  Passive scalar mixing (metals, molecules, etc.) in the turbulent interstellar
medium (ISM) is critical for abundance patterns of stars and clusters, galaxy
and star formation, and cooling from the circumgalactic medium. However, the
fundamental scaling laws remain poorly understood in the highly supersonic,
magnetized, shearing regime relevant for the ISM. We therefore study the full
scaling laws governing passive-scalar transport in idealized simulations of
supersonic turbulence. Using simple phenomenological arguments for the
variation of diffusivity with scale based on Richardson diffusion, we propose a
simple fractional diffusion equation to describe the turbulent advection of an
initial passive scalar distribution. These predictions agree well with the
measurements from simulations, and vary with turbulent Mach number in the
expected manner, remaining valid even in the presence of a large-scale shear
flow (e.g. rotation in a galactic disk). The evolution of the scalar
distribution is not the same as obtained using simple, constant "effective
diffusivity" as in Smagorinsky models, because the scale-dependence of
turbulent transport means an initially Gaussian distribution quickly develops
highly non-Gaussian tails. We also emphasize that these are mean scalings that
only apply to ensemble behaviors (assuming many different, random scalar
injection sites): individual Lagrangian "patches" remain coherent
(poorly-mixed) and simply advect for a large number of turbulent flow-crossing
times.
