Libration-induced mean flow in a spherical shell

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 ε 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(ε ² E5/ 42 ) in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations, and a good agreement is demonstrated.