In addition to the well-known gas phase mass-metallicity relation (MZR), recent spatially-resolved observations have shown that local galaxies also obey a mass-metallicity gradient relation (MZGR) whereby metallicity gradients can vary systematically with galaxy mass. In this work, we use our recently-developed analytic model for metallicity distributions in galactic discs, which includes a wide range of physical processes -- radial advection, metal diffusion, cosmological accretion, and metal-enriched outflows -- to simultaneously analyse the MZR and MZGR. We show that the same physical principles govern the shape of both: centrally-peaked metal production favours steeper gradients, and this steepening is diluted by the addition of metal-poor gas, which is supplied by inward advection for low-mass galaxies and by cosmological accretion for massive galaxies. The MZR and the MZGR both bend at galaxy stellar mass $sim 10^{10} - 10^{10.5},rm{M_{odot}}$, and we show that this feature corresponds to the transition of galaxies from the advection-dominated to the accretion-dominated regime. We also find that both the MZR and MZGR strongly suggest that low-mass galaxies preferentially lose metals entrained in their galactic winds. While this metal-enrichment of the galactic outflows is crucial for reproducing both the MZR and the MZGR at the low-mass end, we show that the flattening of gradients in massive galaxies is expected regardless of the nature of their winds.
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