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H$_2$O distribution in the disc of HD 100546 and HD 163296: the role of dust dynamics and planet--disc interaction

[Abridged] Far-infrared observations with Herschel revealed a surprisingly low abundance of cold-water reservoirs in protoplanetary discs. On the other hand, a handful of discs show emission of hot water transitions excited at temperatures above a few hundred Kelvin. In particular, the protoplanetary discs around the Herbig Ae stars HD 100546 and HD 163296 show opposite trends in terms of cold versus hot water emission: in the first case, the ground-state transitions are detected and the high-J lines are undetected, while the trend is opposite in HD 163296. We performed a spectral analysis using the thermo-chemical model DALI. We find that HD 163296 is characterised by a water-rich (abundance $\gtrsim 10^{-5}$) hot inner disc (within the snowline) and a water-poor ($< 10^{-10}$) outer disc: the relative abundance may be due to the thermal desorption of icy grains that have migrated inward. Remarkably, the size of the H$_2$O emitting region corresponds to a narrow dust gap visible in the millimeter continuum at $r=10\,$au with ALMA. The low-J lines detected in HD 100546 instead imply an abundance of a few $10^{-9}$ in the cold outer disc ($> 40$ au). The emitting region of the cold H$_2$O transitions is spatially coincident with that of the H$_2$O ice previously seen in the near-infrared. Notably, millimetre observations with ALMA reveal the presence of a large dust gap between nearly 40 and 150 au, likely opened by a massive embedded protoplanet. In both discs, we find that the warm molecular layer in the outer region (beyond the snow line) is highly depleted of water molecules, implying an oxygen-poor chemical composition of the gas. We speculate that gas-phase oxygen in the outer disc is readily depleted and its distribution in the disc is tightly coupled to the dynamics of the dust grains.