Transition from inertial wave turbulence to geostrophic turbulence in rotating fluids - an experimental study

A broad array of geophysical and astrophysical flows, be it in planetary cores, the Earth's ocean and atmosphere, or stellar interiors, are turbulent and strongly affected by rotation. These flows are very different from homogenous and isotropic turbulence owing to the presence of structures influenced by the Coriolis force and evolving on very different time scales. On one hand, the restoring effect of the Coriolis force leads to the existence of inertial waves that oscillate at frequencies smaller than or equal to twice the rotation rate. On the other hand, vortices aligned with the rotation axis and persisting over much longer time scales are also ubiquitously observed in rotating turbulence; they are associated to a balance between the pressure and Coriolis forces and are often called "geostrophic vortices". In the vast majority of rotating turbulence experiments and simulations, in which flows are excited by random fields, alternative jets or pairs of vortices, geostrophic eddies are dominant compared to inertial waves, the latter being advected and deformed by these persistent flows.

We propose to re-evaluate this standard observation via examining the turbulent saturation of parametric instabilities in rotating fluids, a problem of great interest for planetary and stellar interiors. Gravitational attraction exerted by other astrophysical bodies on a planet or a star produces a tidal deformation of its shape and causes periodic alterations of its rotation rate called librations. The combination of these two effects excites the resonance of two inertial waves at a precise frequency that eventually breaks down into turbulence. To reach this turbulent state, the forcing is very different compared to more classical works on rotating turbulence. First, energy is transfered to the flow only via a couple of inertial waves; besides, in the geophysical regimes we consider, both the forcing amplitude and dissipation are weak (i.e. both the Rossby and the Ekman numbers are small).

The experimental set-up we use to study the turbulent saturation of this inertial wave instability is designed to mimic tidal forcings in astrophysical bodies. An ellipsoidal container is mounted on a rotating table with a secondary motor forcing harmonic perturbations of the mean rotation rate. We characterise the turbulent saturation flow with particle image velocimetry (PIV). With temporal analysis of the velocity fields, we show that at the lowest forcing amplitudes, the resonant waves excite daughter waves with different frequencies via non-linear resonant interactions. Conversely, at large forcing amplitudes, the resonant waves force a more classical geostrophic turbulence where the flow is dominated by strong vortices.

Our study suggests that, in the regimes of low forcing amplitude and dissipation, it is possible to excite an inertial wave turbulence, i.e. a flow with many waves that are non-linearly, resonantly, interacting. It is the first time a transition between a wave-dominated and a geostrophic-dominated regime is observed experimentally. This result has strong implications for instance on tidal dissipation and dynamo action in planets and stars. In particular, generation of a magnetic field by a set of inertial waves has seldom been considered and remains to be studied in detail.