Non-linear internal waves breaking in stellar-radiation zones: Parametrisation for the transport of angular momentum: Bridging geophysical-to-stellar fluid dynamics

Context. Internal gravity waves (IGWs) are one of the mechanisms that can play a key role in efficiently redistributing angular momentum in stars along their evolution. The study of IGWs is thus of major importance since space-based asteroseismology reveals a transport of angular momentum in stars, which is stronger by two orders of magnitude than the one predicted by stellar models ignoring their action or those of magnetic fields. Aims. IGWs trigger angular momentum transport when they are damped by heat or viscous diffusion, when they meet a critical layer where their phase velocity in the azimuthal direction equals the zonal wind or when they break. Theoretical prescriptions have been derived for the transport of angular momentum induced by IGWs because of their radiative and viscous dampings and of the critical layers they encounter along their propagation. However, none have been proposed for the transport of angular momentum triggered by their non-linear breaking. In this work, we aim to derive such a physical and robust prescription, which can be implemented in stellar structure and evolution codes. Methods. We adapted an analytical saturation model - which has been developed for IGWs' nonlinear convective breaking in the Earth's atmosphere and has been successfully compared to in situ measurements in the stratosphere - to the case of deep spherical stellar interiors. This allowed us to derive the saturated amplitude of the velocity of IGWs breaking in stellar radiation zones through convective overturning of the stable stratification or the instability of the vertical shear of IGWs motion and of the angular momentum transport they trigger. In a first step, we neglected the modification of IGWs by the Coriolis acceleration and the Lorentz force, which are discussed and taken into account in a second step. Results. We derive a complete semi-analytical prescription for the transport of angular momentum by IGWs that takes into account both their radiative damping and their potential nonlinear breaking because of their convective and vertical shear instabilities. We show that the deposit of angular momentum by breaking waves increases with their latitudinal degree, the ratio of the Brunt-Va & iuml;s & auml;l & auml; frequency and the wave frequency; and when the density decreases or the Doppler-shifted frequency vanishes. This allows us to bring the physical prescription for the interactions between IGWs and the differential rotation to the same level of realism as the one used in global circulation models for the atmosphere.