The combination of tidal deformations and rotation can destabilize planetary cores via elliptical instability, a mechanism also observed in localized strained vortices. It has been shown to generate turbulent flows composed of non-linearly interacting waves and strong columnar vortices with varying respective amplitudes, depending on control parameters and geometry. This nonlinear saturation is reminiscent of classical homogeneous rotating turbulence sustained by an arbitrary forcing where coherent structures invariant along the rotation axis naturally emerge from initially isotropic conditions.

We present a suite of direct numerical simulations to investigate the saturation regime of the elliptical instability relevant for planetary applications. This corresponds to a weak axial deformation (or equivalently a weak ellipticity of the base flow streamlines) and weak dissipation (or equivalently low Ekman numbers). We use the classical shearing-box approximation in a tri-periodic domain using the open-source pseudo-spectral code SNOOPY. We recover theoretical results obtained with a local WKB type approach and study the fully nonlinear saturation of the instability. The main focus of our study concerns the transition from vortex-dominated to wave-dominated regimes. This is achieved by simulating the growth and saturation of the elliptical instability and adding frictional damping to the geostrophic component only, to mimic its interaction with boundaries.
For weak friction, we recover the formation of geostrophic vortices saturating at the size of numerical domain. The initial resonance conditions at the origin of the elliptical instability can disappear due to the presence of these large-scale flows.
When friction is large enough, a wave-dominated regime that exhibits many signatures of inertial wave turbulence is observed and characterized for the first time. Finally, we clarify which regime is expected in geophysics by combining recent experimental and numerical results.