The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regions: I. Free waves

Context. Quantifying tidal interactions in close-in two-body systems is of prime interest since they have a crucial impact on the architecture and the rotational history of the bodies. Various studies have shown that the dissipation of tides in either body is very sensitive to its structure and to its dynamics. Furthermore, solar-like stars and giant gaseous planets in our Solar System experience differential rotation in their outer convective envelopes. In this respect, numerical simulations of tidal interactions in these objects have shown that the propagation and dissipation properties of tidally excited inertial waves can be strongly modified in the presence of differential rotation. Aims. In particular, tidal inertial waves may strongly interact with zonal flows at the so-called co-rotation resonances, where the wave's Doppler-shifted frequency is cancelled out. The energy dissipation at such resonances could deeply modify the orbital and spin evolutions of tidally interacting systems. In this context, we aim to provide a deep physical understanding of the dynamics of tidal waves at co-rotation resonances in the presence of differential rotation profiles that are typical of low-mass stars and giant planets. Methods. In this work, we have developed an analytical local model of an inclined shearing box that describes a small patch of the differentially rotating convective zone of a star or a planet. We investigate the propagation and the transmission of free inertial waves at co-rotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model. Results. We find that inertial waves can be fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux in the fluid when damped at critical levels, or they can extract action flux from the fluid when amplified at critical levels. Both situations can lead to significant angular momentum exchange between the tidally interacting bodies.