Latitudinal libration driven flows in triaxial ellipsoids

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 resonance 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 & Cébron 2013). This theoretical approach is consistent with the results of Chan et al. (2011) andZhang 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 stability analysis of this uniform vorticity flow. To this end, we adopt different methods (Lifschitz & Hameiri 1991;Gledzer & 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.

Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability

“…This agreement is promising, particularly given that the simulations were initialised with random noise (rather than a "clean" initialisation using the eigenfunction of the fastest growing mode), and they also include non-negligible viscosity. Similar agreement between simulations and theoretical predictions has been found for the related problem of elliptical instability driven by latitudinal libration (Vantieghem et al 2015).…”

Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

“…An equivalent set has also been adopted by Wu & Roberts (2011) (where the basis vectors up to ℓ = 5 are written down explicitly) for studying instabilities of precessional motions in the Earth's core, and by Vantieghem (2014) for studying the free IMs of a triaxial ellipsoid in a rigid container (which was subsequently applied to study latitudinal libration-driven elliptical instability by Vantieghem et al 2015).…”

Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits

“…The latter studies look at the effects of various mechanical forcings in steady spheroidal containers (a = b), using the explicit formula of spheroidal inertial modes (Greenspan 1968;Zhang et al 2004). On the other hand, Vantieghem et al (2015) study global instabilities driven by latitudinal libration in triaxial ellipsoids, but only considering the inertial modes of degree n 3 (Vantieghem 2014). The explicit spatial dependence of inertial modes is not available in triaxial ellipsoids for higher degrees (Vantieghem 2014;Backus & Rieutord 2017a), and also in spheroidal containers as soon as the time dependence of the figure axes (2.10) is taken into account.…”

“…Yet, the elliptical instability may be a viable alternative as a strong source of dissipation (Cébron et al 2010a; Le Bars et al 2010;Barker 2016a). The libration-driven elliptical instability also occurs in synchronised moons (Kerswell & Malkus 1998;Cébron et al 2012b;Vantieghem et al 2015). Finally the elliptical instability is the first ingredient to explain the observed transition towards turbulence in experiments and simulations (Grannan et al 2014;Favier et al 2015;Grannan et al 2017;Le Reun et al 2017).…”