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 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.

💡 Research Summary

The paper investigates the generation of a mean (time‑averaged) flow in a rotating spherical shell that is subjected to longitudinal libration, i.e., a harmonic modulation of its rotation rate. The focus is on the regime where the libration frequency ω exceeds twice the mean rotation rate Ω (ω > 2Ω). In this high‑frequency range inertial waves are not directly forced, so the origin of any mean flow must be sought in nonlinear mechanisms.

Using a perturbation expansion in the small libration amplitude ε and the small Ekman number E (viscous effects are weak), the authors first solve the linear problem (order ε). The linear response is confined to Ekman boundary layers of thickness δ_E ∼ E^{1/2} on both the inner and outer spheres. At second order (ε²) the quadratic nonlinear term (u₁·∇)u₁ produces a non‑zero time‑average, which acts as a body force driving a steady flow in the bulk.

The analytical solution shows that the leading‑order mean flow is purely azimuthal (zonal) and depends only on the cylindrical radius s. Its amplitude scales as ε² and is independent of the axial coordinate z. The functional form differs inside and outside the “tangent cylinder” that touches the inner sphere; consequently the azimuthal velocity exhibits a discontinuity at the cylinder radius s_c. This discontinuity is a direct consequence of the different Ekman‑layer dynamics on the inner and outer boundaries.

To resolve the jump, the authors invoke Stewartson layers—thin, multi‑scale shear layers that develop around s = s_c. These layers have thicknesses ranging from E^{1/4} to E^{1/3} and provide a smooth connection between the inner‑ and outer‑region solutions. Within the Stewartson layers a weak axial flow appears, scaling as ε² E^{5/42}; although its magnitude is far smaller than the azimuthal component, it is required for mass conservation and for the internal consistency of the layered structure.

The theoretical predictions are validated against axisymmetric numerical simulations of the full Navier‑Stokes equations. Simulations are performed for ε = 0.05–0.2 and E = 10^{-6}–10^{-4}. Time‑averaged azimuthal velocity profiles match the analytical expressions with high accuracy, and the predicted Stewartson‑layer thickness and smoothing of the discontinuity are clearly observed. The weak axial flow is also detected at the expected ε² E^{5/42} magnitude.

The study concludes that, even in the absence of directly forced inertial waves, longitudinal libration at high frequency can generate a robust mean zonal flow through nonlinear interactions in the Ekman layers. The flow is essentially geostrophic (cylindrically symmetric) and its structure is governed by the geometry of the spherical shell, especially the tangent cylinder. The identification of Stewartson layers as the mechanism that regularizes the discontinuity extends classical boundary‑layer theory to librating rotating systems. These results have implications for planetary interiors where libration of the mantle or core may drive large‑scale zonal currents, for astrophysical bodies experiencing tidal or rotational oscillations, and for engineering applications involving rotating machinery with periodic speed modulation.