Homogenization and mixing measures for a replenishing passive scalar field

The efficiency with which an incompressible flow mixes a passive scalar field that is continuously replenished by a steady source-sink distribution has been quantified using the suppression of the mean scalar variance below the value it would attain in the absence of the stirring. We examine the relationship this mixing measure has to the effective diffusivity obtained from homogenization theory, particularly establishing precise connections in the case of a stirring velocity field that is periodic in space and time and varies on scales much smaller than that of the source. We explore theoretically and numerically via the Childress–Soward family of flows how the mixing measures lose their linkage to the homogenized diffusivity when the velocity and source field do not enjoy scale separation. Some implications for homogenization-based parametrizations of mixing by flows with finite scale separation are discussed. © 2010 American Institute of Physics. doi:10.1063/1.3456726