Libration-induced mean flow in a spherical shell

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 it can generate through non-linear interactions in the Ekman layers a strong mean zonal flow in the interior. An analytical theory is developed using a perturbative approach in the limit of small libration amplitude $\epsilon$ and small Ekman number $E$. The mean flow is found to be at leading order an azimuthal flow which scales as the square of the libration amplitude and only depends 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 which scales as $O({\epsilon }^2{E}^{5/42})$ in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations and a good agreement is demonstrated.

Accepted for publication in Journal of Fluid Mechanics

References in corpus (4)

• Five Planets Orbiting 55 Cancri
• Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity
• Fluid flows in a librating cylinder
• Experimental study of libration-driven zonal flows in non-axisymmetric containers

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• Libration-driven inertial waves and mean zonal flows in spherical shells
• Spontaneous generation of inertial waves from boundary turbulence in a librating sphere
• Mean zonal flows induced by weak mechanical forcings in rotating spheroids
• Libration driven multipolar instabilities
• Zonal flows driven by libration in rotating spherical shells: the case of periodic characteristic paths