  Mixing a passive scalar field by stirring can be measured in a variety of
ways including tracer particle dispersion, via the flux-gradient relationship,
or by suppression of scalar concentration variations in the presence of
inhomogeneous sources and sinks. The mixing efficiency or efficacy of a
particular flow is often expressed in terms of enhanced diffusivity and
quantified as an effective diffusion coefficient. In this work we compare and
contrast several notions of effective diffusivity. We thoroughly examine the
fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by
a steady sinusoidal source-sink distribution to explore apparent quantitative
inconsistencies among the measures. Ultimately the conflicts are attributed to
the noncommutative asymptotic limits of large P$\acute{\text{e}}$clet number
and large length-scale separation. We then propose another approach, a
generalization of Batchelor's 1949 theory of diffusion in homogeneous
turbulence, that helps unify the particle dispersion and concentration variance
suppression measures.
