Transport by gravito-inertial waves in differentially rotating stellar radiation zones I - Theoretical formulation

Context. We examine the dynamics of low-frequency waves in differentially rotating stellar radiation zones, the angular velocity being taken as generally as possible depending both on radius and on latitude in stellar interiors. The associated induced transport of angular momentum, which plays a key role in the evolution of rotating stars, is derived. Aims. We focus on the wave-induced transport of angular momentum, taking into account the Coriolis acceleration in the case of strong radial and latitudinal differential rotation. We thus go beyond the "weak differential rotation" approximation, where rotation is almost a solid-body one plus a residual radial differential rotation. As has been shown in previous works, the Coriolis acceleration modifies such transport. Methods. We built analytically a complete formalism that allows the study of rotational transport in stellar radiation zones taking into account the wave action modified by a general strong differential rotation. Results. The different approximations possible for low-frequency waves in a differentially rotating stably stratified radiative region, namely the traditional and the JWKB approximations, are examined and discussed. The complete bidimensional structure of regular elliptic gravito-inertial waves, which verify these approximations, is derived and compared to those in the "weak differential rotation" case. Next, associated transport of energy and of angular momentum in the vertical and in the horizontal directions and the dynamical equations, respectively for the mean radial differential rotation and the latitudinal one, are obtained. Conclusions. The complete formalism, which takes into account low-frequency wave action, is derived and can be used for the study of secular hydrodynamics of radiative regions and of the associated mixing. The modification of waves due to general strong differential rotation and their feed-back on the angular momentum transport are treated rigourously. In a forthcoming paper (Paper II), this formalism will be applied to the case of solar differential rotation. However, the case of hyperbolic gravito-inertial waves should be carefully studied.