  The mixing efficiency of a flow advecting a passive scalar sustained by
steady sources and sinks is naturally defined in terms of the suppression of
bulk scalar variance in the presence of stirring, relative to the variance in
the absence of stirring. These variances can be weighted at various spatial
scales, leading to a family of multi-scale mixing measures and efficiencies. We
derive a priori estimates on these efficiencies from the advection--diffusion
partial differential equation, focusing on a broad class of statistically
homogeneous and isotropic incompressible flows. The analysis produces bounds on
the mixing efficiencies in terms of the Peclet number, a measure the strength
of the stirring relative to molecular diffusion. We show by example that the
estimates are sharp for particular source, sink and flow combinations. In
general the high-Peclet number behavior of the bounds (scaling exponents as
well as prefactors) depends on the structure and smoothness properties of, and
length scales in, the scalar source and sink distribution. The fundamental
model of the stirring of a monochromatic source/sink combination by the random
sine flow is investigated in detail via direct numerical simulation and
analysis. The large-scale mixing efficiency follows the upper bound scaling
(within a logarithm) at high Peclet number but the intermediate and small-scale
efficiencies are qualitatively less than optimal. The Peclet number scaling
exponents of the efficiencies observed in the simulations are deduced
theoretically from the asymptotic solution of an internal layer problem arising
in a quasi-static model.
