Metallicity distributions of mono-age stellar populations of the Galactic disc from the LAMOST Galactic spectroscopic surveys

We have investigated the metallicity distributions of mono-age stellar populations across the disc of $6 \lesssim R \lesssim 12$\,kpc and $|Z| \lesssim 2$\,kpc using samples selected from the main-sequence turn-off and sub-giant (MSTO-SG) stars targeted by the LAMOST Galactic Spectroscopic surveys. Both the mean values and the profiles of the distributions exhibit significant variations with age and position. We confirm that the oldest ($>11$\,Gyr) stars have nearly flat radial [Fe/H] gradients at all heights above the disc but show negative vertical [Fe/H] gradients. For stars younger than 11\,Gyr, the radial [Fe/H] gradients flatten with $|Z|$, while the vertical [Fe/H] gradients flatten with $R$. Stars of 4--6\,Gyr exhibit steeper negative radial [Fe/H] gradients than those of either younger or older ages. Values of [$\alpha$/Fe] of mono-age stellar populations also show significant radial and vertical gradients, with patterns varying with age. The [Fe/H] distribution profiles of old ($>8$\,Gyr) stars vary little with $R$, while those of younger stars exhibit strong radial variations, probably a consequence of significant radial migration. The [$\alpha$/Fe] radial distribution profiles show opposite patterns of variations with age compared to those of [Fe/H]. We have also explored the impacts of stellar mixing by epicycle motions (blurring) on the [Fe/H] and [$\alpha$/Fe] distributions, and found that blurring mainly change the widths of the distribution profiles. Our results suggest that the disc may have experienced a complex assemblage history, in which both the "inside-out" and "upside-down" formation processes may have played an important role.