Kelvin waves over a differentially rotating spherical shell

Be stars are currently viewed as B-type stars surrounded by a disc fuelled by the star itself during episodic excretion events. The origin of these events is poorly understood. This study aims to determine whether surface equatorial Kelvin waves can be unstable and therefore can play a role in triggering the Be phenomenon. We first derived an analytical expression for gravito-inertial modes in the shallow water framework. We then numerically investigated the evolution of equatorial Kelvin modes as the system parameters varied. We extended the study to thick-layer configurations with a constant-density fluid. We then analysed the stability of these modes under differential rotation and viscous effects. We show that equatorial Kelvin waves still exist in a spherical shell of finite thickness, but their equatorial confinement is weaker. At low azimuthal wave numbers, Kelvin waves lie in the inertial-wave frequency band and therefore exhibit specificities of inertial waves, such as shear layers associated with singularities of the Poincaré equation. These shear layers constitute new dissipative structures for Kelvin waves. When a radial (shellular) differential rotation is imposed, we show that equatorial Kelvin waves can be destabilised, provided that differential rotation and viscosity are in an appropriate range. We trace back the non-monotonic behaviour of the instability growth rate to the rise of a critical layer where the fluid azimuthal velocity equals the phase speed of the surface waves. This study provides new insights into the behaviour of equatorial Kelvin waves in astrophysics, particularly in rapidly rotating stars. The results reinforce the idea that gravito-inertial waves, and more specifically the equatorial Kelvin waves, can be unstable and thus constitute key components in the mechanisms leading to the Be phenomenon.