In the framework of the GIARPS High-resolution Observations of T Tauri stars (GHOsT) project, we aim to characterize the atomic and molecular winds in a sample of classical T Tauri stars (CTTs) of the Taurus-Auriga region. We analyzed the flux calibrated [OI] 630 nm and $rm H_2$ 2.12 $rm mu m$ lines in a sample of 36 CTTs observed at the Telescopio Nazionale Galileo with the HARPS and GIANO spectrographs. We decomposed the line profiles into different kinematic Gaussian components and focused on the most frequently detected component, the narrow low-velocity (v$rm_p < 20$ $rm km$ $rm s^{-1}$) component (NLVC). We found that the $rm H_2$ line is detected in 17 sources ($sim 50 %$ detection rate), and [OI] is detected in all sources but one. The NLV components of the $rm H_2$ and [OI] emission are kinematically linked, with a strong correlation between the peak velocities and the full widths at half maximum of the two lines. Assuming Keplerian broadening, we found that the [OI] NVLC originates from a disk region between 0.05 and 20 au and that of $rm H_2$ in a region from 2 and 20 au. We did not find any clear correlation between v$rm_p$ of the $rm H_2$ and [OI] NVLC and the outer disk inclination. This result is in line with previous studies. Our results suggest that molecular and neutral atomic emission in disk winds originate from regions that might overlap, and that the survival of molecular winds in disks strongly depends on the gas exposure to the radiation from the central star. Our results demonstrate the potential of wide-band high-resolution spectroscopy in linking tracers of different manifestations of the same phenomenon.
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