Simulations of fluid motion in ellipsoidal planetary cores driven by longitudinal libration

Deformed by tidal forces, the cavity of a planetary fluid core may be in the shape of a biaxial ellipsoid x 2 / a 2 + y 2 / b 2 + z 2 / a 2 = 1 , where a and b are two different semi-axes and z is in the direction of rotation. Gravitational interaction between a planet and its parent star exerts an axial torque on the planet and forces its longitudinal libration, a periodic variation of its angular velocity around its rotating axis. Longitudinal libration drives fluid motion in the planetary core via both viscous and topographic coupling between the mantle and fluid. For an arbitrary size of the equatorial eccentricity E = a 2 - b 2 / a , direct numerical simulation of the fully nonlinear problem is carried out using an EBE (Element-By-Element) finite element method. It is shown that fluid motion driven by longitudinal libration vacillates between two different phases: a prograde phase when the planet’s rotation speeds up and a retrograde phase when it slows down. For weak longitudinal libration, the fluid motion is laminar without exhibiting noticeable differences between the two phases and a multi-layered, time-independent, nearly geostrophic mean flow can be generated and maintained by longitudinal libration in a biaxial or triaxial ellipsoidal cavity. For strong slow libration, there are profound differences between the two different phases: the retrograde phase is usually marked by fluid motion with instabilities and complex spatial structure while in the prograde phase the flow is still largely laminar.