Fluid dynamics in a librating triaxial ellipsoidal planet

Abstract

A planetary body can be deformed by both tidal force and rotational effect. In consequence, the cavity of a planetary fluid core is usually in the shape of a triaxial ellipsoid ${x^2}/{a^2} {y^2}/{b^2} {z^2}/{c^2}=1$, where $a, b$ and $c$ are three different semi-axes and $z$ is in the direction of rotation. Gravitational interaction between a planet and its parent star can force longitudinal libration by exerting an axial torque on the planet. When the equatorial eccentricity $e$ is small, $E^{1/4} << e=sqrt{a^2-b^2}/a^2 << 1$, where $E$ is the Ekman number, we derive an analytical solution describing librationally driven flows in the mantle frame of reference valid for any librating frequency. When the equatorial eccentricity $e$ is arbitrary, we carry out direct numerical simulation of the fully nonlinear problem in the same frame of reference, using an EBE (Element-By-Element) finite element method. A satisfactory agreement between the analytical solution and the nonlinear numerical simulation is achieved for small librating amplitude and new interesting nonlinear phenomena are revealed for large librating amplitude.