseille University, 49 rue F. Joliot-Curie, F-13013 Marseille, France
Stéphane Le Dizès*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS, and Aix-Marseille University, 49 rue F. Joliot-Curie, F-13013 Marseille, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
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Abstract

We investigate the flow in a spherical shell subject to a time harmonic oscillation of its rotation rate, also called longitudinal libration, when the oscillation frequency is larger than twice the mean rotation rate. In this frequency regime, no inertial waves are directly excited by harmonic forcing. We show, however, that, through nonlinear interactions in the Ekman layers, it can generate a strong mean zonal flow in the interior. An analytical theory is developed using a perturbative approach in the limit of small libration amplitude $\varepsilon $ and small Ekman number $E$ . The mean flow is found to be at leading order an azimuthal flow that scales as the square of the libration amplitude and depends only on the cylindrical radius coordinate. The mean flow also exhibits a discontinuity across the cylinder tangent to the inner sphere. We show that this discontinuity can be smoothed through multi-scale Stewartson layers. The mean flow is also found to possess a weak axial flow that scales as $O({\varepsilon }^{2} {E}^{5/ 42} )$ in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations, and a good agreement is demonstrated.

JFM classification

Boundary Layers: Boundary Layers Geophysical and Geological Flows: Geophysical and Geological Flows

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 181 - 209
Copyright
©2013 Cambridge University Press

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