Libration-induced mean flow in a spherical shell

TL;DR

This paper studies how longitudinal libration in a spherical shell induces a mean zonal flow through non-linear interactions in the Ekman layers, with analytical and numerical analysis revealing the flow structure and discontinuities.

Contribution

It develops an analytical perturbative theory for libration-induced mean flow in a spherical shell, including flow discontinuities and smoothing mechanisms, validated by numerical simulations.

Findings

Mean flow scales as the square of libration amplitude

Discontinuity across the tangent cylinder can be smoothed by Stewartson layers

Good agreement between analytical theory and numerical simulations

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.