We consider the hydrodynamic stability of homogeneous, incompressible and rotating ellipsoidal fluid masses. The latter are the simplest models of fluid celestial bodies with internal rotation and subjected to tidal forces. The classical problem is the stability of Roche–Riemann ellipsoids moving on circular Kepler orbits. However, previous stability studies have to be reassessed. Indeed, they only consider global perturbations of large wavelength or local perturbations of short wavelength. Moreover many planets and stars undergo orbital motions on eccentric Kepler orbits, implying time-dependent ellipsoidal semi-axes. This time dependence has never been taken into account in hydrodynamic stability studies. In this work we overcome these stringent assumptions. We extend the hydrodynamic stability analysis of rotating ellipsoids to the case of eccentric orbits. We have developed two open-source and versatile numerical codes to perform global and local inviscid stability analyses. They give sufficient conditions for instability. The global method, based on an exact and closed Galerkin basis, handles rigorously global ellipsoidal perturbations of unprecedented complexity. Tidally driven and libration-driven elliptical instabilities are first recovered and unified within a single framework. Then we show that new global fluid instabilities can be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits. Their existence is confirmed by a local analysis and direct numerical simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital eccentricity may have a strong destabilising effect in celestial fluid bodies, which may lead to space-filling turbulence in most of the parameters range.
Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits