Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry

Abstract

We report on axisymmetric numerical simulations of rapidly rotating spherical shells in which the axial rotation rate of the outer shell is modulated in time. This allows us to model planetary bodies undergoing forced longitudinal libration. In this study we systematically vary the Ekman number, 10−7≤E≲10−4, which characterizes the ratio of viscous to Coriolis forces in the fluid, and the libration amplitude, Δϕ. For libration amplitudes above a certain threshold, Taylor–Görtler vortices form near the outer librating boundary, in agreement with the previous laboratory experiments of Noir et al. [Phys. Earth Planet. Inter. 173, 141 (2009)]. At the lowest Ekman numbers investigated, we find that the instabilities remain spatially localized at onset in the equatorial region. In addition, nonzero time-averaged azimuthal (zonal) velocities are observed for all parameters studied. The zonal flow is characterized by predominantly retrograde flow in the interior, with a stronger prograde jet in the outer equatorial region. The magnitude of the zonal flow scales as the square of the librational forcing, ϵ2, where ϵ=Δϕf and f is the dimensionless libration frequency defined as the ratio between the libration frequency and the mean angular rotation rate. In addition, the zonal flow is primarily independent of the Ekman number, implying that the zonal flow does not depend on the viscosity of the fluid. The simulations show that the zonal flow is driven by nonlinearities in the Ekman boundary layer; it is not driven by Taylor–Görtler vortices or by inertial waves in the fluid interior. Application of our results suggests that many librating bodies in the solar system are above the onset for centrifugal instabilities, with values up to ∼30 times supercritical. However, the spatial localization of the instabilities at onset in our simulations suggests that their effects are limited on the global dynamics of librating bodies. We find that the zonal flows driven by libration in axisymmetric spherical shells are unlikely to produce significant planetary magnetic fields, but will likely generate nonzero mean torques on the bounding surfaces.