  Metal production in galaxies traces star formation, and is highly
concentrated toward the centers of galactic discs. This suggests that galaxies
should have inhomogeneous metal distributions with strong radial gradients, but
observations of present-day galaxies show only shallow gradients with little
azimuthal variation, implying the existence of a redistribution mechanism. We
study the role of gravitational instability-driven turbulence as a mixing
mechanism by simulating an isolated galactic disc at high resolution, including
metal fields treated as passive scalars. Since any cylindrical field can be
decomposed into a sum of Fourier-Bessel basis functions, we set up initial
metal fields characterized by these functions and study how different modes
mix. We find both shear and turbulence contribute to mixing, but the mixing
strongly depends on the symmetries of the mode. Non-axisymmetric modes have
decay times smaller than the galactic orbital period because shear winds them
up to small spatial scales, where they are erased by turbulence. The decay
timescales for axisymmetric modes are much greater, though for all but the
largest-scale inhomogeneities the mixing timescale is still short enough to
erase chemical inhomogeneities over cosmological times. These different
timescales provide an explanation for why galaxies retain metallicity gradients
while there is almost no variation at a fixed radius. Moreover, the
comparatively long timescales required for mixing axisymmetric modes may
explain the greater diversity of metallicity gradients observed in high
redshift galaxies as compared to local ones: these systems have not yet reached
equilibrium between metal production and diffusion.
