Latitudinal libration driven flows in triaxial ellipsoids

TL;DR

This paper investigates latitudinal libration-driven flows in triaxial ellipsoids, deriving laminar solutions, analyzing stability, and validating with numerical simulations to understand planetary core dynamics.

Contribution

It provides the first linear stability analysis of libration-driven flows in triaxial ellipsoids, including resonance and instability mechanisms relevant to planetary cores.

Findings

Resonance occurs when libration frequency matches spin-over mode.

Flow is prone to inertial instabilities via parametric resonance.

No instabilities are expected in the liquid cores of Moon, Io, and Mercury.

Abstract

Motivated by understanding the liquid core dynamics of tidally deformed planets and moons, we present a study of incompressible flow driven by latitudinal libration within rigid triaxial ellipsoids. We first derive a laminar solution for the inviscid equations of motion under the assumption of uniform vorticity flow. This solution exhibits a res- onance if the libration frequency matches the frequency of the spin-over inertial mode. Furthermore, we extend our model by introducing a reduced model of the effect of viscous Ekman layers in the limit of low Ekman number (Noir and C\'ebron 2013). This theoretical approach is consistent with the results of Chan et al. (2011) and Zhang et al. (2012) for spheroidal geometries. Our results are validated against systematic three-dimensional numerical simulations. In the second part of the paper, we present the first linear sta- bility analysis of this uniform vorticity flow. To this end, we adopt different methods (Lifschitz and Hameiri 1991; Gledzer and Ponomarev 1977) that allow to deduce upper and lower bounds for the growth rate of an instability. Our analysis shows that the uniform vorticity base flow is prone to inertial instabilities caused by a parametric resonance mechanism. This is confirmed by a set of direct numerical simulations. Applying our results to planetary settings, we find that neither a spin-over resonance nor an inertial instability can exist within the liquid core of the Moon, Io and Mercury.