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Quantifying tidal interactions in close-in two-body systems is of primeinterest since they have a crucial impact on the architecture and on therotational history of the bodies. Various studies have shown that thedissipation of tides in either body is very sensitive to its structure and toits dynamics, like differential rotation which exists in the outer convectiveenveloppe of solar-like stars and giant gaseous planets. In particular, tidalwaves may strongly interact with zonal flows at the so-called corotationresonances, where the wave's Doppler-shifted frequency cancels out. We aim toprovide a deep physical understanding of the dynamics of tidal inertial wavesat corotation resonances, in the presence of differential rotation profilestypical of low-mass stars and giant planets. By developping an inclinedshearing box, we investigate the propagation and the transmission of freeinertial waves at corotation, and more generally at critical levels, which aresingularities in the governing wave differential equation. Through theconstruction of an invariant called the wave action flux, we identify differentregimes of wave transmission at critical levels, which are confirmed with aone-dimensional three-layer numerical model. We find that inertial waves can beeither fully transmitted, strongly damped, or even amplified after crossing acritical level. The occurrence of these regimes depends on the assumed profileof differential rotation, on the nature as well as the latitude of the criticallevel, and on wave parameters such as the inertial frequency and thelongitudinal and vertical wavenumbers. Waves can thus either deposit theiraction flux to the fluid when damped at critical levels, or they can extractaction flux to the fluid when amplified at critical levels. Both situationscould lead to significant angular momentum exchange between the tidallyinteracting bodies.

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