Asymptotic regimes of rotating flows driven by mechanical forcing in planetary cores: turbulent saturation and large-scale organisation

Many terrestrial bodies, including the Earth, are surrounded by a magnetic field protecting them from high energy stellar particles. It originates in the turbulent motion of the liquid, conducting iron core of these planets and moons. The complex motion of liquid iron in planetary cores is often thought to be driven by thermal and solutal convection, but such a model is sometimes hard to conciliate with the heat budget of terrestrial planets, especially the smaller ones. To explain the existence of magnetic fields surrounding small moons such as Ganymede and Io, mechanical forcing induced by tides has been proposed as an alternative source of turbulence in planetary cores. Tidal interaction between a terrestrial body and a companion results in a distortion of the shape of the body, a deformation that remains mostly directed towards the companion and may rotate at a different rate compared to the planet or the moon spinning rate. This is the case for instance of the Earth-Moon system: the Earth’s tidal bulge rotates at the Moon’s orbiting rate (in 27 days) whereas the Earth’s spinning rate is much larger (1 day). Another effect of tidal interaction is to force periodic variations of the length of the day, an oscillation called “libration.” These two effects (differential rotation and libration) have been shown to excite parametric resonance of inertial waves, the latter being spontaneous oscillations of rotating fluids interiors induced by the restoring action of the Coriolis force. This resonance is called the “elliptical instability.” The inertial waves grow exponentially and eventually collapse into turbulence. Although the saturation of the instability is the most important state for dynamo action and orbital evolution of planets, it remains poorly understood. The work presented throughout this dissertation aims at better characterising the turbulence resulting from the elliptical instability, in particular in regimes that are relevant to geo- and astrophysics when both the tidal forcing amplitude and the viscous dissipation are weak. This investigation of the non-linear saturation of the parametric resonance is carried out with experiments and idealised numerical simulations, complemented by theoretical investigations. In the experiment, we reveal that two regimes exist in the saturation of the instability. The first one, which is classical of turbulence in rotating fluids, is dominated by strong vortices invariant along the rotation axis, or “geostrophic.” Additionally, we exhibit a new regime which is dominated by inertial waves in non-linear resonant interactions, a state called “inertial wave turbulence.” To extend our understanding of these two states and to fully characterise the inertial waves interactions, we proceed to idealised numerical simulations in a local cartesian model of tidal flows. It allows producing the two regimes of saturation and exploring the weak forcing and dissipation regime. With this ideal model, we show that the transition between the two regimes mentioned earlier is caused by an instability that vanishes below a finite forcing amplitude. We also explore the possibility for direct forcing of strong geostrophic motion by the resonant waves directly, but our simulations suggest that they should not dominated in the geophysical limit. We therefore conclude that the superposition of inertial waves type of saturation is the relevant one for planetary cores. We finally investigate the stability of stably stratified planetary cores undergoing tidal distortion. Similarly to the elliptical instability, we exhibit a resonance of internal waves, which are oscillations caused by the stable density stratification. We show with idealised numerical simulations that the resonant waves give rise to internal wave turbulence in the non-linear saturation of the instability.