Diagnoses to unravel secular hydrodynamical processes in rotating main sequence stars

Context. Recent progress and constraints brought by helio and asteroseismology call for a better description of stellar interiors and an accurate description of rotation-driven mechanisms in stars. Aims. We present a detailed analysis of the main physical processes responsible for the transport of angular momentum and chemical species in the radiative regions of rotating stars. We focus on cases where meridional circulation and shear-induced turbulence all that are included in the simulations (i.e., no either internal gravity waves nor magnetic fields). We put special emphasis on analysing the angular momentum transport loop and on identifying the contribution of each of the physical process involved. Methods. We develop a variety of diagnostic tools designed to help disentangle the role of the various transport mechanisms. Our analysis is based on a 2-D representation of the secular hydrodynamics, which is treated using expansions in spherical harmonics. By taking appropriate horizontal averages, the problem reduces to one dimension, making it implementable in a 1D stellar evolution code, while preserving the advective character of angular momentum transport. We present a full reconstruction of the meridional circulation and of the associated fluctuations of temperature and mean molecular weight, along with diagnosis for the transport of angular momentum, heat, and chemicals. In the present paper these tools are used to validate the analysis of two main sequence stellar models of 1.5 and 20 M(circle dot), for which the hydrodynamics has previously been extensively studied in the literature. Results. We obtain a clear visualisation and a precise estimation of the different terms entering the angular momentum and heat transport equations in radiative zones of rotating stars. This enables us to corroborate the main results obtained over the past decade by Zahn, Maeder, and collaborators concerning the secular hydrodynamics of such objects. We focus on the meridional circulation driven by angular momentum losses and structural readjustments. We confirm quantitatively for the first time through detailed computations and separation of the various components that the advection of entropy by this circulation is balanced very well by the barotropic effects and the thermal relaxation during most of the main sequence evolution. This enables us to simplify for the thermal relaxation on this phase. The meridional currents in turn advect heat and generate temperature fluctuations that induce differential rotation through thermal wind, thus closing the transport loop. We plan to make use of our refined diagnosis tools in forthcoming studies of secular (magneto-) hydrodynamics of stars at various evolutionary stages.