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Galaxy clusters are unique laboratories for studying astrophysical processes and their impact on halo gas kinematics. Despite their importance, the full complexity of gas motion within and around these clusters remains poorly known. This paper is part of a series presenting the first results from the new TNG-Cluster simulation, a suite comprising 352 high-mass galaxy clusters including the full cosmological context, mergers and accretion, baryonic processes and feedback, and magnetic fields. Studying the dynamics and coherence of gas flows, we find that gas motions in galaxy cluster cores and intermediate regions are largely balanced between inflows and outflows, exhibiting a Gaussian distribution centered at zero velocity. In the outskirts, even the net velocity distribution becomes asymmetric, featuring a double peak where the second peak reflects cosmic accretion. Across all cluster regions, the resulting net flow distribution reveals complex gas dynamics. These are strongly correlated with halo properties: at a given total cluster mass, unrelaxed, late-forming halos with fewer massive black holes and lower accretion rates exhibit a more dynamic behavior. Our analysis shows no clear relationship between line-of-sight and radial gas velocities, suggesting that line-of-sight velocity alone is insufficient to distinguish between inflowing and outflowing gas. Additional properties, such as temperature, can help break this degeneracy. A velocity structure function (VSF) analysis indicates more coherent gas motion in the outskirts and more disturbed kinematics toward halo centers. In all cluster regions, the VSF shows a slope close to the theoretical models of Kolmogorov (∼1/3), except within 50 kpc of the cluster centers, where the slope is significantly steeper. The outcome of TNG-Cluster broadly aligns with observations of the VSF of multiphase gas across different scales in galaxy clusters, ranging from ∼1 kpc to megaparsec scales. 

ABSTRACT Stellar and supernova nucleosynthesis in the first few billion years of the cosmic history have set the scene for early structure formation in the Universe, while little is known about their nature. Making use of stellar physical parameters measured by GALAH Data Release 3 with accurate astrometry from the Gaia EDR3, we have selected ∼100 old main-sequence turn-off stars (ages ≳12 Gyr) with kinematics compatible with the Milky Way stellar halo population in the Solar neighbourhood. Detailed homogeneous elemental abundance estimates by GALAH DR3 are compared with supernova yield models of Pop III (zero-metal) core-collapse supernovae (CCSNe), normal (non-zero-metal) CCSNe, and Type Ia supernovae (SN Ia) to examine which of the individual yields or their combinations best reproduce the observed elemental abundance patterns for each of the old halo stars (‘OHS’). We find that the observed abundances in the OHS with [Fe/H] > −1.5 are best explained by contributions from both CCSNe and SN Ia, where the fraction of SN Ia among all the metal-enriching SNe is up to 10–20 per cent for stars with high [Mg/Fe] ratios and up to 20–27 per cent for stars with low [Mg/Fe] ratios, depending on the assumption about the relative fraction of near-Chandrasekhar-mass SNe Ia progenitors. The results suggest that, in the progenitor systems of the OHS with [Fe/H] > −1.5, ∼ 50–60 per cent of Fe mass originated from normal CCSNe at the earliest phases of the Milky Way formation. These results provide an insight into the birth environments of the oldest stars in the Galactic halo.